schrödinger theory from the perspective of classical fields derived from quantal sources

Schrodinger theory from the perspective of classical fields derivedfrom quantal sourcesq

V. Sahnia, b

aDepartment of Physics, Brooklyn College of the City University of New York, Brooklyn, NY 11210, USAbThe Graduate School and University Center, City University of New York, 365 Fifth Avenue, New York, NY 10016, USA

Abstract

In this paper we describe Schro¨dinger theory from the perspective of classical fields whose sources are quantal expectationsof Hermitian operators. As such these fields may be considered as being intrinsic to and thereby descriptive of the quantumsystem. The perspective is valid for both ground and bound excited-states. The fields, whose existence is inferred via thedifferential virial theorem, are as follows: (i) an electron-interaction fieldEee(r ,t) derived via Coulomb’s law from the pair-correlation density; (ii) a kinetic-fieldZ(r ,t) derived as the derivative of the kinetic-energy–density tensor obtained from thespinless single-particle density matrix; (iii) a differential density fieldD(r ,t) which is the gradient of the Laplacian of theelectron density; and (iv) a current-density fieldJ(r ,t) derived as the time derivative of the current density whose source too isthe spinless single-particle density matrix. The total energy and its components may be expressed in terms of these fields: theelectron-interaction potential and kinetic energies via the fieldsEee(r ,t) andZ(r ,t), respectively, and the external potentialenergy via a conservative field which is the sum of all the fields present. The field perspective is illustrated by application to theexactly solvable stationary ground-state of the Hooke’s atom, its extension to the time-dependent case being surmised via theHarmonic Potential theorem. Finally, we note that both Schro¨dinger and Kohn–Sham density-functional theory are nowdescribable in terms of classical fields derived from quantal sources.q 2000 Elsevier Science B.V. All rights reserved.

Keywords: Schrodinger theory; Classical fields; Quantum sources

1. Introduction

The Schro¨dinger equation[1] for a many-electronsystem in a real local external potential of the formv(r ,t) is (in atomic units:e� " � m� 1)

i2C�X; t�

2t� H�t�C�X; t�; �1�

whereC (X,t) is the wavefunction,X � x1; x2;…; xN;

x � rs; r ands the spatial and spin coordinates, andwhere the Hamiltonian operatorH�t� is a sum of thekinetic T, external potentialV�t�, and the electron-

interaction potentialU energy operators:

T � 212

Xi

72i ; �2�

V�t� �X

i

v�r i ; t�; �3�

and

U � 12

X0i; j

1ur i 2 r j u

: �4�

In quantum mechanics, properties of systems aredetermined in terms of the position probabilitydensity, or equivalently as expectation values of the

Journal of Molecular Structure (Theochem) 501–502 (2000) 91–99

0166-1280/00/$ - see front matterq 2000 Elsevier Science B.V. All rights reserved.PII: S0166-1280(99)00417-0

www.elsevier.nl/locate/theochem

q Dedicated to Professor R. Ga´spar on the occasion of his 80thyear.

corresponding operators taken with respect to thewavefunction. These expectations are functions oftime since the wavefunction depends on time, andthe spatial and spin coordinates are integrated out.Thus, the energyE(t) is the expectation

E�t� � C�t�h ji 22t

C�t�j i � kC�t� uHuC�t�l: �5�

The energy in turn may be written in terms of itskinetic T(t), externalEext(t), and electron-interactionEee(t) energy components:

E�t� � T�t�1 Eext�t�1 Eee�t�; �6�where

T�t� � kC�t�uTuC�t�l; �7�

Eext�t� � kC�t�uVuC�t�l; �8�and

Eee�t� � kC�t�uUuC�t�l: �9�The purpose of the paper is to provide an alternate,

possibly more physically tangible, description of aquantum system and thereby its energy and energycomponents in terms ofclassical fields. This differentperspective, nonetheless, still lies within the rubric ofSchrodinger theory and its probabilistic interpretation,in that the sources of these fields are quantum-mechanical expectations of Hermitian operatorstaken with respect to the wavefunction. Thus, thesefields may be thought of as beinginherent to thequantal system, with each field, (or sum of fields),contributing to a specific energy component. Itfollows that the description in terms of fields is alsovalid for bound excited states as well as for stationarystate Schro¨dinger theory. A similar field description ofboth time-dependent [2] and stationary state [3–9]density-functional theory [10–13] already exists.

In Sections 2–4, respectively, we define the quantalsources, the fields to which they give rise, and theenergy component expressions in terms of thesefields. To demonstrate the field perspective we applyit in Section 5 to the exactly solvable Hooke’s atom inits ground state. We end in Section 6 with concludingremarks and a brief discussion of the correspondingfield description of density-functional theory.

2. Definition of quantal sources

In this section we define the quantum-mechanicalsources of the fields intrinsic to the system. Thesesources are the electron densityr(r ,t), the spinlesssingle-particle density matrixg(r ,r 0,t), and the pair-correlation densityg(r ,r 0,t). These sources are definedas follows:

2.1. Electron densityr (r ,t)

The electron densityr (r ,t) is N times the proba-bility of a particle being ar :

r�r ; t� � NXs

ZC* �rs;XN21

; t�C�rs;XN21; t� dXN21

;

�10�where XN21 � x2; x3;…; xN; dXN21 � dx2;…;dxN;

andR

dx � Ps

Rdr : The density is also the expecta-

tion of the density operator

r �X

i

d�r i 2 r j�; �11�

so that

r�r ; t� � kC�t�ur uC�t�l: �12�The total electronic charge is

Rr�r ; t� dr � N: This

charge density islocal in that its structure remainsunchanged as a function of electron position foreach instant of time.

2.2. Spinless single-particle density matrixg (r ,r 0,t)

The single-particle density matrixg (r ,r 0,t) which isdefined as

g �r ; r 0; t� � NXs

ZC p�rs;XN21

; t�C�r 0s;XN21; t� dXN21

;

�13�may also be expressed as the expectation value of theHermitian operatorX:

g�r ; r 0; t� � kC�t�uXuC�t�l; �14�where

X � A 1 iB; �15�

A� 12

Xj

�d�r j 2 r �Tj�a�1 d�r j 2 r 0�Tj�2a��; �16�

V. Sahni / Journal of Molecular Structure (Theochem) 501–502 (2000) 91–9992

B� i

2

Xj

�d�r j 2 r �Tj�a�2 d�r j 2 r 0�Tj�2a��; �17�

Tj(a) is a translation operator such thatTj�a�C�…; r j ;…; t� � C�…; r j 1 a;…; t�; and a�r 0 2 r :

2.3. Pair-correlation density g(r ,r 0,t)

The pair-correlation densityg(r ,r 0,t) is the densityat r 0 for an electron atr , and is defined as the ratio ofthe expectations

g�r ; r 0; t� � kC�t�uP�r ; r 0�uC�t�l=r�r ; t�; �18�where P�r ; r 0� is the Hermitian pair-correlationoperator

P�r ; r 0� �Xi;j

0d�r i 2 r �d�r j 2 r 0�: �19�

The total charge independent of electron position isRg�r ; r 0; t� dr 0 � N 2 1. The pair-correlation density

is anonlocalcharge distribution in that its structure foreach instant of time changes as a function of electronposition for nonuniform electron density systems. Thepair-correlation density may also be expressed interms of its local and nonlocal components as

g�r ; r 0; t� � r�r 0; t�1 rxc�r ; r 0; t�; �20�whererxc�r ; r 0; t�; the Fermi–Coulomb hole charge, isthe reduction in density atr 0 which results as a conse-quence of correlations due to the Pauli exclusionprinciple and Coulomb’s law. The total charge ofthe Fermi–Coulomb hole, independent of electronposition, is

Rrxc�r ; r 0; t� dr 0 � 21

3. Definitions of classical fields

The different fields within the quantum system arethe electron-interactionEee(r ,t) field which is a sum ofthe HartreeEH(r ,t) and Pauli–CoulombExc(r ,t) fields,the differential densityD(r ,t), kinetic Z(r ,t), andcurrent-densityJ(r ,t) fields. These fields are asdefined below.

3.1. Electron-interaction fieldEee(r ,t)

The electron-interaction field is derived viaCoulomb’s law from the pair-correlation density

g(r ,r 0,t) as

Eee�r ; t� �Z g�r ; r 0; t��r 2 r 0�

ur 2 r 0u3dr 0: �21�

With the pair density expressed as in Eq. (20), the fieldEee(r ,t) may be written as a sum of its HartreeEH(r ,t)and Pauli-CoulombExc(r ,t) components as

Eee�r ; t� � EH�r ; t�1 Exc�r ; t�; �22�where

EH�r ; t� �Z r�r 0; t��r 2 r 0�

ur 2 r 0u3dr 0; �23�

and

Exc�r ; t� �Z rxc�r ; r 0; t��r 2 r 0�

ur 2 r 0u3dr 0: �24�

The Hartree fieldEH(r ,t) is conservative as it is due toa local charge distributionr(r ,t) so that 7 ×EH�r ; t� � 0:

3.2. Differential density fieldD(r ,t)

The differential density fieldD(r ,t) is defined as

D�r ; t� � d�r ; t�=r�r ; t� �25�where

d�r ; t� � 214772r�r ; t�: �26�

This field also arises from the electronic density whichis a local charge, so that it too is conservative and7 ×D�r ; t� � 0:

3.3. Kinetic fieldZ(r ,t)

The source of the kinetic fieldZ(r ,t) from whichthe kinetic energy is obtained is the single particledensity matrixg (r ,r 0,t). The field is defined as

Z�r ; t� � z�r ; t; �g��=r�r ; t�; �27�where the kinetic fieldz(r ,t) is defined by its componentza(r ,t) as

za�r ; t� � 2Xb

2

2rbtab�r ; t�; �28�

and wheretab(r ,t) is the kinetic-energy–density tensor

V. Sahni / Journal of Molecular Structure (Theochem) 501–502 (2000) 91–99 93

defined in turn as

tab�r ; t� � 14

22

2r 0a2r 00b1

22

2r 0b2r 00a

" #g�r 0; r 00; t�ur 0�r 00�r :

�29�The fieldZ(r ,t) is classical in the sense that it is derivedas the derivative of a tensor. It is the sourceg(r ,r 0,t) thatis quantum mechanical.

3.4. Current density fieldJ(r ,t)

The single particle density matrixg(r ,r 0,t) is alsothe source for the current-density fieldJ(r ,t) which isdefined as

J�r ; t� � 1r�r ; t�

2

2tj�r ; t�; �30�

wherej(r ,t) is the current density:

j�r ; t� � i

2�7 0 2 7 00�g�r 0; r 00; t�ur 0�r 00�r : �31�

The fieldJ(r ,t) too is classical from the perspectiveof the hydrodynamic continuity and force equations.(Typically, these are 2r=2t 1 7·�rv� � 0; andr�2v=2t� � 7p; where r (r ,t) is a matter density,v(r ,t) a velocity, andp(r ,t) a (scalar) pressure).

The fieldsEee(r ,t) [or Exc(r ,t)], Z(r ,t), andJ(r ,t)are in general not conservative. However, their sumalways is, so that 7 × �Eee�r ; t�1 Z�r ; t�1J�r ; t�� 0: However, for systems of spherical orcylindrical symmetry, or systems treated within thecentral field approximation, or those with translationalsymmetry in two directions etc., for which theinhomogeneity of the electron density is one-dimensional, these fields are separately conservative.For such systems 7 × Eee�r ; t� � 7 × Z�r ; t� �7 × J�r ; t� � 0:

4. Energy components in terms of fields

The kinetic, external, and electron-interactionenergies as defined by the expectations of Eqs.(7)–(9), can now be expressed directly in termsof the respective classical fields described above.

4.1. Electron-interaction potential energy Eee(t)

The electron-interaction potential energyEee(t) in

terms of the fieldEee(r ,t) is expressed in virial form as

Eee�t� �Zr�r ; t�r ·Eee�r ; t� dr : �32�

With Eee(r ,t) expressed as a sum of its HartreeEH(r ,t)and Pauli–CoulombExc(r ,t) components, Eq. (22),the energy may be written as

Eee�t� � EH�t�1 Exc�t�; �33�whereEH(t) is the Hartree or Coulomb self-energy

EH�t� �Zr�r ; t�r ·EH�r ; t� dr ; �34�

andExc(t) the exchange-correlation energy

Exc�t� �Zr�r ; t�r ·Exc�r ; t� dr : �35�

The energyEee(t) may also be expressed directly interms of the source charge distribution which givesrise to it as the energy of interaction between thedensity r(r ,t) and the pair-correlation densityg(r ,r 0,t):

Eee�t� � 12

ZZ r�r ; t�g�r ; r 0; t�ur 2 r 0u

dr dr 0: �36�

Similarly, the Coulomb self-energy is

EH�t� � 12

ZZ r�r ; t�r�r 0; t�ur 2 r 0u

dr dr 0; �37�

and the exchange-correlation energy is the energy ofinteraction between the density and the Fermi–Coulomb hole charge distribution

Exc�t� � 12

ZZ r�r ; t�rxc�r ; r 0; t�ur 2 r 0u

dr dr 0: �38�

4.2. Kinetic energy T(t)

The kinetic energyT(t) too is expressed in terms ofthe kinetic fieldsZ(r ,t) or z(r ,t) in virial form as

T�t� � 212

Zr�r ; t�r ·Z�r ; t�dr � 2

12

Zr ·z�r ; t� dr :

�39�

4.3. External potential energy Eext(t)

The external energyEext depends on all the fieldspresent in the quantal system. To see this, the


expectation of Eq. (8) is rewritten as

Eext�t� �Zr�r ; t�v�r ; t� dr : �40�

From the differential virial theorem [2,14], theexternal potentialv(r ,t) at eachinstant of timeis thework done to move an electron from some referencepoint at infinity to its position atr in the force of aconservative fieldF(r ,t):

v�r ; t� � 2Zr

∞F�r 0; t�· dl 0; �41�

where

F�r ; t� � 2Eee�r ; t�1 D�r ; t�1 Z�r ; t�1 J�r ; t�:�42�

This work done is path independentsince 7 ×F�r ; t� � 0: That this must be the case is consistentwith our implicit assumption in the construction of theHamiltonian that the external potential at each instantof time is path-independent. The physical inter-pretation of the external potentialv(r ,t) in terms ofthe various fields follows from the differential virialtheorem [2,14] which states that7v�r ; t� � 2F�r ; t�;whereF(r ,t) is given by Eq. (42).

For stationary states with a time-independentexternal potentialv(r ), the Schro¨dinger equation is

i2Cn�X; t�

2t� HCn�X; t� � EnCn�X; t�; �43�

where the wavefunctionsCn(X, t) are eigenfunctionsof the Hamiltonian operator, andEn are the eigen-values of the energy. The solutions of Eq. (43) areof the form

Cn�X; t� � cn�X� e2iEnt; �44�

where the functionsc (X) and eigenvaluesE of theenergy are determined by the equation

Hc�X� � Ec�X�: �45�The description of the stationary state system in termsof fields is the same as for the time-dependent case,but with the sources and fields determined bythe functionsc (X), the phase factor vanishing viathe definition of the source distributions. Further, thecurrent density fieldJ�r ; t� � 0; so that the energycomponents and the external potentialv(r ) are definedas before but by the static fieldsEee(r ), D(r ), Z(r ). Itis also evident that the field concept is extendable tobound excited states.


Fig. 1. Densityr(r) and radial probability densityr2r(r).

5. Example of field perspective: the Hooke’s atom

To demonstrate the concept of representing aquantum system by classical fields, we consider theexactly solvable stationary ground-state of theHooke’s atom [15–17]. The model systemHamiltonian is (in atomic units)

H � 21272

1 21272

2 118�r2

1 1 r22�2

e2

ur1 2 r2u; �46�

and the wavefunction and eigenenergy of the corre-sponding time-independent Schro¨dinger equation Eq.(45) are

c�r1; r2� � C e2R2=2 e2r2

=8�1 1 r=2�; �47�and

E � 2; �48�where r � r 2 2 r1;R � �r1 1 r2�=2; C � 1=�2p5=4

�5 ��pp

1 8�1=2� � 0:0291122: This wavefunctionsatisfies the electron–electron but not the electron–nucleus cusp condition [18]. The analytical expres-sions for the sources: the densityr (r ), single-particledensity matrix g (r , r 0), and the pair-correlation

densityg(r , r 0); the fields: the HartreeEH(r ), Pauli-Coulomb Exc(r ), electron-interaction Eee(r ) andkinetic z(r ); and the energies: the HartreeEH,exchange-correlationExc, electron-interactionEee,kinetic T and externalEext, are given in Ref. [5].

In Fig. 1 we plot the densityr (r ) which is thesource for the differential density fieldD(r ), as wellas for the Hartree fieldEH(r ) and energyEH. Theradial probability densityr2r�r � is also plotted.

In Fig. 2, the Fermi–Coulomb hole chargerxc(r , r 0)is graphed for different electron positions atr �0; 0:4;5 and 25 a:u: The electron is assumed to be onthe z-axis corresponding tou � 08. It is the crosssection through the hole corresponding tou 0 � 08with respect to the electron–nucleus direction that isplotted. (The part of the figure corresponding tor 0 ,0 corresponds to the structure foru � p and r 0 > 0).The nucleus is at the origin. The nonlocal nature of thischarge is clearly evident. For an electron at the nucleusr � 0; the charge is spherically symmetric about theorigin. For other electron positions it is not. The elec-tron–electron cusp at the elctron position atr � 0 and0.4 a.u. is also clearly exhibited. Observe that as theelectron position from the nucleus increases�r � 5; 25 a:u:�, the hole is more spread out. For


Fig. 2. The Fermi-Coulomb hole charge distributionr xc(r ,r 0) for electron positions atr � 0;0:4; 5 and 25 a:u: The electron is on thez axiscorresponding tou � 08. The cross-section corresponding tou 0 � 08 with respect to the electron nucleus direction is plotted. The part of thegraph forr 0 , 0 corresponds to the structure foru � p; r 0 > 0.

electron positions in the far asymptotic region (notplotted), the Fermi-Coulomb hole charge becomes anessentially static charge with unchanging structure thatis spherically symmetric about the nucleus.

The HartreeEH(r ), Pauli–CoulombExc(r ), andelectron-interactionEee(r ) fields are plotted in Fig.3. Note that all these fields vanish at the origin:

EH(r ) becauser(r ) is spherically symmetric, andExc(r ) due to the fact that for an electron at thenucleus, its sourcer xc(r ,r 0) is spherically symmetricabout the electron. The signs of these fields (andthereby of the corresponding energies) is a conse-quence of the various charge distributions, with eachfield exhibiting shell structure. Their asymptotic


Fig. 3. HartreeEH(r ), Pauli–CoulombExc(r ) and electron-interactionEee(r ) fields.

Fig. 4. Kinetic fieldz(r ).

structure is also governed by the total chargeand structure of the various sources. Thecomponent energies are:EH � 1:030250 a:u:; Exc �20:582807 a:u:; andEee� 0:447443 a:u:

The kinetic field z(r ) is plotted if Fig. 4. (Theanalytical expression for the tensortab (r ;[g ]) aswell as graphs of its diagonal and off-diagonalelements are given in Ref. [5]). This field tooexhibits shell structure and is principally positive,decaying asymptotically as a negative function. Theresulting kinetic energy due to this field isT �0:664418 a:u:

Finally, in Fig. 5 we plot the differential densityfield d(r ). It is interesting to note that this fieldmirrors that of the kinetic field z(r ). It tooexhibits shell structure, but decays symptotically as apositive function. Theseparatecontributions of thefields Eee(r ), D(r ), andZ(r ) to the external energymay be determined via Eqs. (41) and (42). The valueof the total external potential energyEext �0:888141 a:u:

The time-dependent structure of the sources, fieldsand energies of the Hooke’s atom in the presence of atime-dependent potential of the form2F(t)·r isreadily obtained as a result of the Harmonic PotentialTheorem (HPT) [19]. According to the HPT, thetime-dependent wavefunction is simply the time-independent solution (multiplied by a phase factor)

shifted by the functiony(t) which satisfies theclassical equation of motionm�y�t� � 2k·y�t�1 F�t�;wherek is the spring constant matrix. In determiningthe expectation for the sources, the phase factorscancel. Thus, the time-dependent structure of thesources and fields at each instant of time is the sameas that of the time-independent case, but translated bya finite value. The current density field would,however, have to be determined and its structurewould depend on the form of external forceF(t).

6. Conclusions

In this paper we provide an alternative descriptionof quantum systems within Schro¨dinger theorywhich is in terms ofclassical fields whose sourcesare quantal expectationsof Hermitian operators.Thus, for example, the electron-interactionpotential energy is defined in Schro¨dinger theory asthe expectation value of the electron-interactionoperator

U � 12

X0i; j

e2

ur i 2 r j u:

(The term on the right hand side of the operatorequation is the electron-interaction potential energy


Fig. 5. Density fieldd(r ).

of classical physics. It may further be defined in termsof an electric field.) The quantal expectation value,however, can also be defined in terms of an elec-tron-interaction fieldEee(r ,t) obtained by Coulomb’slaw from the time-dependent pair-correlation densitywhich constitutes the source charge distribution forthe field. Thus, this quantal potential energy isdescribed from the classical perspective of a dynami-cal charge distribution and the field to which it givesrise. Similarly, the kinetic energy, the expectation ofthe kinetic energy operator, is derived in terms of akinetic fieldZ(r ,t) obtained from the kinetic-energy–density tensor, the source of the field being the time-dependent spinless single-particle density matrix.These fields, together with the densityD(r ,t) andcurrent-densityJ(r ,t) fields, exhibit characteristicproperties of a quantum system such as shellstructure in atoms or the Bardeen–Friedel oscillationsat a metal surface etc., and are thereby descriptive ofthe system. As such the field perspective ofSchrodinger theory provides further physical insightinto the properties of a quantal system.

There is another interesting aspect of this descrip-tion in terms of fields. In writing the Schro¨dingerequation, one begins with an assumption for thestructure of the external potential, whether it beCoulomb, screened-Coulomb, harmonic or anhar-monic etc. It can now be interpreted that this potentialarises in a self-consistent manner from the sum ofthese fields. As such the contribution ofeach ofthese fields to the structure of the external potentialand to the external potential energy can be explicitlydetermined.

The existence of the various fields within aquantum system is arrived at via the differential virialtheorem [2,14]. This theorem further leads to amathematically rigorousphysical description oftime-dependent [2] and stationary-state [3–9]Kohn–Sham density-functional theory in terms offields. This description is independent of anyelectron-interaction action or energy functionals ofthe density or their derivatives. By deriving the corre-sponding differential virial theorem for the system ofnoninteracting fermions with the same density as thatof Schrodinger theory, it is possible to explicitlydefine the local (multiplicative) time-dependentelectron-interaction potential of these fermions. Thepotential is the work done in a conservative field

which is the sum of the electron-interactionEee(r ,t),a correlation-kinetic Ztc(r ,t), and a correlationcurrent-densityJc(r ,t) field. The latter two fieldsare a consequence of the fact that the kineticenergy and current density of the noninteractingsystem differ from that of Schro¨dinger theory. Thesources of these fields is the difference betweenthe time-dependent spinless single particle andDirac density matrices, the latter constructed via theKohn–Sham orbitals. The energy components instationary-state density-functional theory may alsobe described in terms of these component fields.Thus, Schro¨dinger theory as explained in thepresent work, and its density-functional theorymanifestation, can both be described from theperspective of classical fields derived from quantalsources.

Acknowledgements

The author thanks Professors A. Nagy and I.G.Csizmadia for the invitation to contribute this paperin honour of Professor R. Ga´spar.

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schrödinger theory from the perspective of classical fields derived from quantal sources

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