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HAL Id: jpa-00246406

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Diffusion of a quantum particle in time-dependent random potential

N. Lebedev, D. Sokoloff, A. Kaganovich

To cite this version:

N. Lebedev, D. Sokoloff, A. Kaganovich. Diffusion of a quantum particle in time-dependent random potential. Journal de Physique I, EDP Sciences, 1991, 1 (9), pp.1213-1215. �10.1051/jp1:1991202�.

�jpa-00246406�

(2)

J Phys. Ifrance 1

(1991)

1213-1215 sBPrBMBRB1991, PAGE 1213

Classification

PhysicsAbstnwt£

05.40-05.50

Shom Communication

Diffusion of

a

quantum particle in time-dependent random

potential

N.I.

Lebedev(I),

D-D-

Sokolofl(2)

and A-S.

Kaganovich(3)

(1) Moscow Institute of

Radioengineering,

Electronics and Automation, 117454, Moscow, U-S-S-B-

(2) Dcpartment

of

Physics,

Moscow State

University,

l19899, Moscow, U-S-S-R-

(~) Theoretical

Physics Dcpartment,

EN. l~cbedcv

Physical

Institute, l17924, Moscow, l~cninsky Prospcct 53, USSR and

Vniigcoinformsystcm~

l13105 Mowcw, Wrshavskoe shosse 8, U.S.S.R.

(Received

20Mqy

lwl,

ttccepted11 Ju~

lwl)

Abstract. For

Schrodinger eqttation

for a

particle moving

in random,

time-dependent potential

with white noise correlation, we prove that perturbation

theory

result for mean square

displacement

X ~-

t~'~

is

asymptotically

exact for a large time t. This is in contrast with the same

eqttation

with imaginary time.

In recent years there raised a

significant

interest in systems described

by Schrbdillger-type

equa-

tion vith random, uncorrelated in time

potenthl:

i~

= hit +

V(x, t)~ (l)

@

=

0;

V

(xi,ti)

V

(x2,t2)

= U

(xi x2)

6(11

t2) (2)

Here x is a d-dimensiofial vector.

Equation (I) relates,

in

particular,

to the

problem

of wave

propagation

in media vith random index of refraction

V(z, t) (in parabolic approximation)

[3, 4]. The same

equation

vith

imaginary

time characterizes thermal fluctuations of directed

polymer

in random media

[I]

and variations of some animal's

population

due to

randomly changing

environment [2].

A group of

analytical

methods have been

applied

for calculation of

polymer's displacement scaling dependence [5-7j

(x~(t))

'~

t~"

means thermal average and we suppose

z(0)

=

0).

At the same time

only

numerical estimates [4,8] are knoAvn

analogous

variable

obeying Schr0dinger equation

vith real time

(I).

In

fac~

the

computations

in[4,8]

give

the same value of a as in

polymer problem.

At first

sight

it looks

natural,

(3)

1214 JOURNALDEPIIIfSIQUEI N°9

but let us note that even in the absence of disorder a

=

1/2

for directed

polymers

while a = I for

quantum

case

(1)

[9]. 'Ib our mind it is essential that iteration scheme uwd in [4, 8] did not

conserve nomalization of wave function in contrast with

equation (I)

itsev

(that

was

specially

taken into account ±±

[4]).

Our

analytical

calculatio~ cited below indicate that weH kwwn

perturbation

result a

=

3/2

(see

e.g.

?

and Refh.

therein)

holds

asymptotically

exact in the Iirnit t - cc for

Schrodinger equation (I).

The starting

point

of our

anafysb

is

equation

for variable

m

(xi, x2,t)

= ill

(xi,t)

~V*

(x2,I) (3)

cited h book [3]. First of all we derive I: in a manner different from that of [3] and

possessing

some

advantages.

The solution of

equation (I)

written in a form of

Feynman integral

is

i~

(xi,t)

=

f Dx(t) e's(x~'))i~(x,o) (4)

S(x(t))

=

(~

dr

((

V(x,t)) (5)

Integration

in

(4)

is over an

trajectories passing through point

xi.

Supposing V(x, t)

Gaussian vith correlator

(2),

the

following

result for~m form

(3)

is

m =

/

DV

P(V) / /

Dx Dx'

e'~~~)~~(~') fit(x, 0)fit* (x', 0) (6) Mitl~statisticalweight

P(V)

= exp

/ ~

dr

/

dk

~V(k, r)V(k, r) (7)

and

U(k) being

Fourier transform of

U(x)

~om

(2).

After

integration

over

V(x, t)

in

(6)

m =

f f

Dx Dx'

e'~(~~') ~(x, 0ji1° (x', 0) (8)

with

9 (x, x')

=

(~

dr

(( (

+

(U (x x') (0))j

(9) Expressions (8)

and

(9) gbve

us

Feynman

version for solution of equation:

= I

(Ax, Azz)

Tn +

lU (xi x2) U(°)I

m

(lo)

This

equation

coincides

(except designations)

vith formula

(44.24)

of book [3]. The above derivation is sirnflar to standard method for

obtaining replica

Hamiltonians in the

theory

of dis- ordered systems

(see

e.g. [7]). Its

advantage

is a

possible application

to random

potenthls

with

long-range

correlations.

(4)

N°9 QUANTUMDIFFUSIONINTIME-DEPENDENTRANDOMPOTENIIAL 1215

Hereafter we assume z to be a

scalar, generafisation

to multidimensional case b

straightfor-

ward and the same results are obtained. it is convenient to rewrite

(10) using

new corrdinates

" ~l ~21 fl " ~l + 22

'~~~~j~'~~

=

21()m(<, n,t)

+

iu(t) u(o)im(<, n,t) (11)

and tO inUOdUCe moments

"l(f, f)n

"

/ dn n~"~(f,

n,

t) (12)

Note that

~

=

m2(f,t) (13)

f-o

We now derive recursive relations for mn.

Multiplying

both

parts

of

(11) by q"

and

integrating

over q one gets

~~" ~~~

~"~~ ~ ~~~~~

~~~~~~"

~~~~

When n

= 0 the first term on r.h,s. b

equal

to zero.

computing for1no,

mi, m2 one

by

one and

accounting (13)

we obtain

(omitting unimportant constant)

(~~(f))

~ ~~j)2

~~ ~~ ~

~~"~°~~

~~

$ j

~~ ~~~~

j- -

Here the first term

reproduces

the answer of

equation (I)

without

potential,

while the second one b

perturbation theory

result.

In

conclusion,

we deduce that the

problems

with

conserving

and

nonconserving

normalization

belong

to different classes of

universality.

We think that the results of

computations

[4,8] are cor- rect for a wave

propagating

in media with random refraction and

>homogeneous absorption (We

latter can also

phenomenologicafly

model

backscattering, neglected

in

parabolic approximation

[3,

4]).

In the cases when normalization is

precisely

conserved, as e.g. for quantum

panicle

in

uncorrelated

potential,

one should await answer

(15)

for t - cc.

References

[I] KARDAR M. and ZHANG Y.-C., Phys. Rev Lea. 58

(1987)

2087.

[2] ZELmVICH Ya.B., MoLcHANov SA., RUzMAIitIN A-A- and SOKOLCFF D.D., ZkETP 89

(1985)

2~6l.

[3]RYTov

S.M., KRAVTSOV Yu.A., TATARSKY VI., Introduction to statistical

radiophysics Q4auka,

Moscow, 1978) (in

russian).

[4] FENG S., GOLUBOVICH L. and ZHANG Y.-C., Phys. Rev Let 65

(1990)

1028.

[~

HUSH D.A., HENLEY C-L-, FISHER D.S., Phys. Rev LetL 5S

(1985)

2924.

[fl

HALI'iN-HEALEY T, Phys. Rev LetL 62

(1989)

442 [7l MEzARD M., PARISI G., J Phys.A 23

(1990)

L1229.

[8] LEBEDEV N-I-, SiGov A.S.,L Stat. Phys.,in press.

[9] FEYNMAN R~P and HIBBS A~R., Quantum mechanics and

path integrals (J.

1Ailey& Sons, N-Y-, 1965).

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