PHYSICAL REVIEW A

VOLUME 53, NUMBER 4

APRIL 1996

Exponential divergence of neighboring quantal trajectories Gonzalo Garcı´a de Polavieja* Departamento de Quı´mica, C-IX, Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain ~Received 23 February 1995; revised manuscript received 19 September 1995! Despite the dynamical stability and absence of chaos in quantum wave functions, it is demonstrated that the quantal trajectories of the de Broglie–Bohm quantum theory of motion are capable of showing linear and exponential divergences of neighboring particle orbits for a general nonintegrable system. A quantal trajectory is influenced by the whole phase space and can be associated with a set of Lyapunov-like exponents which separate time scales of different stability PACS number~s!: 03.65.2w, 05.45.1b

Great progress has been made in the study of the quantum-classical correspondence of classically chaotic systems @1#; yet the existence of quantum chaos is still unclear. Several arguments have been used to dismiss chaos in quantum mechanics. Recurrence of wave functions in bounded systems @2#, the dynamical stability of wave functions @3#, and the logarithmic compressibility of eigenvalues and wave functions in a broad class of bounded systems @4# are some of these arguments. However, considering quantum mechanics a statistical theory @5#, it is not surprising that all familiar tests of chaotic motion fail, as they do in classical Liouville mechanics. Comparison of the quantum and Liouville mechanics have yielded important results in the study of the quantum dynamics of classically chaotic systems. Schack and Caves @6# found hypersensitivity to perturbations in the quantum baker’s map analogous to that of its Liouville representation. Mendes @7# has determined the classical definition of Lyapunov exponent for classical densities and quantum wave functions using a directional derivative of d as the deformation direction in the space of functions. Other approaches to the quantum chaos problem have also recently been given in the literature. Zurek and Paz @8# have considered the implications of decoherence for quantum chaos. Blu¨mel @9# has found the existence of exponential sensitivity in a spin-precession apparatus, but with no meaningful classical analog. Kanno and Ishida @10# discovered that a quantum-mechanical path that contributes to Feynman’s path integral has the property of the baker’s map. We believe that the problem of the existence of quantum chaos is closely related to the problem of finding the quantum analog of a classical particle orbit in phase space. We will not argue here for the necessity of the trajectory concept in quantum mechanics @11#, but it has been shown to be consistent with all quantum-mechanical results in the form of the de Broglie–Bohm ~BB! @12# quantum theory of motion, the Bohm-Vigier @13# stochastic mechanics and Nelson stochastic mechanics @14#. In this paper, we use the BB trajectories in phase space as the quantum analog of classical phase-space orbits. And we apply the basic definition of chaos, the exponential divergence of trajectories initially at arbitrary close phase-space

*Present address: The Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OX1 3UB, United Kingdom. 1050-2947/96/53~4!/2059~3!/$10.00

53

points, to the BB quantum theory as a test of the existence of chaotic motion in quantum mechanics. In the BB quantum theory of motion the complete state of the system is specified by the wave function C that obeys the Schro¨dinger equation and by the configuration of its particles that obeys, for the one-particle case,

S

D

dx \ C ~ x! 5 ¹ln . dt 2mi C * ~ x!

~1!

The BB theory and conventional quantum mechanics predict the same results, provided the initial particle configuration obeys Born’s statistical law for the probability density, i.e., r 5 u C u 2 . This equality has been justified by Bohm @15#, Bohm and Vigier @13#, Valentini @16#, and Du¨rr, Goldstein, and Zanghi @17#. Before analyzing the general case of a nonintegrable system, we present the simplest case of instability, the parabolic barrier, with potential energy V(x)52 21 m m 2 x 2 . The classical solution for this case is x ~ t ! 5x ~ 0 ! coshm t1

p~ 0 ! sinhm t, mm

p ~ t ! 5 p ~ 0 ! coshm t1m m x ~ 0 ! sinhm t.

~2a! ~2b!

The quantum-mechanical wave function with initial minimum uncertainty has the closed form:

X

C ~ x,t ! 5N ~ t ! exp 2

1 $ a ~ t !@ x2q¯ ~ t !# 2 \

C

2ip¯ ~ t !@ x2q¯ ~ t !# % ,

~3!

with N(t) a normalization factor, q¯ (t) and p¯ (t) the position and momentum of the center of the wave packet that obey the classical solution ~2!, and

a~ t !5

im m mm 2 tanh2 m t. 2 cosh2 m t 2

~4!

The BB trajectories for this case are the solution of Eq. ~1! for the wave packet in ~3!: x ~ t ! 5q¯ ~ t ! 1 @ x ~ 0 ! 2q¯ ~ 0 !#~ cosh2 m t ! 1/2 2059

~5!

© 1996 The American Physical Society

GONZALO GARCI´A DE POLAVIEJA

2060

FIG. 1. Natural logarithm of the Euclidean distance between two de Broglie–Bohm orbits initially close to the center of a minimum uncertainty wave packet centered at (x 01 ,x 02 ,p 01 ,p 02 ) 5(0,0,0,5.48) in case ~a! and centered at (x 01 x 02 ,p 01 ,p 02 ) 5(5.28,0.44,0,5.44) in case ~b!.

Although the classical and BB dynamics, Eqs. ~2! and ~5!, respectively, are different; they both show an exponential separation of neighboring orbits with the same Lyapunov exponent m . We have also calculated the BB trajectories for a nonintegrable bounded system with a smooth potential which classically shows linear or exponential divergence of phasespace trajectories, depending on the initial conditions. Its Hamiltonian ~natural units will be used throughout, i.e., \5m51) is 1 a b H5 ~ p 21 1p 22 ! 1 ~ x 21 x 22 ! 1 ~ x 41 1x 42 ! , 2 2 4

~6!

with ~a,b!5~0.04, 0.01!. The classical mechanics of this system has been extensively analyzed @18#. It has also been used in quantum-classical correspondence studies concerning eigenvalue statistics @19#, transport @20#, localization of eigenfunctions @21#, and of groups of eigenfunctions @22#. We have variationally calculated the first 900 eigenstates and eigenvalues of the system using a harmonic oscillator basis set converging its eigenvalues to five significant figures. We have computed the time-dependent wave function as C~ t !5

(n u n &^ n u C ~ 0 ! & exp~ 2iE n t ! ,

53

FIG. 2. Same as case ~a! in Fig. 1 for longer times.

classical-mechanical behavior associated with the two wavepacket centers corresponds to regular and chaotic motion, respectively. The quantal BB trajectories have been numerically calculated integrating Eq. ~1! using a forth-order Runge-Kutta method with a time step Dt5331023 . For each wave packet, we present results for two BB trajectories located at points very close to the center of the wave function, 0 28 x6 i (0)5x i 610 . As a measure of the separation of the BB trajectories in configuration space we present in Fig. 1 the natural logarithm of the Euclidean distance

S( 2

D~ t !5

i51

2 2 @x1 i ~ t ! 2x i ~ t !#

D

1/2

versus time. Case ~a!, which corresponds to the wave packet centered at (x 01 ,x 02 ,p 01 ,p 02 )5(0,0,0,5.48), displays a general behavior that corresponds to a nonexponential separation of the BB trajectories. On the contrary, in case ~b!, corresponding to the wave packet centered at (x 01 ,x 02 ,p 01 ,p 02 ) 5(5.28,0.44,0,5.44), the general separation of the BB orbits is exponential until saturation at t543, due to the effective boundedness of the motion. We have also calculated the corresponding separation in phase space that shows the same behavior but with superimposed noise. Unlike classical trajectories, BB orbits cannot in general be classified as completely regular or chaotic. They may

~7!

where u n & is the nth eigenstate of the system. For mathematical convenience we have used the initial state as a minimum uncertainty wavepacket centered at the phase-space point (x 01 ,x 02 ,p 01 ,p 02 ), 2

C~x1 ,x2 ;x01 ,x02 ,p01 ,p02!5~p!21/4

F

1

G

) exp 2 2 ~xj2x0j !21i~p0j xj! . j51 ~8!

We present results for two wave packets. One it is centered at the phase-space point (x 01 ,x 02 , p 01 , p 02 )5(0,0,0,5.48) and the other at (x 01 ,x 02 , p 01 , p 02 )5(5.28,0.44,0,5.44). The

FIG. 3. l(t), Eq. ~5!, for cases ~a! and ~b! of Figs. 1 and 2.

53

EXPONENTIAL DIVERGENCE OF NEIGHBORING QUANTAL . . .

show instances of regular and chaotic motion as they tunnel into regions of different stabilities. For example, any initially regular orbit may show exponential divergence with neighboring orbits at some value of time. To illustrate this behavior, we present in Fig. 2 the same data as in Fig. 1 for case ~a! but for longer times. Clearly the separation of the BB orbits has become exponential after t'45. For the reasons adduced above, BB trajectories cannot in general be associated with a particular Lyapunov exponent but a set of Lyapunov-like exponents that separate different time scales of motion. We have calculated

S D

1 d~ t ! l ~ t ! 5 ln t d~ 0 ! for cases ~a! and ~b!, where

S( 2

d~ t !5

i51

2 1 2 2 2 @x1 i ~ t ! 2x i ~ t !# 1 @ p i ~ t ! 2p i ~ t !#

~9!

D

1/2

is the Euclidean distance of the BB trajectories in phase space. These results are presented in Fig. 3. Note that in case

@1# M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics ~Springer-Verlag, Berlin, 1990!. @2# T. Hogg and B. A. Huberman, Phys. Rev. Lett. 48, 711 ~1982!. @3# R. Blu¨mel and U. Smilansky, Phys. Rev. Lett. 52, 137 ~1984!. @4# J. Ford, G. Mantica, and G. H. Ristow, Physica D 50, 493 ~1991!; J. Ford and M. Ilg, Phys. Rev. A 45, 6165 ~1992!; J. Ford and G. Mantica, Am. J. Phys. 60, 1086 ~1992!. @5# L. E. Ballentine, Quantum Mechanics ~Prentice Hall, Englewood Cliffs, NJ, 1990!. @6# R. Schack and C. M. Caves, Phys. Rev. Lett. 71, 525 ~1993!. @7# R. Vilela Mendes, Phys. Lett. A 171, 253 ~1992!. @8# W. H. Zurek and J. P. Paz, Phys. Rev. Lett. 72, 2508 ~1994!. @9# R. Blu¨mel, Phys. Rev. Lett. 73, 428 ~1994!. @10# R. Kanno and A. Ishida, J. Phys. Soc. Jpn. 63, 2902 ~1994!. @11# D. Bohm and B. J. Hiley, The Undivided Universe ~Rouletge, London, 1993!; P. R. Holland, The Quantum Theory of Motion ~Cambridge University Press, Cambridge, England, 1993!. @12# L. de Broglie, C. R. Acad. Sci. Paris 183, 447 ~1926!; 184, 273

2061

~a! l converges to a value l 1a very close to zero, which corresponds to regular motion but starts to increase at t'45 to a value l 2a '0.3 corresponding to an exponential divergence of the BB trajectories. In case ~b! we show the value of l until saturation stabilized to a nonzero value of l b '0.5 that corresponds to an exponential separation of the orbits. To conclude, we have demonstrated the exponential separation of neighboring BB orbits in the simple case of the parabolic barrier and we have presented an explicit calculation of de Broglie–Bohm trajectories for a system that classically shows exponential divergences of neighboring trajectories in phase space and chaos, depending on its initial conditions. We have found regular and chaotic motion in the BB quantum theory of motion, applying the same definitions as those applied to classical mechanics. Unlike a classical trajectory, a general BB orbit is influenced by the whole phase-space structure and displays regular or chaotic behavior, depending on the value of time. Thus, a BB orbit can be associated with several Lyapunov-like exponents that distinguish time scales of different stabilities.

@13# @14# @15# @16# @17# @18#

@19# @20# @21# @22#

~1927!; D. Bohm, Phys. Rev. 85, 166 ~1952!; 85, 180 ~1952!. D. Bohm and J.-P. Vigier, Phys. Rev. 96, 208 ~1954!. E. Nelson, Phys. Rev. 150, B1079 ~1966!. D. Bohm, Phys. Rev. 89, 458 ~1953!. A. Valentini, Phys. Lett. A 156, 5 ~1991!; 158, 1 ~1991!. D. Du¨rr, S. Goldstein, and N. Zanghi, Phys. Lett. A 172, 6 ~1992!. A. Carnegie and I. C. Percival, J. Phys. A 17, 801 ~1984!; W.-H. Steeb, C. M. Villet, and A. Kunick, ibid. 18, 3269 ~1985!; P. Dahlquist and G. Russberg, Phys. Rev. Lett. 65, 2837 ~1990!. B. Eckhardt, Phys. Rep. 163, 205 ~1987!. O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 2, 43 ~1993!. B. Eckhardt, G. Hose, and E. Pollak, Phys. Rev. A 39, 3776 ~1989!. G. G. de Polavieja, F. Borondo, and R. M. Benito, Phys. Rev. Lett. 73, 1613 ~1994!.

Exponential divergence of neighboring quantal ...

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