10 February 1997

ELSEVIER

PHYSICS LETTERS A

Physics Letters A 226 (1997) 14-16

A quantum mechanical angle anholonomy

Erik SjGqvist 1 Department of Mathematical Sciences, University of Durham, Durham DHl 3LE. UK Sub-Faculty of Philosophy, Oxford University, 10 Merton Street, Oxford OXI 4JJ, UK2

Received 27 April 1996; revised manuscript received 4 December 1996; accepted for publication 5 December 1996 Communicated by P.R. Holland

Abstract

It is shown that adiabatic averaging for a de Broglie-Bohm particle moving on a ring guided by a slowly changing wave function leads to an anholonomic shift.

The concept of anholonomy has proved to be of importance for understanding the dynamics of physi- cal systems driven by slowly changing environments. Berry [ l] showed that adiabatic evolution of a quan-

tum state generates an anholonomic phase factor. Han-

nay [ 21 found the corresponding classical quantity in terms of a geometric shift of the angle variable. In

the context of classical dissipative systems Kepler and

Kagan [ 31 (see also Ref. [4] ) showed that having adiabatic invariants, i.e. quantum numbers or classical

action variables, is not essential for the existence of anholonomic effects.

In this Letter a quantum mechanical analogue of

Kepler and Kagan’s geometric shift is discussed. The aim is to show that the concept of anholonomy applies to quantum mechanical quantities not directly related

to quantum numbers. In order to give meaning to the anholonomy we invoke the de Broglie-Bohm pilot- wave formulation of nonrelativistic quantum theory

[5-71. Let us first briefly recall the elements of pilot-wave

theory. A physical system consists of a wave and a

’ E-mail: erik.sjoqvist@philosophy.oxford.ac.uk. 2 Present address.

point particle. The wave is described by the wave func- tion @(q, t), which is a solution of the time dependent Schrijdinger equation. The particle follows a trajectory

Q(t) in configuration space given by the equation

(1)

where J* is the quantum mechanical current and p@ the probability density fulfilling the continuity equa-

tion

(2)

The density p+ expresses the impossibility to actually observe the trajectory Q(t) .

Adiabaticity in this framework can be defined as follows. If both the particle and the wave function move in cycles of periods 7 and T, respectively, i.e.

Q(7) = Q(0) and t,b(q,T) = e’+t,b(q,O), 4 being an arbitrary real number, then we say the dynamics is

adiabatic if r < T. We now derive the anholonomy. Consider the an-

gular equation of motion

0375-9601/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00930-9

E. Sjiiqvist/Physics Letters A 226 (1997) 14-16 15

&” P* 0=@

(3)

for a particle on a ring. Assume J” and p@ depend on time through a set of slowly changing parameters R = R(EI), E > 0 being small. Define the equal time parametrisation of the ring [ 31

B(f)

d@(c), R(E~)) = w(R(~t)) s ,,P~(UW)) J"(ttR(a))

0

(4)

where w(R) is the frozen angular frequency for the

circulation of the particle around the ring at fixed R. Thus 40( 2~, R) = 27r. Using (3) and (4) we obtain

dq dR -=w+VRrp.z. dt

(5)

Assume adiabaticity so that the &dependence on the

r.h.s. of (5) can be eliminated by adiabatic averaging

technique [ 81 replacing (5) with the averaged equa-

tion

(6)

From the continuity equation we deduce J* (8, R) = js(R)+O(E),i.e.w/27r=j@+O(E),whichtogether with (3) and (5) yields

2 =p+ddB+O(e).

Thus the adiabatic average (6)) which is a time av-

erage over a single trajectory, equals the average over

an ensemble of particles spread out around the ring

according to p@. Integrating (6) over a closed path C : t E

[0, l/e] - R(d), where R( 1) = R(O), and using (7)) we obtain (neglecting terms of O(E) )

Aq=/wdl+{*.

0 C

(8)

The one-form A is given by

257

A = s

dBp@ dR9 = (dRp),

0

(9)

where dR = dR.VR and ( ) means ensemble average.

The first term on the r.h.s of (8) is the T-dependent dy- namic shift A%. The second term is the anholonomic shift Asia, depending on the path C in parameter space

but independent of T. The dynamic and anholonomic shifts, being of order E-’ and E’, respectively, are the two lowest terms in an e-expansion of the total angle

shift. Higher order terms are nonadiabatic corrections.

It is useful to rewrite AP~ using Stokes’ theorem:

AR, = s dtz A (dw) 9 (10)

where dS = C. From ( IO) it follows directly that an R- dependent shift of the origin, cp - q + A(R), A be-

ing globally differentiable, does not change Aspa. Thus the angle anholonomy is gauge invariant. Furthermore

since p@ depends on R, dRA (dRp) it (dRAdRp) = 0,

i.e. the anholonomic shift is in general nonzero.

As an illustration consider a particle of unit charge

and unit mass moving on a slowly rotating ellipse in

the presence of a weak, constant and homogeneous magnetic field. Assume the eccentricity is small so that we can take the radius of the ellipse to be independent of 8 and equal to one. This leads to the Hamiltonian

operator (h = 1)

H=-~($+iB/2)2 +kcos[28-Q(t)]. (11)

Here B is the component of the magnetic field per- pendicular to the plane of the ellipse, k measures the

eccentricity (the “circle” corresponds to k = 0) and @ is the angle between the semi-major axis and the 0 = 0 direction. The potential k cos( 28 - @) [ 91 is the low-

est term in an expansion of the potential V(20 - @), being such that N@ = -~V/&J, where Ne is the 8- component of the normal force acted on the particle

by the ring. The appearance of a factor 2 in front of 6’

is due to the C2 symmetry of the ellipse.

Let us concentrate on the first (&) and second

(4+) excited states arising from the coupling between the e’ie/&- f 7r ree rotator functions (for simplicity we neglect coupling to rotator functions with higher

angular momentum). Diagonalising the Hamiltonian operator in this basis yields

16 E. Sjiiqvist/Physics Letters A 226 (1997) 14-16

++(e,@) = &[cos(fr)ei~o-@~2) + sin( .!r)e-i(e-@/2)],

qS_(O,@) = &[sin( +r)ei(B-@/2)

_ cos(.!~)e-i(~-V)~, (12)

where cos r = B/I/~. Now assume @J = 2mt, where E is much smaller than the inverse of the

energy difference dm between c,$ and c#-.

According to the quantum adiabatic theorem [lo] we then have +(e,O) =&(0,@(O)) - @(B, t) = e’qh ( ‘)& (0, Q(t) > yielding (neglecting the small B/2 term in the velocity)

L&e,@) = cos r sinrcos(28 - @) * 1’

(13)

which entails that adiabaticity holds for all B and CD

provided

E < &I cosq. (14)

This follows from r = 1 &?( l/&) de]. If (14) is

satisfied then a direct calculation gives

Apoa,* = it sin2 r (15)

and

cos r A%,* = *T. (16)

We note that ( 15) only depends on the parameter r, i.e. is independent of the cyclic time 1 /e (as long as

the adiabaticity requirement ( 14) holds). On the other hand ( 16) depends explicitly on l/e.

In conclusion, we have shown that the generalisa- tion of the angle shift A.50 = wT = 2rJ@T for a sta- tionary current to time dependent environments con-

tains an anholonomic contribution. Thus the quantum

mechanical current on a ring exhibits an anholonomic

shift analogous to the geometric effect found by Kepler

and Kagan [ 31 for classical dissipative systems.

I would like to thank an anonymous referee for in- valuable comments. This work was supported by a scholarship from The Royal Swedish Academy of Sci-

ences (Per Erik Lindahl’s Foundation).

References

111 M.V. Berry, Proc. R. Sot. A 392 (1984) 45. [2] J.H. Hannay, J. Phys. A 18 (1985) 221. 131 T.B. Kepler and M.L. Kagan, Phys. Rev. Len. 66 (1991)

841. [4] M.L. Kagan. T.B. Kepler and I.R. Epstein, Nature 349

(1991) 506. 151 L. de Broglie, J. Phys. (Paris) 8 ( 1927) 225. [6] D. Bohm, Phys. Rev. 85 (1952) 166, 180. [7] P.R. Holland, The quantum theory of motion (Cambridge

Univ. Press, Cambridge, 1993). [S] V.I. Arnold, Mathematical methods of classical mechanics,

2nd Ed. (Springer, Berlin, 1989) Ch. 10. [9] H.C. Longuet-Higgins, U. Opik, M.H.L. Pryce and R.A.

Sack, Proc. R. Sot. A 244 ( 1958) 1. [IO] A. Messiah, Quantum mechanics, Vol. 2 (North-Holland,

Amsterdam, 1962) Ch. 17.