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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�.
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J Phys. Ifrance 1
(1991)
1213-1215 sBPrBMBRB1991, PAGE 1213Classification
PhysicsAbstnwt£
05.40-05.50
Shom Communication
Diffusion of
aquantum 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
ofPhysics,
Moscow StateUniversity,
l19899, Moscow, U-S-S-R-(~) Theoretical
Physics Dcpartment,
EN. l~cbedcvPhysical
Institute, l17924, Moscow, l~cninsky Prospcct 53, USSR andVniigcoinformsystcm~
l13105 Mowcw, Wrshavskoe shosse 8, U.S.S.R.(Received
20Mqy
lwl,ttccepted11 Ju~
lwl)Abstract. For
Schrodinger eqttation
for aparticle moving
in random,time-dependent potential
with white noise correlation, we prove that perturbation
theory
result for mean squaredisplacement
X ~-
t~'~
isasymptotically
exact for a large time t. This is in contrast with the sameeqttation
with imaginary time.In recent years there raised a
significant
interest in systems describedby 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(11t2) (2)
Here x is a d-dimensiofial vector.
Equation (I) relates,
inparticular,
to theproblem
of wavepropagation
in media vith random index of refractionV(z, t) (in parabolic approximation)
[3, 4]. The sameequation
vithimaginary
time characterizes thermal fluctuations of directed
polymer
in random media[I]
and variations of some animal'spopulation
due torandomly changing
environment [2].A group of
analytical
methods have beenapplied
for calculation ofpolymer's displacement scaling dependence [5-7j
(x~(t))
'~
t~"
means thermal average and we suppose
z(0)
=
0).
At the same timeonly
numerical estimates [4,8] are knoAvnanalogous
variableobeying Schr0dinger equation
vith real time(I).
Infac~
thecomputations
in[4,8]give
the same value of a as inpolymer problem.
At firstsight
it looksnatural,
1214 JOURNALDEPIIIfSIQUEI N°9
but let us note that even in the absence of disorder a
=
1/2
for directedpolymers
while a = I forquantum
case(1)
[9]. 'Ib our mind it is essential that iteration scheme uwd in [4, 8] did notconserve nomalization of wave function in contrast with
equation (I)
itsev(that
wasspecially
taken into account ±±
[4]).
Our
analytical
calculatio~ cited below indicate that weH kwwnperturbation
result a=
3/2
(see
e.g.?
and Refh.therein)
holdsasymptotically
exact in the Iirnit t - cc forSchrodinger equation (I).
The starting
point
of ouranafysb
isequation
for variablem
(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 ofFeynman integral
isi~
(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 antrajectories passing through point
xi.Supposing V(x, t)
Gaussian vith correlator(2),
thefollowing
result for~m form(3)
ism =
/
DVP(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 ofU(x)
~om(2).
After
integration
overV(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
usFeynman
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 forobtaining replica
Hamiltonians in thetheory
of dis- ordered systems(see
e.g. [7]). Itsadvantage
is apossible application
to randompotenthls
withlong-range
correlations.N°9 QUANTUMDIFFUSIONINTIME-DEPENDENTRANDOMPOTENIIAL 1215
Hereafter we assume z to be a
scalar, generafisation
to multidimensional case bstraightfor-
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
bothparts
of(11) by q"
andintegrating
over q one gets
~~" ~~~
~"~~ ~ ~~~~~~~~~~~"
~~~~When n
= 0 the first term on r.h,s. b
equal
to zero.computing for1no,
mi, m2 oneby
one andaccounting (13)
we obtain(omitting unimportant constant)
(~~(f))
~ ~~j)2
~~ ~~ ~
~~"~°~~
~~$ j
~~ ~~~~
j- -
Here the first term
reproduces
the answer ofequation (I)
withoutpotential,
while the second one bperturbation theory
result.In
conclusion,
we deduce that theproblems
withconserving
andnonconserving
normalizationbelong
to different classes ofuniversality.
We think that the results ofcomputations
[4,8] are cor- rect for a wavepropagating
in media with random refraction and>homogeneous absorption (We
latter can also
phenomenologicafly
modelbackscattering, neglected
inparabolic approximation
[3,
4]).
In the cases when normalization isprecisely
conserved, as e.g. for quantumpanicle
inuncorrelated
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 statisticalradiophysics Q4auka,
Moscow, 1978) (inrussian).
[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