Chaotic quantum motion in a space-time periodic potential: an exactly solvable model

PHYSICA ELSEVIER

Physica D 82 (1995) 371-381

Chaotic quantum motion in a space-time periodic potential: an exactly solvable model Ralf Nasilowski, Hans-Joachim Schellnhuber Universiffit Oldenburg, FB Physik, PosOeach2503, 26111 Oldenburg, Germany Received 19 October 1994; accepted 8 December 1994 Communicated by B.V. Chirikov

Abstract We study the quantum mechanical motion of a particle in a space-time periodic potential of the Chirikov-Taylor type, 3(t - n). It is well known that the classical mechanical motion can be chaotic, where the chaotic character of the motion manifests itself by a macroscopic momentum diffusion (Brownian motion): ((p (t) - p (0)) 2) ~ t. We here show that the quantum mechanical version of this system, under certain conditions, is exactly solvable. It turns out that the momentum diffusion occurs also in the quantum system. This is remarkable, as it seems to contradict the widespread belief that "quantum chaos is impossible".

V(x, t) = (K/4~r 2) cos(2~rx) ~ - o o

1. I n t r o d u c t i o n

i.e., one then observes that the macroscopic quadratic change o f momentum grows linearly with t,

Let us consider the motion o f a particle in a onedimensional, time-dependent potential, described by the Hamiltonian function

H ( x , p , t) = lp2 + V(x, t), where the potential is periodic both in x and in t,

V(x + l,t) = V(x,t),

V ( x , t + l) = V ( x , t ) .

where D is the diffusion constant. Let us now further assume that the s p a c e - t i m e periodic potential has the special form

V(x,t)=V(x)

~

6(t-n),

n=--cx3

( W e here have set, without loss o f generality, m = A = 7. = 1, where m is the particle mass, h the spatial wavelength, and 7- the time period o f the potential.) This is normally a nonintegrable problem, which means that the classical mechanical motion can be chaotic [ 1 3 ]. The chaotic nature o f the motion o f our particle, in turn, manifests itself macroscopically by a diffusive behavior o f the m o m e n t u m (Brownian motion),

where V(x) = V(x + l ) is a periodic function, and denotes the Dirac delta function. The Hamilton equations can then be integrated explicitly over a unit time interval, and one obtains the difference equations

x ( t + 1) = x ( t ) + p ( t + 1), p ( t + 1) = p ( t ) + F (\ x ( t ) ]/ ,

0167-2789/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0167-2789 (94)00246-0

(2)

372

R. Nasilowski, H;-J. Schellnhuber / Physica D 82 (1995) 371-381

for t ~ Z, where dV(x)

F(x) .-

- dx

,

and p ( t ) is to be understood as the left-sided limit at t C Z. The dynamical system defined in this way is a slightly generalized version of the well-known Chirikov-Taylor model [4,5,3,6-8]. The standard Chirikoy-Taylor system is obtained by setting K

V(x) = ~

cos(2~rx),

F(x) =-V'(x)

K = ~ sin(27rx),

(3)

where K is a parameter. Despite this "piecewise integrability", the ChirikovTaylor system behaves like a typical nonintegrable system; in particular, one observes chaotic behavior and macroScopic momentum diffusion, as described above. For some specially chosen (nonstandard) potential functions V ( x ) , one can even obtain exact results [9,12]. The value of the diffusion coefficient normally depends on the initial condition. In fact, one typically observes a coexistence of chaotic and nonchaotic regions in phase space, interwoven with each other in a complicated fractal structure. Nevertheless, one may say that, roughly speaking, if the parameter K in (3) is large ( K >> 1 ), then the nonchaotic regions of the phase space are negligible, and the diffusion coefficient is practically independent of the initial condition. All these classical mechanical facts are well known. We now ask: what happens in the quantum mechanical case? At first glance, the answer seems obvious: "It is a generally accepted fact that quantum chaos does not exist; consequently, there is no macroscopic momentum diffusion in our quantum system". In fact, our system is formally identical with the popular "kicked rotator", and it is well known that there is no (angular) momentum diffusion in the quantum mechanical kicked rotator [ 13-19]. There is, however, a subtle difference between the kicked rotator and our system: in the kicked rotator, the coordinate x has the meaning of an angle, i.e., the values x and x + 1 refer to the same physical'configuration, which implies that the wavefunction must be periodic in x. In our system, on the

other hand, x and x + 1 are physically different and, consequently, arbitrary nonperiodic wavefunctions are allowed. In other words, the Hilbert space of our system, which one might call a "kicked translator", is larger than that of the kicked rotator. As far as we know, our quantum mechanical system (i.e., without the periodic boundary condition on the wavefunction) has not yet been considered in the literature. The rest of this paper is organized as follows. In Section 2 we consider the Schr~Sdinger equation of our system and show how it can be exactly solved if a certain "quantum resonance" condition is satisfied. (This quantum resonance condition has already been discussed, in the context of the kicked rotator, in Refs. [ 13,14] .) We here also point out a connection to another popular solvable system of the "kicked quantum rotator" type, introduced by Fishman, Grempel, and Prange [20-22]. In Section 3 our system is discussed in the Heisenberg picture and a quantum mechanical fluctuation-dissipation formula for the diffusion coefficient is derived. In Section 4 we show how the fluctuation-dissipation formula can be evaluated exactly if the quantum resonance condition is satisfied. In Section 5 we apply our theory to a standard Gaussian wavepacket and obtain an exact formula for the diffusion coefficient as a function of the wavepacket parameters.

2. The Schri~dinger equation Let us turn to the quantum mechanical version of our system. Its Schrtdinger equation reads - i h O ~9(x, t) =/-)(t) ~9(x, t), where ¢ is the wavefunction and Oo

[I(t)=l/32+V(2)

E n=--

8(t-n)

(4)

0<9

is the Hamiltonian operator. Here, as usual, the operator 2 means multiplication by x, and fi = -iI~3/3x. By formally integrating the Schr0dinger equation from t to t + 1, one obtains a difference equation of the form ~b(x, t + 1) = O 0 ( x , t)

(5)

R. Nasilowski, H.-J. Schellnhuber / Physica D 82 (1995) 371-381

for t E Z, where 0 is the so-called monodromy operator. The formal solution of (5) with initial condition ~ ( x , 0 ) = ~p(x) reads

~(x, t) = Ot ¢ ( x ) , where the propagator,/)t, is obtained by t-fold iteration of the monodromy operator 0. In particular, one obtains the "stationary states" of the system by the ansatz ~ ( x , t) = e -i2rrvt ~t(x),

(6)

Note that the "eigenfrequencies" v are well defined only modulo 1. For our special Hamiltonian (4), the monodromy operator can be computed explicitly: it reads (7)

= e -ip2/2h e - i v ( ~ ) / h ,

where we follow the usual convention that the wavefunction ~ ( x , t) is to be understood as left-sided limit with respect to t. The action of the operator (7) on a wavefunction g,(x) is given by Og,(x) = (i 2rrh)-l/2

× f dyei ( x - y ) z / z h

e -iv(y)/h ~(y).

simultaneously eigenfunctions of S. Such solutions are of the form ¢ ( X , t) = e i2~rqx q~(x, t ) ,

(9)

where q is a parameter, and q~ is a spatially periodic function ("Bloch function"), ~b(x, t) = ~b(x + 1, t). Inserting (9) and (8) into (5) yields a monodromy equation for the Bloch function,

qS(x,t + 1) = U ( q ) ~ix, t),

which leads to the eigenvalue problem Lr~//(x) = e -i2~rv 0 ( x ) .

373

(8)

--00

Thus, (6) becomes an integral eigenvalue equation. Unfortunately, one cannot normally solve this eigenvalue problem in closed form. Until now we have not yet exploited the periodicity property

V(x) = V ( x + 1) of our potential function. This spatial periodicity implies that the monodromy operator 0 commutes with the unitary symmetry operator

where the Bloch monodromy operator ~r(q) is given by

U(q) ~b(x) =

0<3

E

ei2~rkx e--i2:c2h(q+k)2

k=--oo

1/2 × f

d y e -i2r&y e -iv(y)/h ~b(y).

( 10)

, J

-1/2 An interesting special case now arises if the Planck constant happens to be a natural number, i.e., if

h=2crh E

{1,2,3 .... }.

(Note that our "Planck constant" h is a dimensionless parameter, which in ordinary physical units is to be replaced by hr/mA2, where m is the particle mass, r is the time period and A the spatial wavelength of our space-time periodic potential.) If this "quantum resonance" condition [ 13,14] is satisfied, the Bloch monodromy operator (10) reduces to the simple formula U(q) ~b(x)

=exp(-iqrhq2-1--~-'2¢rV(x-(q+½,h)) ×gb(x-(q+l)h).

(11)

In deriving this expression from (10), we have used the periodicity of ~b(x) and V(x), arid the fact that e 4-irrhk2 = e +i~rhk = ( -- 1 ) hk

for h, k E Z.

= ei~/h = eO/e~. The commutativity of S and 0 enables us to construct solutions of the monodromy equation (5) which are

By iterating the operator (11) t times, one obtains the following explicit formula for the time-dependent Bloch function:

R. Nasilowski, H.-J. Schellnhuber / Physica D 82 (1995) 371-381

374 ¢ ( x , t) = U(q) t ¢ ( x )

t

=exp(_izrq2ht_. i_~_z ~2~r V(x-(q+½)hs))

ture. In our case, the Heisenberg operators 2(t) and /3 (t) can be expressed in terms of the SchrSdinger operators, 2 and/3, by

s=l

2(t) = 0 - t 2 0 ',

x¢(x-(q+½)ht), where ¢ ( x ) = ¢ ( x , t = 0). We want to conclude our discussion of the SchrSdinger equation with the following remark, which could be of interest: our Bloch monodromy operator, (11), can be interpreted as the monodromy operator of a SchrSdinger equation with the Hamiltonian

fflq( t ) = ce/3 + ~ -~ V ( 2 )

~

(t - n ) ,

(12)

nm--OO

where a=(q+½)

h,

/3(0 = 0 - t / 3 0 ',

(14)

for t C Z, where 0 is our monodromy operator, while the time-dependent SchrSdinger wavefunction I~/,(t)) is related to the time-independent Heisenberg wavefunction I•) by [¢,(t)) = 0'10).

Since the Heisenberg operators £ (t) and/3 (t) behave formally in many respects like the corresponding classical mechanical variables, x(t) and p (t), one might suppose that the classical difference equations (2) are also satisfied by the Heisenberg operators. This is indeed the case, i.e., we have

fl=½(hq) 2.

Note that (12) differs from our original Hamiltonian, (4), only in that the original kinetic energy operator, p2/2, has been replaced by a/3 + ft. Hamiltonians of the form (12) have been considered, in a different context, by Fishman, Grempel, and Prange [20-22]. These authors alSO discuss the eigenfunctions and the spectrum of the monodromy operator of such systems.

Theorem 1 (Heisenberg equations). The Heisenberg operators 2 ( 0 and/~(t) satisfy the difference equations 2(t + 1) = 2 ( t ) +/3(t + 1), p ( t + 1) = p ( t ) + F(2(t)).

(15)

Proof. Using the explicit form (7) of U, we obtain 0 .~ 0 -1 = e-ip2/2h2 e it)2/2h

3. The Heisenberg equations and a fluctuation-dissipation theorem

= 2 + e -ip2/ah [2,e ip2/2h] =2 -p

The problem we are interested in reads now: can a macroscopic momentum diffusion process occur in our quantum system, like it does in the classical mechanical case? In other words, does (1) also hold quantum mechanically? To find this out, we first need a quantum mechanical prescription to evaluate the expectation value on the left hand side of (1). Such a prescription is given by

(16)

and

O-Ip 0 =eiV(SO/hp e -iv(~)/h =p + eiV(~)/h [fi,e-iV(~)/h] =/3 + F ( 2 ) .

(17)

Here, the commutators in (16) and (17) have been evaluated according to the formula

< @ ( t ) - / 3 ( 0 ) ) 2 > = <~b/(/3(t) - fi(o))Z g'>(~3 )

[A,f(B)a=tA, B]f'(a) where/3 (t) and Ig') are the momentum operator and the (normalized) wavefunction in the Heisenberg pic-

if [[fi~,B],J~] = 0

(see Appendix, Lemma 1). We now multiply (16) and (17) by 0 t from the left, and by 0 -t from the right,

R. Nasilowski, H.-J. Schellnhuber / Physica D 82 (1995) 371-381

375

and obtain, using (14) and the fact that ~J-tF(2) ~jt =

Theorem 2 (Fluctuation-dissipation formula). The

F ( tJ-t 2~Jt ),

momentum diffusion coefficient is

2(t-

D : = l,i-m ~ ~l ( @ ( t ) - p ( o ) )2 /

1) = 2 ( t ) - f i ( t ) ,

f i ( t + 1) = f ( t ) + F ( 2 ( t ) ) , oo

[]

= 31 ~

C(t).

t=--oo

From the second equation in (15) it now follows that t--1

s=0 so that the expectation value (13) can be expressed as

To evaluate our fluctuati0n-dissipation formula, we now need, in some form, an explicit expression for the Heisenberg force operator

F(t) := F ( 2 ( t ) ) .

r-~ s=O

where

C(s, t) := (¢ IF(2(s) ) F(~c(t) )l 0 ) -

4. Solution of the Heisenberg equations and evaluation of the fluctuation-dissipation formula

(18)

Since the force function F(x) is periodic, it can be expanded in a Fourier series, o<)

We call (18) the "force autocorrelation matrix", since the quantities C (s, t) may be regarded as elements of an infinite quadratic matrix. With the aid of this autocorrelation matrix C(s, t), one can now define an "autocorrelation function" C (t) by

C(t) = iim C ( s , s + t),

(19)

S----->CX)

provided this limit exists. Even if the limit (19) should not exist, one can still define an autocorrelation function by the more general formula

Fn ei2zrnx.

Thus, F ( t ) is a linear combination of the Heisenberg operators ei2~rn2( t).

These operators, in turn, satisfy the difference equation ei21rnYc(t - 1) = ei2~rn(~(t) -p(t) ) --_ e-irrhn ei2Crn~(t) e-i2zrnp(t) = e iTrhn e-i2zrnp(t) ei2~rn-t(t),

T-1 C(t)=r__.o~lim 1 Z C ( s , s + t ) . s=0

(20)

If we assume that the limit (20) exists and that the convergence is uniform with respect to t, then

13(t) - f i ( O )

F (x ) = ~

=

C(r,s)

e ~+~ = e-C~2 e ~ eB = eC/2 e~ e,~

r,s---O oo

~_t

where the first equality sign follows from (15), while the second and third equality sign follow by application of the well-known Baker-Campbell-Hausdorff formula

~ C(s) s= -- oo

for t ~ oo, provided, of course, that the latter infinite sum converges. Thus, we have

if [A,/~] = c E C (where C := the set of complex numbers; see Appendix, Lemma 2). In a similar manner, the second equation of (15) yields

376

R. Nasilowski, H.-J. Schellnhuber / Physica D 82 (1995) 371-381

ei27rn/~(t+l) = ei2rrn(p(t)--V'(~c(t)))

ei2"n'np(TM) = ei2rrn/~(t).

= ei2~nP ( t )

Consequently,

× exp ( i2¢rV( Sc(t) - hn)h - V( 2( t) ) ) (i2¢r V(2(t)) - V(2(t)h

=exp

ei2~n/~(t) = ei2~nP,

where/3 = /3(0) is the usual Schr6dinger operator. Thus, (21) becomes

+ hn))

X ei2rrnp(t).

ei2~rnYc(t+ 1) = ( - - 1 ) hn ei2m~2(t) ei2~rnp

Here, we have used the following, slightly more general version of the Baker-Campbell-Hausdorff formula:

= ( - - l ) hn e i2rrnp e i2~rn2(t) ,

(23)

where we have used the trivial fact that e ±irrhn = ( -- 1 ) hn

exp(fi,+

f ' ( B ) ) = e 'i exp( f ( ~ ) -

f(~-c)c From (23) we conclude that ei2~rn~(t) = ( - 1 ) hnt

ei2rrn2 ei2rrnt/~

= ( -- 1) hnt ei2~rnt/~ei2Crn2.

if[A.,B] = c E C

(24)

(see Appendix, Lemma 2). Thus, we have

The action of (24) on a wavefunction is

Theorem 3 (Exponential Heisenberg equations). The operators exp(i27rn2(t)) and exp(i2crn/3(t)) satisfy the difference equations

Consequently, the Heisenberg force operator P ( t ) is explicitly given by

e i2~n~(t) ~ ( x ) =

co ~

P(t) ~(X)=

eiZ~n~(t) =eirrh n ei27rn~(t) ei2~-np(t+l)

F n e i2¢rnyc(t) ~ ( x )

n=--o0 O0 = ~ ( - - 1 ) h n t F n e i2z'nx ~ ( X -}- h n t ) .

= e -i~rhn ei2~rn/~(t+l) ei2rrn2(t),

(21)

(-1) hnt ei2~nx @(x + hnt).

:

II=--O0 ei2~rnp( t+ l )

This simple fact now enables us to calculate various interesting expectation values. In particular, the timedependent force expectation value is

=exp ( i27rV(2(t) ) - V(2(t) + hn) × ei2~rnP(t)

-if(t) = (~O I P( t) ~0)

= ei2rrn!5(t )

× exp (i2~r V ( 2( t) - hn) - V( 2( t) ) ) h

OQ = ~ (--1)hntFn n=-oo co

(22)

× / dxei2~rnx¢*(x)~p(xWhnt),

Let us now again assume that the "quantum resonance" condition

--OO

and the force autocorrelation matrix is h c {1,2,3 .... } is satisfied. In this case, the difference quotients of V in (22) vanish due to the periodicity of V, i.e., (22) reduces to

C(s, t) =
of) ~ m,n=--f3o

( - - 1 ) h(ms+nt) Fm Fn

(25)

R. Nasilowski,H.-J. Schellnhuber/ PhysicaD 82 (1995)371-381 0(3

X S dxei2~rx(m+n) lis*(x)+(x+ h(ms+ nt)), --00

(26) where the asterisk denotes the complex conjugate. In the standard case (3), the only nonvanishing Fourier coefficients are F4-1 = TiK/47r, so that (25) and (26) reduce to

t-1 lim 1 Z y ( s t-~oo t s--O

) =0.

From (27) and (29), on the other hand, we conclude that the force autocorrelation function is

C(t) = lim C(s,s + t) s----+ o o

K2 = (--1)ht ~ 2

oo

F ( t ) = ( - 1 ) h, K Im S dx ei2~rx~ * ( x ) ~ ( x

Re

q- ht )

O<3

(27)

377

f dx O*(x) ~(x + ht). --00

(30)

Consequently, the momentum diffusion coefficient is

and

D=½ ~

C(s,t) = (-1) h(s+t) K2

C(t)

l=--O0

8¢r2 _

K2

167r2

( - - 1 ) ht

~ t=--O<3

--OO

i dx ei4~rx~*(x) O(x + h(t + s))),

- Re

-oo

(28)

where "Re" and "Im" mean "real part" and "imaginary part". Let us now additionally assume that our wavefunction ~ has the property O<3

P

(31)

Note that condition (29) ensures that both C (t) and D are finite and well defined. We further remark that the expression (31) can be brought by elementary algebraic manipulations to the form

K2

D - 167r2h

~ s=-

[~ ( s / h -

½)12

,

(32)

oo

where

OO

/ t=--oo

i dx O*(x) O(x + ht). --00

dx [~(x)[

IO(x + ht)[ <

oo.

(29)

,.I oO

oo

~ ( q ) := /

We think that this condition is satisfied by any "physically reasonable" (normalized!) wavefunction. Under the condition (29), we may conclude from (27) that the time-dependent momentum expectation value t--1

p(t) := (fi(t)) =-p(O) + ~-]-ff(s) s=O converges to a well-defined, finite limit value

dx e -i2~rqx ¢ ( x )

--OO

is the Fourier transform of 0- Eq. (32) confirms explicitly the fact that D is real and nonnegative. From the above considerations, we conclude that 0 < D < oo. This means that a quantum momentum diffusion actuallytakesplace. Only in exceptional cases (namely if all terms of the infinite sum (32) simultaneously vanish) can it happen that D = 0.

oo

p(oo) =p(0) + ~-~F(t).

5. Fate of a w a v e p a c k e t

t=0

This implies, in particular, that the long-time averaged expectation value of the force vanishes,

Formula (28) permits us to calculate in a simple way the force autocorrelation matrix C (s, t), and thus

R. Nasilowski, H.-J. Schellnhuber / Physica D 82 (1995) 371-381

378

also the force autocorrelation function C(t) and finally the diffusion coefficient D, for arbitrary wavefunctions ~O. As an example, let us consider a "standard wavepacket", described by the normalized Ganssian wavefunction

+(x) = (27-ro-2)-1/4exp (

i

(x-x)2 ipx)i 4 ° - ~ + ---if- 33)

2

(34)

while o- is the localization width of the wavepacket, ~: = ((~ _ (~))2).

S'---~(X)

~,~

o-/h=0.4 -+--

T

~/h=0.6 ~r/h=0.8 -x--

[ 1

./~=10~ -

~lh=0.2 "0--

(35)

Note that the expectation values (34) and (35) all refer implicitly to time t = 0, because in our case ~b(x) is a Heisenberg wavefunction, i.e., qz(x) is the Schr6dinger wavefunction O(x, t) at t = 0. One easily checks that (33) satisfies our condition (29). Inserting (33) into (30), we find the force autocorrelation function C ( t ) = lim

I

a/h=0.0

j:-2

DI-D

~ = (/3),

[

I

~

3

Here, 2 and p are the mean position and mean momentum of the wavepacket, = {2),

i

-0.4

-0.2

0

0.2

0.4

-if+hi2

Fig. 1. The momentum diffusion coefficient D as a function of the mean momentum ff and the localization width o-. Shown is not the value D itself, but the scaled value D/-D, where D = K2/16~ -2 is the "canonical value". Note that the curves continue periodically outside the shown range of ft.

(where f is an arbitrary analytic function), however, we can transform (37) to

C(s,s+t)

;

K2

= 8¢r2 exp \-~-~-~2] (36)

D = 1--~ '/gg~ Zh ~ × ~_~

from which we extract the momentum diffusion coefficient

e x p ( - 8~-2°-2 ( h 2 ~ + ½h - s ) 2 )

I

$=----OO

The latter expression shows explicitly that 0 < D <

OO

E c(t)

0<3.

t~--O0

= 16~r2 E

exp

-~2

+i2~'t(p+½h

/)

.

Obviously, the diffusion coefficient D depends on the momentum expectation value, p, and the localization width, tr, of the wavepacket, i.e.,

/=--OO

(37) Unfortunately, this infinite sum cannot be summed up in closed form. By means of the Poisson resummation formula

E f(t)= E t = -- O0

$-~ - - 0 0

dte-i2~stf(t) 00

D =D(~,o-), where D is a periodic function of ~, D ( ~ + 1 , o - ) = D ( p , tr). The function D = D(ff, tr) is depicted graphically in Fig. 1. Averaging (37) over p yields

R. Nasilowski, H.-J. ScheUnhuber / Physica D 82 (1995) 371-381 I

--

D :=

f

K2 d ~ D ( f f , o') - 16¢r2 .

(38)

0

Note that this averaged value D is independent of o-. We may therefore justly call D the "canonical value" of the diffusion coefficient. It is this canonical value D that one would predict if no information about the system's state (such as x, p, or o-) were available at all. In the limit o- ~ 0, the wavefunction (33) degenerates to a 6-function: 0 ( x ) ~ 8 ( x - Y), and formula (37) reduces to

We finally remark that the canonical value (38) coincides exactly with the value predicted by the classical "random phase approximation". In this approximation, one assumes that the force autocorrelation function is

C ( t ) ,~

=0

(40)

0

otherwise.

One so obtains 1

D = 71

Z

C ( t ) ,.~ 5

dx

F(x)

0

as o- ---~ 0.

K2 C(t)"°-ff-~23 t

Inserting F ( x ) = (K/2cr)sin(27rx) yields D K2/16~ -2, which agrees exactly with our canonical value D. This agreement is not quite accidental, of course; in fact, as we have seen above, the approximation (40) becomes asymptotically exact in the limit o- --+ 0 in our quantum model.

as o- --+ 0,

6. Concluding remarks

where 1 0

if t = 0 , otherwise

is the discrete 6-function. In other words, the force then behaves effectively like an ideal "white noise". In the limit o- ---+ o~, on the other hand, our wavefunction (33) becomes a plane wave: 0 ( x ) e x p ( i f f x / h ) , and we have asymptotically

ON_ -

i,

t=--oo

Thus, we have found that the momentum diffusion coefficient D takes its canonical value D, if the quantum state at t = 0 was sharply localized. We remark that the autocorrelation function (36) then becomes

6t :=

{ 0

K2 D _~ 16~-2

379

16~-2

~

,~ -p + h / 2 -

s

aso-~oo.

S=--OO

That is, the momentum diffusion is suppressed if the initial state has a sharply defined momentum, p = g. In this limit case, the autocorrelation function (36) becomes K2 ( ) C ( t ) ~-- ~ cos 27rt (p + h / 2 )

as o" --+ co. (39)

Note that (39) is quasiperiodic in t and, in particular, does not decay as Itl --+ oo.

We have obtained various interesting exact results for our quantum Chirikov-Taylor system; in particular, we could explicitly demonstrate that there is a macroscopic momentum diffusion, like in the classical mechanical case. This behavior is quite different from that of the quantum mechanical kicked rotator, in which such a momentum diffusion does not take place. That seems paradoxical, as the kicked rotator is in fact formally identical to our system. The paradox can be resolved as follows: the wavefunctions of the kicked rotator satisfy the periodicity condition ~/,(x + 1) = ~/,(x),

(41)

whereas the wavefunctions in our system satisfy the normalization condition f

dx I~p(x)I 2

= 1.

(42)

Obviously, the conditions (41) and (42) are mutually exclusive, i.e., the wavefunctions of the rotator are not

R. Nasilowski, H.-J. ScheUnhuber / Physica D 82 (1995) 371-381

380

permissible in our system, and vice versa. This simple fact explains how the two "formally identical" systems can behave so differently. Unfortunately, our exact results are valid only if the "quantum resonance" condition h ~ {1,2, 3 .... } is satisfied. Thus we cannot study the highly interesting classical limit (h --+ 0) within our model. Nevertheless, we think that our results give us some insight into the quantum behavior of classically chaotic systems.

Lemma 2 (Baker-Campbell-Hausdorff formula).

Let

,4 and/7 be two operators with

[~,~1 =c ~ 0, where c is a complex number. Then

e~+D= e-C~2 e~ e~ = e C/2 e t? e a,

(44)

and eA+f'(D) =e/i

Appendix: Commutator formulas

= exp

We here prove some useful commutator formulas, which are referenced from the text.

Lemma

1. Let ~ and/7 be two operators whose com-

mutator

exp ( f(~) - f ( ~ - c ) (f(D+c)--f(~))

^

e A ,

C

(45) where f is an arbitrary analytic function.

d:=[~,/~] := 7, D - D ~

Proof.

We first note that setting f(x) = x2/2 in (45) yields (44); it therefore suffices to prove (45). Let

commutes with/~, i.e.,

J?(t):=etA

[D,O] = 0 ,

exp( f ( ~ ) -

f(~-tc))c

'

(46)

and let f be an analytic function. Then

where t is a real parameter. Derivation with respect to t yields

[ ~ , f ( D ) ] = f'(D) d,

d x(t) = X(t) (A + f ' ( ~ - tc))

where f ' is the derivative of f .

Proof.

Since f is an analytic function, it can be expanded in a Taylor series, so it suffices to prove the assertion for the functions f(x) = x". Thus we have to show that [.4, B"] = r i b " - 1 C

We now apply Lemma 1 to evaluate the commutator in (47):

et,i [ e { , e x p ( f ( B ) - f ( B - t c ) ) l

(43)

f'(B-tc)).

=X(t) (f'(J~)-

for n E {0, 1,2 .... }. We prove this by induction on n. Assuming that (43) holds for a particular value of n, we find

This inserted into (47) leads to the differential equation

[/~,n n-I-1 ] = [A.,B n ] D-+-D n r~,i, ~i~]

dt

+

=nD"-ldD+D"~

Consequently,

= ( n + 1)D"¢,

2(t) =2(0) e t~+i'(~)~,

i.e., the assertion holds also for n + 1.

[]

which,, according to the definition (46), means that

R. Nasilowski, H.-J. Schellnhuber / Physica D 82 (1995) 371-381 etA exp ( f ( B ) -- f _ ( ~ - t c ) ) =et(.4+f'(~)) c (48) We now obtain the assertion (45) from (48) by setting t=±l.

[]

Acknowledgements This

work

was

supported

by

the

Deutsche

Forschungsgemeinschaft.

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