Quantum Trajectories

Edited by

Keith H. Hughes

School of ChemistryUniversity of Wales BangorBangorLL57 2UWUnited Kingdom

and

Gerard Parlant

Institut Charles GerhardtUniversite Montpellier 2, CNRSEquipe CTMM, Case Courrier 1501Place Eugene Bataillon34095 MontpellierFrance

Suggested Dewey Classification: 541.2

ISBN 978-0-9545289-9-7

Published by

Collaborative Computational Projecton Molecular Quantum Dynamics (CCP6)

Daresbury LaboratoryDaresburyWarringtonWA4 4ADUnited Kingdom

c©CCP6 2011

ii

Preface v

Mixed Quantum/Classical Dynamics:Bohmian and DVR Stochastic TrajectoriesC. Meier, J. A. Beswick, T. Yefsah 1

Trajectory-Based Derivation of Classical and Quantum MechanicsBill Poirier 6

The Bipolar Reaction Path Hamiltonian (BRPH) Approach for Multi-Dimensional Reactive Scattering SystemsJeremy B. Maddox and Bill Poirier 9

An Iterative Finite Difference Method for Solving the Quantum Hydro-dynamic Equations of MotionBrian K. Kendrick 13

Kinematic Quantum TrajectoriesTimothy M. Coffey 20

Complex Trajectories and Dynamical Origin of Quantum ProbabilityMoncy V. John 25

Energy Rays for Electromagnetic Pulses Scattering from Metal-Dielectric StructuresRobert E. Wyatt 28

Quantum Dynamics through Quantum PotentialsS. Duley, S. Giri, S. Sengupta and P. K. Chattaraj 35

Conceptual Issues, Practicalities and Applications of Bohmian and otherQuantum Trajectories in NanoelectronicsJohn R. Barker 43

Principles of Time Dependent Quantum Monte CarloIvan P. Christov 49

Types of Trajectory Guided Grids of Coherent States for Quantum Prop-agationDmitrii V. Shalashilin and Miklos Ronto 54

iii

Accurate Deep Tunneling Description by the Classical Schrodinger Equa-tionXavier Gimenez and Josep Maria Bofill 62

The Bohmian Model, Semiclassical Systems and the Emergence of Clas-sical TrajectoriesAlex Matzkin 66

Quantum Trajectories for Ultrashort Laser Pulse Excitation DynamicsGerard Parlant 69

Quantum Dynamics and Super-Symmetric Quantum MechanicsEric R. Bittner and Donald J. Kouri 72

Bohmian Trajectories of Semiclassical Wave PacketsSarah Romer 79

Quantum Trajectories in Phase SpaceCraig C. Martens 83

The Semiclassical Limit of Time Correlation Functions by Path IntegralsG. Ciccotti 86

Path Integral Calculation of (Symmetrized) Time Correlations Functions

S. Bonella 91

On-the-fly Nonadiabatic Bohmian DYnamics (NABDY)Ivano Tavernelli, Basile F. E. Curchod and Ursula Rothlisberger 96

Quantum Many-Particle Computations with Bohmian Trajectories: Ap-plication to Electron Transport in Nanoelectronic DevicesA. Alarcon, G.Albareda, F.L.Traversa and X.Oriols 103

An Account on Quantum Interference from a Hydrodynamical Perspec-tiveA. S. Sanz 111

Quantum, Classical, and Mixed Quantum-Classical HydrodynamicsI. Burghardt and K. H. Hughes 116

iv

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Preface

This booklet was produced as a result of the CCP6 Workshop on “QuantumTrajectories” held at the Bangor University, UK, between July 12-14, 2010.The workshop was sponsored by the UK Collaborative Computation Project 6(CCP6) on molecular quantum dynamics. Details of CCP6 and its activities canbe found at http://www.ccp6.ac.uk.The main focus of the workshop was the hydrodynamic formulation of quan-

tum mechanics. Over the last 10-13 years the hydrodynamic formulation ofquantum mechanics has grown into a quantum trajectory methodology withwidespread applications in physics and chemistry. Furthermore, theoretical in-vestigations into the hydrodynamic form of quantum mechanics have spawned,or inspired, a number of other trajectory based approaches to quantum mechan-ics. The workshop brought together researchers involved in the computationaland numerical development of quantum trajectories and researchers involved inuse of quantum trajectories as an insightful and intuitive approach for studyingand interpreting a wide range of quantum phenomena in a diverse range of fields.Each speaker was asked to provide a brief article which could be collected into

a workshop booklet that reviews their work and the topics covered in their talk.This booklet should be of interest to both the specialist and non-specialist inthis field. The booklet covers computational approaches to the topic of quantumtrajectories and should serve as a guide to many of the recent developmentsin this field. The editors would like to thank all those who participated andcontributed to the workshop.

K. H. HughesG. ParlantAugust 2011

v

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Mixed Quantum/Classical Dynamics:Bohmian and DVR Stochastic Trajectories

C. Meier, J. A. Beswick and T. Yefsah∗

Laboratoire Collisions, Agregats, Reactivite, IRSAMC

Universite Paul Sabatier, Toulouse, France

I. INTRODUCTION

In many systems comprising a large number of particles, even though a de-tailed quantum treatment of all degrees of freedom is not necessary, there mayexist subsets that have to be treated quantum mechanically under the influ-ence of the rest of the system treated classically. In these cases, mixed quan-tum/classical approaches have to be used to describe its dynamics [1]. Themost popular of these are the mean-field approximation [2], the surface hoppingtrajectories [4, 5] or the methods based on quantum/classical Liouville spacerepresentations [6–12].In the mean-field treatment the force for the classical motion is calculated by

averaging over the quantum wavefunction. In the surface hopping scheme theclassical trajectories move according to a force derived from a single quantum

state with the possibility of transitions to other states. Based on these ideas,Bastida et al [13] have proposed a scheme (MQCS for mixed quantum classical

steps) in which the quantum wave function is expanded in a discrete variable rep-resentation (DVR) rather than in the usual finite (spectral) basis representation(FBR). The interpretation of the dynamics of the quantum degree of freedom isprovided in terms of stochastic hops from one grid point to another, governed bythe time evolution of the wave packet. Another way to mix quantum mechanicswith classical mechanics, proposed in refs. [14–20], is based on Bohmian quan-tum trajectories for the quantum/classical connection. In this approach, whichwas called MQCB (Mixed Quantum/Classical Bohmian) trajectories, the wavepacket is used to define de Broglie-Bohm quantum trajectories [21–24] which inturn are used to calculate the force acting on the classical variables. The main

∗ present address: Laboratoire Kastler Brossel, Departement de Physique de l’Ecole NormaleSuperieure, Paris, France

1

difference between MQCB and MQCS lies in the way the force on the classi-cal variables is calculated. In the MQCS method, the force is evaluated usingdifferent discrete points of the quantum degree of freedom, while in the MQCBmethod, it is calculated at the continuous points of the quantum trajectory.Hence, the underlying question is to determine what is the connection between

the sequence of grid points obtained by stochastic hopping methods and the de-terministic, continuous quantum trajectory. In his paper Quantum mechanics in

terms of discret beables [25], Jeroen C. Vink has shown that indeed a connectionexists for a three-point approximation of the kinetic energy operator.In this contribution, we generalize Vink’s formulation to an arbitrary N -point

expression. The paper is organized as follows. In Sec. (II) we give a briefreview of MQCS method; in Sec. (III) we address the problem of representing aquantum wave packet by stochastic hops between discrete points in space, andunder which conditions the sequence of hopping trajectories converge towards adeterministic quantum trajectory. Finally Sec. (IV) gives a general discussionof the relationship between all these treatments.

II. STOCHASTIC DVR AND MQCS

Considering a Hamiltonian with x being the quantum degree of frredom andX(t) the classical trajectory. Choosing a one-dimensional (diabatic) DVR basis|xn〉 with eigenvalue xn of the position operator, for the quantum degree offreedom, the wavefunction can be expressed by ψ(x,X(t), t) =

∑

n cn(t) |xn〉,where the wave packet Ψ(x, t) propagates with time along the x coordinate, andis a DVR basis function with eigenvalue xn of the position operator, such thatcn(t) = 〈xn|ψ(t)〉 = ψ(xn,X(t), t). The values of the time-dependent coefficientscan be obtained by solving the time-dependent Schrodinger equation using anyconventional method (split operator, etc...).Initially the wave packet is a coherent superposition of DVR states. The time

evolution of the coefficients of the wave packet in these DVR states is given by

dcndt

= −i

~

∑

m

cm(t)〈xn|H|xm〉 = −i

~

[

∑

m

cm(t)〈xn|T |xm〉+ V (xn,X(t))

]

(1)

where we have used the fact that the non diagonal matrix elements of the Hamil-tonian reduce to the matrix elements of the kinetic energy operator T in the DVRbasis and the potential energy matrix is diagonal.The classical degrees of freedom are calculated using the force evaluated at

one single DVR point,

P = −1

M

∂ (V (x,X))

∂X

∣

∣

∣

∣

x=xn(t),X=X(t)

(2)

2

the dynamics in the quantum subspace is accounted for by the possiblity ofquantum jumps between these points. Hence, xn is a sequence of DVR points,forming a stochastic quantum trajectory, defined as follows: Eq.(1) determinesthe transition probabilities between DVR states. Let us introduce the diagonaldensity matrix elements Pn = |cn|

2 = |ψ(xn,X(t), t)|2 representing the proba-bility for the system to be in state |xn〉 at time t. The time evolution of Pn(t)is deduced from Eq.(1) can be written as [26]

dPn

dt=

∑

m

(Tnm Pm − Tmn Pn) (3)

with

Tnm =~

MD(2)

nmℑ

c∗m cn|cm|2

if Tnm > 0 (4)

where we have denoted by D(2)nm = 〈xn|∂

2/∂x2|xm〉 the matrix elements of thesecond derivative. In this form, it can be viewed as a rate equation, which canbe solved by a stochastic approach using Pn→m = Tnmδt as probability to hopfrom DVR point |xn〉 to |xm〉 during a timestep δt.This approach is similar to the way the quantum potential energy surface

for the classical particle dynamics is selected in Tully’s fewest switches method[3] for mixed quantum-classical dynamics. MQCS method can thus be thoughtas the implementation of the surface hopping method in the discrete variablerepresentation, rather than in the usual finite basis representation.

III. DISCRET BEABLES

In this section, we discuss the connection between a continuous quantum tra-jectory and stochastic hopping between discrete grid points first proposed byVink in 1993 [25]. To this end, it is enough to consider only one quantum degreeof freedom. We consider a particle of massM in a one dimensional space extend-ing from −∞ < x < +∞. This space is discretized as xn = ǫ n; n = −∞,+∞where ǫ is the grid spacing. If we denote by Pn = |ψ(xn, t)|

2 = |ψn|2 the proba-

bility to find the particle at xn, the evolution in time of Pn is given by

∂Pn

∂t= −

~

Mℑ

ψ∗n

∂2ψ

∂x2

∣

∣

∣

∣

∣

xn

(5)

Using the three-point centered expansion of the second derivative and a Taylorexpansion of the wavefunction according to ψn±1 = ψn± ǫ ψ

′n, this equation can

again be written in the form (3) with [25, 27]

3

Tnm =S′m

M ǫδm,n−1 if S′

m ≥ 0 ; Tnm = −S′m

M ǫδm,n+1 if S′

m ≤ 0 (6)

with ψn = An exp(i Sn/~) and the prime denoting a first derivative with respectto x. Finally Tnn is determined by the normalization Tnn dt = 1−

∑

n 6=m Tnm dt

with dt ≤ M ǫ|S′

n−1−S′

n+1| in order to have Tnn dt ≥ 0. Vink’s has shown [25], that if

one uses of (6) it is possible to recover the causal Bohmian trajectories througha stochastic algorithm with minimal choice [27] for the transition density. Theproof is straightforward [26]. For a trajectory starting in point n at t = 0 theprobability to jump to the position n+ 1 is, if we assume that S′

n 6= 0,

Pn+1 =S′n

M ǫdt (7)

The mean average displacement after dt will then be given by dx = ǫPn+1 whichusing (7) gives in the limit ǫ→ 0

∂x

∂t=

1

M

∂S

∂x(8)

which is precisely the Bohmian equation for the quantum trajectories. Further-more, one can also show that the second moment of the distribution, i.e., thedispersion in the average displacement, vanishes in the limit ǫ→ 0 [26].

These results can be generalized to an arbitrary number of points to expressthe second derivative by a discrete N -point formula. Using the expressions givenin Fornberg [28], one can show that eq. (5) can always be written in the formof eq. (3), however, the Tnm now couple sites beyond nearest neighbors [26].Again, in the limit of ǫ → 0, the mean-squared dispersion vanishes, and thesequence of hopping points converges towards the Bohmian trajectory.

IV. DISCUSSION

Comparing MQCB and MQCS, we see that both methods calculate the back-reaction of the quantum system onto the classical system not by averaging theforce over the quantum wave function (as does the mean field method), but bychoosing to evaluate this force only at one single point. The time evolutionof this point, however, is different in both methods: the MQCB method usesthe deterministic Bohm trajectory associated with the quantum wave function,the MQCS method generated these points by stochastic jumps between discreteDVR points of an underlying DVR basis. On the other hand, generalizing theVink’s formulation, we have shown that Bohmian trajectories can be obtainedby stochastic jumps between discrete points in the limit of a large number of

4

points. Clearly, the MQCS method is based on the same idea except that thediscret points are chosen according to the DVR representation. In the limit ofa large number of DVR points the MQCS trajectories should converge towardsthe Bohmian MQCB trajectories.

[1] Classical and Quantum Dynamics in Condensed Phase Simulations (B. Berne andG. Cicotti and D. Coker Eds.; World Scientific, Singapour,1998)

[2] G. Billing, Int. Rev. Phys. Chem. 13,309 (1994).[3] J. C. Tully, J. Chem. Phys. 93, 1061 (1990)[4] J.C. Tully, Int. J. Quantum Chem. 25, 299 (1991).[5] J. C. Tully. Nonadiabatic dynamics. In Modern methods for multidimensional

dynamics computations in chemistry, edited by D. L. Thompson, page 34. (WorldScientific, Singapore, 1998)

[6] D. A. Micha and B. Thorndyke Adv. Quantum Chem. 47, 292-312 (2004).[7] B. Thorndyke and D. A. Micha, Chem. Phys. Lett. 403 (2005) 280-286.[8] A. Donoso, C. C. Martens, J. Chem. Phys. 102, 4291 (1998)[9] R, Kapral, G. Cicotti, J. Chem. Phys. 110, 8919 (1999)

[10] S. Nielsen, R, Kapral, G. Cicotti, J. Chem. Phys. 115, 5805 (2001)[11] I. Horenko, C. Salzmann, B. Schmidt, C. Schutte, J. Chem. Phys. 117, 11075

(2002)[12] I. Horenko, M. Weiser, B. Schmidt, C. Schutte, J. Chem. Phys. 120, 8913 (2004)[13] A. Bastida, J. Zuniga, A. Requena, N. Halberstadt, and J.A. Beswick, PhysChem-

Comm 7, 29 (2000)[14] E. Gindensperger and C. Meier and J.A.Beswick, J. Chem. Phys. 113, 9369 (2000).[15] E. Gindensperger, C. Meier and J. A. Beswick, Adv. Quantum Chem. 47, 331-346

(2004).[16] E. Gindensperger and C. Meier and J.A.Beswick,J. Chem. Phys. 116,8 (2002).[17] E. Gindensperger and C. Meier and J.A. Beswick and M.-C. Heitz, J. Chem. Phys.

116, 10051 (2002).[18] O.V. Prezhdo and C. Brooksby Phys. Rev. Lett. 86, 3215 (2001).[19] L.L. Salcedo Phys. Rev. Lett. 90, 118901 (2003).[20] O.V. Prezhdo and C. Brooksby Phys. Rev. Lett. 90, 118902 (2003).[21] The quantum theory of motion P. R. Holland (Cambridge University Press, 1993).[22] L. de Broglie, C. R. Acad. Sci. Paris 183, 447 (1926); 184, 273 (1927).[23] D. Bohm, Phys. Rev. 85, 166 (1952); 85, 180 (1952).[24] R. E. Wyatt, Quantum Dynamics with trajectories, Springer, New York (2005)[25] J.C. Vink, Phys. Rev. A 48, 1808 (1993)[26] C. Meier, J. A. Beswick, T. Yefsah, in Quantum Trajectories, edited by P. K.

Chatteraj, Taylor & Francis, (2010)[27] J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge Univer-

sity Press, (1987)[28] B. Fornberg, A practical guide to pseudospectral methods, Cambridge University

Press (1998)

5

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Trajectory-Based Derivation of Classical and Quantum

Mechanics

Bill PoirierDepartment of Chemistry and Biochemistry, Texas Tech University,

Lubbock, Texas 79409-1061, United States

Why are the laws of nature what they are? It is well known that a relativelysmall change in any of the fundamental physical constants would lead to a rad-ically different and far less complex universe, incapable of sustaining intelligentlife that could make such queries [1]. Interpretations of this anthropic princi-ple vary. But what of the fundamental laws themselves, i.e. the mathematicalequations that govern dynamics? Much less attention has been paid here.

Fundamentally, there are only two dynamical equations of profound impor-tance, i.e. Newton’s (classical) laws, and the Schrodinger (quantum) equation.On the surface, these appear nothing like one another—which is interesting ofitself (e.g., why should there be a classical limit?) Yet connections have beenknown since the earliest days of the quantum theory. D. Bohm [2] developedthese into a full-fledged quantum formulation, a hybrid approach in which thestate of a system is represented as a wavefunction plus a quantum trajectory.The latter obeys Newton-like equations with a modified potential, V (x)+Q(x, t)[one-dimensional (1D) systems are presumed here], with the quantum correctionQ obtained from the wavefunction. More recently, it has been shown that thewavefunction may be dispensed with entirely [3, 4]—i.e., one may construct acomplete formulation of (spin-free) quantum mechanics based solely on trajecto-ries, thus enabling true comparisons between classical and quantum mechanics.In a recent article [4], the author derived a 4th-order but otherwise Newton-

like ordinary differential equation (ODE), describing the quantum trajectoriesfor 1D stationary scattering states. What is it that makes this particular ODE—together with the Newton ODE—so special? Are there certain properties thatonly these satisfy, or could a viable dynamical law be constructed from essen-tially any ODE? Some clues are provided in Ref. [4], and extended there also tothe non-stationary case. In this paper, we show that the classical and quantumODEs are the simplest that can be constructed so as to satisfy the two bedrockphysical principles of energy conservation and action extremization. Some otherODE’s satisfying these conditions—representing “alternative dynamical laws”—are also presented, although they are very rare, and only a handful have beendiscovered thus far. The dynamical laws of nature are evidently extremely spe-

6

cial.We begin with a derivation of the classical dynamical law. Consider a candi-

date trajectory, x(t), together with potential V [x] and kinetic T [x] “functionals”(term used somewhat loosely). Neither functional form is yet specified; they arefor the moment regarded as unknowns. For essentially any choice of V [x] andT [x], an allowed dynamical trajectory x(t) is defined to be one which extrem-izes the action, i.e. the time integral of the Lagrangian, L[x, x] = T [x] − V [x].This condition leads (via Euler-Lagrange) to a 2nd-order ODE for x(t), whoseprecise form depends on the specification of the V and T functionals. If inaddition to action extremization, one also imposes the condition that energy,E[x, x] = T [x]+V [x] be conserved, this results in a constraint on the functionalsthemselves. Specifically, though V [x] is still found to be unconstrained, T [x]must take the form of a constant times x2. Identification of that constant withone half the mass, m/2, leads at once to Newton’s ODE. Thus, only the standardforms of classical physics emerge as permissible, i.e., no other candidate kineticenergy (e.g., T [x] = Ax4) would lead to a consistent dynamical law satisfyingboth conditions above.One might argue that the above analysis is overly simplistic, in that a fairly

specific decomposition of the Lagrangian functional into a pure x componentplus a pure x component is presumed. In particular, this implies a coordinatex in terms of which space is homogeneous (albeit another standard principle ofphysics), as a result of which the kinematic kinetic energy quantity T [x] must bex-independent. In any case, a generalized coordinate treatment changes nothingfundamental about the above conclusions. With q denoting the generalizedcoordinate, one has the more general form T [q, q] = T [f [q]q], where both T andf are essentially arbitrary. The two conditions above then imply that

E[q, q] = H[q, q], (1)

where the Hamiltonian H is the usual (minus) Legendre transform of L[q, q] inthe variable p = ∂L/∂q. Equation (1) is not satisfied in general; but the allowedforms that do satisfy this equation correspond precisely to those of the standardgeneralized Lagrangian formulation of mechanics (i.e., encompassing arbitrarypoint transformations). A further generalization—the broadest possible—wouldconsider completely arbitrary L[q, q] functionals; but in this case, T and V con-tributions (and therefore E) can no longer be defined (though H still can be).The functionals above involve only 1st-order time derivatives of x(t); deriving

additional dynamical laws therefore clearly requires higher-order time deriva-tives. A kinematic quantum correction functional, Q[x, x,

...x , . . .], is posited, in

terms of which the Lagrangian and energy functionals become:

L[x, x, x, . . .] = T [x]− V [x]−Q[x, x,...x , . . .] (2)

E[x, x, x, . . .] = T [x] + V [x] +Q[x, x,...x , . . .] (3)

Application of the action extremization and energy conservation conditions then

7

does two important things. First, it imposes an extremely severe constraint onthe allowed functional form for Q. Second, it leads to a 2nd-or-higher-order ODEfor x(t)—i.e., the new dynamical law—whose solutions are the correspondingquantum trajectories.A systematic determination of all permissible dynamical laws has thus been

performed, for Q[x, x,...x , . . .] functionals of successively higher orders, up to 3rd

order. The complete set of meromorphic solutions is presented below:

V [x] = completely unconstrained

T [x] =m

2x2

Q[x, x, . . .] =

A order 0 (classical mechanics)

no solutions order 1

no solutions order 2

B(

xn

x2n −

2n

4n+2

xn−2...

x

x2n−1

)

order 3

(4)

To date, no solutions at higher than 3rd order have been found, although theirexistence has not been disproven. Any linear combination of the above solutionsis also a solution, as the dynamical law problem itself is linear (as opposed tothe corresponding quantum trajectory ODEs). With regard to the value of theparameter n, meromorphicity requires that this must be an integer. For n < 2,undesirable singular trajectories result. The simplest solution with well-behavedtrajectories is therefore n = 2. This choice leads exactly to the trajectories ofstandard quantum mechanics [4], with the identification B = −(5/4)(~2/2m).

A 1st-order non-meromorphic solution has also been discovered, which may berelevant for bipolar quantum trajectory methods [5, 6]. Other future work willinclude generalization for spin, non-stationary states, and multiple dimensions.

Acknowledgments. B. Poirier gratefully acknowledges a grant from The RobertA. Welch Foundation (Grant No. D-1523) and a Small Grant for ExploratoryResearch from The National Science Foundation (Grant No. CHE-0741321).

[1] N. Bostrum, Anthropic Bias: Observation selection effects in science and philosophy(Routledge New York, 2002).

[2] D. Bohm, Phys. Rev. 85 (1952) 166; Ibid. 180.[3] P. Holland, Ann. Phys. 315 (2005) 505.[4] B. Poirier, Chem. Phys. 370 (2010) 4.[5] N. Froman and P. O. Froman, JWKB Approximation (North-Holland, 1965).[6] C. Trahan and B. Poirier, J. Chem. Phys. 124 (2006) 034115; Ibid. 034116.

8

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

The Bipolar Reaction Path Hamiltonian (BRPH)Approach for Multi-Dimensional Reactive Scattering

Systems

Jeremy B. Maddox(a) and Bill Poirier(b)

(a) Department of Chemistry, Western Kentucky University,

Bowling Green, Kentucky 42101-1079, United States and

(b) Department of Chemistry and Biochemistry,

Texas Tech University, Lubbock, Texas 79409-1061, United States

One generic goal of chemical physics is to predict the reaction probabilities ofelementary chemical reactions using quantum theory. Fundamentally, these re-active collisions can be viewed as quantum scattering processes, and the reactionprobabilities are related to the elements of the scattering matrix, or S-matrix.The S-matrix may be calculated from the solutions of either the time-dependentor time-independent Schrodinger equations (TDSE and TISE, respectively) [1].In the present work we follow the latter approach and describe a new method forcalculating stationary wave functions of a multi-dimensional (multi-D) reactivescattering system. We build upon the bipolar counter-propagating wave method-ology (CPWM) developed by Poirier and co-workers for various applications inchemical physics [2–5]. The term “bipolar” refers to a specific representationof a stationary wave function in terms of two traveling waves that propagatein opposite directions. We invoke an adiabatic representation of the scatter-ing system’s Hamiltonian that effectively reduces the multi-D problem to a 1Dproblem involving multiple scattering amplitudes. This approach also provides aconvenient framework to connect CPWM algorithms [6] with the reaction pathHamiltonian formalism developed by Miller and co-workers for applications inpolyatomic reactive scattering [7]. Hence, we refer to the method as the bipolarreaction path Hamiltonian (BRPH) approach to reactive scattering.To illustrate the BRPH we consider a 2D scattering problem with a linear

reaction coordinate x and a vibrational coordinate y. The Hamiltonian of thesystem is given by

H = −(~2/2m)(∂2x + ∂2

y) + V (x, y) (1)

where m is a reduced mass and V is a potential energy function. For simplicitywe assume that V is asymptotically (x → ±∞) symmetric, such that the re-actants and products have the same eigenfunctions with energies Ei, which are

9

identified with the different scattering channels of the system. Furthermore, wedefine a set of adiabatic eigenfunctions φi(x, y) and energies ǫi(x) that dependparametrically on the reaction coordinate via the x-dependence of V . It is im-plied that V (x, y) → V (y), ǫi(x) → Ei, and φi(x, y) → φi(y), asymptotically. Agiven stationary state with energy E is expanded as a sum over scattering chan-nels φE(x, y) =

∑n

i ai(x, t)φi(x, y), where n is the number of open channels (i.e.,φi) satisfying E > Ei. The scattering amplitudes ai will satisfy different bound-ary conditions, thus distinguishing between members of the n-fold degenerateset of total scattering states, i.e., φE .

We invoke the so-called “constant velocity” bipolar decomposition of the am-plitudes ai = ai+ + ai−, where ai± are referred to as the BRPH components.These are associated with a pair of counter-propagating traveling waves

ai±(x, t) = αi±(x) exp[(i/~)(±pix− Et)] (2)

where pi =√

2m(E − Ei) is the momentum of a free particle with kinetic en-ergy E − Ei. We adopt a scattering convention, such that the ai+’s representincident/transmitted waves that move with constant positive momenta, and theai−’s represent reflected waves with negative momenta. The αi±’s depend on thescattering potential and are determined numerically using CPWM algorithms.The fact that the superposition of the BRPH components must be a solution

of the TISE constrains the relationship between the various ± amplitudes. Sometime ago, Froman and Froman (FF) derived this constraint in the context of ageneralized semiclassical theory [8]. In the present work, the FF condition isgiven by ∂xai = (i/~)pi(ai+ − ai−) and may be combined with the TISE toconstruct a set of Lagrangian-like hydrodynamic equations of motion for theBRPH components

dtai± = ∂tai± ± vi∂xai± = Fiai± +Gi(ai+ + ai−) +Hi (3)

where

Fi =i

~(E − 2Ei) (4)

Gi = −i

~(ǫi − Ei) (5)

Hi =i~

2m

∑

j

[

2

(∫

φ∗

i ∂xφjdy

)

∂xaj +

(∫

φ∗

i ∂2xφjdy

)

aj

]

(6)

The BRPH equations and their solutions can be qualitatively understood interms of competing fluxes. The scattering boundary conditions impose a flux ofincident probability amplitude. The Fi term is associated with the flux carriedby a plane-wave, such that both incident/transmitted and reflected amplitudepropagate through the scattering region. The Gi term couples the ± components

10

and leads to intra-channel flux that essentially converts incident/transmittedamplitude into reflected amplitude, and vice versa, within the same scatteringchannel. The Hi term depends on all of the BRPH components and leads tointer-channel flux. Both Gi and Hi will vanish asymptotically. The competitiveaction of the scattering boundary conditions and various fluxes eventually leadsto a steady state at long times, such that both the FF condition and the TISEare satisfied.In Figure 1 we present benchmark results for a 2D scattering problem defined

by an Eckart barrier along x and a Morse oscillator along y

V (x, y) = V0 sech2(αx) +D(x)(1− exp[−β(y − Y (x))])2 (7)

D(x) = D0 + V0 sech2(αx) (8)

Y (x) = y0 +∆y sech2(αx) (9)

where V0=0.01, D0=0.05, α=3, β=1.2, ∆y=1.25, and m=2000; all quantitieshave atomic units. The left panel shows slices of the potential at x → −∞ andx = 0. The potential is not separable in x and y. The right panel shows state-to-state reaction probabilities for the lowest three scattering channels. For eachenergy value we have used independent calculations using finite differences totest whether the calculated total stationary states satisfy both the FF conditionand the TISE. The numerical test for the percent error in the energy is evaluatedas ((HφE)/φE − E)/E over the spatial grid. The median percent error, withrespect the grid, in the calculated total energies were less than 0.002 for allenergy points shown. We are presently extending the BRPH approach to morerealistic scattering problems with curvilinear reaction coordinates.Acknowledgments. J. B. Maddox gratefully acknowledges start-up support

from the Department of Chemistry and the Applied Research and TechnologyProgram (ARTP) at Western Kentucky University. B. Poirier gratefully ac-knowledges a grant from The Welch Foundation (Grant No. D-1523) and a SmallGrant for Exploratory Research from The National Science Foundation (GrantNo. CHE-0741321).

[1] D. J. Tannor, Introduction to quantum mechanics: A time-dependent perspective(University Science Books, 2007).

[2] B. Poirier, J. Chem. Phys. 121 (2004) 4501.[3] C. Trahan and B. Poirier, J. Chem. Phys. 124 (2006) 034115; Ibid. 034116.[4] B. Poirier and G. Parlant, J. Phys. Chem. A 111 (2007) 10400.[5] B. Poirier, J. Chem. Phys. 128 (2008) 164115; Ibid. 129 (2008), 084103.[6] B. Poirier, J. Theor. Comput. Chem. 6 (2007) 99.[7] W. H. Miller, N. C. Handy and J. E. Adams, J. Chem. Phys. 72 (1980) 99.[8] N. Froman and P. O. Froman, JWKB Approximation (North-Holland, 1965).

11

0 1 2 3 4 50

2

4

6

8

10

12

y HbohrL

VE

0

æ ææ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

æ

ææ æ æ

ææ

ç ç ç ç ç ç ç ç ç ç ç ç ççç

ç

ç

ç

ç çç

ç

ç

ç

ç

ç

ç

+ + + + + + + + +

+

+

+

+

+

+

+

+

+

+

+

+

+

+

++ +

+

1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

EE0

prob

abili

ty

FIG. 1: (Left) Potential energy slices for the Eckart+Morse oscillator problem. (Right)State-to-state reaction probabilities: (filled) 0-0, (empty) 1-1, and (cross) 0-1. Horizon-tal/vertical lines indicate the lowest eigenenergies of the asymptotic Morse oscillator.

12

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

An Iterative Finite Difference Method for Solving the

Quantum Hydrodynamic Equations of Motion

Brian K. KendrickTheoretical Division (T-1, MS-B268)

Los Alamos National LaboratoryLos Alamos, NM 87545 USA

I. INTRODUCTION

The quantum hydrodynamic equations of motion associated with the deBroglie-Bohm[1–5] description of quantum mechanics describe a time evolvingprobability density whose “fluid” elements evolve as a correlated ensemble ofparticle trajectories. These equations are intuitively appealing due to theirsimilarities with classical fluid dynamics and the appearance of a generalizedNewton’s equation of motion in which the total force contains both a classi-cal and quantum contribution. However, the direct numerical solution of thequantum hydrodynamic equations (QHE) is fraught with challenges: the proba-bility “fluid” is highly-compressible, it has zero viscosity, the quantum potential(“pressure”) is non-linear, and if that weren’t enough the quantum potentialcan also become singular during the course of the calculations. Collectivelythese properties are responsible for the notorious numerical instabilities asso-ciated with a direct numerical solution of the QHE. The most successful andstable numerical approach that has been used to date is based on the MovingLeast Squares (MLS) algorithm.[6–8] The improved stability of this approach isdue to the repeated local least squares fitting which effectively filters or reducesthe numerical noise which tends to accumulate with time. However, this methodis also subject to instabilities if it is pushed too hard. In addition, the stabil-ity of the MLS approach often comes at the expense of reduced resolution orfidelity of the calculation (i.e., the MLS filtering eliminates the higher-frequencycomponents of the solution which may be of interest).Recently, a promising new solution method has been developed which is based

on an iterative solution of the QHE using finite differences.[9] This method (re-ferred to as the Iterative Finite Difference Method or IFDM) is straightforwardto implement, computationally efficient, stable, and its accuracy and convergenceproperties are well understood. A brief overview of the IFDM will be presentedfollowed by a couple of benchmark applications on one- and two-dimensional

13

Eckart barrier scattering problems.

II. METHODOLOGY

In the de Broglie-Bohm description of quantum mechanics, the complex wavefunction is written in polar form ψ(x, t) = exp[C(x, t)] exp[i S(x, t)/~] and sub-stituted into the time dependent Schrodinger equation. The velocity field isdefined as v(x, t) = ∇S/m where m is the mass. The resulting quantum hydro-dynamic equations (QHE) are given by (in a fixed Eulerian frame)

dC

dt= −

1

2∇ · v − v · ∇C , (1)

mdv

dt= −∇ (V +Q)−mv · ∇v . (2)

The classical potential is denoted by V and the quantum potential Q is given by

Q = −~2

2m(∇2C + |∇C|2) . (3)

The classical and quantum forces are given by fc = −∇V and fq = −∇Q,respectively. The probability density is ρ = e2C and the quantum trajectoriesare obtained by solving the equation x = v.The IFDM is based on an iterative solution (Newton’s method) of the non-

linearly coupled QHE using an implicit second-order central differencing in bothspace and time. A “control volume” approach is used for which the velocity gridis “staggered” relative to the spatial grid. The use of a staggered velocity gridis important in order to prevent the appearance of spurious oscillations in thenumerical solutions.[10] In one-dimension, the x coordinate is discretized using aset of N computational cells with centers located at xj where j = 0, . . . N+1 andwidth ∆x. The velocities vj+1/2 are defined on the “cell walls” which lie half-waybetween the cell centers (i.e., xj+1/2 = (xj + xj+1)/2). The time coordinate isdiscretized using a set of evenly spaced time intervals tn (n = 0, 1, . . .) separatedby ∆t.The 2nd-order finite difference expressions for the QHE (Eqs. 1 and 2) are

given by[9]

(Cn+1j − Cn

j )/∆t = −[

(vn+1j+3/2 − vn+1

j−1/2) + (vn+1j+1/2 − vn+1

j−3/2)

+(vnj+3/2 − vnj−1/2) + (vnj+1/2 − vnj−3/2)]

/(16∆x)

−[

(vn+1j+1/2 + vn+1

j−1/2)(Cn+1j+1 − Cn+1

j−1 )

+ (vnj+1/2 + vnj−1/2)(Cnj+1 − Cn

j−1)]

/(8∆x) , (4)

14

and

m (vn+1j+1/2 − vnj+1/2)/∆t = (fn+1

j+1/2 + fnj+1/2)/2− m[

vn+1j+1/2(v

n+1j+3/2 − vn+1

j−1/2)

+ vnj+1/2(vnj+3/2 − vnj−1/2)

]

/(4∆x) , (5)

where f = fc+fq denotes the total force. Equations 4 and 5 are solved iterativelyby rewriting these two equations as

f1(C, v) = Cn+1j − Cn

j −∆tRHSCeq , (6)

f2(C, v) = vn+1j+1/2 − vnj+1/2 −∆tRHSveq/m , (7)

where RHSCeq and RHSveq denote the right-hand-side of Eqs. 4 and 5, re-

spectively. Using an abbreviated notation where C = Cn+1j and v = vn+1

j+1/2,

Newton’s method in two variables consists of expanding f1 and f2 in a Taylorseries expansion with respect to C = Co +∆C and v = vo +∆v where Co andvo represent an initial guess (typically the solution from the previous time step:Co = Cn and vo = vn)

f1(C, v) = f1(Co, vo) +∂f1∂C

∆C +∂f1∂v

∆v , (8)

f2(C, v) = f2(Co, vo) +∂f2∂C

∆C +∂f2∂v

∆v . (9)

By setting f1(C, v) = 0 and f2(C, v) = 0, the ∆C and ∆v can be expressedin terms of the known quantities f1(Co, vo) and f2(Co, vo) and their derivativeswith respect to C and v. Analytic expressions for the derivatives of f1 and f2with respect to C and v can be derived by taking the appropriate derivatives ofEqs. 6 and 7.[9] Once the ∆C and ∆v are computed, the values of C and v areupdated and these updated values become the new guess (i.e., the Co and vo).Equations 8 and 9 are then solved for new values of ∆C and ∆v and the processis repeated until convergence is achieved. In practice, the absolute values of the∆C, ∆v, f1, and f2 are monitored until all four are simultaneously less than someuser specified convergence threshold at all of the grid points. The converged Cand v are now the correct values at the future time step tn+1. The time indexis incremented n → n + 1 and the process is repeated to find the values of Cand v at the next time step and so on. The iterative solution method convergesexponentially with respect to the iteration count which typically varies between10 - 20 iterations depending upon the size of the time step (∆t) and the desiredlevel of convergence. At each time step, the values of the C0, CN+1, v−1/2, andvN+3/2 at the edges of the grid are determined by applying the appropriate finite-

difference expressions for the Gaussian-like boundary conditions: ∂2C/∂x2 =constant and ∂2v/∂x2 = 0.[9]

15

A stability analysis of the IFDM was performed using the method of trunca-tion error analysis.[9] In this approach, Taylor series expansions of the C andv with respect to ∆x and ∆t are substituted into the finite difference Eqs. 4and 5 and the truncation error terms (i.e., the terms multiplying ∆x2 and ∆t2)are explicitly derived. Some of these error terms represent numerical diffusionand take the form γv ∂

2v/∂x2 and γC ∂2C/∂x2 (the explicit functional forms for

γv and γC are given in Ref. 9). The effective diffusion coefficients are givenby Γv = ∆t2 γv and ΓC = ∆t2 γC which can be positive or negative and canalso change sign during the course of the calculation. Even a temporary occur-rence of a negative diffusion coefficient can lead to instabilities and a completebreakdown of the calculations. To ensure a stable propagation, the overall dif-fusion coefficient must remain positive. This can be accomplished using thewell-known hydrodynamic technique called “artificial viscosity”.[11–13] In thisapproach, a positive diffusion term with the same functional form as that derivedin the truncation error analysis is introduced into the finite difference equations.The positive diffusion coefficient multiplying this new diffusive term is chosenlarge enough to ensure that the overall diffusion coefficient is always positive sothat the propagation will remain stable. In other words, the overall diffusion(truncation error + artificial viscosity) is always dissipative or stabilizing. Itis important to note that the diffusive artificial viscosity term is of the same

order in ∆t2 as the truncation error. Thus, the stability of the calculation canbe maintained even as the grid spacing and time step are decreased (i.e., theresolution of the calculation is increased).

III. APPLICATIONS

The IFDM has been applied to three different test cases: (1) a one-dimensionalfree Gaussian wave packet, (2) a one-dimensional Gaussian wave packet scat-tering off an Eckart barrier, and (3) a two-dimensional Gaussian wave packetscattering off an Eckart barrier coupled with a harmonic oscillator potential inthe other degree of freedom.[9, 14, 15] The IFDM results for these three differ-ent applications were compared to the results of two other numerical methods:the Crank-Nicholson algorithm[11] (which is based on a finite difference solutionof the time-dependent Schrodinger equation) and the MLS/ALE approach[8](which is another quantum hydrodynamic method). These comparisons providean initial assessment of the convergence properties, numerical accuracy, compu-tational efficiency, and stability of the IFDM.The accuracy of the IFDM solution for the one-dimensional free Gaussian

wave packet was found to be in excellent agreement with the analytic resultwith an average percent difference of less than 5 × 10−5 % using a grid spacingof ∆x = 0.05 au. In contrast the Crank-Nicholson method required a muchsmaller grid spacing of ∆x = 0.0005 au to obtain an average percent difference

16

of 6 × 10−3 %. This is due to the fact that the Crank-Nicholson algorithmis directly solving for the highly oscillatory real and imaginary parts of thewave function. The accuracy of the 4-th order MLS/ALE solution for the freeGaussian wave packet was found to be similar to the IFDM. The stability ofthe IFDM was verified out to the edges of the fixed (Eulerian) grid where theprobability density is extremely small ρ < exp(−40, 000).For the one- and two-dimensional Eckart barrier tunneling problems, the

IFDM transmission probabilities were found to be in excellent agreement withboth the Crank-Nicholson and MLS/ALE methods. In all three applications,the IFDM proved to be more computationally efficient than either the Crank-Nicholson or MLS/ALE approach. For the one-dimensional Eckart barrier tun-neling problem, the IFDM transmission probabilities were computed 1.8 and6.9 times faster than those computed using the Crank-Nicholson and MLS/ALEmethods, respectively.[9] For the two-dimensional Eckart barrier tunneling prob-lem, the IFDM transmission probabilities were computed in 15 minutes com-pared to the 1 hour required by the ALE/MLS approach.[15]The latest version of the IFDM requires no filtering of the second derivative

of C in the evaluation of the quantum force.[15] The original version used amoving window average of ∂2C/∂x2 to help reduce the numerical noise andimprove stability.[9] We have found that this window averaging can be eliminatedall together by simply increasing the artificial viscosity coefficient to ν1o = 1 ×105 and truncating the fixed (Eulerian) grid to some reasonable value of thedensity (i.e., ρ > 1×10−8). A “smart” Eulerian grid is then implemented which“activates” or adds new edge points to the grid when the density at the edgepoint increases above a user specified threshold (i.e., ρ > 1× 10−8). The valuesof C and v at the edge point are initialized in a stable way by using a finitedifference representation of the appropriate Gaussian like boundary conditionsat the edge (i.e., ∂2C/∂x2 = constant and ∂2v/∂x2 = 0). These modificationsnot only improve the accuracy and resolution of the method by eliminating thewindow averaging of the second derivative of C, they also significantly improvethe computational efficiency of the method by reducing the size of the grid. Thecalculations are performed only at those “active” Eulerian grid points where thedensity is significant. In this way, the fixed Eulerian grid is optimized so that itexhibits similar computational advantages as a Lagrangian or ALE grid.

IV. CONCLUSIONS AND FUTURE WORK

The Iterative Finite Difference Method (IFDM) is a promising new approachfor the direct numerical solution of the quantum hydrodynamic equations ofmotion.[9] The set of non-linearly coupled finite differenced quantum hydrody-namic equations are solved iteratively at each time step using Newton’s method.The IFDM converges exponentially with respect to the iteration count and is

17

second order accurate in both space and time. The stability of the methodwas investigated using truncation error analysis and the functional form of thepotentially negative (unstable) numerical diffusion terms were identified. Thestability of the IFDM is ensured by introducing a positive (stabilizing) numeri-cal diffusion term (“artificial viscosity”) into the finite difference equations. Thisstabilizing term is of the same order as the errors which are already present inthe calculation. Thus, the stability of the method can be maintained even as theresolution of the calculation is increased. The methodology was briefly reviewedand the results of several initial benchmark calculations discussed. These initialapplications show that the IFDM is computationally efficient, stable, and nu-merically accurate. Recent improvements to the IFDM include the eliminationof all filtering or averaging of the second derivative of C and a truncation of thefixed (Eulerian) grid. A “smart” Eulerian grid is used for which the calculationsare performed only at active grid points which contain significant probabilitydensity.In its present form, the IFDM does not solve the “node problem” (i.e., the

singularities in the quantum potential and force which can occur in the regionswhere the density approaches zero due to the interference effects in the reflectedcomponent of a wave packet scattering off a barrier).[16] However, an importantfeature of the IFDM is that it remains stable even in the presence of nodes whichallows the propagation to continue and accurate transmission probabilities canbe computed. Also, the artificial viscosity technique can be used to mitigatethe “node problem” by preventing the formation of nodes (at the expense ofsmoothing out the solution in the region near the node).[8, 9] Future work willinclude investigation of a possible analytic treatment of the “node problem”within the IFDM, its extension to higher dimensions, and most importantly itsapplication to real molecular systems.

V. ACKNOWLEDGMENTS

This work was done under the auspices of the US Department of Energy at LosAlamos National Laboratory. Los Alamos National Laboratory is operated byLos Alamos National Security, LLC, for the National Nuclear Security Adminis-tration of the US Department of Energy under contract DE-AC52-06NA25396.

[1] L. de Broglie, C. R. Acad. Sci. Paris. 183, 447 (1926); 184, 273 (1927).[2] D. Bohm, Phys. Rev. 85, 166 (1952); 85, 180 (1952).[3] P. R. Holland, “The Quantum Theory of Motion”, Cambridge University Press,

New York, 1993.

18

[4] R. E. Wyatt, “Quantum Dynamics with Trajectories: Introduction to QuantumHydrodynamics”, Springer, New York, 2005.

[5] “Quantum Trajectories”, edited by P. K. Chattaraj, CRC Press/Taylor & FrancisGroup, 2010.

[6] C. Lopreore and R. E. Wyatt, Phys. Rev. Lett. 82, 5190 (1999).[7] R. E. Wyatt, Chem. Phys. Lett. 313, 189 (1999).[8] B. K. Kendrick, J. Chem. Phys. 119, 5805 (2003).[9] B. K. Kendrick, J. of Molec. Struct: THEOCHEM 943, 158 (2010).

[10] S. V. Patankar, “Numerical Heat Transfer and Fluid Flow”, Hemisphere Publish-ing Co., New York, 1980.

[11] W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, “NumericalRecipes; The Art of Scientific Computing”, Cambridge University Press, NewYork, 1986.

[12] J. VonNeumann and R. D. Richtmyer, J. Appl. Phys. 21 (1950) 232.[13] E. Scannapeico and F. H. Harlow, Los Alamos National Laboratory report LA-

12984, (1995).[14] D. K. Pauler and B. K. Kendrick, J. Chem. Phys. 120 (2004) 603.[15] B. K. Kendrick, unpublished.[16] R. E. Wyatt and E. R. Bittner, J. Chem. Phys. 113 (2000) 8898.

19

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Bangor

Kinematic Quantum Trajectories

Timothy M. Coffey∗

Department of Physics and Center for Complex Quantum Systems,

1 University Station C1600, University of Texas, Austin, TX 78712, USA

The typical route to quantum trajectories by Bohm [1, 2] uses Schrodinger’swave equation. The complex wave function is written in a polar form ψ(~x, t) =R(~x, t) exp[−iS(~x, t)] for real and single-valued functions R and S. The waveequation then implies a continuity equation and a quantum Hamilton-Jacobi’sequation. The particles evolve according to a guidance law, ~p = ∇S, and inaddition to the classical potential, they are also influenced by a quantum poten-tial Q = −(~2/2mR)∇2R. Bohm’s construction is dynamical in nature since itprovides causes for the particle motions: masses, forces, potentials. In addition,for Bohm these trajectories are ontological, that is, an electron, for example,actually is a particle with a specific position and momentum that evolves alonga trajectory [2].The kinematic route to quantum trajectories, on the other hand, doesn’t

utilize or solve any equations of motion, but rather develops a geometricalor informational description of only the evolution of the probability densityρ = ψ∗ψ. By design kinematic particle trajectories are conserved, and for alltimes are distributed according to the quantum probability density. There-fore, by their respective continuity equations the velocity fields are related,~vkinematic = ~vBohm + ~va/ρ with ∇ · ~va = 0. In one-dimension the gauge ve-locity va is necessarily zero, and thus in one-dimension all kinematic trajectorieswill be identical to Bohm’s trajectories.One possible way to achieve kinematic trajectories is by simply sampling the

density, sorting the sampled points, and then connecting the points in order [3].This sampling method, however, is limited to one-dimensional problems (and inhigher dimensions if the wave function is separable in each coordinate) due to thelack of a natural sorting order in higher dimensions. A recent method [4] thatemploys the geometrical construction of centroidal Voronoi tessellations(CVT)[5, 6], though, overcomes this ordering problem by maintaining particle identitythroughout the calculation. Using the CVT method for separable and non-

∗Email: tcoffey@physics.utexas.edu

20

separable two-dimensional examples suggest that the velocity gauge ~va, mighttypically be zero, which means that the CVT and Bohm trajectories typicallymight be identical. The only disparity known at this time between the CVTmethod and Bohm trajectories is around a persistent stationary node. Aroundthe node the Bohm trajectories might have circular orbits or be at rest, while theCVT trajectories are always at rest. Neither case is experimentally verifiable,however, but the CVT method seems to provide a more consistent descriptionof the situation.The CVT approach realizes that the particles themselves are some sort of

representation of the probability density. For a finite number of particles, N ,the measure of the quality of the representation is done by introducing a noveldistortion functional,

D =

N∑i=1

∫Ci

(~x− ~xi)2ρ(~x)γ d~x, (1)

where γ = (k + 2)/k, k is the number of dimensions (the length of each po-sition vector ~xi), and each integration is taken over the Voronoi volume [7],Ci = ~x | ‖ ~x − ~xi ‖≤‖ ~x − ~xj ‖ for all j 6= i.. This distortion functionalis similar to the distortion measure in the field of vector quantization or signalcompression [5], though the word ‘quantization’ in this field has nothing to dowith quantum mechanics. In vector quantization the distortion functional hasthe same form as (1) except with γ = 1. The best representation of the proba-bility density, the set of particle ~xi’s, is defined to be the one which minimizesthe distortion. The characteristic of the minimum particle configuration is thatthey form a CVT. The global minimization of the distortion function introducesa non-locality behavior of the CVT trajectories, which is also present for Bohmtrajectories.Frequently, the CVT’s are computed using the Lloyd-Max iterative algorithm

(also known in the literature simply as the Lloyd algorithm) [8, 9]. The Lloyd-Max algorithm typically begins with a random sampling of the density as shownby the dots in Figure 1(a); the density has two high dense regions in the lower-left and upper-right corners of the figure. The lines in the figure are the Voronoitessellation of this particle configuration. During each iteration the algorithmcomputes the Voronoi tessellation of the particle positions, then each particle ~xiis moved to the center of mass or centroid of its particular Voronoi cell Ci by, ~x

′

i =∫Ci

ρ(~x)~x dV/∫Ci

ρ(~x) dV. The algorithm continues until some stopping criterion

is satisfied; typically either a fixed number of iterations, or the maximum distanceany particle moves during the iteration is less than some small predeterminedvalue. Shown in Figure 1(b) is the resulting CVT after 200 such iterations.Notice the uniformity of the structure in part (b) as opposed to the tessellationin part (a). The Lloyd-Max algorithm is beneficial since it is easy to implement,and has several non-degeneracy and global minimum or fixed-point convergenceproofs in one and many dimensions [10–12]. More importantly, the algorithm

21

(a) (b)

FIG. 1: (a) A Monte-Carlo sampling of a probability density with high dense regions inthe lower-left and upper-right. The single dots represent the possible particle positions.The straight lines are the Voronoi tessellation of these particle positions. (b) Thecentroidal Voronoi tessellation after 200 iterations of the Lloyd-Max algorithm thatbegan from the initial sampling.

keeps track of each particle’s position during the entire computation, which isnecessary for identifying each particle to build its trajectory.The CVT trajectory method begins by sampling the probability distribution at

t = 0 to get N particle positions. The initial sampling, then, is used to constructan initial CVT at t = 0. Time is then advanced a small amount δt, and ratherthan resampling the probability density, the CVT at t = 0 is recycled and usedas the initial particle configuration for the CVT computation at the new time.The calculation of the center of mass of each Voronoi cell Ci during the Lloyd-Max algorithm is done not with the usual ρ(~x; t), but instead with ρ(~x; t)(k+2)/k

in keeping with the distortion functional in (1). Recall that the Lloyd-Maxalgorithm keeps track of each particle’s position during the CVT computation,therefore, one can chain the i-th particle’s positions at the various times to forma trajectory. In Figure 2 is shown an example of a two-dimensional CVT atthree times, and the construction of a particular trajectory. Notice that theCVT method will never have trajectory intersections, which again is a familiarbehavior of the Bohm trajectories as well.In Figure 3 is a random subset of 20 trajectories from the total ensemble of

N = 400 trajectories computed for a two-dimensional free gaussian packet. Thepacket begins concentrated at the center of the box, and then spreads out in time.For each CVT trajectory (+) the corresponding Bohm trajectory (solid line) iscalculated. In the figure, we can see that the CVT trajectories match the Bohmtrajectories quite well. In fact, for the whole ensemble the Pearson correlation

22

FIG. 2: At each time a centroidal Voronoi tessellation(CVT) is computed using theparticle positions from the previous time as input. The Lloyd-Max algorithm beginswith the old positions, but uses the probability distribution at the new time. Eachparticle’s position is tracked during each iteration of the Lloyd-Max algorithm. Afterthe algorithm stops the trajectories are constructed by mapping a particle’s old positionto the new position as shown in the figure.

coefficients between the components of the CVT and Bohm trajectories wererx = 0.996 and ry = 0.997.Recall that the Bohm trajectories are derived from dynamical equations of

motion, while the CVT method utilizes a geometrical depiction of the density.By their nature, the CVT trajectories are simply kinematically portraying theevolution of the probability density. In the cases in which the CVT methodresults in Bohm trajectories, it would seem it is unreasonable to interpret theBohm trajectories as ontological. Rather, the Bohm trajectories, like the CVTtrajectories, should be thought to simply be kinematically depicting the evolu-tion of the probability density as well.

23

+++++++++++

+++++++++++

+++++++++++

+++++++++++

+

+

+

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

+++++++++++

++++++++++++

+

+

++

+

+

+

+

+

+

+++++++++++

+

+

+

+

+

+

+

+

+

+

+

+++++++++++

+

+

+

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

+++++++++++

+++++

++++

++

+++++++++++

++

+

+

+

+

+

+

+

+

+

+++++++++++

+++++++++++ ++

+++++++++

20 30 40 50 60 70 80xHtL20

30

40

50

60

70

80

yHtL

FIG. 3: A comparison of the Bohm trajectories (solid line) and the CVT trajectories(+) for a two-dimensional gaussian wave packet. The gaussian begins concentratedat the middle of the figure, and as time progresses the gaussian spreads. The figureshows a random subset of 20 trajectories from the total of 400 particles used in thecalculation.

[1] D. Bohm, Phys. Rev. 85 (1952) 166, 180.[2] D. Bohm and B. J. Hiley, The Undivided Universe (Routledge, New York, 1993).[3] T. M. Coffey, R. E. Wyatt and W. C. Schieve, J. Phys. A: Math. Theor. 41 (2008)

335304.[4] T. M. Coffey, R. E. Wyatt and W. C. Schieve, J. Phys. A: Math. Theor. 43 (2010)

335201.[5] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression (Kluwer

Academic, Boston, 1992).[6] Q. Du, V. Faber and M. Gunzburger, SIAM Rev. 41 (1999) 4 637.[7] A. Okabe, B. Boots and K. Sugihara, Spatial Tessellations (John Wiley & Sons,

New York, 1992).[8] S. P. Lloyd, IEEE Trans. Inf. Theo. IT-28 (1982) 129, Reprinted in Quantization,

edited by P.F. Swaszek (Van Nostrand Reinhold, New York, 1985).[9] J. Max, IEEE Trans. Inf. Theo. IT-6 (1960) 7, Reprinted in Quantization, edited

by P.F. Swaszek (Van Nostrand Reinhold, New York, 1985).[10] Q. Du, M. Emelianenko and L. Ju, SIAM J. Num. Anal. 44 (2006) 1 102.[11] Q. Du and M. Emelianenko, SIAM J. Num. Anal. 46 (2008) 3 1483.[12] M. Emelianenko, L. Ju and A. Rand, SIAM J. Num. Anal. 46 (2008) 3 1423.

24

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Complex Trajectories and Dynamical Origin of Quantum

Probability

Moncy V. JohnDepartment of Physics, St. Thomas College,

Kozhencherry, Kerala 689641, INDIA

Complex quantum trajectories were first obtained and drawn by adopting amodified de Broglie-Bohm approach to quantum mechanics [1]. First we notice

that substituting eiS/~ for Ψ in the Schrodinger equation gives the quantumHamilton-Jacobi equation [2]. Inspired by its similarity with the correspondingclassical equation, we postulate an equation of motion similar to that in thepilot-wave theory of de Broglie [3],

mx ≡∂S

∂x=

~

i

1

Ψ

∂Ψ

∂x, (1)

for the particle. To use this, one needs to find S from the standard solution Ψof the Schrodinger equation. The trajectories x(t) are obtained by integratingthe above equation with respect to time. In general they will lie in a complexx-plane. The eigentrajectories x(t) ≡ xr(t)+ ixi(t) in the free particle, harmonicoscillator and potential step problems and trajectories for a wave packet solutionwere obtained in [1]. As an example, some of the complex trajectories in then = 1 harmonic oscillator are shown in figure 1. These appear as the famousCassinian ovals. The Jacobi lemniscate that passes through xr = 0, xi = 0 is aspecial case of these ovals.One of the challenges before this complex quantum trajectory representation

is to explain the quantum probability axiom. In a recent work which exploresthe connection between probability and complex quantum trajectories [4], theBorn’s probability density to find the particle around some point on the real axisx = xr0 was found to be given by

Ψ⋆Ψ(xr0, 0) ≡ P (xr0) = N exp

(

−2m

~

∫ xr0

xidxr

)

, (2)

where the integral is taken along the real line. Since it is defined and usedonly along the real axis, the continuity equation for probability in the standard

25

FIG. 1:

quantummechanics is valid here also, without any modifications. This possibilityof regaining the quantum probability distribution from the velocity field is aunique feature of the complex trajectory formulation. For instance, in the deBroglie approach, the velocity fields for all bound eigenstates are zero everywhereand it is not possible to obtain a relation between velocity and probability.

In addition, we consider it desirable to extend the probability axiom to thexrxi-plane and hence look for the probability of a particle to be in an areadxrdxi around some point (xr, xi) in the complex plane. Let this quantity bedenoted as ρ(xr, xi)dxrdxi. Here it is natural to impose a boundary conditionthat ρ agrees with Born’s rule along the real line. Then one must see whethera continuity equation for probability holds everywhere in the xrxi-plane. Anextended probability density that satisfies these two conditions in most regionsof the complex plane was proposed in [4] as

ρ(xr, xi) = ρ0 exp

[

−4

~

∫ t

t0

Im

(

1

2mx2 + V (x)

)

dt′]

, (3)

where the integral is taken along the trajectory [xr(t′), xi(t

′)]. One needs toknow ρ0 at some initial point (xr0, xi0) on the trajectory and if we choose thisas (xr0, 0), the point of crossing of the trajectory on the real line, then ρ0 maytake the value P (xr0) and may be found using (2). It was shown in [4] that ρ in(3) satisfies the desired continuity equation for the particle, as it moves along.However, it is seen that a solution of the continuity equation cannot satisfy

the boundary condition for some regions of the complex plane, such as the regioninside the lemniscate where trajectories do not enclose the poles of x. In thecontext of solving the continuity equation, it is easy to see that this is due tothe boundary condition overdetermining the problem. In the trajectory integralapproach, one can explain it as a disagreement of the values of ρ at two consecu-tive points of crossing of the trajectory on the real axis, with that prescribed byBorn’s rule. Given this situation, we look for a trajectory integral definition forρ that can agree with the Born’s rule (on the real line) in such regions, even if it

26

does not obey the continuity equation in the extended region. Such a definition,similar to that in equation (3), was found in [5] as

ρ′(xr, xi) ∝ P (xr0) exp

[

−4

~

∫ t

t0

Im

(

1

2mx2

)

dt′]

. (4)

Comparing with (3), we note the absence of the potential term V (x) in theintegrand. This trajectory integral definition for such regions can be seen togive Ψ⋆(x)Ψ(x) with x complex. This is independently considered in [6, 7], butfor the entire complex plane. Since here it is defined only for the subnests whichdo not enclose the poles of x, there is no difficulty in normalising the combinedprobability for the entire plane, which is shown in figure 2.

FIG. 2:

Another important observation we make in this connection is regarding theclassical limit of quantum mechanics. It may be noted that the probabilityaxiom in this modified de Broglie formulation helps to distinguish the classicallimit of quantum harmonic oscillator as one in which the oscillator is probableto be found only very close to the real axis [5]. This result is very significant forcomplex quantum trajectories, for it explains why the complex extension is notobservable even indirectly in the classical limit. We anticipate that this propertyis generally true.

[1] M.V. John, Found. Phys. Lett. 15 (2002) 329; quant-ph/0102087 (2001).[2] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1980.[3] G. Bacciagaluppi, A. Valentini, Quantum Theory at the Crossroads, Cambridge

University Press, Cambridge, (2009); quant-ph/0609184v1 (2006).[4] M.V. John, Ann. Phys. 324 (2009) 220; arXiv:0809.5101 (2008).[5] M.V. John, Ann. Phys. DOI. 10.1016/j.aop.2010.06.008 (2010)[6] C.-C. Chou, R.E. Wyatt, Phys. Rev. A 78 (2009) 044101.[7] C.-D. Yang, Chaos, Solitons Fractals 42 (2009) 453.

27

28

K. H. Hughes and G. Parlant (eds.) Quantum Trajectories

© 2011, CCP6, Daresbury

Energy Rays for Electromagnetic Pulses Scattering from Metal-Dielectric

Structures

Robert E. Wyatt

Department of Chemistry and Biochemistry

University of Texas

Austin, Texas 78712

In this study, time-dependent electromagnetic (EM) energy distributions

are described for femtosecond pulses scattering from a micron-scale metal-

dielectric structure. Associated with these distributions are ensembles of energy

rays, the integral curves of the Poynting vector field. The energy rays

demonstrate how- and when- EM energy is redistributed in space and time.

Typically, energy rays show bending and undulations, even in ‘free space’ and in

regions where the refractive index is invariant. Energy rays for the EM field are

analogous to Bohm trajectories in quantum mechanics [1], which in the

Hydrodynamic Formulation of Quantum Mechanics [2, 3] are integral curves for

the probability flux vector field. By analogy, energy rays in EM theory form a

central element in the Hydrodynamic Formulation of Electrodynamics [3, 4].

Several earlier investigations have reported computational results on energy rays

[5-10]. Unlike the results reported here, these studies dealt with monochromatic

scattering processes.

Energy rays for EM scattering generally have very different properties

from the rays in geometrical optics (GO). In the traditional approach to GO [11],

geometrical rays are solutions to the ray equation, d / ds(ndr / ds) n, where

29

s is the arc length along the ray. It is important to note that these rays bend only

when the refractive index changes. The ray equation is equivalent to a set of 1-st

order equations [12]: 0/ ( / 2 )(2 ),dr dt c k k / ,dk dt G where

k S and the geometrical potential is given by 2

0 0( ) ( / 2 )( ) .G r c k nk

Not very well known is the approach to optical rays that Luneburg developed in

the mid-1940s [13]. By applying the method of characteristics to the two

Maxwell curl equations, he showed that the ray equation can be obtained without

approximation. As a result, it can be stated that the short wavelength

approximation involved in the traditional GO derivation is a sufficient but not

necessary condition. More recently, an alternate derivation of these results has

been presented [14] which does not rely on the method of characteristics. Also,

Orefice et al [12] recently derived exact ray tracing equations from the

Helmholtz equation. These equations are similar to the 1-st order equations

mentioned previously, except that the total potential in the second equation is the

sum of G and the wave potential, defined by 2

0( / 2 )( ) / ,W c k R R where

R is the wave amplitude

In this study, energy rays and EM energy densities will be presented for

a time-dependent scattering environment involving a micron-scale composite

structure having both dielectric and metallic regions. The fields used to generate

these results were obtained by numerically integrating the Maxwell curl equations

using the FDTD (finite-difference time-domain) algorithm [15-17]. The current

study, along with the recent work of Chu and Wyatt [18], are the first to blend the

accurate computation of time-dependent EM with synchronous generation of

energy rays. For the model studied here, the grating problem, there are three

dielectric segments, ‘optical holes’, in the grating and they are separated by

30

reflective metallic segments. Circular electromagnetic wave fronts emanating

from the source point first impinge on the central dielectric segment and then

encounter the two outer segments at slightly later times.

Maxwell’s two curl equations are given by:

E B/ t, H J D/ t. The following constitutive relations for

linear isotropic non-magnetic media are assumed in these studies, 0 rD E,

0B H, where 0 and 0 are the vacuum permittivity and permeability, and

r is the relative permittivity (dielectric constant). For these studies of scattering

in two dimensions, zˆE E k is perpendicular to the x-y plane and the

components of H in this plane are given by x y(H ,H ). At one grid point, a

modulated Gaussian source was used to create the electric field. The perfectly

matched layer method [19] was used to prevent echoes from the edges of the grid

(also see refs. [16, 17]).

Equations for the energy rays may be introduced by first considering

Poynting’s theorem [20], which is given by U / t S J E, in which

J is the current density. In this equation, the Poynting vector and energy density

are given by 0S E B/ , 0 0U (1/ 2) E E B B/ , where S is the

energy flux and U is the energy density. From S and U, the equation of

motion for an energy ray, given the starting position 0r , is dr / dt S/ U. In

the current study, an ensemble involving several hundred trajectories were

launched at a delayed starting time, chosen so that the field at the source point

had decayed to a negligible value. The equation for energy rays is analogous to

31

one of the equations that can be used to compute quantum trajectories [2] in terms

of the probability flux and the probability density.

FIGURE 1. Energy rays for the grating problem superimposed upon a contour

map of the electric field at the final time step. An ensemble of rays is launched

from the grid (green dots) near the left-center region of the figure and some of

these rays (red) transmit through the dielectric segments in the grating. The

reflected rays (yellow) on the left side of the grating experience large deflections.

32

FIGURE 2. Enlargement of the lower-left region of Fig. 1. One transmitted ray

(red) and three reflected rays (pink and green) are highlighted. The pink reflected

ray makes several bounces near the metal surface, and the two green rays make

abrupt turns before departing from the reflection zone.

Computational results for grating scattering problem are illustrated in Figs. 1

and 2. The grating is shown as the vertical strip in Fig. 1, with the metal

segments (gray) separating three dielectric windows. The Gaussian pulse was

created at a source point on the left side of the figure and an ensemble of rays was

launched from a set of grid points (green dots) located between the source point

and the grating. Rays that transmit the dielectric segments are plotted in red at

the final time (800 time steps, or 136 fs) and rays that reflect from the grating are

shown in yellow. These rays are superimposed upon a contour map of the electric

field. Although it appears that rays may intersect, they never pass through the

33

same spatial point at the same time. Rays that transmit the outer dielectric

segments are focused toward larger values of the y coordinate when they

emerge from the grating. In addition, there is extensive bending of the reflected

rays and the turning points for many of these rays occur to the left of the metal

segments. These rays are reflecting in ‘free space’, relatively far from the

reflective metal segments. An enlargement of the lower-left region of this figure

is shown in Fig. 2. One transmitted ray (red) and three reflected rays (pink and

green) are highlighted. The transmitted ray first bounces from the metal surface,

then makes it into the dielectric segment. The pink reflected ray makes several

bounces near the metal surface, and the two green rays make sharp turns before

departing from the reflection zone. Many of the reflected rays bounce several

times from the metal surface before finally escaping from this region.

For monochromatic scattering, we have already mentioned that a system

of coupled 1-st order equations has been derived for the energy rays [12]. In an

extension of earlier work dealing with spin 1/2 particles, Holland replaced the

Maxwell curl equations by a matrix equation for the Riemann-Silberstein vector

[29], which specifies the state of the fields. This state function was then

represented in polar form and Hamilton-Jacobi (HJ) and continuity equations

were derived for the amplitude and action functions. The HJ equation contains an

effective potential having a structure similar to the Bohm quantum potential.

Further investigation of the computation and properties of wave potentials would

be very informative.

A novel approach to the computation of energy rays has recently been

described by Coffey and Wyatt [30]. These rays were generated without

employing the Poynting vector and no equations of motion were assumed or used.

A similar method was used previously to generate quantum trajectories directly

from the time evolving probability density [31].

34

Acknowledgment

This work was supported in part by a research grant from the Robert Welch Foundation

(grant number F-0362). We thank Tim Coffey and Chia-Chun Chou for many informative discussions.

[1] D. Bohm, Phys. Rev. 85, 166 (1952).

[2] R. E. Wyatt, Quantum Dynamics with Trajectories (Springer, New York, 2005).

[3] A. S. Sanz and S. Miret-Artes, to be published. [4] I. Bialynicki-Birula, Nonlinear Dynamics, Chaotic and Complex Systems, E. Infeld, R. Aelazny,

and A. Galkowski (Cambridge University Press, Cambridge, 1997), p. 64.

[5] R. D. Prosser, Int. J. Theor. Phys. 15, 169 (1976); H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, Phys. Rev. E 62, 7330 (2000); T. Wunscher, H. Hauptmann, and F. Herrmann, Am. J. Phys.

70, 600 (2002); E. Hesse, J. Quant. Spect. & Rad. Trans. 109, 1374 (2008); A. S. Sanz, M.

Davidovic, M. Bozic, S. Miret-Artes, Ann. Phys. 325, 763 (2010); M. Gondran and A. Gondran, Am. J. Phys. 78, 598 (2010).

[6] M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 2003).

[7] A. Orefice, R. Giovanelli, and D. Ditto, Found. Phys. 39, 256 (2009). [8] R. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

[9] C. Belevier-Cebrerus and M. Rodriguez-Danta, Am. J. Phys. 69, 360 (2001).

[10] K. S. Yee, IEEE Transactions on Antennas and Propagation, 14, 302 (1966). [11] A. Taflove and S. C. Hagness, Computational Electrodynamics, The Finite-Difference Time-

Domain Method (Artech House, Boston, 2005). [12] C. C. Chu and R. E. Wyatt, to be published

[13] J. P. Berenger, J. Comp. Phys. 114, 185 (1994).

[14] F. Richter, M. Floian, and K. Henneberger, Europhys. Lett. 81, 67005 (2008) [15] P. R. Holland, Proc. Roy. Soc. A 461, 3659 (2005).

[16] T. M. Coffey and R. E. Wyatt, to be published.

[17] T. M. Coffey, R. E. Wyatt, and W. C. Schieve, J. Phys. A 43, 335301 (2010).

35

K. H. Hughes and G. Parlant (eds.) Quantum Trajectories

© 2011, CCP6, Daresbury

Quantum Dynamics through Quantum Potentials

S. Duley, S. Giri, S. Sengupta and P. K. Chattaraj*

Department of Chemistry and Centre for Theoretical Studies, Indian Institute of Technology

Kharagpur, Kharagpur 721302, India

* Author for correspondence: Email: pkc@chem.iitkgp.ernet.in

The hydrodynamic interpretation of quantum mechanics was first

formulated by Madelung [1] soon after the introduction of Schrodinger’s wave

equation. In this formalism the Schrödinger time dependent equation for a single

particle is transformed into two fluid dynamical equations: an equation of

continuity, and an Euler type equation of motion wherein the probability density

defined by 2

is interpreted as the charge density, 2

and the velocity v

of this charge fluid is obtained from the phase of the complex-valued

wavefunction. The introduction of the concept of a fluid with associated density

and velocity to study time evolution of a quantum system provides a “classical”

approach to describe quantum phenomena. It was through the works of Bohm

[2,3], de Broglie [4,5] and Takabayashi [6,7] the hydrodynamic interpretation has

gained acceptance and has seen wide use in different areas of physics and

chemistry.

To obtain the quantum fluid dynamical equations, the time dependent

single-particle wavefunction is expressed as an ansatz (polar form)

),(exp),(),( 2

1trSi

trtr (1)

Where 22R ; R and S being real functions of position and time.

Substituting this ansatz in the single particle time dependent Schrödinger equation

tiV

m

22

2 (2)

and separating the real and imaginary parts, the two quantum fluid dynamical

equations are obtained as follows, the continuity equation

36

0)(.

v

t

(3)

and the equation of motion;

)( quVVdt

dvm (4)

where

Sm

v 1 (5)

vvt

v

dt

dv).(

(6)

and

2

1

2

1

22

/2

m

Vqu

(7)

Like the time dependent Schrödinger equation from which they are derived, the

fluid dynamical equations essentially constitute an initial value problem. Since

the wavefunction is a single valued function of position, the associated fluid

dynamical variables R or equivalently 2

1

and S need to satisfy certain conditions.

The single valuedness of requires that R be single-valued as well, while the

phase function S is not so constrained. In fact due to the polar form of the ansatz

(equation 1), S can be multivalued so that any two S functions differing by an

integral multiple of 2 may give rise to the same . In a nodal region the

phase function is not well defined and may undergo a discontinuous jump. It has

been pointed out by Wallstrom [8,9] that this condition on the multivalued S

function has to be introduced properly in the solution of the Schrödinger

equation. It has been shown [8] that unless this condition on S is imposed for

regions separated by nodes where 0 an infinite number of different

solutions to the fluid dynamical equations may arise.

The fluid dynamical equation of motion (equation 4) contains two potential terms,

viz. the classical potential V and a quantum potential or the so called Bohm

potential [2,10] quV . So unlike classical hydrodynamics the quantum fluid is

subjected to an additional potential which is of purely quantum origin, so that the

dynamics of local fluid dynamical quantities are dependent on this quantum

potential [11]. However the global behaviour of physically observable quantities

of the quantum systems are found to be unaffected by this quantum potential [11],

as the expectation values of position and momentum coordinates for the particle

are dependent only on the externally applied potential V. In this regard the basic

37

character of the fundamental fluid dynamical variables, viz., charge density

and the fluid velocity v requires some attention. While the charge density is a

physical observable the local velocity field v cannot be so regarded. It has been

shown explicitly by Kan and Graffin [12] that the current density vj being

a physical observable the fluid velocity v cannot be represented by a linear

Hermitian operator so that it cannot be a physical observable.

Although the quantum fluid velocity field v is not a physical observable

it offers a significant advantage in carrying over the common classical pictures

into the quantum domain and to “classically” explain certain quantum processes.

The quantum trajectories [13] introduced in the de Broglie-Bohm causal

interpretation of quantum mechanics play a key role. The quantum trajectories are

obtained from the equation 5 by solving

Smdt

drv

1 (9)

to obtain the function )(tr . It should be noted that the fluid velocity m

Sv

is

related to the complex momentum p as

2

1

lni

vmp . (10)

and the individual trajectories computed from the equation 9 for different initial

positions do not represent the actual single particle quantum system. The quantum

trajectories have been computed by Hirschfelder et al [14] for a Debye-Picht

[15,16] wavepacket incident on a two dimentional squre potential barrier and

other [17,18] dynamical systems. The single-valuedness of the wavefunction ,

implies in the Madelung transcription a similar single valuedness on the

probability amplitude R and a quantization condition on the gradient of the phase

function S. Since the fluid velocity is expressed as m

Sv

, then this condition

can be expressed as

m

nhdrv

C

. (11)

where ......2,1,0 n . Using Stoke’s theorem this can be related to a

surface integral over the area bounded by C as

m

nhdAv

A

).( (12)

38

where ......2,1,0 n . But the quantity )( v is the well known

hydrodynamical vorticity allowing the equation 12 to be recast as

m

nhdA

A

. (13)

with the quantization condition, ......2,1,0 n . For an area containing a

node of the wavefunction 0, n implies a nonzero value of the vorticity .

Since satisfies the quantization condition as given by equation 12, the

vorticity around wavefunction nodes is quantized, as first discovered by Dirac

[19, 20]. Furthermore it has been pointed out that [21] since the wavefunction can

be generally divided into a real and an imaginary component, a wavefunction

node requires that both components are simultaneously equal to zero.

In their study of a Hydrogen atom colliding with a collinear Hydrogen

molecule McCullough and Wyatt [22] have reported occurrence of a vortex in the

probability current plot, which have been termed by them as “the quantum

whirlpool effect”. This has been further confirmed by studies of Kuppermann and

coworkers [23]. A detailed study of nodal topology of wavefunctions giving rise

to such vortices has been carried out by Reiss [24-27] and Heller et al [28]

The quantum fluid dynamical equations for a single particle have been

extended to situations where there is a magnetic field along with the scalar field.

If E and B are the electric and magnetic fields, respectively, the fluid dynamical

equations read [29] as

0)(.

v

t

(14)

)()( quVVvc

ee

t

vm

BE (15)

where the fluid velocity is

)(1

Ac

eS

mv . (16)

The spin-magnetic interaction term has been included [28] and the modified fluid

dynamical equations have been obtained. The electron spin has been incorporated

more rigorously using the Pauli spin and the relevant equations [7,30,31] have

been derived, while the quantum fluid dynamical equations for the relativistic

case have been obtained [32].

As in the case of single particle systems, the Schrödinger equation can

be transformed into a continuity equation and an Euler-type equation of motion

for many particle systems as well. For an N- particle system these equations have

the general form

39

N

i

ivt 1

0)(. (17)

N

i

quNcliiii

i VrrVvvt

v

1

1 ).....,,().( (18)

where i denotes the ith particle of the N- particle system. However, the fluid-

dynamical quantities like the density and velocity field correspond to a fluid in

the 3N dimensional configuration space, and do not have any direct physical

significance for N>1. In this case the interpretation of the essentially fluid

dynamical features like vortices and streamlines are no longer germane and the

power of visualization so unique to the single particle quantum fluid dynamics

appear to be lost. Attempts to directly project the fluid dynamical equations for

such many-particle systems onto the 3D Euclidean space have not seen much

success [33], except within some approximate single body theories like the

Hartree and the Hartee-Fock theories. While Takabayashi [6] formulated the

quantum fluid dynamical equations for a many-particle system using Hartree

theory, Wong and coworkers [34, 35] have applied the time dependent Hartree-

Fock theory for the N- body quantum fluid system. Using the concept of natural

orbital theory [36], a many-electron system subject to electric and magnetic fields

has been treated within a 3D quantum fluid dynamics [37].

The formulation of density functional theory (DFT) [38, 39] for many

particle systems has been very successful in explaining the electronic structure,

bonding and properties of atoms and molecules for time independent situations

and for ground states in terms of single-particle density ),( tr as the

fundamental variable. The essence of the density functional theory is that the

electron density contains all information about the ground state of a many-

electron system and the true charge density distribution minimizes the related

energy functional [38]. The density functional theory has been further extended to

a time dependent situation [40, 41]. To treat the dynamical problem for many

electron systems the quantum fluid dynamics (QFD) [42,43] and the time

dependent density functional theory (TDDFT) have been combined to obtain the

quantum fluid density functional theory (QFDFT) [44-56]. In the quantum fluid

density functional theory an N- electron system is mapped onto a system of N

noninteracting particles moving under the influence of an effective potential

),( tveff r obeying a generalized nonlinear Schrödinger equation as (in au)

t

tittveff

),(),(),(

2

1 2 rrr (19)

40

which is solved to yield the time dependent single “orbital” ),( tr for the many-

electron system considered. The quantum fluid dynamical quantities of this

system viz. ),( tr and the current density ),( trj or the velocity is obtained as

2 (20)

j (21)

where

)(exp),( 2

1

it r (22)

Recently QFDFT has been successfully applied in solving the ion-atom collision

problems [46-50, 55,56] and atom-field interaction problems [51,52, 54].

The quantum fluid dynamics of nonlinear oscillators have been

investigated [57-59] to study the signatures of nonintegrability through the

hydrodynamic formalism. Certain distinctive features in the processes involving

many particle systems such as an ion colliding with a many – electron atom [60]

have been studied using the quantum fluid density functional theory in order to

understand the associated charge transfer process and related electronic structure

principles.

Counter to the conventional quantum mechanics, de Broglie [61-65]

introduced the idea that the wavefunction should instead of replacing the concept

of material point be in coexistence with the point particle thus extending the

classical concepts into quantum domain. It was suggested that in the

nonrelativistic situation an ensemble of identical particles located in different

places may be attached to the Schrödinger wavefunction, so that their probability

of finding is governed by 2

as proposed by Born. Essentially de Broglie

assigned a dual role to be played by the Schrödinger wavefunction so that

along with its conventional interpretation mentioned above, it also causally

directs the trajectory of an ensemble element. This guidance principle was used

by de Broglie [4,10] to construct the stationary state orbits of hydrogen atom.

In summary, various quantum potential based approaches like quantum

fluid dynamics, quantum theory of motion and quantum fluid density functional

theory compliment the conventional quantum dynamics through its classical

interpretation.

Acknowledgment

We want to thank CSIR, New Delhi for financial support and Professor A. S. Sanz for going through

the ms.

41

[1] E. Madelung, Z. Phys. 40 (1926) 322

[2] M. Jammer, “The Philosophy of Quantum Mechanics” (Wiley, New York, 1974).

[3] E. Squires, “The Mystery of the Quantum World” (Adam Hilger, Bristol, 1986). [4] L. de Broglie, “Nonlinear Wave Mechanics: A Causal Interpretation” (Elsevier Amsterdam,1960).

[5] L. de Broglie, “The current Interpretation of wave Mechanics” (Elsevier Amsterdam, 1964).

[6] T. Takabayashi, Prog. Theor. Phys. 8 (1952) 143. [7] T. Takabayashi, Prog. Theor. Phys. 9 (1953) 187.

[8] T. C. wallstrom, Phys. Lette. A 184 (1994) 229.

[9] T. C. wallstrom, Phys. Lette. A 49 (1994) 1613. [10] F. J. Belinfante, “A Survey of Hidden Variable Theories” (Pergamon Press, New York, 1973).

[11] L. Janossy, M. Ziegler-Naray, Acta. Phys. 86 (1974) 262.

[12] K. K. Kan, J. J. Griffin, Phys. Rev. C 15 (1977) 1126. [13] (a) P. R. Holland, "The Quantum Theory of Motion" (Cambridge University Press, Cambridge

UK, 1993); (b) R. E. Wyatt, "Quantum Dynamics with Trajectories: Introduction to Quantum

Hydrodynamics" (Springer, New York, 2007).(c) "Quantum Trajectories", ed. P. K. Chattaraj, (Taylor & Francis/CRC Press, Florida, 2010 (in press))

[14] J. O. Hirschfelder, A. C. Christoph, W. E. Palke, J. Chem. Phys. 61 (1974) 5435.

[15] J. Picht, Ann. Phys. (Leipz.) 30 (1909) 755. [16] P. Debey, Ann. Phys. (Leipz.) 77 (1925) 685, 785.

[17] J. O. Hirschfelder, K. T. Tang, J. Chem. Phys. 64 (1976) 760.

[18] J. O. Hirschfelder, K. T. Tang, J. Chem. Phys. 65 (1976) 470. [19] P. A. M. Dirac, Proc. Roy. Soc. A 133 (1931) 60.

[20] I. Bialynichi-Birula, Z. Bialynichi-Birula, Phys. Rev. D 3 (1971) 2410.

[21] J. O. Hirschfelder, J. Chem. Phys. 67 (1977) 5477. [22] E. A. McCullogh, Jr., R. E. Wyatt, J. Chem. Phys. 54 (1971) 3578.

[23] A. Kuppermann, J. T. Adams, D. G. Truhlar, Seventh Intern. Conf. Physics of Electronic and

Atomic Collisions, Belgrade, Yugoslavia. [24] J. Riess, Ann. Phys. 67 (1971) 347.

[25] J. Riess, Helv. Phys. Acta. 45 (1972) 1066.

[26] J. Riess, Phys. Rev. D 2 (1970) 647. [27] J. Riess, Phys. Rev. B 13 (19676) 3862.

[28] D. F. Heller, J. O. Hirschfelder, J. Chem. Phys. 66 (1977) 1929.

[29] L. Janossay and M. Ziegler-Naray, Acta. Phys. Hung. 16 (1964) 345. [30] L. Janossay and M. Ziegler-Naray, Acta. Phys. Hung. 20 (1966) 233.

[31] L. Janossay, Found. Phys. 3 (1973) 185.

[32] T. Takabayashi, Prog. Theor. Phys. Japan 14 (1955) 283. [33] L. Janossy, Found. Phys. 6 (1976) 341.

[34] C. Y. Wong, J. Math. Phys. 16 (1976) 1008.

[35] C. Y. Wong, J. A. Marhun, T. A. Welton, Nucl. Phys. A 253 (1975) 469.

[36] W. H. Adams, Phys. Rev. 183 (1969) 31, 37.

[37] S. K. Ghosh, B. M. Deb. Int. J. Quant. Chem. 22 (1982) 871.

[38] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [39] W. Kohn, L. J. Sham, Phys. Rev. A 140 (1965) 1133.

[40] E. Runge, E. K. U. Gross, Phys, Rev. Lett. 52 (1984) 997.

[41] A. K. Dhara, S. K. Ghosh, Phys. Rev. A 35 (1987) 442. [42] B. M. Deb, S. K. Ghosh, J. Chem. Phys. 77 (1982) 342.

[43] L. J. Bartolotti, Phys. Rev. A 24 (1981) 1661.

[44] B. M. Deb, P. K. Chattaraj, Chem. Phys. Lett. 148 (1988) 550.

42

[45] (a) B. M. Deb, P. K. Chattaraj, “Solution: Introduction and Applications” (ed. M. Lakshmanan, Springer-Verlag, Berlin, 1988); (b) P. K. Chattaraj, “Symmetries and Singularity Structures:

Integrability and Chaos in Nonlinear Dynamical Systems”, pp. 172-182 (eds. M. Lakshmanan,

M. Daniel, Springer Verlag, Berlin 1990). [46] B. M. Deb, P. K. Chattaraj, Phys. Rev. A 39 (1989) 1696.

[47] B. M. Deb, P. K. Chattaraj, S. Mishra, Phys. Rev. A 13 (1991) 1248.

]48] S. Nath, P. K. Chattaraj, Pramana-J. Phys. 45 (1995) 65. [49] P. K. Chattaraj, S. Nath, Chem. Phys. Lett. 217 (1994) 342.

[50] P. K. Chattaraj, S. Nath, Proc. Indian Acad. Sci. (Chem. Sci) 106 (1994) 229.

[51] P. K. Chattaraj, S. Nath, Int. J. Quant. Chem. 49 (1991) 705. [52] P. K. Chattaraj, Int. J. Quant. Chem. 41 (1992) 845.

[53] P. K. Chattaraj, S. Sengupta and A. Poddar Int. J. Quant. Chem., DFT Spl. Issue 69 (1998)279.

[54] P. K. Chattaraj and B. Maiti J. Phys. Chem. A 105 (2001) 169.

[55] P. K. Chattaraj and B. Maiti J. Am. Chem. Soc. 125 (2003) 2705.

[56] P. K. Chattaraj, B. Maiti J. Phys. Chem. A 108 (2004) 658.

[57] (a) P. K. Chattaraj, S. Sengupta, Phys. Lett. A 181 (1993) 225. (b) S. Sengupta and P. K. Chattaraj Phys. Lett. A. 215 (1996) 119. (c) P. K. Chattaraj, S. Sengupta and B. Maiti Int. J. Quantum Chem.,

100 (2004) 254.

[58] P. K. Chattaraj, Ind. J. of Pure and App. Phys. 32 (1994) 101. [59] P. K. Chattaraj, S. Sengupta, Ind. J. of Pure and App. Phys. (1996).

[60] P. K. Chattaraj and S. Sengupta J. Phys. Chem. A 103 (1999) 6122.

[61] L. de Broglie, C. R. Acad. Sci. Paris 183 (1926) 447. [62] L. de Broglie, Nature 118 (1926) 441.

[63] L. de Broglie, C. R. Acad. Sci. Paris 184 (1927) 273.

[64] L. de Broglie, C. R. Acad. Sci. Paris 185 (1927) 380. [65] L. de Broglie, J. de. Phys. 8 (1927) 225.

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

© 2011, CCP6, Daresbury

Conceptual Issues, Practicalities and Applications of Bohmian and other Quantum Trajectories in Nanoelectronics

John R. BarkerDepartment of Electronics and Electrical Engineering

University of Glasgow, Glasgow G12 8LT, UK

1. INTRODUCTION

The study of quantum transport in semiconductor devices, particularly in nano-structured systems, has benefited from the interpretive power of the concept of quantum trajectories either in direct space or in phase-space. Here we briefly review progress since the 1980s.

2. WAVE-PACKET TRAJECTORIES AND TRAVERSAL TIMES

The simple classical notion of a particle arriving at a particular place at a particular time is problematic in quantum mechanics and led to considerable practical interest in the concept of a tunnelling time [1]. The rapid development of novel quantum devices that exploited semiconductor heterostructures, such as resonant tunnelling diodes, led to several detailed studies of the traversal time problem based on generating trajectories derived from monitoring features of scattered wave-packets obtained from direct solution of the time-dependent Schrödinger equation [2-5]. Further interest followed the advent of single-electron devices [6-9]. The concept of traversal/tunnelling time was reviewed under the PHANTOMS programme [10] and has been resurrected again [11] in studies of SiGe devices and in the context of experimental data in [12].

3. BOHMIAN TRAJECTORIES

For many years the theory of quantum transport in nanostructures has benefited from the interpretive power of Bohmian trajectories [13] by the post-processing [11, 14-29] of direct quantum calculations for either wave-functions or non-equilibrium Green’s Functions . The resulting trajectories are deterministic [20] with interesting topological properties. Non-deterministic trajectories have also been advocated in an extension of Bohmian mechanics to a stochastic form

43

but have had limited application. The possibilities for using ab initio Bohmian mechanics as a direct self-contained simulation tool has been discussed in the light of modern semiconductor device simulation but runs into difficulties when vortex motion occurs [20, 23]. Time-dependent studies of transport in semiconductor quantum waveguide structures have revealed a rich source of vortex flows corresponding to highly complex quantum potentials [24]. The Bohm picture has proved useful in interpreting novel logic devices that rely on non-invasive measurement [25].

4. QUANTUM POTENTIAL/DENSITY GRADIENT

In modern semiconductor device modelling many groups now routinely use the Quantum Potential or Density Gradient as a calibratable function for incorporating effects of quantum confinement, quantum transport and tunnelling within ab initio drift-diffusion modelling [29-36] where the band structure may have a profound impact [23, 36, 65] on the standard form [13] of the quantum potential. This approach generally compares quite well with full non-equilibrium Green function methodology except where macro-vortex formation occurs. As with any attempt to forward integrate the Bohmian equations of motion there are difficulties when strong vortex flows occur because the quantisation of vorticity stems from an integrability condition on the phase [20]. However, there are severe limitations to using the quantum potential alone, particularly if vortex motion is present in the current flow. It is then not possible to derive the velocity field from the carrier density alone (through the continuity equation) because the flow is not irrotational. Generalisations, based on gauge invariance, to include a quantum vector potential[37] have been suggested and explicit examples constructed. These simple pictures have been corroborated by full self-consistent Non-Equilibrium Green’s function studies of nano-structured devices where the quantum hydrodynamic velocity field is obtained directly(reviewed[29]). Various important generalisations of the quantum potential are reviewed in [38].

5. QUANTUM HYDRODYNAMIC FLOWS

The Bohm picture may also be viewed as a quantum hydrodynamic picture as foreseen by Madelung [39]. In the quantum transport modelling of devices whether by solving the time dependent Schrödinger equation or by the full power of non-equilibrium Green’s function (NEGF) methodology, a well-defined charge density and current density may be computed with the aid of Poisson’s equation from which device modellers extract the current-voltage characteristics. A velocity field may be extracted from the ratio of current density

44

to the charge density. The resulting density field, velocity field pair may be energy resolved in steady state problems or may be computed for the macroscopic density and velocity fields. This construction is essentially a post-processing method that reveals trajectories (in Bohmian view) or streamlines (quantum hydrodynamic view). We have performed extensive studies of quantum transport in realistic nano-devices (where one assumes the presence of non-self-averaging) atomistic micro-variability using this approach under-pinned by NEGF methods [28-29, 40-59]. An approach to a full many body field theoretic quantum hydrodynamics is outlined in [66]

6. HEURISTIC AND TOPOLOGICAL METHODS

Topologically-based methods have been used to give post-processing interpretations of velocity fields and density fields in inhomogeneous devices [21]. For example, the presence of strong nodes in the density field gives rise to quantised vortex flow in the velocity field [21, 24, 27, 29, 60-63]. From these studies we have proposed heuristic topologically-based methods to derive ab initio velocity fields (trajectory families) which allow the incorporation of dissipative processes such as inelastic scattering, charge capture and de-trapping [22-24] using non-Hermitian effective potentials [64]. A quasi-string formalism has been devised as a computational model to show that coherent quantum states are reconstructible from a variational principle for quantum trajectories [23]. These tools have been particularly useful in qualitative predictions of the effects of atomistic micro-variability in semiconductor devices [24, 29, 65].

7. TRAJECTORIES BASED ON QUANTUM DISTRIBUTIONS

Applications of the Wigner function for quantum electronic transport appeared in the 1980s [67-74], mostly of a formal nature. The first computation for a realistic time-dependent physical semiconductor system was made in [75]. Quantum Monte Carlo for device simulation has been extensively investigated since then [76-79]. In many cases the non-positive definiteness of the Wigner distribution was overcome by using damping theoretic methods and by averaging procedures. The equations of motion for carriers have been obtained by moment expansions of the non-local Wigner integral equation of motion and solved using Ensemble Monte Carlo simulation by trajectory tracking [78]. Recently a powerful method [79] has emerged for handling the non-positive definite Wigner function by particle tracking but it is highly compute-intensive. Unfortunately even this method suffers from second problem that follows because the Wigner function does not have compact support in phase space [17]. This problem

45

derives from the geometric centre-of-mass construction. Thus a wave-packet incident on a simple 1D tunnel barrier may split into two well-defined reflected and transmitted packets; but in the Wigner representation this situation leads to well-defined exit packets in phase space plus a wildly oscillating structure midway between the exiting packets in a region where the position distribution and momentum distributions are essentially zero. This problem has been overcome by our introduction of a new unique quantum distribution [80, 81] which we have called a C-distribution that has manifest compact support in phase space (C for compact support and complex valued). The C-distributions may be derived from the density matrix using a mixed real space-momentum representation and the approach generalises to double-time double-space non-equilibrium Green’s functions. The equations of motion and possible Monte Carlo trajectory computational schemes are discussed within exactly soluble models [81] that illustrate the formalism and its interpretation. The near-classical limit is easily obtained and lends itself to path-variable iterative methods including Monte Carlo trajectory schemes. The formalism has well-defined phase space trajectories for stationary states, time-dependent states and open systems.

[1] M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982).[2] J. R. Barker, "Physics and Fabrication of Microstructures and micro-devices", (Springer-

Verlag), Proc. in Physics 13, 210 (1986).[3] S. Collins, D. Lowe and J. R. Barker, J.Phys.C 20, 6213 (1987).[4] S. Collins, D. Lowe and J. R. Barker, J.Phys.C 20, 6233 (1988).[5] S. Collins, D. Lowe and J. R. Barker, J. Applied Physics 63, 142 (1988).[6] J.R. Barker, in “Granular Nanoelectronics”, NATO ASI Series B: Physics 251

(Plenum Press: New York) 327 (1991).[7] K. K. Likhaerev, in “Granular Nanoelectronics”, NATO ASI Series B: Physics 251

(Plenum Press: New York) 371 (1991).[8] J. Cluckie, and J.R. Barker, Semiconductor Sci.Tech. 9, 930 (1994).[9] J R Barker, “Hot Electrons in Semiconductors”, ed N. Balkan (Clarendon Press: Oxford)

Ch 14, 321 (1997).[10] J.R. Barker, S. Brouard, V. Gasparian, G. Iannaccone, J.P. Jauho, C.R. Leavens, J.G.

Muga, R. Sala, and D. Sokolovsky, Report on the first European Workshop on Tunnelling Times, Phantoms Newsletter 7, 5 (1994).

[11] J R Barker and J.R. Watling, Microelectronic Engineering 63, 97 (2002).[12] G. Nimtz, Foundations of Physics 39, 1346 (2009). [13] D. Bohm, Phys. Rev. 85, 166–193, (1952).[14] J.R. Barker, “Granular Nanoelectronics”, NATO ASI Series B 251 (Plenum Press, New York), 19 (1991).[15] J.R. Barker, “Granular Nanoelectronics”, NATO ASI Series B 251

(Plenum Press, New York), 327 (1991).[16] J.R. Barker, S. Roy and S. Babiker,”Science and Technology of Mesoscopic Structures”,

(London:Springer Verlag), Ch 22, 213 (1992).[17] J.R. Barker in “Handbook on Semiconductors, volume 1” (Elsevier-North Holland)

Ch 19, 1079 (1992).[18] J R Barker, Semiconductor Sci.Tech. 9, 911 (1994).

46

[19] J R Barker, “Quantum transport in ultra-small devices”, (Plenum Press, New York) 171 (1995).

[20] J R Barker, Semiconductor Science and Technology 13A, 93 (1998). [21] J. R. Barker , D. K. Ferry, and R. Akis, Superlattices and Microstructures 27, 319 (2000).[22] J R Barker and J.R. Watling, Superlattices and Microstructures 27, 347 (2000).[23] J R Barker, VLSI Design 13, 237 (2001)[24] J.R. Barker, Microelectronic Engineering 63, 223 (2002).[25] J. R. Barker, Semiconductor Science and Technology 13 A, 93 (1998).[26] J R Barker , "Progress in Non-equilibrium Green's Functions II", (World Scientific Publ.,

Singapore), 198 (2003).[27] J.R. Barker, Physica E 19, 62 (2003).[28] J. R. Barker, Semiconductor Science and Technology 19S, 56, (2004).[29] J.R. Barker, A. Martinez, A. Svizhenko, M.P. Anantram and A. Asenov,

J. Phys. Conf. Ser. 35, 233 (2006).[30] M. G. Ancona, Phys. Rev. B 42, 1222 (1990).[31] C. S. Rafferty, B. Biegel, Z. Yu, M. G. Ancona, J. Bude, and R. W. Dutton, in “Simulation

of Semiconductor Processes and Devices”, (K. De Meyer and S. Biese- mans, Eds. Berlin, Germany: Springer), 137 (1998).

[32] A. G. Ancona, Z. Yu, R. W. Dutton, P. J. Vande Vorde, M. Cao, and D. Vook, in Proc. SISPAD ’99, 235 (1999).

[33] A. Asenov, G. Slavcheva, A. Brown, J. H. Davies and S. Saini, IEEE Transactions on Electron Devices 48, 722 (2001).

[34] M.G. Ancona et al, J. App. Phys. 104, 073726 (2008).[35] A. Asenov, A. R. Brown, G. Roy, B. Cheng, C. L. Alexander, C. Riddet, U. Kovac, A.

Martinez, N. Seoane and S. Roy, Journal of Computational Electronics 8, 349 (2009).[36] J.R. Watling, J.R. Barker, S. Roy, J. Computational Electronics 1, 279 (2002).[37] J.R. Barker, J. Computational Electronics 1, 17 (2002).[38] D. Vasileska, H.R. Khan, A. S. Ahmed, C. Ringhofer, C. Hetzinger,

Int. J. Nanoscience 4, 305 (2005).[39] E. Madelung, Z. Phys. 40, 322 (1926). [40] A. Martinez, N. Seone, A. R. Brown,

J. R. Barker and A. Asenov, IEEE Transactions on Electron Devices 57, 1626 (2010). [41] A. Martinez, N. Seoane, A. Brown, J. Barker, and A. Asenov,

J. Phys. Conf. Ser. 220, (2010).[42] A. Martinez, N. Seone, A. R. Brown, J. R. Barker and A. Asenov,

IEEE Transactions on Nanotechnology 8, 603 (2009).[43] N. Seoane, A. Martinez, A.R. Brown, J.R. Barker, A. Asenov,

IEEE Transactions on Electron Devices 56, 1388 (2009).[44] A. Martinez, K.Kalna, P.V.Sushko, A.L. Schluger, J.R. Barker and A. Asenov, IEEE Transactions on Nanotechnology 8, 159 (2009).[45] A. Martinez, M. Bescond, A. R. Brown, J. R. Barker and A. Asenov,

J. Computational Electronics 7, 359 (2008).[46] A Martinez, J R Barker, M Bescond, A R Brown and A Asenov,

J. Phys. Conf. Ser. 109, 012026 (2008).[47] K.Kalna , A. Martinez, A. Svizhenko, M. P. Anantram, J. R. Barker and A. Asenov,

J. Computational Electronics 7, 288 (2008).[48] A. Martinez, K. Kalna, J. R. Barker, and A. Asenov,

Physica Status Sol.C-Current topics in Solid State Physics 5, 47 (2008).[49] A. Martinez, M. Bescond, J.R. Barker, A. Svizhenko, M. P. Anantram, C. Millar and A.

Asenov, IEEE Transactions Electron Devices 54, 2213 (2007).[50] A. Martinez, J. R. Barker, A. Svizhenko, M. P. Anantram, and A. Asenov,

47

IEEE Transactions on Nanotechnology 6, 438 (2007).[51] A. Martinez, K. Kalna, J.R. Barker and A. Asenov, Physica E 37, 168, (2007)[52] A. Martinez, J. R. Barker, A. Asenov, A. Svizhenko and M.P. Anantram,

J. Computational Electronics 6, 215 (2007).[53] A. Martinez, J.R. Barker, A. Svizhenko, M.P. Anantram and A. Asenov,

Springer Proc. in Physics, 110 (August 15th ) (2006).[54] A. Martinez, A. Svizhenko, M.P. Anantram, J.R. Barker, and A. Asenov, A.,

J. Phys. Conf. Ser. 35, 269 (2006).[55] A. Martinez, J.R. Barker , A. Svizhenko, M.P. Anantram, A. Brown, B. Biegel, and A.

Asenov, J. Phys. Conf. Ser. 38, 192 (2006).[56] A. Martinez, A. Svizhenko, M.P. Anantram, J.R. Barker, A.R. Brown and A. Asenov,

IEDM 2005, IEDM Technical Digest, San Francisco, December, 613 (2005).[57] J.R. Barker, J. Computational Electronics 2, 153 (2003).[58] J. R. Barker, Superlattices and Microstructures 34, 361 (2004).[59] J. R. Barker, Semiconductor Science and Technology 19S, 56, (2004).[60] J.R. Barker, Physics of Semiconductors: Proceedings of the 26th International Conference

on the Physics of Semiconductors, Edinburgh, 2002, Institute of Physics Conference Series 171, ed A R Long and J H Davies, IoP Publishing, Bristol (UK), P231 (2003).

[61] J. R. Barker and A. Martinez, J. Computational Electronics 3, 401, (2004).[62] J.R. Barker, Physics of Semiconductor, ed J. Menedez and C.G. Van de Walle, AIP Press

27 1493 (2005).[63] J.R. Barker, AIP Conference Proceedings 995,

Nuclei and Mesoscopic Physics, 104 (2008).[64] D.K. Ferry and J. R. Barker, Applied Physics Letters 74, 582 (1999).[65] J. R. Barker and J. R. Watling, VLSI Design 13, 453 (2001).[66] J.R. Barker , J. Computational Electronics 1, 23 (2002). [67] J.R. Barker, “Physics of Non-linear Transport in Semiconductors”, NATO ASI Series B 52

(Plenum Press, New York), Ch. 5, 126 (1980).[68] J. R. Barker, J. Physique 42 245 (1981).[69] J. R. Barker, J. Physique 42 293 (1981).[70] J. R. Barker, "Handbook of Semiconductors" 1, (North Holland: Oxford)

Ch. 13, 617 (1982).[71] J. R. Barker and S. Murray, Phys. Letters A 93, 271 (1983). [72] J. R. Barker, D. Lowe and S. Murray, in "Physics of Sub-Micron Structures" ed H.L.

Grubin, D K Ferry and K Hess, (Plenum Press:New York), 277 (1984).[73] J. Lin and L.C. Chiu, J. Applied Physics 57, 1373 (1984)[74] W.R. Frensley, Phys. Rev. B 36, 1570 (1987). [75] J. R. Barker, Physica B 134, 22 (1985).[76] J. R. Barker, "Physics and Fabrication of Microstructures and micro-devices",Springer-

Verlag, Proc. in Physics 13, 210 (1986).[77] J.R. Barker, “Granular Nanoelectronics”, NATO ASI Series B 251

(Plenum Press, New York), 19 (1991). [78] H. Kosina, International Journal of Computational Science and Engineering 2, 100 (2006).[79] L. Shifren, C. Ringhofer, and D. K. Ferry,

IEEE Transactions on Electron Devices 50, 769 (2003).[80] J.R. Barker, Physica E 42, 491 (2010).[81] J.R. Barker, J. Computational Electronics,A new approach to modelling quantum

distributions and quantum trajectories for density matrix and Green functionsimulation of nano-devicesaccepted for publication, on-line from October 9th, (2010).

48

49

K. H. Hughes and G. Parlant (eds.) Quantum Trajectories

© 2011, CCP6, Daresbury

Principles of Time Dependent Quantum Monte Carlo

Ivan P. Christov Physics Department, Sofia University

1164 Sofia, Bulgaria

1. INTRODUCTION

The recent advent of lasers that produce pulses with duration below one

femtosecond (attosecond pulses) [1] has allowed to probe events on the

charateristic time scale of correlated electronic motion in atoms, molecules and

solid state. Since the computational cost for directly solving many-body problems

in quantum mechanics scales exponentially with the system dimensionality the

realistic description of correlated electrons in attosecond time scale requires the

development of new efficient methods for solving time-dependent Schrödinger

equation (TDSE). It is generally believed that the exponential-time scaling

inherent to the many-body quantum systems is related to the non-local quantum

correlation effects. Some of the existing methods to treat the electron correlation

effects approximately include time-dependent density functional theory (TDDFT)

[2] where the many-body problem is reduced to single-body problems of non-

interacting electrons moving in an effective exchange-correlation potential, which

is however generally unknown. More reliable, but computationally very

expensive is the multiconfiguration time dependent Hartree-Fock method [3].

Recently, we have introduced a new approach to solve many-body

quantum problems which uses both particles and waves and reduces the many-

body TDSE to a set of coupled single-body TDSE and equations of motion for

Monte Carlo particles where each particle (walker) is attached to separate guiding

wave (de Broglie-Bohm pilot wave). In this way the new method named time

dependent quantum Monte Carlo (TDQMC) [4-7] recovers the symmetry that is

due to the particle-wave dualism in quantum mechanics. It is important to stress

that in TDQMC all calculations are performed in physical space for both the

particles and the associated guiding waves, unlike in traditional Quantum Monte

Carlo techniques where the evolution occurs in configuration space. It is assumed

in TDQMC that the walker distribution in space reproduces the electron density

where each individual particle samples its own distribution given by the modulus

square of the corresponding guide wave. Thus the many-body probability

distribution in space is an intersection of the single-body distributions sampled

50

by the individual Monte Carlo particles, which is consistent with the standard

interpretation of quantum mechanics. As a matter of fact, the TDQMC method

can be considered to be a reconciliation of some aspects of the Copenhagen and

the de Broglie-Bohm theories, with direct application to quantum calculations.

Some advantages of TDQMC as sompared to other QMC and particle

methods are that the density function is positive everywhere and therefore the

“fermion sign” problem is avoided. Also, TDQMC does not use Bohmian

quantum potential which usually causes numerical difficulties. Since TDQMC

uses particles and waves in a symmetric way it allows to treat self cositently

complex quantum systems of different kinds of particles (e.g. electrons and

nuclei) without invoking the Born-Oppenheimer approximation.

2. TDQMC DYNAMICS

For a non-relativistic system consisting of nuclei and electrons TDQMC

reduces the many-body TDSE to a set of coupled single-body TSDE for the

different replicas of the quantum sub-system. For example, for the electronic

degree these equations read [4-7]:

2

2

1

( , ) [ ( )]2

Kk eff ki i i e n i J

Je

i t V tt m

r r R

[ ( )] ( , ) ( , )N

eff k ke e i j ext i i i

j i

V t V t t

r r r r , (1)

and similarly for the nuclear degree. The effective potentials are introduced in

Eq.(1) in order to incorporate various local and non-local quantum correlation

effects. For the electron-electron interaction we have:

1

( ) ( )1[ ( )] [ ( )]

,

l kM j jeff k l

e e i j e e i jk k klj j j

t tV t V t

z t

r r

r r r rr

Κ , (2) ,

where K is a smoothing kernel, kjZ are weighting factors, and M is the total

number of Monte Carlo walkers. To account for the nonlocality, the effective

potentials in Eq.(2) involve weighted nonlocal Coulomb interactions experienced

by a given trajectory from the i-th electron ensemble from the trajectories that

51

belong to the j-th electron ensemble. The width of the kernel ,k kj j t r plays the

role of a characteristic length of the nonlocal quantum correlations that depends

on the density of the walkers in the quantum system. It is calculated at each instant

of time using kernel density estimation procedure. The limit , 0k kj j t r we call

the ultra-correlated case where each walker from a given electron ensemble

interacts with only one walker from each of the rest of the ensembles. The motion

of the Mote Carlo walkers can be calculated in the simplest case using the de

Broglie-Bohm guidance equation:

1( ) Im ( , )

( , )

k ki i ik

e i

tm t

v r rr

(3)

The TDQMC algorithm involves no free parameters.

3. RESULTS

As a simple illustration of TDQMC method we show here the results for

the ground state walker distribution and ionization probability of one-dimensional

helium atom exposed to a strong femtosecond laser pulse. After propagation over

300 complex time steps, the initial random walker ensemble evolves towards

steady state, with distribution in (2D) configuration space shown in Figure 1.

FIGURE 1. Walker distribution in configuration space for 1D Helium for

symmetric ground state.

52

It is seen from Fig.1 that the walker’s density exhibits a butterfly shape

where the particles are pushed away from the region of equal coordinates (X1=X2

in Fig.1) due to the repulsive Coulomb potential. Another exmple of the

correlated electron dynamics is shown in Fig.2 where the helium atom is ionized

in the field of ultashort laser pulse. It is seen that the TDQMC prediction is very

close to the exact result for the ionization probability while the time dependent

Hartree-Fock (TDHF) and the ultra-correplated calculations significatly

underestimete and overestimate the ionization yield, respectively.

FIGURE 2. Time dependent ionization probability for 1D helium.

4. CONCLUSION

The TDQMC method presented describes the evolution of quantum

systems by using ensembles of classical particles and quantum waves. The guide

waves obey a set of coupled linear Schrödinger equations where the use of

effective potentials accounts for the local and nonlocal correlation effects

between the particles. Unlike other many-body quantum methods TDQMC does

not involve calculation of overlap, exchange and correlation integrals, which

significantly improves its scaling properties. Since particles and waves are used in

a symmetric manner in TDQMC approximations such as Born-Oppenheimer are

obsolescent.

Acknowledgment

The author gratefully acknowledges support from the National Science Fund of Bulgaria under

contracts DO-02-115/2008 and DO-02-167/2008.

[1] M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P.

Corkum, U. Heinzmann, M. Drescher and F. Krausz, Nature 414, 511 (2001).

53

[2] M. Petersilka, U. J. Gossmann, and E. K. U. Gross, Phys. Rev. Lett. 76, 1212 (1996). [3] J. Zanghellini, M. Kitzler, C. Fabian, T. Brabec, and A. Scrinzi, Laser Phys. 13, 1064 (2003). [4] I. P. Christov, Opt. Express 14, 6906 (2006).

[5] I. P. Christov, J. Chem. Phys. 127, 134110 (2007). [6] I. P. Christov, J. Chem. Phys. 128, 244106 (2008).

[7] I. P. Christov, J. Chem. Phys. 129, 214107 (2008).

54

K. H. Hughes and G. Parlant (eds.) Quantum Trajectories

© 2011, CCP6, Daresbury

Types of Trajectory Guided Grids of Coherent States for Quantum

Propagation

Dmitrii V. Shalashilin and Miklos Ronto School of Chemistry University of Leeds, LS2 9JT, UK

* Author for correspondence: Email: d.shalashilin@leeds.ac.uk

1. INTRODUCTION

Several methods have been suggested in the literature, which employ various

trajectory guided grids of Frozen Gaussian wave packets as a basis set for

quantum propagation. They exploit the same idea that a grid can follow the wave

function. Therefore a basis does not have to cover the whole Hilbert space of the

system. Among the methods considered in this article are the method of

variational Multiconfigurational Gaussians (vMCG) technique and related

Gaussian Multiconfigurational Hartree method (G-MCTDH) [1], the method of

Coupled Coherent States (CCS) [2] and the recently developed

Multiconfigurational Ehrenfest approach (MCE) [3,4]. The methods differ in

exact way their grids are guided. The goal of this article is to present all four

techniques from the same perspective and compare their mathematical structure.

2. THEORY

It is well known that the time dependence of a wave function n ,...,, 21

is simply that of its parameters, and equations for the “trajectories” tj can be

worked out from the variational principle

0S (1)

55

by minimizing the action LdtS of the Lagrangian

nn

nnnn

Ht

i

L

,...,,ˆˆ

*,...,*,*

,...,,,...,,*,...,*,*,...,*

2121

1111

(2)

The variational principle (1) straightforwardly leads to the Lagrange equations of

motion

0

αα

L

dt

dL , (3)

which can also can be written in Hamilton’s form.

α

αD

H (4)

which is a system of linear equations for the time derivatives of the parameters of

the wave function and D is a matrix

n

l

n

jj

jl ip

D i

,...,,*,...,*,*

**2121

(5)

The importance of Eq.(4) obtained by Kramer and Saraceno [5] is in showing

that mathematical structures of quantum and classical mechanics are identical and

therefore all methods and achievements of classical dynamics can be directly

applied to quantum mechanics. For example the introduction of quantum

ergodoicity becomes very straightforward.

The vMCG method applies the above approach to the wave function

which is a superposition of several Gaussian Coherent States.

Nl

ll tta,1

z (6)

56

In some versions of vMCG the Coherent State can also be “distributed” among

two (or more) electronic potential energy surfaces 1 and 2 .

Nk

kkk ttatat,1

2211 )()()()( z (7)

The wave function (7) is similar in spirit with G-MCTDH approach which

describes the „system“ ...)()( 2211 tata kk on a regular basis and

the „bath“ by Gaussian wave packets )(tkz . Here we use Klauder’s z-

notations for the Coherent states which labels a 1D Gaussian Coherent State

22exp

24

1

ipqqxp

iqxzx

(8)

characterized by its position q and momentum p with a single complex number z

2

2

1

12

1

piqz

(9)

The equations of vMCG, which are simply the Lagrange Eq.(3) or Hamilton’s

Eq.(4) written for the vector of parameters is

),...,,,...,,,...,,,...,( 1111221111 NMNMNN zzzzaaaaα of the wave function

(7).

The equations of vMCG are sometimes numerically unstable. Figures 1

and 2 illustrate such instability on the example of the wave function propagation

in the simple 1D Morse potential on the basis of only 3 basis Coherent States.

Figure 1 shows a jump of the norm due to numerical instability of the equations,

which occurs when two trajectories z1 and z3 overlap in phase space as shown at

the Fig.2.

57

Figure 1 Discontinuity in the norm due to numerical instability and

stiffness of the equations. The jump can be eliminated by decreasing the time

step, which makes propagation rather inefficient

Figure 2 Two trajectories of states in phase-space intersect and numerical

instability occurs at the moment of their intersection.

58

In this very simple calculation the problem can be eliminated by decreasing the

time step. More generally the vMCG approach achieves numerical stability by

regularization techniques so that good and well converged results can be obtained

[1].

The vMCG approach simplifies if only one configuration is used for the

wave function (7)

)(...)()()( 2211 ttatat z (10)

which yields the well known Ehrenfest approach. For the wave function (10) the

Kramer-Saraceno equations are simply those of the standard Ehrenfest method

where the trajectory for z is the one by average Hamiltonian

*zz

EhrHi (11)

where

2211

1221211222221111

**

****ˆaaaa

aaHaaHaaHaaHHH Ehr

(12)

and trajectory for the amplitudes a are given by a system of coupled equations

1212222

2121111

2

2

aiHaHiia

aiHaHiia

z*z*zz

z*z*zz

(13)

59

which can be rewritten in a more compact form if the amplitudes are presented as

a product of oscillating exponent of the classical action and relatively smooth

preexponential factor 2,12,12,1 exp iSda , for which the equations (13)

become

211212

122121

exp

exp

SSidiHd

SSidiHd

(14)

Ehrenfest dynamics is very stable and robust but the wave function (10)

is not flexible enough to converge to exact quantum result. Recently a new

method of Multi-Configuration Ehrenfest dynamics was suggested, which

combines the robustness and stability of Ehrenfest method with the flexibility and

accuracy of vMCG. The idea of MCE is to use the wave function (7) but

assuming predetermined Ehrenfest trajectories (11) for z to guide the basis and

apply variational principle to the amplitudes a only. Then a system of coupled

equations for a is obtained [3]. The most recent version of MCE replaces Eq.(7)

with the ansatz

Nk

kkkk

Nk

kk ttatatDttDt,1

2211

,1

)()()()( z

(15)

representing the wave function as a superposition of Ehrenfest configurations [4].

Assuming Ehrenfest trajectories (11-14) for a and z the coupled equations for the

amplitudes D of each Ehrenfest configuration are obtained and can be found in.

For the case when only single state 1 is present in each configuration in the

wave function (15) the MCE is equivalent to the Coupled Coherent States

method. Both MCE and CCS are very robust and numerically stable and have

60

been used in a number of simulations. In addition they reduce the number of

expensive variational equations thus economizing computational cost.

2. CONCLUSIONS

This article shows that

(1) The equations of vMCG technique can be obtained as those of Kramer

and Saraceno theory for multiconfigurational wave function.

(2) The Ehrenfest method can also be obtained variationally for the single

configurational wave function.

(3) The MCE technique uses Ehrenfest trajectories to guide trajectories of

Gaussians and fully variational equations for their amplitudes. On a single

potential energy surface MCE becomes equivalent to to CCS.

(4) The advantage of CCS and MCE is that they reduce the number of

expensive variational equations and improve numerical stability without any

regularization

(5) Using “approximate” trajectories instead of those yielded by fully

variational approach is not an approximation but simply a choice of a basis

set.

[1] I.Burghardt, H.-D.Meyer, and L.S.Cederbaum, J.Chem.Phys., 111, 2927 (1999);. G.A.Worth and

I.Burghardt, Chem.Phys.Lett. 368, 502 (2003); I. Burghardt, M. Nest, and G.A. Worth, J.Chem.Phys.

119, 5364 (2003); I. Burghardt, K. Giri, and G. A. Worth,

J.Chem.Phys.. 129 174101 (2008).

[2] D. V. Shalashilin and M. S. Child, J. Chem. Phys., 113, 10028 (2000), .D. V. Shalashilin and M.

S. Child, J. Chem. Phys., 114, 9296 (2001).; D. V. Shalashilin and M. S. Child, J. Chem. Phys., 115,

5367 (2001); D. V. Shalashilin and M. S. Child, J. Chem. Phys., 121, 3563 (2004).; P.A.J. Sherratt,

61

D.V. Shalashilin, and M.S. Child, Chem. Phys., 322, 127 (2006);. D. V. Shalashilin and M. S. Child,

Chem. Phys., 304, 103 (2004);

[3] D.V. Shalashilin, J.Chem.Phys., 130 244101 (2009) ;

[4] D.V. Shalashilin, J.Chem.Phys., 132, 244111 (2010) ;

[5] P. Kramer and M. Saraceno, Geometry of the Time-Dependent Variational Principle in Quantum

Mechanics (Springer, NewYork, 1981).

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

© 2011, CCP6, Daresbury

Accurate Deep Tunneling Description

by the Classical Schrödinger Equation

Xavier Giménez1,3 and Josep Maria Bofill2,3

1)Departament de Química Física, 2)Departament de Química Orgànica, 3)Institut de Química Teòrica i Computacional. Universitat de Barcelona.

Martí i Franquès, 1. 08028 Barcelona, Spain.

The description of classical processes in quantum terms, is actually known for

almost five decades, the Classical Schrödinger Equation (CSE).1 It is a non–linear

differential equation arising from real quantities, as a limiting case of the Time–

Dependent Schrödinger Equation (TDSE), i.e. a linear differential equation for

complex functions. The CSE provides an equation of motion for classical

particles, in the closest possible language to quantum mechanics. It was

introduced, in perhaps the most extensive work on the subject, by Schiller, 2-4 who

develops a complete class of classical analogs of quantum algebra objects, in an

analysis of van Vleck’s classical limit of quantum mechanics.5 The author

explores as well a number of specific cases analytically solvable, such as the

hydrogen atom or asymptotic Rutherford scattering. Shortly thereafter, N. Rosen

rederived CSE in a rather heuristic form1,6, emphasizing its non–linear character,

as well as discussing the use of mixtures, rather than combinations, for correct

classical–limit densities.6 Holland7 did an extensive analysis of the implications

of the non–linearity of CSE, pointing out the interest in having available generic

solutions, based on expansion schemes for the wavefunction. Ghose,8 and Ghose

and Samal9 used CSE to study a continuous transition from quantum to classical

62

mechanics. They added to the later term in CSE a multiplicative factor

accounting for decoherence, such that for short times it displays quantum

behaviour, whereas for later times the system naturally relaxes with a

phenomenological bath and undergoes classical. These authors develop a

numerical algorithm to solve CSE and compute individual trajectories, but they

assume that the later term in CSE equation arises from TDSE (thus avoiding an

iteration process, see next chapter). Later on, Wyatt10 stresses the connection

between CSE and classical trajectories, whereas Nikolic,11 on the contrary,

analyses CSE in probabilistic terms.

To the best of our knowledge, no practical, sufficiently general algorithm appears

to be available for solving CSE. One may actually consider, in advance, that the

non–linearity of CSE prevents any gaining in computational capability from such

calculations. However, there have been some remarkable advances in solving a

generic class of non–linear Schrödinger equations,12 a class that includes CSE.

Therefore, a new impetus might be ready to understanding classical mechanics in

quantum terms, by solving CSE, as applied to problems going beyond those

having solutions in closed form. It is our hope to provide, in addition, different

standpoints for looking at the classical limit of quantum mechanics, as well as

compelling evidence on quantitative indicators for this classical limit, at least for

some representative systems and selected physical quantities.

In the present work, a numerical test considers a scattering process across an

Eckart potential barrier. This is a well–known system, used to test the ability of

semiclassical methods to reproduce the so–called deep tunneling regime.13-16

Figure 1 shows several time snapshots of a coherent state wavepacket colliding

against an Eckart barrier. This makes the central wavepacket energy to be one

half of the barrier height, so that any transmission is due solely to tunneling.

63

Figure 1 shows that accurate and CSE values for the wavepacket density

remarkably agree for all times. In particular, it is outstanding the ability of CSE

to reproduce the oscillatory fringes of the reflected packet, at intermediate times,

and the tunneling transmitted packet. In addition, Classical Schrödinger Equation

results essentially capture the whole deep tunneling behaviour, at the conditions

of the present study. Other physical conditions, corresponding to smaller masses

64

Figure 1 Time snapshots for tunneling transmission of a wavepacket traversing an Eckart barrier. Continuous red line: Accurate quantum mechanical results; continuous blue: present Classical Schrödinger Equation calculations.

and smaller widths (not shown here), have also been explored. The observed

trend is that quantum tunneling is again fairly well reproduced, but convergence

rates are much slower and, in some cases, the number of iterations dramatically

increases for large collision times. Nevertheless, these results for tunneling

Eckart transmission come to a bit of surprise, mainly when one recalls the well–

known difficulties that this physical system caused to several other semiclassical

techniques.17,18

Even though CSE is formally equivalent to Hamilton–Jacobi, it is clear that its

specific implementation in the present work does allow for the occurrence of

quantum effects. More specifically, the present algorithm is based on a complex

canonical transformation, a matrix equation cast, an allowance for complex

momentum, as well as the application of quantum mechanical continuity

conditions. All these features contribute to a remarkable change from the

equivalent conditions imposed to the classical equations of motion, so that

dwelling inside the classically forbidden region is allowed and quantum wave–

like behaviour is certainly taken into account.

[1] N. Rosen, Am. J. Phys. 32, 597 (1964).[2] R. Schiller, Phys. Rev. 125, 1100 (1962).[3] R. Schiller, Phys. Rev. 125, 1109 (1962).[4] R. Schiller, Phys. Rev. 128, 1402 (1962).[5] J.H. van Vleck, Proc. Nat. Acad. Sci. 14, 178 (1928).[6] N. Rosen, Am. J. Phys. 33, 146 (1965).[7] P.R. Holland, The Quantum Theory of Motion, Cambridge Univ. Press., Cambridge, 1993.[8] P. Ghose, Found. Phys. 32, 871 (2002).[9] P. Ghose, K. Samal, Found. Phys. 32, 893 (2002).[10] R.E. Wyatt, Quantum dynamics with trajectories, Springer, New York, 2005.[11] H. Nikolic, Found. Phys. Lett. 19, 553 (2006).[12] F.W. Strauch, Phys. Rev. E 76, 046701 (2007).[13 S. Keshavamurthy, W.H. Miller, Chem. Phys. Lett. 218, 189 (1994).[14] N.T. Maitra, E.J. Heller, Phys. Rev. Lett. 78, 3035 (1997).[15] F. Grossmann, Phys. Rev. Lett. 85, 903 (2000).[16] M. Saltzer, J. Ankerhold, Phys. Rev. A 68, 042108 (2003).[17] F. Grossmann, Phys. Rev. Lett. 85, 903 (2000).[18] D.J. Tannor, S. Garaschuck, Annu. Rev. Phys. Chem. 51, 553 (2000).

65

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

The Bohmian Model, Semiclassical Systems and the

Emergence of Classical Trajectories

Alex MatzkinLab. de Physique Theorique et Modelisation (CNRS Unite 8089),Universite de Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France

I. INTRODUCTION

The de Broglie–Bohm interpretative framework [1] (abbreviated as BM, for’Bohmian model’) is the main alternative to standard quantum mechanics (QM).While BM and QM give equivalent results, the trajectory-based representationand flow-based numerical methods specific to BM have given rise to an increas-ing number of works employing BM as a tool to compute and interpret thephysics of various systems [2]. Notwithstanding, BM was not introduced as acomputational scheme, but rather as manner of providing QM with an ontolog-ical framework [3]: the aim of BM is to give a realist [4] description of quantumphenomena in terms of the motion of point-like particles following well-definedtrajectories, allowing to unify the classical and quantum descriptions of nature.Although the programme is attractive, we argue it cannot work. Our argumentsare based on the investigation of semiclassical systems. These systems, while be-ing purely quantum, display a striking correspondence between their propertiesand those of the equivalent classical system. The properties of the Bohmianparticle are radically different and spoil this quantum-classical correspondence.An important consequence is that Bohmian trajectories can never become clas-sical in these systems, raising the problem of whether BM can account for theemergence of classical dynamics.

II. SEMICLASSICAL SYSTEMS AND THE QUANTUM-CLASSICAL

CORRESPONDENCE

The investigations of the quantum-classical correspondence, which has its ori-gins in the early days of quantum mechanics were revived in the 1990’s in thecontext of quantum chaos [5, 6]. It is today well-established that several typesof quantum systems – known generically as semiclassical systems – display themanifestations of properties belonging to the classical analog of these systems.

66

This is due to the fact that the wavefunction propagates essentially along thetrajectories of the corresponding classical system; indeed in these cases the semi-classical approximation to the path integral propagator, given by [7]

K(x0,x, t) =∑

k

1

2iπℏ

∣

∣

∣

∣

det∂2Sk

∂x∂x0

∣

∣

∣

∣

1/2

exp (iSk(x0,x, t)/ℏ+ iφk) , (1)

becomes excellent. Here the sum runs on all the classical trajectories k connect-ing x0 to x in the time t. Sk is the classical action for the kth trajectory and thedeterminant is the inverse of the Jacobi field familiar from the classical calculusof variations, reflecting the local density of the paths; φk is a phase accountingfor reflections and conjugate points encountered along the kth trajectory. As aresult individual classical trajectories are ”visible” in these quantum systems (inthe wavefunction or through observable quantities), and their spectral propertiessuch as the distribution of the energy levels depend on the average properties offamilies of classical orbits. This is why the quantum counterpart of classicallychaotic systems have specific universal features quite different from the quantumanalogue of a classically regular system.

III. BOHMIAN TRAJECTORIES AND THE DYNAMICAL

MISMATCH

Bohmian trajectories are defined from the probability density current, thatdepends most crucially on the choice of the initial distribution. Hence Bohmiantrajectories in a given system can be chaotic or regular depending on the cho-sen initial distribution. This is also the case in semiclassical systems: Bohmiantrajectories bear no relation with the underlying classical dynamics, spoiling thequantum-classical correspondence characteristic of semiclassical systems. Thefact that by way of the propagator (1) the wavefunction evolves along classicaltrajectories does not mean that the current density will necessarily do so. Thereason is that an initial localized wavefunction spreads and interferes yieldingBohmian trajectories quite different from the classical ones, even though eachpart of the wavepacket moves along a classical trajectory. This ”dynamical mis-

match” was discussed at length in Ref. [8], and specific examples have beengiven for Rydberg atoms [9], a hydrogen atom in a magnetic field [8], and asquare billiard [10].The dynamical mismatch between Bohmian trajectories and classical motion

has serious implications [8] concerning the empirical acceptability of the deBroglie-Bohm theory as describing the real behaviour of the quantum world.The usual hand-waving argument – quantum and classical particle motions be-long to different domain, the former approaching the latter in some limit as thequantum potential vanishes – is clearly not applicable: it seems indeed untenable

67

to explain the appearance of classical structure in the wave and in the statisticaldistribution of the particles while upholding that the real dynamics of the par-ticle remains highly non-classical, without any type of correspondence with theclassical dynamics that should emerge at some point. From within BM, the onlyoption to account for the emergence of classical trajectories from the quantumBohmian dynamics is to rely on decoherence and non-spreading wavepackets.These two ingredients, however, are not specific to BM (and therefore do notneed any of its ontological propositions), but form part of the practical recipestandard QM employs in order to account for the quantum-classical transition.It is well-known that this practical recipe is inconsistent [11, 12], since it givesincompatible meanings to the reduced density matrices.The study of the BM properties in semiclassical systems leads us to conclude

that despite its attractive appeal and high interest including in practical com-putations, BM is unable to account for the emergence of classical trajectories.

[1] P. R. Holland, ”The Quantum Theory of Motion” (Cambridge University Press,Cambridge, 1993).

[2] R. E. Wyatt, ”Quantum Dynamics with Trajectories” (Springer, Berlin, 2005).[3] D. Bohm. and B. J. Hiley, ”The Undivided Universe: an ontological interpretation

of quantum theory” (Routledge, London, 1993).[4] A. Matzkin Eur J Phys 23 285 (2002).[5] M. C. Gutzwiller, ”Chaos in Classical and Quantum Mechanics” (Springer, Berlin,

1990).[6] M. Brack M and R. Badhuri, ”Semiclassical Physics” (Westview Press, Boulder,

USA, 2003).[7] W. Dittrich and M. Reuter ”Classical and Quantum Dynamics” (Springer, Berlin,

2001).[8] A. Matzkin and V. Nurock, Studies In History and Philosophy of Science B 39, 17

(2008).[9] A. Matzkin, Phys. Lett. A 361, 294 (2007).

[10] A. Matzkin, Found. Phys. 39, 903 (2009).[11] S. L. Adler, Studies In History and Philosophy of Science B 34, 135 (2003).[12] A. Leggett, J. Phys.: Condens. Matter 14, R415-R451 (2002).

68

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Quantum trajectories for ultrashort

laser pulse excitation dynamics

Gerard ParlantInstitut Charles Gerhardt, Universite Montpellier 2, CNRS,

Equipe CTMM, Case Courrier 1501,

Place Eugene Bataillon, 34095 Montpellier, France

The Quantum Trajectory Method [1] (QTM) is a computational implementa-tion of Madelung’s hydrodynamical approach to quantum mechanics based onthe ansatz ψ = A exp(iS/~). The probability density A2 is partitioned into afinite number of “particles” which evolve along quantum trajectories guided bythe gradient of the action ∇S. Quantum trajectories obey Newton-like equa-tions with a modified potential V (x) +Q(x, t), where the quantum potential Qcouples trajectories to one another. In QTM, coupled equations of motion forthe densities and action functions of all particles are propagated in time, andfrom these the wave function can be synthesized at each instant.A computational drawback of QTM is that the quantum potential Q may

become very large, or even singular, in some situations, thus giving rise to nu-merically unstable trajectories. This is the case, in particular, of quantum in-terferences that may induce very small amplitude in the wave function. Thisso-called node problem [1] is currently the major obstacle that prevents the de-velopment of QTM.QTM has been extended to the dynamics of electronic nonadiabiatic colli-

sions [1–3]. In the multisurface QTM, a constant number of quantum trajectoriesare propagated on each electronic state (there is no “trajectory hop” betweensurfaces) and probability density and phase information are transferred from onestate to another through communications between trajectories.By substituting the polar form ψj(x, t) = Aj (x, t) exp [iSj(x, t)/~] into the

time-dependent Schrodinger equation (TDSE), one can derive [1, 2] the hydro-dynamic equations for the nuclear motion of a two-state system, where for eachstate j, Aj is the (positive) amplitude, and Sj is the action function, both realquantities, and Pj(x, t) = ∇Sj(x, t) is the momentum associated with the flowvelocity vj(x, t) = Pj(x, t)/m of the probability fluid. Like their single-statecounterparts, the continuity equations for the densities ρj = A2

j ,

dρ1/dt = −ρ1∇v1 − λ12; dρ2/dt = −ρ2∇v2 − λ21, (1)

express the conservation of the probability fluid on each individual surface; in

69

addition, the extra source/sink terms, λ12 = −λ21 = (2V12/~) (ρ1ρ2)1/2

sin (∆),take into account the transfer of probability density from one state to the otherone, with ∆ = (S1 − S2)/~. (n.b. we assume an observer “going with theflow of probability” [1], so that Eqs. (1), (2), and (4) involve the total timederivative d/dt.) In the Newtonian equations,

dp1/dt = −∇ (V11 +Q11 +Q12) ; dp2/dt = −∇ (V22 +Q22 +Q21) , (2)

one can see that any given particle evolving on state j is subject to three forcecomponents: (i) the classical force, (ii) the quantum force, that reflects theinfluence of all other particles on the same surface, and (iii) an extra couplingforce, that derives from the off-diagonal quantum potential Qjk,

Q12 = V12 (ρ2/ρ1)1/2

cos (∆) , with ρ1Q12 = ρ2Q21. (3)

Finally, equations for the rate of change of the action functions read:

dS1/dt = mv21/2− V11 −Q11 −Q12; dS2/dt = mv2

2/2− V22 −Q22Q21. (4)

In the hydrodynamical equations of motion for nonadiabatic dynamics,Eqs (1)–(4), it appears that particle motions and interstate transitions areclosely interconnected. In particular, interstate transfer forces, deriving from Qij

[Eq. (3)] can noticeably modify the course of trajectories, and eventually changethe result of the calculation. Unfortunately, the multistate QTM inherits thenumerical drawbacks of its single-state counterpart mentioned above. In addi-tion, extra propagation difficulties, related to the interstate electronic couplingper se, may be anticipated, especially in case of large values of V12.

In this work, we use a form of the multisurface QTM that formally separatesinterstate transitions from single-state nuclear motions [6, 7] (see also [4]). Oneach state, quantum trajectories are propagated in a moving frame attached toa decoupled Gaussian wave packet, while transition probabilities are obtainedfrom coupled continuity equations similar to Eqs (1). In practice, this is doneby substituting into the TDSE, for each state j = 1, 2, a split polar form of thenuclear wave function, ψj (x, t) = φj (x, t) × χj (x, t), with φj = aj exp (isj/~)and χj = αj exp (iσj/~), where the φj ’s refer to single-state motion while the χj ’scorrespond to inter-state exchange of density and phase.For each state j, the “decoupled” wave function φj(x, t) is solution to the

single-state TDSE

−~2

2m∇

2φj + Vjjφj = i~∂

∂tφj , (5)

while the “coupled” wave function χj (x, t) satisfies a modified TDSE

−~2

2m∇

2χj −~2

m

∇φjφj

∇χj = i~∂

∂tχj − Vjk

φkφjχk, (6)

70

where one can notice an interstate coupling term on the right-hand sideof Eq. (6).Bohmian equations of motion for φj(x, t) and χj(x, t), solutions to Eqs (5)

and (6), are easily derived [6, 7]. It must be mentioned that while this decouplingprocedure is formally exact, in the laser excitation example given below, theradiative coupling is so large that the source/sink terms λjk and the off-diagonalquantum potentials Qjk dominate the behavior of the “coupled” part of thewave function, so that the Bohmian equations of motion for χj(x, t) can besimplified [7] to give:

dωj/dt ≈ −λjk/fj ; dσj/dt ≈ −Qjk, (7)

where fj = a2j and ωj = α2

j are the probability densities relative to φj and χj ,respectively.The decoupled-representation QTM approach described above has been ap-

plied to a model problem [5] of two adiabatic potential energy curves coupledthrough a laser pulse. The system is initially in its ground vibrational state andis promoted to a repulsive state by the laser field. Due to large values of thelaser field, electronic state populations exhibit several oscillations (Rabi flops)during the short laser pulse. A small number of trajectories (typically 30 to 100)is sufficient to obtain an excellent agreement with exact quantum calculations,a number that can be contrasted with the 105 semiclassical trajectories usedin Ref. 5. Moreover, an interesting “hole” structure in the initially Gaussianwave function is observed. It has been previously interpreted as a momentumkick from the laser pulse to the wave packet [8]. We suggest that it might beof interest to investigate this dynamical effect in terms of the interstate transferforces deriving from the off-diagonal quantum potentials Qjk.

[1] R. E. Wyatt, Quantum Dynamics with Trajectories: Introduction to Quantum Hy-

drodynamics, Springer, New York, 2005.[2] R. E. Wyatt, C. L. Lopreore, and G. Parlant, J. Chem. Phys., 114 (2001) 5113.[3] I. Burghardt and L. S. Cederbaum, J. Chem. Phys., 115 (2001) 10312.[4] S. Garashchuk, V. A. Rassolov, and G. C. Schatz, J. Chem. Phys., 123 (2005)

174108; V. A. Rassolov and S. Garashchuk, Phys. Rev. A, 71 (2005) 032511.[5] F. Grossmann, Phys. Rev. A, 60 (1999) 1791.[6] G. Parlant, Quantum Trajectories, Chap. 17, CRC Press/Taylor & Francis, Boca

Raton, 2010.[7] J. Julien and G. Parlant, unpublished.[8] U. Banin, A. Bartana, S. Ruhman, and R. Kosloff, J. Chem. Phys., 101 (1994)

8461.

71

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Quantum dynamics and super-symmetric quantum

mechanics.

Eric R. Bittner∗ and Donald J. KouriDepartment of Chemistry, University of Houston, Houston, TX 77204

In my talk I will present an overview of our recent work involving the use of supersymmetric quan-tum mechanics (SUSY-QM). I begin by discussing the mathematical underpinnings of SUSY-QMand then discuss how we have used this for developing novel theoretical and numerical approachessuitable for studying molecular systems. I will conclude by discussing our attempt to extend SUSY-QM to multiple dimensions.

I. INTRODUCTION

We advance a new approach which casts the problem of excited states ofa physical Hamiltonian in terms of nodeless ground states associated with ahierarchy of “auxiliary Hamiltonians”. This hierarchy of Hamiltonians resultsfrom the implementation of the “super-symmetric quantum mechanics” (SUSY-QM) formalism introduced some years ago in particle physics and quantum fieldtheory[1, 2]. In high-energy physics (HEP), SUSY is a fundamental symmetrywhich relates elementary particles of one spin to another particle with identicalmass/energy but whose spin differs by ±~/2 . In essence, SUSY predicts thatfor every boson there exists a corresponding fermion with the same mass/energy.As of date, superpartners of the particles of the Standard Model have not beenobserved; supersymmetry, if it exists, must be a broken symmetry allowing the’sparticles’ to be heavy. Supersymmetric quantum mechanics, on the other hand,borrows the basic formal ideas from HEP to solve quantum mechanics problems.In short, SUSY-QM introduces pairs of Hamiltonians that share a particularmathematical relationship (termed partner Hamiltonians) such that for everyeigenstate of one Hamiltonian, its partner has a corresponding eigenstate withthe same energy but with a lower quantum number. In brief, one introduces a“superpotential”, W , such that the original Hamiltonian can be written in the

∗corresponding author

72

form

H = (−∂x +W )(+∂x +W ) = Q+Q (1)

where Q+ and Q are “charge operators” analogous to the creation/annihilationoperators in the quantum treatment of the 1D harmonic oscillator. From ahistorical context, it is interesting to note that Schrodinger used a SUSY-liketechnique in his original solution of the hydrogen atom. In fact the final resultfor the H-atom spectrum and wavefunctions can be derived with greater easyusing SUSY than with the more traditional approach involving the recursionrelation for Laguerre polynomials.While SUSY-QM has also been explored for one dimensional, non-relativistic

quantum mechanical problems[3–8], thus far these studies have focused on theformal aspects and on obtaining exact, analytical solutions for the ground statefor specific classes of problems. In several recent papers[9–12] we have begun ex-ploring the SUSY-QM approach as the basis of a general computational schemefor bound state problems. Of course, all this begs the question: Can this ap-proach be generalized to higher numbers of dimensions and to more than a singleparticle? There has been substantial effort in the past to do just this.[4–7, 13–24] However, until now, no such generalization has been found that is able togenerate all the excited states and energies even for so simple a system as a pairof separable, 1-D harmonic oscillators (HO) or equivalently, for a separable 2-Dsingle HO.Here we briefly report on our generalization of SUSY-QM to higher dimensions

and showed that it does, in fact, yield the correct analytical results for separableand non-separable problems. [12] We present a succinct summary of our ap-proach as applied to simple problems and discuss extensions to more complexsystems such as atomic clusters.

II. BASIC FORMALISM

We begin with a brief summary of our new generalization of SUSY-QM to treathigher dimensionality and more than one particle. Previous attempts generallyinvolved introducing additional, “spin-like” degrees of freedom.[5, 7, 16–19, 21,22, 25–27] However, no practical extension to higher dimensions or reduction tonumerical method has been produced to date following these lines. The extensionto more general systems involves introducing separate coordinate systems foreach particle in the system. Consequences of this include: 1) a simple expressionfor the Laplacian for the system 2) vector generalizations of the SUSY super-potential and of the two charge operators that generate the Hamiltonians for thefirst and second sectors 3) the original Hamiltonian remains exactly the sameas before except that it is now in the SUSY factored form 4) then the firstsector problem can be addressed using any of the standard approaches 5) the

73

(a) (b)

FIG. 1: (a) Convergence of first excitation energy for a model anharmonic potentialusing a n-point DVR. Gray squares: standard variational approach. Here we plotthe error between the numerical and exact values ǫ = log

10|E(n) − E(exact)|. Black

squares: SUSY. Dashed lines are linear fits used to guide the eye.(From Ref. [11]) (b)Location of excited state node using the Monte Carlo/SUSY approach on a modeldouble well potential.(From Ref. [10]).

second sector Hamiltonian is, however, a second rank tensor (or matrix) andits wave function is a vector. This is analogous to the situation in relativisticquantum mechanics where tensor Hamiltonians and vector wave functions resultfrom the embedding of coordinate systems separately in each particle (with thedifference being that time continues to be a parameter here as opposed to eachparticle having its own proper time) 5) despite this complication, we derived theRayleigh-Ritz variational principle for this sector and were able to solve for thevector energy eigenstates of sector 2. To illustrate the general form of the theorywe write the quantum Hamiltionian (using ~

2/2m = 1)

H1 = ~Q†1 ·

~Q1 + E10 ; ~Q1 = ∇+ ~W1; ~W1 = −∇ lnψ1

0 (2)

where ψ10 is the usual ground-state of H1 and ~∇ is the 3n dimensional gradient

operator. It is straightforward to verify that Eq. 1 yields a factorization of the

original Schrodinger equation. The superpotential ~W1 is a real-valued columnvector. Within our approach, we write the sector 2 Hamiltonian as tensor prod-

uct:←→H 2 = ~Q1

~Q†1 + E1

0

←→I . The degeneracies between sector 1 and sector 2 are

the consequence of the “inter-twining” relations:

~Q1H1 =←→H 2 · ~Q1 & ~Q†

1 ·←→H 2 = H1

~Q†1

74

The full SUSY algebraic structure is obtain by writing this in matrix form bydefining the super-charge operators

Q =

[

0 0~Q1 0

]

& Q† =

[

0 ~Q†1

0 0

]

(3)

and writing

H =

[

H1 0

0←→H 2

]

= Q†,Q (4)

This formalism applies to any number of distinguishable particles and has beenproved to yield correct numerical results for the case of 2 particles (or equiva-lently, for a 2-D 1 particle system) with non-separable interaction.

Then we define the sector 3 scalar Hamiltonian by

H3 = ~Q2 · ~Q+2 + E

(2)0 (5)

with the ground state wave equation

H3ψ(3)0 = E

(3)0 ψ

(3)0 . (6)

It is easily seen that E(3)0 = E

(2)1 −E

20 . This procedure continues until all bound

states of the original Hamiltonian are exhausted. It should also be clear thatthe sector 2 excited state wave function is obtained from the nodeless sector 3ground state by applying ~Q+

2 to it. Then the second excited state for sector

1 results from taking the scalar product of ~Q+1 with ~ψ

(2)1 . The approach thus

leads to an alternating sequence of scalar and tensor Hamiltonians, with thelowest energy state of the nth member of the hierarchy being isoenergetic withthe first-excited state of the n− 1 sector (for n > 0). The critical observation isthat in all cases we need only determine nodeless ground states and we can usethis fact to develop novel numerical approaches for determining the excitationspectrum for complex multi-dimensional systems.

III. SUMMARY AND OUTLOOK

The idea of using “out-side the box” ideas to advance quantum computa-tional techniques certainly is within the spirit of this CCP6 conference. Whilewe have not emphasized the dynamical aspects of our approach and have focusedsolely upon finding stationary states, we have begun to formulate a “quantumtrajectory” approach for determining the eigenstates of the tensor-sector Hamil-tonians in our approach using the Dirac-Frenkel-McLachlan variational principleextended to imaginary time. We next sketch briefly how this may be used.

75

The essential step in implementing ND-SUSY will be in how we choose torepresent the many-body wave function. Let us assume that for an arbitrary Ndimensional system we can write a sector 1 state as

ψo =∏

n

g(~x, λn) (7)

where g(~x; λn) is a multidimensional gaussian parameterized by λn. Forexample we can write

g(~x; λn) = exp

(

−1

2(~x− ~µn)

TS−1n (~x− ~µn) + an

)

(8)

where λn = an, ~µn, S−1n ∈ Re. Here, an is the weight ascribed to a given

gaussian, µn is the center, and S−1n is the covariance matrix. Note that we are

writing this in the most general form since ~x represents the cartesian coordinatesof all the particles in the system. In practice, however, we ascribe a factorizedgaussian to each physical particle so that the total number of coefficients 10×np

where np = N/3 is the number of particles in the system. We can constructthe ground state by requiring that δE = 0 and determining the coefficients viaintegrating the McLachlan/Dirac/Frenkel variational principle in imaginary time(i.e. we make the analytic continuation by taking it/~→ τ/~ ) viz

n

∫(

∂ψ

∂λj

)∗

(Hψ +∂λi∂τ

∂ψ

∂λi)d~x = 0. (9)

Now, let’s write our ground state in the product basis

ψo =∏

k

g(~xk, λk) (10)

where k denotes a given physical particle and λk are the variational coefficients

associated with that particle. Now, construct the superpotential ~W1 (eq. 13 inour JPC paper),

W1kn = −∂

∂xknlnψo (11)

Since ψo is a product of gaussians and taking Sk to be symmetric, one finds asimple expression for the superpotential.

W1kn =∑

m

(S−1k )nm(xkm − µkm) (12)

where m = 1, 2, 3 labels the cartesian coordinates for particle k.

76

This notion can be extended to the tensor sectors by writing Eq. 13 as

∫

(

∂ ~ψ

∂λj

)†

· (←→H 2 · ~ψ +

∂λi∂τ

∂ ~ψ

∂λi)d~x = 0. (13)

where ~ψ is a trial state in sector-2. This results in a series of coupled non-linear“trajectory” equations for the coefficients. Integrating this forward in imaginarytime will result in a robust variational estimate for the sector-2 ground statewhich then can be used to determine the excitation energy or sector-1excitedstate wave function. We are currently exploring this idea.

Acknowledgments

This work was supported in part by the National Science Foundation (ERB:CHE-0712981) and the Robert A. Welch foundation (ERB: E-1337, DJK: E-0608). The authors also acknowledge Prof. M. Ioffe for comments regarding theextension to higher dimensions.

[1] E. Witten, Nuclear Physics B (Proc. Supp.) 188, 513 (1981).[2] E. Witten, J. Differential Geometry 17, 661 (1982).[3] H. Baer, A. Belyaev, T. Krupovnickas, and X. Tata, Physical Review D (Particles

and Fields) 65, 075024 (pages 8) (2002), URL http://link.aps.org/abstract/

PRD/v65/e075024.[4] A. A. Andrianov, N. V. Borisov, M. V. Ioffe, and M. I. Eides, Theoretical and

Mathematical Physics 61, 965 (1984).[5] A. A. Andrianov, N. V. Borisov, M. I. Eides, and M.V.Ioffe, Phys. Lett. A 109,

143 (1985).[6] F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).[7] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Phys. Lett. A 105, 19 (1984).[8] A. Gangopadhyaya, P. K. Panigrahi, and U. P. Sukhatme, Phys. Rev. A 47, 2720

(1993).[9] D. J. Kouri, T. Markovich, N. Maxwell, and B. G. Bodman, J. Phys. Chem. A

113, 7698 (2009).[10] E. R. Bittner, J. B. Maddox, and D. J. Kouri, The Journal of Physical Chemistry

A 113, 15276 (2009), URL http://dx.doi.org/10.1021/jp9058017.[11] D. J. Kouri, T. Markovich, N. Maxwell, and E. R. Bittner, The Journal of Physical

Chemistry A 113, 15257 (2009), URL http://dx.doi.org/10.1021/jp905798m.[12] D. J. Kouri, K. Maji, T. Markovich, and E. R. Bittner, The Journal of Physical

Chemistry A 114, 8202 (2010), http://pubs.acs.org/doi/pdf/10.1021/jp103309p,URL http://pubs.acs.org/doi/abs/10.1021/jp103309p.

77

[13] P. T. Leung, A. M. van den Brink, W. M. Suen, C. W. Wong, and K. Young,Journal of Mathematical Physics 42, 4802 (2001), URL http://link.aip.org/

link/?JMP/42/4802/1.[14] R. de Lima Rodrigues, P. B. da Silva Filho, and A. N. Vaidya, Physical Review

D (Particles, Fields, Gravitation, and Cosmology) 58, 125023 (pages 6) (1998),URL http://link.aps.org/abstract/PRD/v58/e125023.

[15] A. Contreras-Astorga and D. J. F. C., AIP Conference Proceedings 960, 55 (2007),URL http://link.aip.org/link/?APC/960/55/1.

[16] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Theoretical and MathematicalPhysics 72, 748 (1987).

[17] A. A. Andrianov and M. V. Ioffe, Phys. Lett. B 205, 507 (1988).[18] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Theoretical and Mathematical

Physics 61, 1078 (1984).[19] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, JETP Lett. 39, 93 (1984).[20] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Phys. Lett. B 181, 141 (1986).[21] F. Cannata, M. V. Ioffe, and D. N. Nishnianidze, Journal of Physics A: Mathe-

matical and General 35, 1389 (2002), URL http://stacks.iop.org/0305-4470/

35/1389.[22] A. Andrianov, M. Ioffe, and D. Nishnianidze, Phys. Lett. A 201, 103 (2002).[23] A. Das and S. A. Pernice, arXiv:hep-th/9612125v1 (1996).[24] M. A. Gonzalez-Leon, J. M. Gullarte, and M. de la Torre Mayado, SIGMA 3, 124

(2007).[25] A.A.Andrianov, M.V.Ioffe, and V.P.Spiridonov, Phys. Lett. A 174, 273 (1993).[26] R. I. Dzhioev and V. L. Korenev, Phys Rev Lett 99, 037401 (2007), ISSN 0031-

9007 (Print).[27] A. A. Andrianov, M. V. Ioffe, and D. N. Nishnianidze, Theoretical and Mathe-

matical Physics 104, 1129 (1995).

78

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Bohmian Trajectories of Semiclassical Wave Packets

S. Romer

joint work with D. DurrMathematisches Institut, LMU Munchen,

Theresienstr. 39, 80333 Munchen, Germany∗

I. INTRODUCTION

There are many ways to formulate the classical limit of quantum mechanics.The strongest assertion would be about “quantum particle trajectories” becom-ing Newtonian. Particle trajectories, however, are not ontological elements oforthodox quantum theory and thus the “classical limit” must be defined in someoperational way. In contrast, Bohmian mechanics, which for all practical pur-poses is equivalent to quantum mechanics, is a quantum theory of point particlesmoving, so the study of the classical limit becomes a straightforward task [1, 2]:Under which circumstances are the Bohmian trajectories of particles approx-imately Newtonian trajectories? Here “approximately” can be understood invarious manners. The technically simplest but also weakest is that at every timet the Bohmian particle’s position is close to the centre of a “classically moving”very narrow wave packet ψ. This essentially amounts to showing that |ψ(t)|2 ismore or less transported along a Newtonian flow (see [3] for a recent work onthis).The strongest and clearly most direct assertion would be that almost every

Bohmian trajectory converges to a Newtonian trajectory in the uniform topol-ogy. We shall prove here a slightly weaker statement, namely that the uniformcloseness holds in probability. We shall establish this result for a particular classof wave packets which were defined by Hagedorn in [4] and which move alongclassical paths.

∗Electronic address: roemer@math.lmu.de

79

II. BOHMIAN TRAJECTORIES OF HAGEDORN WAVE PACKETS

In Bohmian mechanics the state of a particle is described by a wave functionψ(y, s) (y ∈ R

3, s ∈ R) and by its position Y ∈ R3. The wave function evolves

according to Schrodinger’s equation (~ = m = 1)

i∂

∂sψ(y, s) = Hψ(y, s) =

(

−1

2y + V (y)

)

ψ(y, s) (1)

with the potential V . The wave function governs the motion of the particle by

d

dsY (y0, s) = vψ (Y (y0, s), s) = Im

(∇yψ(Y (y0, s), s)

ψ(Y (y0, s), s)

)

, Y (y0, 0) = y0 . (2)

For a wave function ψ the position Y is a random variable the distribution ofwhich is given by the equivariant probability measure P

ψ with density |ψ(y)|2(Born’s statistical rule; see [2, 5] for a precise assertion). This means that at anytime t the particle will typically be somewhere in the “main” support of |ψ(y, t)|.Thus for a narrow wave packet which, according to Ehrenfest’s theorem, moves– at least for some time – along a classical trajectory, at every instance of timet the position of the particle will typically be close to a classical position. To besure: this does not imply that a typical Bohmian trajectory stays close to theclassical trajectory for the whole duration of a given time interval, since it mayevery now and then make a large excursion.We consider a sufficiently smooth potential and a special class of initial wave

functions where the potential V varies on a much larger scale than the wavefunctions, see e.g. [1] for a physical discussions of the scales. More precisely, wechoose V ε(y) := V (εy) for some small parameter ε, thus defining a microscopic(y, s) and a macroscopic scale (x, t) := (εy, εs). As initial wave functions we

take the semiclassical wave packets Φεk(Xcl(0),P cl(0), ·) defined by Hagedorn in

[4, 6]. They are non-isotropic three dimensional generalised Hermite polynomialsof order k := |k| multiplied by a Gaussian wave packet centred around the

classical phase space point (Xcl(0),P cl(0)). On the macroscopic scale, i.e. onthe scale of variation of the potential, their standard deviation is of order

√ε

both in position and momentum, that is they vary on an intermediate scale.This is the best order of ε allowed, since by Heisenberg’s uncertainty relationσyσp ∼ 1 on the microscopic scale, so on the macroscopic scale σxσp = εσyσpmust be of order ε.In the following, we change to macroscopic coordinates (x, t) = (εy, εs). With

:= x, ∇ := ∇x and ψε(x, t) := ε−3

2ψ(xε, tε) Schrodinger’s equation then

reads

iε∂

∂tψε(x, t) = Hεψε(x, t) =

(

−ε2

2+ V (x)

)

ψε(x, t) . (3)

80

In this setting Hagedorn [4, 6] proved: with an error of order√ε in L2-norm

the solution ψεk(x, t) of (3) with initial data ψεk(x, 0) = Φεk(Xcl(0),P cl(0),x)

is given by Φεk(Xcl(t),P cl(t),x), where (Xcl(t),P cl(t)) is the corresponding

classical phase space trajectory, that is the solution of the Newtonian law ofmotion with initial data (Xcl(0),P cl(0)).

Now consider the Bohmian trajectories on the macroscopic scale, i.e. solutionsof the differential equation

d

dtXε(x0, t) = vψ

ε

k(Xε(x0, t), t) =

= εIm

(∇ψεk(Xε(x0, t), t)

ψεk(Xε(x0, t), t)

)

, Xε(x0, 0) = x0 .

(4)

Our main result [9] is their convergence in probability: For all T > 0 and γ > 0there exists some R <∞ such that

Pψε

k(·,0)(x0 ∈ R

3 | maxt∈[0,T ]

|Xε(x0, t)−Xcl(t)| ≤ R√ε) > 1− γ (5)

for all ε small enough.

III. IDEA OF THE PROOF

Since the Bohmian trajectories Xε(x0, t) (as solutions of (4)) are continu-

ous in t, none of those starting close to Xcl(0) (with x0 ∈ BR√ε(X

cl(0)) =

x ∈ R3 | |Xcl(0) − x| < R

√ε) can leave the vicinity BR

√ε(X

cl(t)) of the

classical trajectory Xcl(t) without crossing the moving sphere ∂BR√ε(X

cl(t)) =

x ∈ R3 | |Xcl(t) − x| = R

√ε. According to Hagedorn’s results [4, 6]

nearly all Bohmian trajectories start in BR√ε(X

cl(0)), so one is left to con-

troll the probability that a Bohmian trajectory crosses ∂BR√ε(X

cl(t)) in thetime interval [0, T ]. The latter is equivalent to controlling the propability thatthe random configuration-space-time trajectory (Xε(·, t) , t) crosses the surface

ΣεT = (x, t) | t ∈ [0, T ], x ∈ ∂BR√ε(X

cl(t)). Invoking the probabilistic

meaning of the quantum probability current density Jψε

k := (jψε

k , |ψεk|2) with

jψε

k := εIm(

(ψεk)∗∇ψεk

)

one can show [7, 8] that an upper bound for this crossingpropability is given by the modulus of the flux across this surface

∫

Σε

T

∣

∣

∣Jψ

ε

k(x, t) ·U∣

∣

∣dσ

where U denotes the local unit normal vector at (x, t). Then the main technicalchallenge is to establish the pointwise estimates for the wave function ψεk and

81

its gradient ∇ψεk that are needed to calculate this integral. For these pointwiseestimates and the precise proof of (5) see [9].

[1] V. Allori, D. Durr, S. Goldstein and N. Zanghı, Journal of Optics B 4 (2002) 482,arXiv: quant-ph/0112005.

[2] D. Durr and S. Teufel, Bohmian Mechanics (Springer, Berlin, 2009), revised trans-lation of Durr, D.: Bohmsche Mechanik als Grundlage der Quantenmechanik,Springer, Berlin, 2001.

[3] P. Markowich, T. Paul and C. Sparber, Journal of Functional Analysis 259 (2010)6 1542 .

[4] G. A. Hagedorn, Ann. Physics 269 (1998) 77.[5] D. Durr, S. Goldstein and N. Zanghı, Journal of Statistical Physics 67 (1992) 843.[6] G. A. Hagedorn, Ann. Inst. H. Poincare Phys. Theor. 42 (1985) 4 363.[7] K. Berndl, Zur Existenz der Dynamik in Bohmschen Systemen, Ph.D. thesis,

Ludwig-Maximilians-Universitat Munchen, (1994).[8] K. Berndl, D. Durr, S. Goldstein, G. Peruzzi and N. Zanghı, Comm. Math. Phys.

173 (1995) 3 647.[9] D. Durr and S. Romer, Journal of Functional Analysis 259 (2010) 9 2404 .

82

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Quantum Trajectories in Phase Space

Craig C. MartensDepartment of Chemistry

University of California, Irvine

Irvine, CA 92697-2025, USA

In this paper we review a method for simulating quantum processes usingclassical-like molecular dynamics in phase space. Our approach is based onsolving the quantum Liouville equation using ensembles of classical trajectories.The nonlocality of quantum mechanics is incorporated in the trajectory rep-resentation as nonclassical interactions between the members of the ensemble,leading to an entanglement of their evolution. The statistical independence ofthe individual trajectories making up an ensemble in the classical limit is lostwhen quantum effects are included, and the entire state of the system must bepropagated as a unified whole. We describe implementations of this approachin the Wigner and Husimi representations of quantum mechanics in phase spaceand apply the method to several model problems. A gauge-like freedom in therepresentation of phase space distribution function evolution with quantum tra-jectory ensembles is described. Finally, we briefly consider the general problemof solving evolution equations using trajectories in contexts beyond quantumdynamics.Quantum mechanics is the proper theoretical framework for describing the

behavior of atoms and molecules [1, 2]. For simple systems, a direct numericalsolution of the time-dependent Schrodinger equation is quite feasable, thanks toadvances in both theoretical methodology and computer performance. This ap-proach ceases to be practical for complex many-body problems, and approximatemethods must be employed. A broad range of such approaches have been devel-oped, including mean-field methods, semiclassical and mixed classical-quantummethods, phenomenological reduced descriptions, and others.One surprisingly effective approach in many cases is to simply ignore quantum

effects altogether and use classical mechanics to describe the motion of atoms inmolecular systems. The result is the method called classical molecular dynam-ics (MD) [3], a commonly used tool for studying many particle systems wherehigh temperatures, large masses, or other factors allow quantum effects in theatomic motion to be neglected. An MD simulation is performed by solving theappropriate Hamilton’s or Newton’s equations of motion for the particles mak-ing up the system given their mutual forces of interaction and appropriate initial

83

conditions. An individual classical trajectory for a multidimensional problem ismuch easier to integrate numerically than the time-dependent wave packet of thecorresponding quantum system. Unless the anecdotal information revealed by asingle trajectory is sufficient, however, collections of trajectories—ensembles—must in general be considered. A distribution of trajectories evolving in phasespace is the most direct classical analogue of an evolving quantum wave packet,and statistical averages of dynamical variables over the classical ensemble par-allel the corresponding quantum expectation values of operators.In order to discuss quantum systems from an analogous ensemble perspec-

tive, we adopt a phase space representation of quantum mechanics—the Wignerrepresentation [4–7].The nonlocality of quantum mechanics forbids arbitrarily fine subdivision of

the quantum distribution into individual independent elements as is possible inclassical mechanics, and insists that the entire state be propagated as a unifiedwhole. If a trajectory ensemble representation of nonlocal quantum motion isto be achieved, the statistical independence of the trajectories must be given upand the individual members of the ensemble must interact with each other.We represent the time-dependent state of the system ρ(q, p, t) as an ensemble

of trajectories. In classical mechanics, the ensemble members evolve indepen-dently of each other. A quantum state, however, is a unified whole, and theuncertainty principle prohibits an arbitrarily fine subdivision and independenttreatment of its constituent parts. We incorporate the non-classical aspects ofquantum mechanics explicitly as a breakdown of the statistical independence ofthe members of the trajectory ensemble. We derive non-classical forces actingbetween the ensemble members that model the quantum effects governing theevolution of the corresponding nonstationary wave packet.The realization of our formalism in the context of a classical molecular dy-

namics simulation is accomplished by generating an ensemble of initial conditionsrepresenting ρW (q, p, 0) and then propagating the trajectory ensemble.The entangled trajectory formalism gives a unique and appealing physical

picture of the quantum tunneling process. Rather than “burrowing” throughthe obstacle, trajectories that successfully escape the metastable well do so by“borrowing” enough energy from their fellow ensemble members to surmountthe barrier. This loan is then paid back, always keeping the mean energy of theensemble a constant.We have outlined an approach to the simulation of quantum processes using

trajectory integration and ensemble averaging [8–12]. The general method hasbeen applied in the context of quantum tunneling through a potential barrier.The basis of the method is the Liouville representation of quantum mechan-ics and its realization in phase space via the Wigner function formalism. Theevolution of the phase space functions is approximated by the motion of thecorresponding trajectory ensembles. In the classical limit, the members of theensemble evolve independently under Hamilton’s equations of motion. When

84

quantum effects are included, however, the corresponding “quantum trajecto-ries” are no longer separable from each other. Rather, their statistical inde-pendence is destroyed by nonclassical interactions that reflect the nonlocality ofquantum mechanics. Their time histories become interdependent and the evolu-tion of the quantum ensemble must be accomplished by taking this entanglementinto account.

[1] C. Cohen-Tannoudji and B. Diu and F. Laloe, Quantum Mechanics, (Wiley, NewYork, 1977).

[2] G. C. Schatz and M. A. Ratner, Quantum Mechanics in Chemistry, (Prentice Hall,Englewood Cliffs, 1993).

[3] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, (ClarendonPress, Oxford, 1987).

[4] E. P. Wigner, Phys. Rev., 40, (1932), 749.[5] K. Takahashi, Prog. Theor. Phys. Suppl., 98, (1989), 109.[6] H. -W. Lee, Phys. Rep., 259, (1995), 147.[7] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, (Oxford University

Press, Oxford, 1995).[8] A. Donoso and C. C. Martens, Phys. Rev. Lett., 87, (2001), 223202.[9] A. Donoso and C. C. Martens, Int. J. Quantum Chem., 87, 2002, 1348.

[10] A. Donoso and C. C. Martens, J. Chem. Phys., 116, (2002), 10598.[11] A. Donoso and C. C. Martens, J. Chem. Phys., 119, (2003), 5010.[12] H. Lopez and C. C. Martens and A. Donoso, J. Chem. Phys., 125, (2006), 15411.

85

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

The Semiclassical Limit of Time Correlation Functions by

Path Integrals

G. CiccottiSchool of Physics, Room 302b UCD-EMSC,

University College Dublin, Belfield, Dublin 4, Ireland∗

I. INTRODUCTION

The exponential scaling of the computational cost of quantum time evolutionwith the number of degrees of freedom motivates current attempts to approx-imate and interpret quantum dynamics via classical trajectories. These canin fact be computed with essentially linear effort and provide a more intuitiverepresentation of the dynamics. In spite of these tempting properties of the tra-jectories, the accuracy and generality of such attempts requires careful analysissince it is unclear whether they can be successful for condensed phase systems.To illustrate this point, we comment on how and when quantum evolution canbe approximated in terms of (generalized) classical dynamics in the calculationof the symmetrized time correlation function [1]

GAB(t, β) =1

ZTrAe

i~Ht∗c Be−

i~Htc (1)

in semiclassical conditions. In the expression above, tc = t − i~β2 , β = 1/kBT

(T is the temperature and kB Boltzmann’s constant), H is the Hamiltonian of

the system and Z = Tre−βH is the canonical partition function. Eq. (1) isequivalent via a relationship in Fourier space to the standard time correlation

function CAB(t, β) =1ZTre−βHAe

i~HtBe−

i~Ht, but it also shares some formal

properties with classical correlation functions, for example it is by constructiona real function, and this suggests that it might be a convenient starting point fordescribing semiclassical systems (see for example[2–8]). The analysis presentedin the following is described in [9] and we refer to that paper for a detailedderivation of the results summarized here.

∗ On leave from Dipartimento di Fisica and CNISM Unita 1, Universita “La Sapienza”,Piazzale Aldo Moro 5, 00185 Rome, Italy

86

II. THEORY

The starting point of our considerations is the Feynman path integral expres-sion (in a mixed coordinate and momenta representation) of the forward andbackward propagators in complex time that appear in eq. (1). To examine thesemiclassical limit of that expression, mean and difference paths in the coordi-nates and momenta are introduced and the exponent of the overall path integralswritten as a Taylor series expansion in the difference paths.

A. First order result

Retaining only terms up to linear order in the Taylor series expansion, thesymmetrized function can be written as

G(1)AB(t, β) =

1

Z

∫

dr0dp1e−β

[

p212m+V (r0)

]

Aw(r0, p1)Bw(rt, pt) (2)

where Ow(r, p) stands for the Wigner transform [10] of operator O, and (rt, pt)are the end points of the classical trajectory evolved from (r0, p1) for a time t.Both the dynamics and the statistical weight in the correlation function abovethus reduce to their fully classical counterparts. The Fourier relationship withthe standard time correlation function mentioned in the Introduction can, how-ever, be used to restore some non-classical properties (such as detailed balance)of this quantity, and it is in fact formally identical to the so-called quantum cor-rection procedure that was introduced by Schofield in ref. [1]. However, it is wellknown that this correction can fail at low temperature even when the systemis non-interacting (see [9] for an explanation of this fact) and, more in general,that the temperature and mass range in which it is valid are quite limited.

87

B. Second order result

The result of a second order truncation of the series expansion of the exponent,instead, can be expressed as

G(2)AB(t, β) =

Zcl(2ǫβ/~)

Z(β)

∫

dr0drN

N−1∏

k=1

∫

drkdpk

dpN2π~

×

e−

2ǫβ~

[

p212m+V (r0)

]

Zcl(2ǫβ/~)

e−∑N

k=11

2σ2 [rk−rk−1−ǫtpkm ]

2

(√2πσ2)N

e−

∑N−1k=1

[pk+1−pk+ǫt∇V (rk)]2

2~ǫβ |∇2V (rk)|

∏N−1k=1

√

2π~ǫβ |∇2V (rk)|

× e−

2ǫβ~

∑Nk=2

[

p2k

2m+V (rk−1)

]

Aw(r0, p1)Bw(rN , pN )FΩ(pk, rk; ǫβ , ǫt)

(3)

where rk−1, pk (k = 1, ..., N) are the positions and momenta along the path,Zcl(2ǫβ/~) is the classical partition function at inverse temperature 2ǫβ/~, andFΩ is discussed below. The factors in the curly bracket are a probability densityand the approximate symmetrized correlation function can then be computed as

the expectation of e−

2ǫβ~

∑Nk=2

[

p2k

2m+V (rk−1)

]

Aw(r0, p1)Bw(rN , pN )FΩ(pk, rk).The variables rk−1, pk can be sampled as follows: The zero time val-ues r0, p1 are obtained from the high temperature classical Boltzmann factor

e−

2ǫβ~

[

p212m+V (r0)

]

/Zcl(2ǫβ/~), while the other variables are generated recursivelyfrom

rk = rk−1 + ǫtpk/m+ σξk k ∈ [1, N ]

pk+1 = pk − ǫt∇V (rk) +√

~ǫβ |∇2V (rk)|ηk+1 k ∈ [1, N − 1](4)

where ξk and ηk+1 are Gaussian white noises. This scheme illustrates how quan-tum mechanical delocalization sets in this semiclassical representation of thecorrelation function. Within this approximation in fact, both the new coor-dinates and momenta are sampled at each complex time step from Gaussiandistributions centered around classically evolved phase space points. The dis-

persion around the classical path is determined by the variances σ2 =~ǫβm

and

~ǫβ∇2V (rk). The classical limit is restored for ~ → 0 and/or β → 0 when thesevariances tend to zero. For finite values of Planck’s constant or of the inversetemperature, the non classical nature of the time evolution of the system ap-pears at each time step in the form of Gaussian random displacements from the

88

”driving” classical propagation. While this interpretation is intriguing, the ac-tual interest of the driving classical trajectory depends crucially on the system.If the potential is everywhere convex, the function FΩ in the integrand reducesto a constant and the estimate of the average as a mean over paths generatedas outlined above is viable. In the more general case of potentials with regionsof negative curvature, on the other hand, this function does not have an ex-plicit form, and there is no reason to expect that it will be localized aroundthe complex paths generated via the sampling scheme of eq. (4). Furthermore,it can be shown that small variations in its argument result in ”explosively”different values for FΩ. Attempts to interpret or estimate the average abovevia a scheme based on localized paths are therefore doomed to failure for tworeasons: first, the integrand is not peaked around the sampling function, secondit is a numerically unstable function. These characteristics are a direct mani-festation of delocalization, an intrinsic property of quantum mechanics that itis very difficult, if not impossible, to represent within this semiclassical scheme.A possible scheme to build upon the considerations presented here and to gobeyond the semiclassical approximation systematically and still employing thepath integral representation of the symmetrized correlation function has beenintroduced in [11] and is summarized in the contribution by S. Bonella.

III. CONCLUSIONS

The path integral expression of the symmetrized correlation function is a usefultool to examine how and when quantum evolution can be approximated via (gen-eralized) classical trajectories. In particular, the second order result presentedin the previous section shows how, in the semiclassical limit, the most relevantcontributions to the path integral localize or, pathologically, de-localize aroundguiding or poorly guiding classical trajectories for general systems. While weemployed the path integral formalism to illustrate how a picture based on classi-cal dynamics is usually not enough to compute quantum properties, the difficultyto account for delocalization appears also in other approximations of quantummechanics (e.g. Wigner-Liouville, semiclassical IVR) pointing to the inherentdifficulty of using a trajectory based picture to represent this phenomenon.

[1] P. Schofield. Phys. Rev. Lett, 4:239, 1960.[2] V.S. Filinov. Mol. Phys., 88:1517, 1996.[3] V.S. Filinov. Mol. Phys., 88:1529, 1996.[4] W.H. Miller, S.D. Schwartz, and J.W. Tromp. J. Chem. Phys., 79:4888, 1983.[5] V. Jadhao and N. Makri. J. Chem. Phys., 129:161102, 2009.[6] J.A. Poulsen, H. Li, and G. Nyman. J. Chem. Phys., 131:024117, 2009.

89

[7] G. Krilov, E. Sim, and B.J. Berne. Chem. Phys., 268:21, 2001.[8] N. Chakrabarti, T. Carrington, and B. Roux. Chem. Phys. Letts., 293:209, 1998.[9] S. Bonella, M. Monteferrante, C. Pierleoni, and G. Ciccotti. J. Chem. Phys.,

133:164104 2010.[10] E. Wigner. Phys. Rev., 40:749, 1932.[11] S. Bonella, M. Monteferrante, C. Pierleoni, and G. Ciccotti. J. Chem. Phys.,

133:164105 2010.

90

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Path Integral Calculation of (Symmetrized) TimeCorrelations Functions

S. BonellaDepartment of Physics, Universita di Roma La Sapienza,

P.la A.Moro 2, 00185 Rome, Italy

I. THEORY

The path integral expression of the symmetrized time correlation function [1],

GA,B(t, β) = 1

ZTrAe

i~Ht∗c Be−

i~Htc can be used to derive a computational

scheme that, in principle, allows to systematically include quantum dynamicaleffects starting from a zero order approximation of this function based uponclassical trajectories. In the following, we summarize this scheme, illustrated indetail in [2]. Introducing resolutions of the identity in the coordinate represen-tation, we can isolate matrix elements of the operators and use the time compo-sition property to rewrite the forward and backward propagators in GAB(t, β)as products of short complex time propagators. Thus (indicating with tilde thevariables referring to the backward propagator)

GAB(t;β) =1

Z

∫

drLdrL < rL|B|rL >

L−1∏

J=1

∫

drJdrJK(rJ+1, rJ+1, rJ , rJ)

×

∫

dr0dr0K(r1, r1, r0, r0) < r0|A|r0 > (1)

In the expression above, we introduced the product of the real and imaginarytime propagators for the (finite) intervals τβ = β/(2L) and τt = t/L, and definedthe short complex time propagator for the leg that evolves the system forwardfrom configuration rJ to configuration rJ+1 and backward from rJ+1 to rJ as

K(rJ+1, rJ+1, rJ , rJ) =

∫

drνJdrνJ < rJ |e

−τβH |rνJ >< rνJ |ei~τtH |rJ+1 >

× < rJ+1|e−

i~τtH |rνJ >< rνJ |e

−τβH |rJ > (2)

The propagators in inverse temperature in the expression above can be computedexactly via relatively straightforward path integral methods for the evaluation

91

of the density matrix [3, 4] in convenient variables (sum and difference). Theproduct of propagators in real time, instead, is approximated by introducing apath integral representation in sum and difference variables [5] (see also contri-bution by G. Ciccotti), and truncating to first order the Taylor series expansionof the representation’s phase in the difference variables. The effect of this lin-earization approximation is to reduce the generation of the path in sum variablesto a time-stepping propagation that is formally identical to a classical evolutionalgorithm. At the same time, the path in difference variables can be almostcompletely integrated analytically and the only remaining contributions fromthese variables appear in two phase factors to be computed at the beginningand the end of the classical propagation along the sum path. The explicit formof the approximate short time propagators can be found in [2]. Substitutingthis form in eq. (1), the symmetrized correlation function can be interpreted asthe ratio of two expectation values over a probability density, P , defined by theproduct of the Green’s functions corresponding to the (approximate) real and(exact) imaginary time dynamics in each short time propagator K (see ref. [2]for the definition). Indicating with I the identity operator, if we use L shorttime propagators, this ratio is

GLAB(t;β) =

< OAB(t;β) >P

< OII(t;β) >P

(3)

where the observable OAB(t;β) contains the matrix elements of the operators,and a product of L − 1 phase factors originated by the surviving differencevariable integrals (see ref. [2] for the explicit definition). Since the lineariza-tion approximation for the real time propagation becomes exact for L → ∞,i.e. as we increase the number of propagators K, the expression above for thesymmetrized correlation function can, in principle, be made as accurate as nec-essary. The averages in eq.(3) can be computed via Monte Carlo sampling of theprobability density P . This sampling is realized by generating paths in complextime according to the propagations in real and imaginary time determined bythe appropriate Green’s functions.The structure of these paths is illustrated in Fig. (1). For L = 1 there is

only one real time leg of duration τt = t while the imaginary time propagationcorresponds to an inverse temperature β/2 for both the mean and differencevariables. The combination of these propagations is illustrated in the top panel.In the figure, the horizontal axis is time, the vertical axis temperature. The ver-tical plane represents the space of configurations associated to the thermal pathintegral, the intermediate states of the path are indicated with the red spheres(in the figure, we use, as an example, six ”thermal beads”). The harmonic inter-actions that represent the kinetic part of Hamiltonian in the thermal paths (seefor example [3]) are indicated with zigzagged lines connecting adjacent beads,while the interactions among the two paths due to the potential are drawn asdashed lines. The propagation in real time is sketched in the figure as the curve

92

FIG. 1: Graphic representation of the propagators in real and imaginary times con-tributing to the approximate Schofield function for the case L = 1 (upper panel) andL = 2 (lower panel). The horizontal axis is real time while the vertical axis is in-verse temperature. The thermal paths are represented as red dots on the verticalplanes. Segments of classical propagation in phase space are represented as continuosred curves in the horizontal planes. The golden circles indicate the connection betweenthe dynamics in real time (horizontal planes) and the representation of the dynamicsin imaginary time (vertical planes).

on the horizontal plane. This plane represents the phase space of the system.The red and golden circle at t = 0 indicates the structure of the initial condi-tions for the evolution: the initial coordinate coincides with the last bead of thethermal path in the mean variables, while the initial momentum is sampled froma Maxwellian at inverse temperature β/2. A phase factor is associated with theinitial and final point of the classical propagation. The structure of the productof the Green’s functions for generic values of L can be inferred from the lowerpart of figure 1 where we show what happens for L = 2. In this case, thereare two segments of classical dynamics, represented as in the previous case bythe curves on the horizontal planes, each of duration t/2 and two propagationsof mean and difference variables in imaginary time, represented in the verticalplanes, each taking the system to an inverse temperature equal to one half ofthe actual inverse temperature. As before, the first segment of dynamics starts,with a Gaussian initial momentum, from the last bead of the mean variables

93

thermal path at t = 0. The end point of this leg of propagation is the initialconfiguration for the mean variable thermal path at t/2, and the second segmentof dynamics has as initial conditions the final coordinate of the mean variablethermal path and a new momentum sampled from a Gaussian. The variances ofthe Gaussians associated to the momentum sampling are doubled with respect tothe case L = 1. Two phase factors at the beginning and end of each classical dy-namics segment are associated with this product of Green’s functions (so L = 2has a total of four phase factors). In general, the product of Green’s functionsin GL

AB(t;β) involves L segments of classical propagation, each of duration t/L,interspersed with L pairs of thermal paths in the mean and difference variables,each at an inverse temperature β/2L. The rules for connecting the coordinateand momenta at the initial and final time of the dynamics with the final andinitial points of the thermal paths, and for constructing the 2L phase factorscontributing to the observable are completely analogous to the L = 2 case.

II. CONCLUSIONS

The method just outlined becomes exact as the number of segments of prop-agation in complex time tends to infinity and, in this respect, it is completelyanalogous to a thermal path integral that becomes exact in the limit of an infinitenumber of ”slices” in the imaginary time propagator. Unfortunately, since in thislimit the number of phase factors in the observable also becomes infinite, thenumerical effort required to compute the Monte Carlo average increases dramat-ically. This is not surprising as any exact expression for the correlation functionwill manifest the well known dynamical sign problem characteristic of quantumdynamics. For semiclassical systems, however, the number of segments requiredto achieve convergence of the approximation can be small enough to make thecomputational strategy described here interesting. The tests performed so farindicate that the L = 1 result provides a viable alternative to standard fullylinearized methods [5–7] with the advantage that the expression for the thermaldensity is simpler than that in those methods. The tests also show that quantumcorrections to the dynamics can indeed be introduced by increasing the numberof short time propagators. However, the numerical cost associated to higherorder iterations of the scheme is very high and more work is necessary to makethe algorithm useful for condensed phase simulations.

[1] P. Schofield. Phys. Rev. Lett, 4:239, 1960.[2] S. Bonella, M. Monteferrante, C. Pierleoni, and G. Ciccotti. J. Chem. Phys.,

133:164105 2010.

94

[3] R. P. Feynman. Statistical Mechanics a Set of Lectures. Addison Wesley, New York,1990.

[4] H. Kleinert. Path Integrals in Quantum Mechanics, Statics, Polymer Physics and

Financial Markets. World Scientific, Singapore, 2004.[5] J.A. Poulsen, G. Nyman, and P.J. Rossky. J. Phys. Chem A, 108:8743, 2004.[6] H. Wang, X. Sun, and W.H. Miller. J. Chem. Phys., 108:9726, 1998.[7] Q. Shi and E. Geva. J. Phys. Chem. A, 107:9070, 2003.

95

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

On-the-fly Nonadiabatic Bohmian DYnamics (NABDY)

Ivano Tavernelli, Basile F. E. Curchod, Ursula RothlisbergerLaboratory of Computational Chemistry and Biochemistry,

Ecole Polytechnique Federale de Lausanne.

I. INTRODUCTION

In the semiclassical description of molecular systems, only the quantum char-acter of the electronic degrees of freedom are considered while the nuclear motionis treated at the classical level [1]. In the adiabatic case, this picture correspondsto the Born-Oppenheimer limit where the nuclei move as point charges on thepotential energy surface (PES) associated with a given electronic state. Thisapproximation relies on the assumption that the electrons follow adiabaticallythe motion of the slow nuclei, and thus that the electronic state of the systemis not affected by the nuclear displacement. Despite the wide success of this ap-proximation, many physical and chemical processes do not fall in a regime wherenuclei and electrons can be considered to be decoupled [2–4]. In particular, foran adequate description of many photochemical and photophysical processesnonadiabatic effects need to be included, e.g. to precisely describe relaxationprocesses via nonradiative internal conversion or intersystem crossing.Within the mixed-quantum classical approaches, the classical path, or Ehren-

fest approximation is the most straightforward one. Here, the classical subsys-tem evolves under the mean field generated by the electrons, and the electronicdynamics is evaluated along the classical path of the nuclei. An important lim-itation of the classical path approach is the absence of a ”back-reaction” of theclassical Degrees of Freedom (DoF) to the dynamics of the quantum DoF. Onthe other hand, these methods are well suited for the study of the nuclear dy-namics in the full phase space (without the need of introducing constraints orlow dimensional reaction coordinates) and can easily be implemented in softwarepackages that allow for the ”on-the-fly” calculation of energies and forces. Oneway is to employ Ehrenfest’s theorem and calculate the effective force on theclassical trajectory through a mean potential that is averaged over the quantumDoF [5–9].Beyond such quasi-classical methods, the semiclassical Wentzel-Kramers-

Brillouin (WKB)-type approach has a long tradition in adding part of the missingnuclear quantum effects to classical simulations. Semiclassical methods [10–12]

96

take into account the phase exp(iS(t)/~) evaluated along a classical trajectoryand are therefore capable - at least in principle - of describing quantum nucleareffects including tunneling, interference effects, and zero-point energies. Themost common semiclassical methods have been extensively reviewed in recentarticles [13–16].The intuitively appealing picture of trajectories hopping between coupled

potential energy surfaces gave rise to a number of quasiclassical implementa-tions [17–24]. The most well-known method is Tully’s ”fewest switches” trajec-tory surface hopping method [17–21, 25] (TSH in this text), which has evolvedinto a widely used and successful technique. In this framework, the nuclearwavepacket is represented by a swarm of independent classical trajectories (in-dependent trajectory approximation, ITA) while the nonadiabatic couplings(NACs) induce hops between different electronic states occurring according to astochastic algorithm, see Fig. 1.

FIG. 1: Left : Trajectory surface hopping in the independent trajectoryapproximation. An initial nuclear wavepacket on an electronic excited state isrepresented by a swarm of independent trajectories (dots and lines). Accordingto a stochastic algorithm, trajectories can jump to other states (vertical dashedlines). The final distribution of a sufficiently large number of such trajectories

is assumed to reproduce the final nuclear wavepacket splitting. Right :Representation of the same process in a nuclear quantum dynamics scheme.

In the adiabatic representation, coupling between electronic and nuclear de-grees of freedom are obtained by calculation of the nonadiabatic coupling terms.The nonadiabatic coupling vectors (NACVs) are defined between state I and Jfor the nuclei γ by:

dγJI(R) =

∫

Φ∗J(r;R) [∇γΦI(r;R)] dr (1)

and the second order nonadiabatic elements by

DγJI(R) =

∫

Φ∗J(r;R)

[

∇2

γΦI(r;R)]

dr. (2)

97

The latter are often neglected due to their usually small size. The complete setof electronic wavefunctions ΦJ(r;R) are the solutions of the time-independentelectronic Schrodinger equation

Hel(r;R)ΦJ(r;R) = EelJ (R)ΦJ(r;R) (3)

In TSH, all trajectories are evolved with forces obtained from an adiabatic sur-face (in contrast to the mean-field approach). In addition, a new equation ofmotion for each trajectory α on state J is propagated, which contains all thenonadiabatic information via a time-dependent amplitude Cα

J (t). The classi-cal trajectories evolve adiabatically according to Born-Oppenheimer dynamicsuntil a hop between two potential energy surfaces (Eel

I and EelJ ) occurs with a

probability given by a Monte Carlo-type procedure. In practice, a swarm of tra-jectories is propagated independently starting from different initial conditions,and the final statistical distribution of all these trajectories is assumed to re-produce the correct time evolution of the nuclear wavepacket (see Fig 1). It isimportant to stress that, at present, no formal justification of Tully’s algorithmhas been formulated. Furthermore, the independent trajectory approximationneglects nuclear quantum effects such as tunneling, interferences between thenuclear wavepackets or (de)coherence effects. All trajectories are independentand do not carry any quantum information like phase or amplitude, which couldbe exchanged with other trajectories.At the end of the 90ies, a new quantum trajectory method (QTM) has been

developed by Wyatt et al. [26] for adiabatic quantum dynamics based on aBohmian (or hydrodynamical) interpretation of quantummechanics. Within thismethod, it is possible to derive formally exact equations of motion for quantumtrajectories (or fluid elements), that explicitly reproduce the nuclear wavepacketdynamics [27]. Trajectories are correlated with one another by means of a quan-tum potential. Different schemes for multisurface QTM have been proposed inthe diabatic representation [28–30].In this work, we propose to solve the non-relativistic quantum dynamics of

nuclei and electrons within the framework of Bohmian dynamics using the adi-abatic representation of the electronic states. An on-the-fly trajectory-basednonadiabatic molecular dynamics algorithm is derived (NABDY, NonAdiabaticBohmian DYnamics), which is able to capture also the nuclear quantum effectsthat are missing in the traditional trajectory surface hopping approach basedon the independent trajectory approximation. The use of correlated trajectoriesproduces a quantum dynamics, which is in principle exact and computation-ally very efficient. Instead of representing the molecular nuclear wavepacketin terms of uncorrelated trajectories, the wavepacket is now split in fluid ele-ments (set of initial molecular configurations) that carry phase and amplitudeinformation. This method allows for on-the-fly dynamics in the adiabatic rep-resentation. Thanks to this formulation, the method can be coupled to the ab

initio code cpmd, which provides electronic energies, classical forces and nonadi-abatic coupling elements for each configuration at DFT/TDDFT level of theory.

98

For the calculations of nonadiabatic coupling elements and more precisely thenonadiabatic coupling vectors between electronic states J and I, dJI(R), withLR-TDDDFT, the main challenge is to express these quantities initially definedin terms of wavefunctions (see Eq. (1)) as functionals of the electronic den-sity d

γJI [ρ](R) or, equivalently, of the set of Kohn-Sham orbitals d

γJI [φ.](R).

Among different techniques, we use auxiliary many electron wavefunctions ob-tained from the ground state and singly excited Kohn-Sham Slater determinants,for which we have shown that the exact results obtained using Many-Body Per-turbation Theory can be recovered [31–33].

II. THEORY AND FIRST APPLICATIONS

Starting from the time-dependent Schrodinger equation, we can use the Born-Oppenheimer Ansatz Ψ(r,R, t) =

∑∞J ΦJ(r;R)ΩJ(R, t) to describe the total

molecular wavefunction Ψ(r,R, t). Here, r gives the positions of all the electronsof the system and R those of the nuclei. J denotes a specific electronic state,ΦJ(r;R) is a solution of the time-independent electronic Schrodinger equationand ΩJ(R, t) can be seen as a nuclear wavefunction. After some manipulationand using the polar representation for the nuclear wavefunctions, we can extractequations of motion for the nuclear amplitude AJ(R, t)

∂AJ(R, t)

∂t=−

∑

γ

1

Mγ

∇γAJ(R, t)∇γSJ(R, t)−∑

γ

1

2Mγ

AJ(R, t)∇2

γSJ(R, t)

+∑

γI

~

2Mγ

DγJI(R)AI(R, t)ℑ

[

eiφ]

−∑

γ,I 6=J

~

Mγ

dγJI(R)∇γAI(R, t)ℑ

[

eiφ]

−∑

γ,I 6=J

1

Mγ

dγJI(R)AI(R, t)∇γSI(R, t)ℜ

[

eiφ]

, (4)

where φ = 1

~(SI(R, t) − SJ(R, t)), and a Newton-like equation for the time

propagation of fluid elements associated to a discretization of the nuclear con-figurational space

Mβ

d2Rβ

(dtJ)2= −∇β

[

EJel(R) +QJ(R, t) +DJ(R, t)

]

(5)

In these equations, SJ(R, t)/~ represents the phase of the nuclear wavefunctionand γ is a general notation for a specific nucleus with massMγ . Eq 5 contains thewell-known quantum potential QJ(R, t) and an additional nonadiabatic quan-tum potential DJ(R, t) which describes all cross-surfaces terms.

99

Numerically, we use a conventional QTM propagation scheme for the adiabaticpart of the dynamics on a specific electronic state (say S1) far from any nona-diabatic regions. When the size of the nonadiabatic couplings and the amountof amplitude transferred reach a certain threshold, the multisurface dynamicsgoverned by the nondiagonal elements dJI(R) and DJI(R) is initiated and anew set of fluid elements is attributed to the new electronic state (S2, see Fig 2).The arbitrary Lagrangian-Eulerian (ALE) frame [34] is used to synchronize thetime propagation of the fluid elements on the different surfaces.

FIG. 2: Schematic view of NABDY. State S2 is activated through a thresholdbased on the incoming amplitude from the initial state S1.

In order to validate our approach, we have first applied it to the study ofthe Tully model 1 system [25]. For initial momenta between 16 and 22 a.u.,the NABDY dynamics compared well with the results obtained from the exactnuclear wavepacket propagation (deviation < 0.6%). On the other hand, thefinal populations obtained from TSH deviate on average by about 2% from theexact result. Interestingly, NABDY dynamics converges with only one tenth ofthe trajectories needed to converge TSH.We have applied NABDY to other nonadiabatic processes, like for instance the

collision of an H atom with a H2 molecule. For this system, we have computedon-the-fly the PESs and the nonadiabatic couplings using DFT/TDDFT withthe LDA exchange-correlation functional as implemented in the plane-wave codeCPMD [35]. The quality of this level of theory for the H-H2 collision has beenassessed in a previous publication [31]. We present here some initial results (seeFig 3), where the H atom is directed almost perpendicular to the H2 bond axiswith two different initial momenta, k = 75 and k = 150 a.u.. In these cases, weobserved a population of the first excited state of 27.9% and 31.4%, respectively.

In conclusion, we report here nonadiabatic Bohmian dynamics, where all theelectronic properties needed are computed on-the-fly by DFT and LR-TDDFT.

100

FIG. 3: Collision of a H atom with a H2 molecule. PESs (blue: ground state,orange: first excited state) and nonadiabatic couplings (black dotted line) arecomputed via DFT/TDDFT. Inset shows the LUMO orbital of the system

close to the avoided crossing.

We are currently working on an efficient adaptation of this scheme in the contextof general nonadiabatic molecular dynamics.

[1] D. Marx and J. Hutter, Modern Methods and Algorithms of Quantum Chemistry,vol. 1 of NIC Series (Forschungszentrum Juelich, 2000) p. 301, p. 301.

[2] A. W. Jasper, S. Nangia, C. Zhu and D. G. Truhlar, Acc. Chem. Res. 39 (2006) 2101.

[3] T. Yonehara, S. Takahashi and K. Takatsuka, J Chem Phys 130 (Jun 2009) 21214113.

[4] K. Takatsuka, International Journal of Quantum Chemistry 109 (2009) 10 2131.[5] R. B. Gerber, V. Buch and M. A. Ratner, J. Chem. Phys. 77 (1982) 32.[6] R. Kosloff and A. Hammerich, Faraday Discuss. 91 (1991) 239.[7] F. A. Bornrmann, P. Nettesheim and C. Schutte, J. Chem. Phys 105 (1996) 1074.[8] C. Zhu, A. W. Jasper and D. G. Truhlar, J. Chem. Phys. 120 (2004) 5543.[9] I. Tavernelli, U. F. Rohrig and U. Rothlisberger, Mol. Phys. 103 (2005) 963.

[10] W. Miller, J. Chem. Phys. 53 (1970) 3578.[11] M. F. Herman, J. Chem. Phys. 81 (1984) 754.[12] E. Heller, J. Chem. Phys. 94 (1991) 2723.[13] M. S. Topaler, T. Allison, D. W. Schwenke and D. G. Truhlar, J. Chem. Phys. 109

(1998) 3321.[14] G. A. Worth and M. A. Robb, Adv. Chem. Phys. 124 (2002) 355.[15] H. D. Meyer and G. A. Worth, Theor. Chem. Acc. 109 (2003) 251.[16] H. D. Meyer, U. Manthe and L. S. Cederbaum, Chem. Phys. Lett. 165 (1990) 73.[17] J. C. Tully and R. K. Preston, J. Chem. Phys. 55 (1971) 562.

101

[18] S. Hammes-Schiffer and J. C. Tully, J. Chem. Phys. 101 (1994) 4657.[19] N. C. Blais, D. Truhlar and C. A. Mead, J. Chem. Phys. 89 (1988) 6204.[20] O. V. Prezhdo and P. J. Rossky, J. Chem. Phys. 107 (1997) 825.[21] I. Krylov and R. Gerber, J. Chem. Phys 105 (1996) 4626.[22] N. L. Doltsinis and D. Marx, Phys. Rev. Lett. 88 (2002) 166402.[23] C. F. Craig, W. R. Duncan and O. V. Prezhdo, Phys. Rev. Lett. 95 (2005) 163001.[24] E. Tapavicza, I. Tavernelli and U. Rothlisberger, Phys. Rev. Lett. 98 (2007)

023001.[25] J. C. Tully, J. Chem. Phys. 93 (1990) 1061.[26] C. L. Lopreore and R. E. Wyatt, Phys. Rev. Lett. 82 (1999) 26 5190.[27] R. E. Wyatt, Quantum dynamics with trajectories: Introduction to quantum hy-

drodynamics (Interdisciplinary applied mathematics, 2005).[28] R. E. Wyatt, C. L. Lopreore and G. Parlant, J. Chem. Phys. 114 (2001) 12 5113.[29] C. L. Lopreore and R. E. Wyatt, J. Chem. Phys. 116 (2002) 4 1228.[30] B. Poirier and G. Parlant, J. Phys. Chem. A 111 (2007) 41 10400.[31] I. Tavernelli, E. Tapavicza and U. Rothlisberger, J. Chem. Phys. 130 (2009)

124107.[32] I. Tavernelli, E. Tapavicza and U. Rothlisberger, J. Mol. Struc. (Theochem) 914

(2009) 22.[33] I. Tavernelli, B. F. E. Curchod and U. Rothlisberger, J. Chem. Phys. 131 (2009)

19 196101.[34] B. K. Kendrick, J. Chem. Phys. 119 (2003) 5805.[35] CPMD (Copyright IBM Corp 1990-2001, Copyright MPI fur Festkorperforschung

Stuttgart, 1997-2001), http://www.cpmd.org.

102

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Quantum Many-Particle Computations with Bohmian

Trajectories:

Application to Electron Transport in Nanoelectronic

Devices

A. Alarcon, G.Albareda, F.L.Traversa and X.OriolsDept. d’Enginyeria Electronica,

Universitat Autonoma de Barcelona (UAB)08193, Bellaterra, Spain

Email: alfonso.alarcon@uab.es

I. INTRODUCTION

We have recently shown [1] that Bohmian trajectories allow a direct treatmentof the many-particle interaction among electrons with an accuracy comparableto Density Functional Theory (DFT) techniques. The computational techniquedeveloped in [1], which allows a computation of the quantum correlations amongparticle without explicitly knowing the many-particle wave-function, can be ap-plied to many different open and closed quantum systems. In particular, in thisarticle, we present a general, versatile and time-dependent 3D quantum elec-tron transport simulator, named BITLLES (Bohmian Interacting Transportin Electronic Structures), based on the computation of many-particle Bohmiantrajectories mentioned in [1]. As a numerical example, we show the ability ofBITLLES simulator to predict the electrical characteristics (DC, AC and fluc-tuations) of a Resonant Tunneling Diode, i.e. a many-particle open quantumsystem far from equilibrium.

II. THE BITLLES SIMULATOR

A. The Monte Carlo nature of the simulator

It is not possible to take into account all degrees of freedom for a close Hamil-tonian of the whole solid-state system (battery, wires, sample,...). Therefore,when we neglect a large part of the degrees of freedom of the circuit, we dis-regard their influence into the N(t) explicitly simulated electrons inside the 3Dactive region of the device, i.e.the simulation box whose volume Ω is depicted in

103

Fig. 1. Thus, we cannot completely specify the initial N(t)-particle wavefunc-tion inside the simulation box because we do not know with certainty the numberof electrons N(t), their energies, their positions.... In the BITLLES simulator,the adaptation of Bohmian mechanics to electron transport in open systems,leads to a quantum Monte Carlo algorithm, where randomness appears becauseof these uncertainties. For this purpose, we take into account two statisticalensembles of the initial properties of the electrons on the numerical simulations.First, a g-distribution that represents the infinite ensemble of all possible distri-butions in the initial positions of Bohmian particles. Second, an h-distribution

that takes into account the uncertainty in the number of electrons in the activeregion N(t), the mean energy associated to the wavepackets of these electronsand the injection times of each electron [2].

B. Self-consistent solution of Poisson equation and many-particleSchrodinger equation

Many-particle time-dependent quantum electron simulators have to providereasonable approximations for handling the many-particle (electron - elec-tron interaction) problem. However, it is well-known that the many-particleSchrodinger equation can be solved for very few -two, three,..- degrees of free-dom. To surpass this computational problem, BITLLES simulator is basedon a novel algorithm for solving the many-particle Schrodinger equation withBohmian mechanics [1]. It can include explicitly the Coulomb and exchangecorrelations (at a level comparable to the time dependent density functionaltheory). Following reference [1] a many-particle Bohmian trajectory ~ra[t] as-sociated to an a-electron can be computed from the following single-particlewavefunction, Ψa(~ra, t), solution of the single-particle Schrodinger equation:

i~∂Ψa(~ra, t)

∂t= −

~2

2m∇2

~ra+ Ua(~ra, ~Ra[t], t) +Ga(~ra, ~Ra[t], t)

+i · Ja(~ra, ~Ra[t], t)Ψa(~ra, t) (1)

where we define: ~Ra[t] = ~r1[t], ~ra−1[t], ~ra+1[t], ~rN [t], t as a vector that containsall Bohmian trajectories except ~ra[t].

The explicit expression of the potentials Ga(~ra, ~Ra[t], t) and Ja(~ra, ~Ra[t], t) areexplained in [1]. However, their numerical values are unknown and need some

educated guesses [1]. Conversely, the term Ua(~ra, ~Ra[t], t) can take into accountCoulomb interaction without any approximation.

In BITLLES simulator the term Ua(~ra, ~Ra[t], t) in Eq.(1) can be computedfrom the following 3D Poisson equation:

∇2~ra

(

ε(~ra)Ua(~ra, ~Ra[t], t))

= ρa(~ra, ~Ra[t], t) (2)

104

where ρa(~ra, ~Ra[t], t) is the the charge density of an a-electron due to the rest ofelectrons (Bohmian trajectories) except itself [3]. Therefore, there is a Coulombpotential (or electric field) for each a-electron. The Poisson equation, Eq.(2) withthe appropriate boundary conditions, provides the electron-electron Coulombinteraction that reproduces accurately the electrostatics of the system. Then,at each simulation time step, dt, we solve N(t) Poisson equations with N(t)different charge densities. These potential energies solution of the N(t) Poissonequation are, then, introduced into the N(t) Schrodinger equations defined inEq. (1). It is precisely in the time-dependence of the potentials of Eq. (1) wherethe correlations with other electrons appear [3].We do also use a novel boundary conditions for solving the Poisson equation

ensuring ’overall charge neutrality’ and ’current conservation’, even for smallsimulating boxes [4].The overall procedure explained in this section provides a self-consistent so-

lution of the Poisson and the many-particle Schrodinger equations beyond themean field approximation. More details are explained in references [2, 3, 5].

C. Time-dependent current computation

In fact, there is an additional argument that justifies the importance of prop-erly introducing the many-particle Coulomb interaction. One has to computetime-dependent variations of the electric field (i.e. the displacement current) toassure that the total time dependent current computed in a surface of the simu-lating box is equal to that measured by an ammeter, i.e. ’current conservation’[2, 5].Therefore, the computation of the total (conduction plus displacement) cur-

rent in BITLLES simulator is made by means of an algorithm based on theRamo-Shockley theorem [6], [7] to compute current in the volume of Fig. 1.

III. NUMERICAL RESULTS WITH THE BITLLES SIMULATOR

In the following subsections, the BITLLES simulator will be used to predictelectrical RTD characteristics. The RTD simulated consists on two highly dopeddrain-source GaAs regions (the leads), two AlGaAs barriers and the quantumwell (the active region).

A. Coulomb interaction in DC scenario

As a first example, we consider the influence of the Coulomb interaction inthe prediction of the current-voltage characteristic of a typical RTD. In Figure

105

FIG. 1: Volume Ω: this is a schematic representation of the arbitrary 3D geometryconsidered in this article as simulation box for the computation of quantum transportwith local current conservation.

FIG. 2: RTD Current-voltage characteristic. Results taking into account the Coulombcorrelations between the leads and the active region are presented in solid circles.Open circles refer to the same results neglecting the lead-active region interaction.Open triangles refer to a wholly non-interacting scenario, i.e. both coulomb interactionbetween the leads and the active region and coulomb interaction among electrons withinthe active region are neglected.

2., we present a comparison between three different I-V characteristics: (i) usingour many-particle Coulomb interacting algorithm with our boundary conditions(solid circles), (ii) using the standard Dirichlet external bias boundary conditions(open circles), (iii) using the Dirichlet conditions and switching off Coulombcorrelations (open triangles). As it can be observed, the differences between

106

these three approaches appear not only in the magnitude of the current but alsoin the position and shape of the resonant region. More technical details can befound in references [1, 3, 4].

B. Coulomb interaction in high frequency scenarios

Next, we provide an example of the computation of the total (conduction plusdisplacement) current in time-dependent scenarios with the BITLLES simula-tor that includes a time-dependent solution of the 3D Poisson equation. Here,we will consider a single electron crossing a RTD to show the accuracy of ourquantum electron transport approach in providing local current conservation, i.e.the sum of the conduction plus the displacement currents is zero when integratedover a closed surface (see Fig. 1) [5].The computation of the trajectory requires the algorithm explained in the Sec.

II. In Fig. 3., we show the total time-dependent current crossing each one of

FIG. 3: Time-dependent total current computed on the six surfaces that form thevolume Ω of Fig. 1.. The computation of the current within the direct method (dashedlines) has spurious effects that are not present when the Ramo-Shockley (solid line) isused.

the six surfaces of volume Ω of Fig. 1.. The numerical evaluation of the totalcurrent through each of the six surfaces is computed from the Ramo-Shockleymethod mentioned in [5]. In general, the results obtained for the total current oneach surface are identical to those obtained from the direct computation of theconduction and displacement current on that particular surface. However, weobserve in the plots of the surfaces 1 and 4 two ’spurious’ peaks when the total

107

FIG. 4: Itran(t) an its Fourier transform in inset a and b respectively. The BITLLESnumerical results are interpreted from RLC circuits.

current is computed from the direct method. The computation of the currentusing the Ramo-Shockley method is free from these spurious numerical peaks.More details in [5].Next, we present the current response in the negative differential conductance

region of the I-V characteristic of a RTD to an input step voltage (see bothinset of Fig. 4.). The results reported in Fig. 4 have been computed includingCoulomb correlation among the N(t) electrons and among these electrons andthose in the leads. As pointed out in the inset 1a), Itran(t) manifests a delayof about 0.1ps with respect to the step input voltage, due to the dynamicaladjustment of the electric field in the leads. After the delay, the current re-sponse becomes an RLC-like (inset 1a), solid line RLC response 2) i.e. purelyexponential. Performing the Fourier transform of Itran(t) (inset b solid line)and comparing with the single pole spectra (Fourier transform of RLC-like re-sponses, inset 1b), dashed and dashed dotted lines) as depicted in the inset b,we are able to estimate the cut-off frequency (about 1.6 THz for this device)and the frequency offset (about 0.76 THz ) due to the delay. More details areexplained in [2].

C. Current-current correlations

As a last example, we show how the BITLLES simulator can compute noisefeatures. In particular, we briefly discuss on how the many-body Coulomb inter-action might affect the Fano factor (zero frequency noise in units of the averagecurrent) for RTDs. Specifically, we investigate the correlation between an elec-

108

tron trapped in the resonant state during a dwell time τd and the ones remainingin the left reservoir. This correlation occurs essentially because the trapped elec-tron perturbs the potential energy felt by the electrons in the reservoir. In thelimit of non-interacting electrons and mean field approximation, the Fano factorwill be essentially proportional to the partition noise. However if the Coulomb

FIG. 5: Fano Factor evaluated using the current fluctuations directly available fromBITTLES

correlation is self-consistently included in the simulations (see Sec. II) this resultis no longer reached (see Fig. 5.). Roughly speaking, an electron tunneling intothe well from the cathode, raises the potential energy of the well by an amountof e/Cequ, where e is the electron charge and Ceq the structure equivalent ca-pacitance. As a consequence, the density of state in the well is shifted upwardsby the same amount and we can obtain sub- and super-poissonian noise. Dueto our treatment of the many-particle Coulomb interaction in the BITLLES

simulator, these and other Coulomb blockade effects are trivially obtained.

IV. CONCLUSION

We have recently shown [1] that Bohmian trajectories allow a direct treatmentof the many-particle interaction among electrons with an accuracy comparable toDensity Functional Theory techniques. The computational technique developedin [1], which allows a computation of the quantum correlations among particlewithout explicitly knowing the many-particle wave-function, can be applied tomany different open and closed quantum systems. In particular, in this article,we present a general, versatile and time-dependent 3D quantum electron trans-port simulator, named BITLLES as an example of the previous algorithm tomany-particle open quantum system far from equilibrium.

109

Acknowledgment

This work was supported through Spanish MEC project MICINN TEC2009-06986.

[1] X. Oriols, Quantum trajectory approach to time dependent transport in mesoscopicsystems with electron-electron interactions, Phys. Rev. Let. 98(6), 066803–066807,(2007).

[2] X.Oriols and J.Mompart, Applied Bohmian Mechanics: From Nanoscale Systemsto Cosmology, Editorial Pan Stanford.

[3] G. Albareda, J. Sune, and X. Oriols, Many-particle hamiltonian for open systemswith full coulomb interaction: Application to classical and quantum time-dependentsimulations of nanoscale electron devices, Phys. Rev. B. 79(7), 075315–075331,(2009).

[4] G. Albareda, H. Lopez, X. Cartoixa, J. J. Sune, X. Oriols, Time-dependent bound-ary conditions with lead-sample Coulomb correlations:Application to classical andquantum nanoscale electron device simulators, Phys. Rev. B 82, 085301, (2010).

[5] A.Alarcon and X.Oriols, Computation of quantum electron transport with localcurrent conservation using quantum trajectories, Journal of Statistical Mechanics:Theory and Experiment. Volume: 2009(P01051), (2009).

[6] S. Ramo, Currents induced by electron motion, Proc. IRE. 27(548), 584–585,(1939).

[7] X.Oriols, A.Alarcon, and E.Fernandez-Dıaz, Time dependent quantum current forindependent electrons driven under non-periodic conditions, Phys. Rev. B. 71,245322–1–245322–14, (2005).

110

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

An Account on Quantum Interference from a

Hydrodynamical Perspective

A. S. SanzInstituto de Fısica Fundamental, Consejo Superior de Investigaciones Cientıficas,

Serrano 123, 28006 - Madrid, Spain

In 1952 David Bohm proposed [1, 2] a physical model to account for thealready long-lasting problem of measurement in quantum mechanics and thecompleteness of the wave function [3]. This physical model, nowadays known asBohmian mechanics, relies on the assumption that a quantum system consists ofa wave and a particle; the wave evolves according to Schrodinger’s equation andrules the particle motion through a guidance condition. This is a very appealingfeature: it allows us to understand quantum processes and phenomena on similargrounds as classical ones, i.e., in terms of the motion (in configuration space)displayed by a swarm of trajectories representing the evolution of a quantumfluid. Actually, this has given rise to a rebirth of Bohmian mechanics alongthe last 10 years, which has passed from being a way to formulate a quantummechanics “without observers” [4] to a well-known and increasingly acceptedresource for new quantum interpretations and computational schemes [5–8].In this report, I briefly summarize part of the most recent research I have

carried out in relation to Bohmian dynamics and quantum phase effects (moredetailed information can be found in the bibliography at the end). In particular,the discussion spins around the physical meaning attributed and attributableto concepts such as coherence, interference or superposition, all of them closelyrelated and inherent to quantum physics. As will be seen, when attention isprimarily paid to the quantum probability density current instead of to theprobability density, very interesting and challenging properties arise. Thoughsuch properties manifest very strikingly through Bohmian mechanics, they aregeneral, usually appearing “masked” within the conventional version of quantummechanics, where we rarely look at quantities dynamically depending on thequantum phase, such as the quantum probability density current.In Bohmian mechanics, the wave function Ψ provides dynamical informa-

tion about the whole available configuration space to quantum particles, whichwill move accordingly [1, 2]. This information is mainly encoded in thephase of Ψ, as can be readily seen through the transformation Ψ(r, t) =ρ1/2(r, t) exp[iS(r, t)/~], where ρ and S are the probability density and phaseof Ψ, respectively, both being real-valued functions. This relation allows us to

111

pass from Schrodinger’s equation to the system of coupled equations:

∂ρ

∂t+ ∇ ·

(

ρ∇Sm

)

= 0, (1)

∂S

∂t+

(∇S)2

2m+ V +Q = 0, (2)

v = r =∇Sm

, (3)

where Q is the so-called quantum potential [1, 2], which depends nonlinearlyon ρ. The first equation is the continuity equation, which rules the ensembledynamics, while (2) and (3) govern the particle’s motion —equation (2) is thequantum Hamilton-Jacobi equation, which describes the phase field evolutionruling the motion of quantum particles through (3). Depending on whether oneis interested in obtaining the quantum trajectories for interpretive purposes ordevising numerical algorithms to synthesize Ψ from them, two different strategiescan be considered, the so-called analytic and synthetic approaches [5].To understand what a Bohmian trajectory is, it is useful to recall the connec-

tion between Bohm’s formulation and the hydrodynamical picture of quantummechanics proposed in 1926 by Madelung [9]. In quantum hydrodynamics, themagnitudes of interest are the probability density, ρ = Ψ∗Ψ, the probability den-sity current, J = ρv, and the drift velocity field, v = J/ρ = ∇S/m. Accordingly,after integration of r, one does not obtain trajectories, but streamlines, whichfollow the flow described by the quantum fluid. In this sense, although Bohmiantrajectories reproduce all features of the quantum process, this could be exactlythe same situation one finds when one puts tracer particles on a classical fluid inorder to determine the properties of the fluid flow: these particles help us to vi-sualize the flow dynamics by moving along streamlines, indicating how the fluidcurrent goes or the energy is transported. For example, in a gaseous fluid onecan use smoke; in liquid fluids, tinny floating particles (e.g., pollen or charcoaldust) or another liquid (e.g., ink); in the hydrodynamical descriptions employedin Cosmology, the tracer particles can be stars, galaxies or clusters. Similarly,Bohmian trajectories can be regarded as the paths described by point-like tracerparticles along quantum streamlines, which allow us to determine the evolution(throughout configuration space) of the quantum flow.Now, in order to understand the implications of quantum coherence, consider a

wave packet superposition, Ψ(r, t) = ψ1(r, t)+ψ2(r, t), where the ψi = ρ1/2i eiSi/~

are given by counter-propagating (i.e., moving with opposite velocities) Gaus-sian wave packets, characterized by a translation velocity (v0) and a spreadingvelocity (vs). Depending on whether their translational motion is faster thantheir spreading or vice versa, we might find two situations [10]: collision-like(v0 ≫ vs) and interference-like (v0 ≪ vs). In the first case, the wave packetsremain well localized after their interference, while in the latter they cannot beresolved due to the permanent presence of interference fringes. If we compute

112

the field v, we find

v =1

m

ρ1∇S1 + αρ2∇S2 +√α√ρ1ρ2∇(S1 + S2) cosϕ

ρ1 + αρ2 + 2√α√ρ1ρ2 sinϕ

+√α

~

m

(

ρ1/21

∇ρ1/22

− ρ1/22

∇ρ1/21

)

sinϕ

ρ1 + αρ2 + 2√α√ρ1ρ2 cosϕ

. (4)

which can also be derived in standard quantum mechanics. This expression con-tains the essence of this theory —i.e., the direct meaning of the concept quantumcoherence— as well as the explanation for the well-known non-crossing propertyof Bohmian mechanics —a direct consequence of the presence of quantum co-herence. However, not much attention is paid to it or, equivalently, to J (exceptwhen computing net fluxes through surfaces, as in processes involving tunnelingor scattering), since one usually focuses on probability densities. To stress theimportance of quantum probability density currents and quantum velocity fields,consider the above superposition describes the head-on collision of two Gaussianwave packets [10]. Though there is no apparent overlapping between the twowave packets initially, the fact that both are present induces very well-definedphase and velocity fields in the region in between which cannot be neglectedregarding the trajectory or phase dynamics [11]. This makes, for example, thatin two-slit experiments one can indeed discern the slit traversed by a particlewithout disturbing it. Indeed, a very important implication readily arises: thisfact has to be taken into account very seriously and properly implemented in anytrajectory-based algorithm where interferences are involved in order to achieve acorrect propagation. Quantum coherence and Bohmian non-crossing give rise toother interesting consequences at a fundamental level, such as the quantum Tal-bot effect [12] or the complexified version of Bohmian mechanics [11, 13], as wellas in more applied problems, such as atom-surface scattering [14] or chemicalreactivity [15], where interference and quantum phase are relevant.The non-crossing property of Bohmian trajectories leads us to think of the

presence of an effective potential, different from Q and coming only from thequantum phase, which can reproduce such an effect. A simple model for thispotential looks like [10]

V (t) =

0, x < xmin(t)−V0(t), xmin(t) ≤ x ≤ 0∞, 0 < x

, (5)

where the border (width) and depth of the square well are

xmin(t) =πσ2

t

2p0σ2

0

~+

~t

2mσ2

0

x0

, V0(t) =2~2

m

1

x2min

(t), (6)

113

respectively. Thus, while the impenetrable wall gives rise to the bouncing of thetrajectories at x = 0, the short-range square well makes the width of the peakcloser to the wall half the width of the remaining peaks —in a wave-packet inter-ference pattern that half of the trajectories giving rise to the central maximumcomes from dynamical regions with opposite quantum phases or, equivalently,opposite velocity fields. Although more refined models than (5) can be consid-ered in order to get a better correspondence, however the important issue hereis to note that, dynamically, a problem involving wave packet superposition isequivalent to the scattering of a single wave packet with an effective potential—classically, something similar happens when two-body collisions are replacedby the collision of an “effective” body, namely the reduced mass system, acted byan “effective” central force. This stresses the difference between the mathemat-ics and the physics of the superposition principle, which becomes more apparentwhen looking at quantum trajectories, though it is incipiently contained in thestandard version of quantum mechanics —one only has to seek for the quantumphase or some associated quantity, such as J or v. Furthermore, notice that theeffective potential (5) has nothing to do with the quantum potential associatedwith a wave-packet superposition (see, for example, Ref. [16] for a picture of thequantum potential associated with the two-slit problem); the origin of (5) is theproperty of quantum coherence (quantum phase), not the probability density.Finally, I would like to mention the fact that the same discussion sustained

above can also be applied when instead of matter wave functions we are dealingwith electromagnetic fields described by Maxwell’s equations. Putting aside thequestion of what a photon (or its wave function) is and regarding it as simply atracer particle that, as before, allows us to “visualize” (trace) the flow of an elec-tromagnetic field, one can proceed as in Bohmian mechanics and try to reproduceinterference patterns by counting single arrivals of such particles. Thus, if theelectromagnetic field is characterized by the electric fieldE(r, t) and the magneticfieldH(r, t), then the three key elements will be: the electromagnetic energy den-sity, U(r, t) = [ǫ0E(r, t) ·E∗(r, t) + µ0H(r, t) ·H∗(r, t)] /4, the electromagneticenergy density current or Poynting vector, S(r, t) = Re [E(r, t)×H

∗(r, t)] /2and a vector velocity field, dr/dt = S(r, t)/U(r, t), which are analogous to thequantities of interest in quantum hydrodynamics. The velocity vector field is setup from the fact that the electromagnetic energy density is transported throughspace in the form of the Poynting vector [17],

S(r, t) = U(r, t)v. (7)

This relations (actually, their time-averaged counterparts) have been used re-cently to carry out a series of studies within the context of polarized light, theArago-Fresnel laws and quantum erasure [18], finding a very nice agreement be-tween the well-known interference patterns and the counting of photon arrivals.

Acknowledgements. First, I would like to thank the people with whom I havedeveloped the different parts of the work here summarized for many interesting

114

and fruitful discussions: Salvador Miret-Artes, Tino Borondo, Bob Wyatt, Chia-Chun Chou, Josep Maria Bofill, Xavier Gimenez, Mirjana Bozic and MilenaDavidovic.Support from the Ministerio de Ciencia e Innovacion (Spain) under Project

FIS2007-62006 to carry out this work and participate at the CCP6 Workshopon Quantum Trajectories is also acknowledged. Moreover, I also want to thankthe Consejo Superior de Investigaciones Cientıficas for a JAE-Doc Contract.

[1] D. Bohm, Phys. Rev. 85, 166, 180 (1952).[2] P. R. Holland, The Quantum Theory of Motion (Cambridge University Press,

Cambridge, 1993).[3] Zurek andWheeler, Quantum Theory of Measurement (Princeton University Press,

Princeton, NJ, 1983).[4] S. Goldstein, Phys. Today 51(3), 42 (1998); Phys. Today 51(4), 38 (1998).[5] R. E. Wyatt, Quantum Dynamics with Trajectories (Springer, Berlin, 2006).[6] P. K. Chattaraj, Quantum Trajectories (Taylor and Francis, New York, ‘to be

published’).[7] X. Oriols and J. Mompart, Applied Bohmian Mechanics: From Nanoscale Systems

to Cosmology (Pan Standford Publishing, Singapore, ‘to be published’).[8] A. S. Sanz and S. Miret-Artes, A Trajectory Description of Quantum Processes,

from the series Lecture Notes on Physics (Springer, Berlin, ‘to be published’).[9] E. Madelung, Z. Phys. 40, 322 (1926).

[10] A. S. Sanz and S. Miret-Artes, J. Phys. A 41, 435303 (2008).[11] A. S. Sanz and S. Miret-Artes, Chem. Phys. Lett. 458, 239 (2008).[12] A. S. Sanz and S. Miret-Artes, J. Chem. Phys. 126, 234106 (2007).[13] C.-C. Chou, A. S. Sanz, S. Miret-Artes and R. E. Wyatt, Phys. Rev. Lett. 102,

250401 (2009); Ann. Phys., doi:10.1016/j.aop.2010.05.009 (2010).[14] A. S. Sanz, F. Borondo and S. Miret-Artes, J. Chem. Phys. 120, 8794 (2004);

Phys. Rev. B 69, 115413 (2004).[15] A. S. Sanz, X. Gimenez, J. M. Bofill and S. Miret-Artes, Chem. Phys. Lett. 478,

89 (2009); Erratum, Chem. Phys. Lett. 488, 235 (2010).[16] A. S. Sanz, F. Borondo and S. Miret-Artes, J. Phys.: Condens. Matter 14, 6109

(2002).[17] M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 2002), 7th

Ed.[18] M. Davidovic, A. S. Sanz, D. Arsenovic, M. Bozic and S. Miret-Artes, Phys. Scr.

T135, 014009 (2009); A. S. Sanz, M. Davidovic, M. Bozic and S. Miret-Artes, Ann.Phys. 325, 763 (2010); M. Bozic, M. Davidovic, T. L. Dimitrova, S. Miret-Artes,A. S. Sanz and A. Weis, J. Russ. Laser Res. 31, 117 (2010).

115

K. H. Hughes and G. Parlant (eds.)Quantum Trajectories

c© 2011, CCP6, Daresbury

Quantum, Classical, and Mixed Quantum-Classical

Hydrodynamics

I. Burghardt∗

Departement de Chimie, Ecole Normale Superieure,

24 rue Lhomond, 75231 Paris cedex 05, France

K. H. Hughes†

School of Chemistry, University of Bangor,

Bangor, Gwynedd LL57 2UW, United Kingdom

I. INTRODUCTION

This contribution reviews the quantum hydrodynamic formulation for mixedstates (density matrices), the classical limit of this formulation, and two mixedquantum-classical hybrid schemes which have been developed in this context.Mixed-state hydrodynamics can be derived by generating the momentum mo-ments of the Wigner distribution [1–6],

〈Pnρ〉(q) =

∫dppnρW (q, p) (1)

The equations of motion for these coordinate-dependent moment quantities rep-resent an infinite hierarchy which necessitates appropriate truncation strategies[7–10]. In the special case of pure states (wavefunctions), mixed-state hydrody-namics reduces to the pure-state formulation of Bohmian mechanics [2, 4, 11].

The first of the hybrid schemes mentioned above involves a hydrodynamic rep-resentation of the quantum subspace [12–15] while the second scheme combinesa discretized representation of the quantum subspace with a hydrodynamic rep-resentation of the classical subsystem [16, 17]. In both cases, the classical limitcorresponds to a linearization approximation which is identical to the approxi-mation by which the classical Liouville equation is obtained from the quantum

∗Electronic address: irene.burghardt@ens.fr†Electronic address: chse04@bangor.ac.uk

116

Liouville equation [13, 15, 18]. (This is to be distinguished from the classicallimit associated with the vanishing of the Bohmian quantum force, see the dis-cussion in Refs. [12, 13, 15]). In the following, a brief summary is given thesetwo quantum-classical schemes.

A. Quantum (hydrodynamic)-classical (Liouvillian) dynamics

The first hybrid scheme combines a quantum hydrodynamic (Bohmian) rep-resentation with a molecular dynamics (MD) like representation for the classicalsubspace. The construction of coupled equations of motion for the quantum andclassical variables proceeds by introducing the following partial hydrodynamicmoment quantities [12, 13],

〈Pnρ〉(q,Q, P ) =

∫dp pnρW (q, p,Q, P ) (2)

where momentum moments are defined with respect to the quantum (q, p) sub-space, with a parametric dependence on the classical (Q,P ) variables. The dy-namical equations for the above moments translate to a Lagrangian picture whichcouples quantum hydrodynamic equations to classical Hamilton’s equations. Aspecific feature is a quantum hydrodynamic force which depends both on thequantum coordinate(s) q and on the classical Liouvillian variables (Q,P ). Firstapplications of this approach, which defines a mixed quantum-classical MD, aredocumented in Refs. [12, 14, 15]. More recently, this scheme has been combinedwith a non-Markovian effective mode representation of the classical subsystem[19], thus yielding a general approach to delayed dissipation.

B. Quantum (Liouvillian)-classical (hydrodynamic) dynamics

While the approach of the preceding section aims at a trajectory (MD-type)representation of the classical subspace, the second hybrid scheme employs aclassical hydrodynamic, mesoscopic representation which can be suitable to de-scribe, e.g., solvation dynamics or transport phenomena. Hybrid moments arenow constructed as follows [16, 17]

〈Pnf〉ξξ′(q) =

∫dp pnfξξ′(q, p) (3)

based upon the quantum-classical distribution f(q, p) =∑

ξξ′ fξξ′(q, p)|ξ〉〈ξ′|,

with a discretized representation of the quantum subspace. Contrary to thequantum hydrodynamic moments of Eq. (2), the moments of Eq. (3) are es-sentially classical hydrodynamic quantities which carry the indices (ξξ′) of the

117

quantum subsystem. The equations of motion for these hybrid moments havefirst been derived in Ref. [16] and have more recently been constructed rigorouslyfrom the microscopic N -particle distribution [17]. Various closure schemes canagain be envisaged, in particular a Grad-Hermite type closure [9, 10], a Gaus-sian closure at the level of a quantum-classical Maxwellian distribution, and adynamical density functional theory (DDFT) approximation by which the hy-drodynamic pressure term is replaced by a free energy functional derivative, as inRef. [16]. The present scheme in its general form [17] thus opens various general-izations of classical-statistical time-dependent density functional approximations[20, 21].

C. Conclusions

Hybrid schemes involving the quantum or classical hydrodynamic picture pro-vide promising dynamical representations which benefit from the Lagrangian tra-jectory dynamics associated with the hydrodynamic description. The connectionof mixed-state hydrodynamics to the underlying phase-space picture allows fora unified description of the classical limit and of dissipative effects. Connectionsto time-dependent density functional methods may open various generalizationsof the latter class of methods.

[1] J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99 (1949).[2] T. Takabayasi, Prog. Theor. Phys. 11, 341 (1954).[3] W. R. Frensley, Rev. Mod. Phys. 62, 745 (1990).[4] J. G. Muga, R. Sala, and R. F. Snider, Physica Scripta 47, 732 (1993).[5] I. Burghardt and K. B. Møller, J. Chem. Phys. 117, 7409 (2002).[6] R. E. Wyatt, Quantum Dynamics with Trajectories: Introduction to Quantum

Hydrodynamics, Springer, Heidelberg, 2005.[7] C. D. Levermore, J. Stat. Phys. 83, 1021 (1996).[8] P. Degond and C. Ringhofer, J. Stat. Phys. 112, 587 (2003).[9] H. Grad, Commun. Pure Appl. Math 2, 331 (19949).

[10] K. H. Hughes, S. M. Parry, and I. Burghardt, J. Chem. Phys. 130, 054115 (2009).[11] I. Burghardt and L. S. Cederbaum, J. Chem. Phys. 115, 10303 (2001).[12] I. Burghardt and G. Parlant, J. Chem. Phys. 120, 3055 (2004).[13] I. Burghardt, J. Chem. Phys. 122, 094103 (2005).[14] K. H. Hughes, S. M. Parry, G. Parlant, and I. Burghardt, J. Phys. Chem. A 111,

10269 (2007).[15] I. Burghardt, K. B. Møller, and K. H. Hughes, in: Quantum Dynamics of Complex

Molecular Systems, Springer, Berlin Heidelberg, 2007, p. 391.[16] I. Burghardt and B. Bagchi, Chem. Phys. 329, 343 (2006).

118

[17] D. Bousquet, K. H. Hughes, D. A. Micha, and I. Burghardt, Extendend hydrody-

namic approach to quantum-classical nonequilibrium evolution I. Theory, J. Chem.Phys., submitted.

[18] R. Kapral and G. Ciccotti, J. Chem. Phys. 110, 8919 (1999).[19] K. H. Hughes and I. Burghardt, in: Quantum Trajectories, Ed. P. Cattaraj, Taylor

and Francis/CRC Press, Chapter 11, p. 163 (2010).[20] A. J. Archer and R. Evans, J. Chem. Phys. 121, 4246 (2004).[21] A. J. Archer, J. Chem. Phys. 130, 014509 (2009).

119