Bohmian trajectory from the “classical” Schrödinger equation Santanu Sengupta, Munmun Khatua, and Pratim Kumar Chattaraj Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 043123 (2014); doi: 10.1063/1.4901034 View online: http://dx.doi.org/10.1063/1.4901034 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The Schrödinger equation with friction from the quantum trajectory perspective J. Chem. Phys. 138, 054107 (2013); 10.1063/1.4788832 Truncation model in the triple-degenerate derivative nonlinear Schrödinger equation Phys. Plasmas 16, 042303 (2009); 10.1063/1.3093394 Chaotic maps derived from trajectory data Chaos 12, 42 (2002); 10.1063/1.1445437 Mixing quantum and classical dynamics using Bohmian trajectories J. Chem. Phys. 113, 9369 (2000); 10.1063/1.1328759 Application of symplectic integrator to stationary reactive-scattering problems: Inhomogeneous Schrödinger equation approach J. Chem. Phys. 106, 4463 (1997); 10.1063/1.473491
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CHAOS 24, 043123 (2014)
€ dinger equation Bohmian trajectory from the “classical” Schro Santanu Sengupta,1 Munmun Khatua,2 and Pratim Kumar Chattaraj2,a) 1
Department of Physics, Veer Surendra Sai University of Technology, Burla 768018, India Department of Chemistry and Center for Theoretical Studies, Indian Institute of Technology, Kharagpur 721302, India 2
(Received 26 June 2014; accepted 23 October 2014; published online 5 November 2014) The quantum-classical correspondence is studied for a periodically driven quartic oscillator exhibiting integrable and chaotic dynamics, by studying the Bohmian trajectory of the corresponding “classical” Schr€odinger equation. Phase plots and the Kolmogorov-Sinai entropy are computed and compared with the classical trajectory as well as the Bohmian trajectory obtained from the time dependent Schr€odinger equation. Bohmian mechanics at the classical limit appears C 2014 AIP Publishing LLC. to mimick the behavior of a dissipative dynamical system. V [http://dx.doi.org/10.1063/1.4901034]
Quantum domain behavior of a classically chaotic oscillator can be understood using Bohmian mechanics. The zero quantum potential limit of the associated quantum trajectories for a periodically driven quartic oscillator is analyzed. It has been observed that the zero quantum potential limit of a Bohmian system does not necessarily exhibit the corresponding classical characteristics.
I. INTRODUCTION
The use of quantum trajectories applying the Bohmian quantum mechanics and its offshoots has now established itself as a bonafide method for studying time evolution of quantum systems as discussed by various investigators.1–6 One of the fundamental appeals of this approach is the possible connection that may exist between the quantum and classical pictures through the associated trajectory concept allowing us to use classical language to describe quantum processes. The fundamental equations of Bohmian mechanics are obtained by using the ansatz iS
W ¼ R exp h ;
(1)
where R and S are real single valued functions in the time dependent Schr€ odinger equation (TDSE) i h
@W h2 2 r W þ VW: ¼ 2m @t
@S 1 h2 r2 R ðrSÞ2 þ V ¼ ; 2m R @t 2m @q 1 ¼ r:ðqrSÞ; @t m
a)
email: [email protected]
1054-1500/2014/24(4)/043123/5/$30.00
q ¼ R2 :
(5)
Equation (3) is what is termed as the Quantum Hamilton Jacobi equation (QHJE) where the usual classical Hamilton Jacobi equation is augmented by the quantum potential h2 r2 R Q ¼ 2m R , while Eq. (4) is interpreted as the equation of continuity for an ensemble of identical noninteracting particles with probability density q. The velocity of these particles at any point is then obtained as v¼
1 rS: m
(6)
Accordingly, the Newton’s equation of motion for an ensemble element can be written as m
dv ¼ rðV þ QÞjr¼rðtÞ : dt
(7)
Van Vleck7 first showed that from a solution of the Hamilton-Jacobi equation one could derive a conserved density in configuration space. This density is the Van Vleck determinant, D, given as 2 @ S D ¼ i k : (8) @q @a D satisfies the equation
(2)
Substitution of Eq. (1) in Eq. (2) and separating the real and imaginary parts, two equations are obtained
where
(9)
@H : vi ¼ @ @S=@qi
(10)
where
(3) (4)
@D @ ð iÞ þ Dv ¼ 0; @t @qi
H is the Hamiltonian and qi , (i ¼ 1, 2, …, N) are the generalized coordinates of the dynamical system. The dependent variable S is known as Hamilton’s principal function. The constants ak are chosen so that the determinant D nowhere vanishes.
24, 043123-1
C 2014 AIP Publishing LLC V
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043123-2
Sengupta, Khatua, and Chattaraj
Chaos 24, 043123 (2014)
Double well systems with Ohmic dissipation have been studied by Dorsey et al.,8 where they have considered a double well potential, which is linearly coupled to a bath of harmonic oscillators. First, the system is coupled to highfrequency oscillators and the second stage consists of coupling the remaining low-frequency oscillators to the modified systems. Gelman and Schwartz9 have also studied dissipative dynamics with the corrected propagator method and tested on model systems like an anharmonic (Morse) oscillator and a double-well potential both coupled to a harmonic bath. Several approaches are followed to derive a quantum theoretical description of dissipation. The two major approaches use either a linear, but time dependent Hamiltonian, or introduce a nonlinear damping potential into the Schr€ odinger equation. To avoid difficulties with nonphysical Hamiltonians, various methods are used to introduce additional friction terms into the linear Schrodinger equation due to which nonlinear Schr€odinger equations are obtained. The first nonlinear approach is due to Kostin10,11 where a nonlinear Schr€odinger-type equation (NLSE) with a logarithmic nonlinearity was prescribed. Several other researchers using different methods12–18 have re-derived this independently or extended by additional friction terms.19 Another type of NLSE’s containing quantum mechanical expectation values has been investigated by Albrecht20 and Hasse and Siissmann.21,22 This formalism has allowed the computation of quantum trajectories and the classification of the systems as regular integrable or chaotic leading to further studies on “quantum chaos.” Perhaps, the most important role played by the idea of quantum potential is to elucidate the topic of quantum and classical correspondence. It is the quantum potential which is responsible for the nonlocal effects of quantum mechanics as seen from the QHJE and discussed in detail by Holland.1 It has been then naturally suggested that the classical limit can be obtained when Q ! 0. The main purpose of this article is to investigate the behavior of Bohmian trajectories in the classical limit of Q ! 0 for a periodically driven quartic oscillator in integrable and chaotic domains. The goal of this paper is to analyze the standard diagnostics of integrability and chaos associated with the driven quartic oscillator in classical and Bohmian domains as well as the corresponding Bohmian dynamics in the classical limit. II. THEORETICAL BACKGROUND AND NUMERICAL DETAILS
An approach to classical mechanics from the TDSE had been introduced by Schiller23 and discussed in this context by Rosen and others,24–26 where a “classical” Schr€odinger equation is obtained by subtracting the quantum potential Q from the TDSE to yield the following classical Schr€odinger equation: i h where
@W h2 2 r W þ Vef f W; ¼ 2m @t
(11)
Vef f ¼ V Q:
(12)
Using the ansatz defined in Eq. (1) as before, we now retrieve the classical Hamilton Jacobi equation
@S 1 ðrSÞ2 þ V; ¼ @t 2m
(13)
while Eq. (7) assumes the form of the usual Newton’s equation of motion m
dv ¼ rðV Þjr¼rðtÞ : dt
(14)
Rosen27 and Berkowitz28 have also explored particular cases of the classical potential when the Bohmian trajectory motion is identical to the classical Newtonian motion and showed that such potentials are quite rare and do not have much physical applications. The introduction of trajectory concepts in the Bohmian quantum mechanics though for an ensemble of identical particles allows one to study the dynamics of classically chaotic systems in the quantum regime. This was first discussed by D€urr29 and investigations have been carried out extensively by various workers.30–35 Specifically, such tools as Lyapunov exponent,33 Kolmogorov-Sinai (KS) entropy,30 surface of section plots,36 power spectra,37 return maps, and bifurcation diagrams38 have been used to characterize the chaos exhibited by the Bohmian trajectories of systems such as the parabolic barrier,39 double well oscillator,40 HenonHeiles oscillator,41 the stadium billiard,42 cubic box,43 and the hydrogen atom in an external field.34 To compute the trajectory, we first obtain the classical wave function from classical Schr€odinger equation through a Peaceman-Rachford finite difference scheme with the Cayley form of the associated unitary operator.44,45 The initial wave packet is taken as 0:25 2 ð xx0 Þ 1 e 2 þik0 x ; (15) Wð x; tÞ ¼ p where the potential parameters are: a ¼ 0.5; b ¼ 10.0; x0 ¼ 6.07 with the amplitude of the driving force g ¼ 10.0 and the particle mass is taken to be 1 a.u. The calculation has been done with a spatial mesh of size Dx ¼ 0.0166667 a.u. and time step Dt ¼ 5.555556 104 and the simulation is run for 52.0 units of time in a spatial grid extending from x ¼ 7.5 to x ¼ þ7.5. After every time step, the Vef f is updated using the value of q at each step. From the “classical” wave function, the velocity field is obtained and the particle position is obtained by solving Eq. (6) to build up the particle trajectory and the corresponding phase plot. As a preliminary physical test of stability, we have solved the “classical” Schrodinger equation where a Gaussian wave packet is moved forward by several time steps and then taken back to its initial position where the original profile is reproduced within the prescribed tolerance limit. In this backward evolution in time, there is no cancellation of errors with those from the forward evolution. Quantum trajectories are obtained for the Bohmian system corresponding to the potential Vcl and compared with
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Sengupta, Khatua, and Chattaraj
Chaos 24, 043123 (2014)
that obtained from the “classical” Schr€odinger equation under the “adjusted” potential Veff discussed above. The initial conditions for the wave packets are given by the position of the center of the packet in position and its initial momentum taken as x0 ¼ 2; p0 ¼ 0 and x0 ¼ þ2; p0 ¼ 0 corresponding to classically integrable and chaotic dynamics, respectively. For each of these phase points, two trajectories with same momentum but differing in position by D ¼ 105 are propagated to compute the phase space distance function defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DðtÞ ¼ ðxðtÞ x0 ðtÞÞ2 þ ðpx ðtÞ p0 x ðtÞÞ2 : (16) A generalized Lyapunov exponent can be defined as 1 D ðt Þ ; ln K ¼ lim Dð0Þ Dð0Þ ! 0 t
(17)
t!1
leading to the introduction of the Kolmogorov-Sinai entropy defined as X h¼ Kþ ; (18) Kþ >0
applied here as a diagnostic tool for chaotic behavior. III. RESULTS AND DISCUSSION
In the present work, we seek a connection between the “classical” trajectories obtained from the trajectories generated using the “classical” wave function as it evolves according to Eq. (6) termed as the “adjusted Bohmian” trajectory and compare with it the Bohmian trajectories obtained from the TDSE for a given system. We consider a one
dimensional double well potential driven by an external periodic force with the classical Hamiltonian having the form p2 H¼ þ ax4 bx2 þ gx cosðx0 tÞ: (19) 2m This driven quartic potential system has been used in various studies related to plasma oscillations,41 inversion of pyramidal molecules like ammonia and phosphine,42 and macroscopic quantum coherence phenomena in SQUIDS.43 This system was also studied to investigate the quantum analogue of Kolmogorov-Arnold-Moser (KAM) transition in field induced barrier penetration using Bohmian trajectories,46 where transition to chaos was observed for a given set of the potential parameters. Here, we look for a connection between the classical trajectories derived as mentioned above using the effective potential Vef f henceforth referred to as the adjusted Bohmian trajectories with the Bohmian trajectories computed using the conventional TDSE. Unlike classical trajectories, quantum trajectories follow the non-crossing rule1 as the gradient of S is a single-valued function of position. The quantum potential prevents crossing of the quantum trajectories. In the present work, we have checked the non-crossing behavior or otherwise for the classical, Bohmian, and adjusted Bohmian trajectories all launched from the same initial positions. The classical trajectories show the crossing phenomenon, whereas the Bohmian trajectories do not cross at all. The adjusted Bohmian trajectories, however, come closer to each other than the Bohmian trajectories at certain times and then again separate. The adjusted Bohmian trajectories also do not cross for the studied systems. The phase plots of the Bohmian and adjusted Bohmian trajectories are compared with that obtained from classical dynamics in Figure 1 for two sets of initial phase points
FIG. 1. Comparison between phase plots for classical, Bohmian, and adjusted Bohmian trajectories; the upper panel corresponds to classically integrable domain, while the lower panel depicts the same for classically chaotic regime.
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043123-4
Sengupta, Khatua, and Chattaraj
FIG. 2. Comparison between Kolmogorov-Sinai entropy for classical, Bohmian, and adjusted Bohmian trajectories for classically integrable and chaotic domains.
(x0 ¼ 2; p0 ¼ 0); (x0 ¼ þ2; p0 ¼ 0) for the amplitude of the driving force g ¼ 10.0. The classical dynamics for the initial phase point (x0 ¼ 2; p0 ¼ 0) is integrable. A spear-head like structure is observed in the phase space. The classical dynamics corresponds to a regular island for the phase point (x0 ¼ 2; p0 ¼ 0). The phase plot for the Bohmian trajectory reflects the underlying classical dynamics, i.e., a regular dynamics for the initial phase point (x0 ¼ 2; p0 ¼ 0). It appears as a cantorus-like structure which is the quantum equivalent of the classical torus. However, the adjusted Bohmian trajectory while more localised in phase space is also less toroidal in nature. For the phase point (x0 ¼ þ2;
Chaos 24, 043123 (2014)
p0 ¼ 0) corresponding to classically chaotic regime, the Bohmian quantum trajectory as well as the adjusted Bohmian trajectory exhibit non integrable features. The classical trajectory shows a chaotic sea in the phase space. The Bohmian trajectory mimicks the classical behavior and also shows a chaotic sea for the phase point (x0 ¼ þ2; p0 ¼ 0). However, the adjusted Bohmian trajectory is restricted to its original potential well and does not cross back and forth between the two wells like the Bohmian or the classical trajectory. The major difference between “classical” and “adjusted Bohmian” trajectories might stem from the fact that momentum is not an independent variable in the latter case and the density conservation is constrained to configuration space and not to phase space. It is interesting to note that the trajectories generated from the “classical” Schr€odinger equation akin to a dissipative dynamical system seem to fall down and move around one of the potential wells. The Kolmogorov Sinai entropy plots are shown in Figure 2, where in the classically integrable regime, the Bohmian quantum trajectory and the classical trajectory exhibit similar lower values of the KS entropy, while the KS entropy value for the adjusted Bohmian trajectory is high. For the classically chaotic domain, the KS entropy plots depict a consistently high value for the adjusted Bohmian
FIG. 3. Comparison between phase plots for classical, Bohmian, and adjusted Bohmian trajectories for a driven quartic oscillator with other initial conditions. For details, see caption of Fig. 1.
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043123-5
Sengupta, Khatua, and Chattaraj
Chaos 24, 043123 (2014)
FIG. 4. Comparison between Kolmogorov-Sinai entropy for classical, Bohmian, and adjusted Bohmian trajectories for a driven quartic oscillator with other initial conditions. For details, see caption of Fig. 2.
trajectory compared to the other two cases. The KS entropy, h, measures the rate of the loss of information for predicting the future course of the trajectory. A dynamical system is said to be chaotic if KS entropy is positive definite. It may, however, be noted that the orders of magnitude of two types of Bohmian trajectories are same, which is different from that of the respective classical trajectory. Figures 3 and 4 present the plots of corresponding diagnostics for different initial conditions, viz., (x0 ¼ 1, p0 ¼ 0), (x0 ¼ 0, p0 ¼ 0), and (x0 ¼ þ1, p0 ¼ 0) for classical, Bohmian, and adjusted Bohmian dynamics encompassing regular and chaotic behavior. These results mimick the same obtained in the above mentioned cases. For the initial phase point (x0 ¼ 1, p0 ¼ 0), the classical trajectory shows integrable features just like that for the initial phase point (x0 ¼ 2, p0 ¼ 0). The Bohmian trajectory shows a cantorus-like structure, which is slightly space filling for the initial phase point (x0 ¼ 1, p0 ¼ 0). For the initial phase points (x0 ¼ 0, p0 ¼ 0) and (x0 ¼ þ1, p0 ¼ 0), the classical as well as the Bohmian trajectory mimick the chaotic dynamics as seen before for the initial phase point (x0 ¼ þ2, p0 ¼ 0). However, the adjusted Bohmian trajectory is more localized in the phase space for the initial phase points (x0 ¼ 1, p0 ¼ 0), (x0 ¼ 0, p0 ¼ 0), and (x0 ¼ þ1, p0 ¼ 0). The KS entropy plots are similar to that observed earlier. Since all initial conditions studied give rise to similar trends in the adjusted Bohmian trajectories, the observed phenomena call for a generalization. IV. CONCLUSION
It appears that the “classical” trajectories obtained here from the “classical” Schr€odinger equation do not exhibit any qualitative similarity with the classical mechanical trajectories and in fact the Bohmian quantum trajectories appear to reflect the dynamical aspects of the classical hamiltonian better albeit with a different order of magnitude. One possible reason might be the fact that the classical Schr€ odinger equation is nonlinear and the solution obtained is a nonlinear superposition of classical solutions as alluded to by Rosen.24 ACKNOWLEDGMENTS
M.K. thanks CSIR, New Delhi for financial assistance. P.K.C. thanks DST, New Delhi for the J. C. Bose National Fellowship. 1
P. Holland, The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993). 2 R. Wyatt, Quantum Dynamics with Trajectories (Springer, 2000).
3
D. D€ urr and S. Teufel, Bohmian Mechanics—The Physics and Mathematics of Quantum Theory (Springer, 2009). 4 P. K. Chattaraj, Quantum Trajectories (CRC Press, Florida, 2010). 5 A. S. Sanz and S. Miret-Artes, A Trajectory Description of Quantum Processes. I. Fundamentals: A Bohmian Perspective (Springer, 2012). 6 J. M. X. Oriols and I. Cirac, Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology (Pan Stanford Publishing, Singapore, 2012). 7 J. H. Van Vleck, Proc. Natl. Acad. Sci. U.S.A. 14, 178 (1928). 8 A. T. Dorsey, M. P. A. Fisher, and M. S. Wartak, Phys. Rev. A 33, 1117 (1986). 9 D. Gelman and S. D. Schwartz, Chem. Phys. 370, 62 (2010). 10 M. D. Kostin, J. Chem. Phys. 57, 3589 (1972). 11 M. D. Kostin, J. Stat. Phys. 12, 145 (1975). 12 B. K. Skagerstam, Phys. Lett. B 58, 21 (1975). 13 B. K. Skagerstam, J. Math. Phys. 18, 308 (1977). 14 K. Yasue, Phys. Lett. B 64, 239 (1976). 15 K. Yasue, J. Stat. Phys. 16, 113 (1977). 16 K. Yasue, Ann. Phys. (N.Y.) 114, 479 (1978). 17 K. K. Kan and J. J. Griffin, Phys. Lett. B 50, 241 (1974). 18 J. D. Immele, K. K. Kan, and J. J. Griffin, Nucl. Phys. A 241, 47 (1975). 19 B. R. Cho and S. G. Mulri, Math. Phys. (Korean) 3, 37 (1980). 20 K. Albrecht, Phys. Lett. B 56, 127 (1975). 21 R. W. Hasse, J. Math. Phys. 16, 2005 (1975). 22 R. W. Hasse, Rep. Prog. Phys. 41, 1027 (1978). 23 R. Schiller, Phys. Rev. 125, 1100 (1962). 24 N. Rosen, Am. J. Phys. 32, 597 (1964). 25 N. Rosen, Found. Phys. 16, 687 (1986). 26 S. Sengupta, M. Khatua, and P. K. Chattaraj, “A quantum potential based density functional treatment of the quantum signature of classical nonintegrability,” in Theoretical and Computational Developments in Modern Density Functional Theory, edited by A. K. Roy (Nova Science Publishers, NY, USA, 2012). 27 N. Rosen, Am. J. Phys. 32, 377 (1964). 28 D. B. Berkowitz and P. D. Skiff, Am. J. Phys. 40, 1625 (1972). 29 D. Durr, S. Goldstein, and N. Zanghi, J. Stat. Phys. 68, 259 (1992). 30 U. Schwengelbeck and F. M. Faisal, Phys. Lett. A 199, 281 (1995). 31 R. Parmenter and R. W. Valentine, Phys. Lett. A 201, 1 (1995). 32 F. M. Faisal and U. Schwengelbeck, Phys. Lett. A 207, 31 (1995). 33 S. Sengupta and P. K. Chattaraj, Phys. Lett. A 215, 119 (1996). 34 G. Iacomelli and M. Pettini, Phys. Lett. A 212, 29 (1996). 35 R. Parmenter and R. W. Valentine, Phys. Lett. A 219, 7 (1996). 36 H. Frisk, Phys. Lett. A 227, 139 (1997). 37 O. de Alcantara Bonfim, J. Florencio, and F. S. Barreto, Phys. Rev. E 58, 6851 (1998). 38 P. K. Chattaraj, S. Sengupta, and A. Poddar, Curr. Sci. 74, 758 (1998). Available at http://www.currentscience.ac.in/Downloads/article_id_074_ 09_0758_0764_0.pdf. 39 G. de Polavieja, Phys. Rev. A 53, 2059 (1996). 40 D. A. Wisniacki, F. Borondo, and R. Benito, Europhys. Lett. 64, 441 (2003). 41 J. Guckenheimer and P. Holmes, Nonlinear Oscillations: Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, New York, 2012). 42 F. Hund, Z. Phys. 43, 805 (1927). 43 J. Kondo, Physica B 125, 279 (1984). 44 A. Goldberg, H. M. Schey, and J. L. Schwartz, Am. J. Phys. 35, 177 (1967). 45 I. Galbraith, Y. S. Ching, and E. Abraham, Am. J. Phys. 52, 60 (1984). 46 P. K. Chattaraj, S. Sengupta, B. Maiti, and U. Sarkar, Curr. Sci. 82, 541 (2002). Available at http://www.currentscience.ac.in/Downloads/article_ id_082_05_0541_0547_0.pdf.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 130.113.86.233 On: Tue, 06 Jan 2015 16:39:32
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CHAOS 24, 043123 (2014)
€ dinger equation Bohmian trajectory from the “classical” Schro Santanu Sengupta,1 Munmun Khatua,2 and Pratim Kumar Chattaraj2,a) 1
Department of Physics, Veer Surendra Sai University of Technology, Burla 768018, India Department of Chemistry and Center for Theoretical Studies, Indian Institute of Technology, Kharagpur 721302, India 2
(Received 26 June 2014; accepted 23 October 2014; published online 5 November 2014) The quantum-classical correspondence is studied for a periodically driven quartic oscillator exhibiting integrable and chaotic dynamics, by studying the Bohmian trajectory of the corresponding “classical” Schr€odinger equation. Phase plots and the Kolmogorov-Sinai entropy are computed and compared with the classical trajectory as well as the Bohmian trajectory obtained from the time dependent Schr€odinger equation. Bohmian mechanics at the classical limit appears C 2014 AIP Publishing LLC. to mimick the behavior of a dissipative dynamical system. V [http://dx.doi.org/10.1063/1.4901034]
Quantum domain behavior of a classically chaotic oscillator can be understood using Bohmian mechanics. The zero quantum potential limit of the associated quantum trajectories for a periodically driven quartic oscillator is analyzed. It has been observed that the zero quantum potential limit of a Bohmian system does not necessarily exhibit the corresponding classical characteristics.
I. INTRODUCTION
The use of quantum trajectories applying the Bohmian quantum mechanics and its offshoots has now established itself as a bonafide method for studying time evolution of quantum systems as discussed by various investigators.1–6 One of the fundamental appeals of this approach is the possible connection that may exist between the quantum and classical pictures through the associated trajectory concept allowing us to use classical language to describe quantum processes. The fundamental equations of Bohmian mechanics are obtained by using the ansatz iS
W ¼ R exp h ;
(1)
where R and S are real single valued functions in the time dependent Schr€ odinger equation (TDSE) i h
@W h2 2 r W þ VW: ¼ 2m @t
@S 1 h2 r2 R ðrSÞ2 þ V ¼ ; 2m R @t 2m @q 1 ¼ r:ðqrSÞ; @t m
a)
email: [email protected]
1054-1500/2014/24(4)/043123/5/$30.00
q ¼ R2 :
(5)
Equation (3) is what is termed as the Quantum Hamilton Jacobi equation (QHJE) where the usual classical Hamilton Jacobi equation is augmented by the quantum potential h2 r2 R Q ¼ 2m R , while Eq. (4) is interpreted as the equation of continuity for an ensemble of identical noninteracting particles with probability density q. The velocity of these particles at any point is then obtained as v¼
1 rS: m
(6)
Accordingly, the Newton’s equation of motion for an ensemble element can be written as m
dv ¼ rðV þ QÞjr¼rðtÞ : dt
(7)
Van Vleck7 first showed that from a solution of the Hamilton-Jacobi equation one could derive a conserved density in configuration space. This density is the Van Vleck determinant, D, given as 2 @ S D ¼ i k : (8) @q @a D satisfies the equation
(2)
Substitution of Eq. (1) in Eq. (2) and separating the real and imaginary parts, two equations are obtained
where
(9)
@H : vi ¼ @ @S=@qi
(10)
where
(3) (4)
@D @ ð iÞ þ Dv ¼ 0; @t @qi
H is the Hamiltonian and qi , (i ¼ 1, 2, …, N) are the generalized coordinates of the dynamical system. The dependent variable S is known as Hamilton’s principal function. The constants ak are chosen so that the determinant D nowhere vanishes.
24, 043123-1
C 2014 AIP Publishing LLC V
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043123-2
Sengupta, Khatua, and Chattaraj
Chaos 24, 043123 (2014)
Double well systems with Ohmic dissipation have been studied by Dorsey et al.,8 where they have considered a double well potential, which is linearly coupled to a bath of harmonic oscillators. First, the system is coupled to highfrequency oscillators and the second stage consists of coupling the remaining low-frequency oscillators to the modified systems. Gelman and Schwartz9 have also studied dissipative dynamics with the corrected propagator method and tested on model systems like an anharmonic (Morse) oscillator and a double-well potential both coupled to a harmonic bath. Several approaches are followed to derive a quantum theoretical description of dissipation. The two major approaches use either a linear, but time dependent Hamiltonian, or introduce a nonlinear damping potential into the Schr€ odinger equation. To avoid difficulties with nonphysical Hamiltonians, various methods are used to introduce additional friction terms into the linear Schrodinger equation due to which nonlinear Schr€odinger equations are obtained. The first nonlinear approach is due to Kostin10,11 where a nonlinear Schr€odinger-type equation (NLSE) with a logarithmic nonlinearity was prescribed. Several other researchers using different methods12–18 have re-derived this independently or extended by additional friction terms.19 Another type of NLSE’s containing quantum mechanical expectation values has been investigated by Albrecht20 and Hasse and Siissmann.21,22 This formalism has allowed the computation of quantum trajectories and the classification of the systems as regular integrable or chaotic leading to further studies on “quantum chaos.” Perhaps, the most important role played by the idea of quantum potential is to elucidate the topic of quantum and classical correspondence. It is the quantum potential which is responsible for the nonlocal effects of quantum mechanics as seen from the QHJE and discussed in detail by Holland.1 It has been then naturally suggested that the classical limit can be obtained when Q ! 0. The main purpose of this article is to investigate the behavior of Bohmian trajectories in the classical limit of Q ! 0 for a periodically driven quartic oscillator in integrable and chaotic domains. The goal of this paper is to analyze the standard diagnostics of integrability and chaos associated with the driven quartic oscillator in classical and Bohmian domains as well as the corresponding Bohmian dynamics in the classical limit. II. THEORETICAL BACKGROUND AND NUMERICAL DETAILS
An approach to classical mechanics from the TDSE had been introduced by Schiller23 and discussed in this context by Rosen and others,24–26 where a “classical” Schr€odinger equation is obtained by subtracting the quantum potential Q from the TDSE to yield the following classical Schr€odinger equation: i h where
@W h2 2 r W þ Vef f W; ¼ 2m @t
(11)
Vef f ¼ V Q:
(12)
Using the ansatz defined in Eq. (1) as before, we now retrieve the classical Hamilton Jacobi equation
@S 1 ðrSÞ2 þ V; ¼ @t 2m
(13)
while Eq. (7) assumes the form of the usual Newton’s equation of motion m
dv ¼ rðV Þjr¼rðtÞ : dt
(14)
Rosen27 and Berkowitz28 have also explored particular cases of the classical potential when the Bohmian trajectory motion is identical to the classical Newtonian motion and showed that such potentials are quite rare and do not have much physical applications. The introduction of trajectory concepts in the Bohmian quantum mechanics though for an ensemble of identical particles allows one to study the dynamics of classically chaotic systems in the quantum regime. This was first discussed by D€urr29 and investigations have been carried out extensively by various workers.30–35 Specifically, such tools as Lyapunov exponent,33 Kolmogorov-Sinai (KS) entropy,30 surface of section plots,36 power spectra,37 return maps, and bifurcation diagrams38 have been used to characterize the chaos exhibited by the Bohmian trajectories of systems such as the parabolic barrier,39 double well oscillator,40 HenonHeiles oscillator,41 the stadium billiard,42 cubic box,43 and the hydrogen atom in an external field.34 To compute the trajectory, we first obtain the classical wave function from classical Schr€odinger equation through a Peaceman-Rachford finite difference scheme with the Cayley form of the associated unitary operator.44,45 The initial wave packet is taken as 0:25 2 ð xx0 Þ 1 e 2 þik0 x ; (15) Wð x; tÞ ¼ p where the potential parameters are: a ¼ 0.5; b ¼ 10.0; x0 ¼ 6.07 with the amplitude of the driving force g ¼ 10.0 and the particle mass is taken to be 1 a.u. The calculation has been done with a spatial mesh of size Dx ¼ 0.0166667 a.u. and time step Dt ¼ 5.555556 104 and the simulation is run for 52.0 units of time in a spatial grid extending from x ¼ 7.5 to x ¼ þ7.5. After every time step, the Vef f is updated using the value of q at each step. From the “classical” wave function, the velocity field is obtained and the particle position is obtained by solving Eq. (6) to build up the particle trajectory and the corresponding phase plot. As a preliminary physical test of stability, we have solved the “classical” Schrodinger equation where a Gaussian wave packet is moved forward by several time steps and then taken back to its initial position where the original profile is reproduced within the prescribed tolerance limit. In this backward evolution in time, there is no cancellation of errors with those from the forward evolution. Quantum trajectories are obtained for the Bohmian system corresponding to the potential Vcl and compared with
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that obtained from the “classical” Schr€odinger equation under the “adjusted” potential Veff discussed above. The initial conditions for the wave packets are given by the position of the center of the packet in position and its initial momentum taken as x0 ¼ 2; p0 ¼ 0 and x0 ¼ þ2; p0 ¼ 0 corresponding to classically integrable and chaotic dynamics, respectively. For each of these phase points, two trajectories with same momentum but differing in position by D ¼ 105 are propagated to compute the phase space distance function defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DðtÞ ¼ ðxðtÞ x0 ðtÞÞ2 þ ðpx ðtÞ p0 x ðtÞÞ2 : (16) A generalized Lyapunov exponent can be defined as 1 D ðt Þ ; ln K ¼ lim Dð0Þ Dð0Þ ! 0 t
(17)
t!1
leading to the introduction of the Kolmogorov-Sinai entropy defined as X h¼ Kþ ; (18) Kþ >0
applied here as a diagnostic tool for chaotic behavior. III. RESULTS AND DISCUSSION
In the present work, we seek a connection between the “classical” trajectories obtained from the trajectories generated using the “classical” wave function as it evolves according to Eq. (6) termed as the “adjusted Bohmian” trajectory and compare with it the Bohmian trajectories obtained from the TDSE for a given system. We consider a one
dimensional double well potential driven by an external periodic force with the classical Hamiltonian having the form p2 H¼ þ ax4 bx2 þ gx cosðx0 tÞ: (19) 2m This driven quartic potential system has been used in various studies related to plasma oscillations,41 inversion of pyramidal molecules like ammonia and phosphine,42 and macroscopic quantum coherence phenomena in SQUIDS.43 This system was also studied to investigate the quantum analogue of Kolmogorov-Arnold-Moser (KAM) transition in field induced barrier penetration using Bohmian trajectories,46 where transition to chaos was observed for a given set of the potential parameters. Here, we look for a connection between the classical trajectories derived as mentioned above using the effective potential Vef f henceforth referred to as the adjusted Bohmian trajectories with the Bohmian trajectories computed using the conventional TDSE. Unlike classical trajectories, quantum trajectories follow the non-crossing rule1 as the gradient of S is a single-valued function of position. The quantum potential prevents crossing of the quantum trajectories. In the present work, we have checked the non-crossing behavior or otherwise for the classical, Bohmian, and adjusted Bohmian trajectories all launched from the same initial positions. The classical trajectories show the crossing phenomenon, whereas the Bohmian trajectories do not cross at all. The adjusted Bohmian trajectories, however, come closer to each other than the Bohmian trajectories at certain times and then again separate. The adjusted Bohmian trajectories also do not cross for the studied systems. The phase plots of the Bohmian and adjusted Bohmian trajectories are compared with that obtained from classical dynamics in Figure 1 for two sets of initial phase points
FIG. 1. Comparison between phase plots for classical, Bohmian, and adjusted Bohmian trajectories; the upper panel corresponds to classically integrable domain, while the lower panel depicts the same for classically chaotic regime.
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FIG. 2. Comparison between Kolmogorov-Sinai entropy for classical, Bohmian, and adjusted Bohmian trajectories for classically integrable and chaotic domains.
(x0 ¼ 2; p0 ¼ 0); (x0 ¼ þ2; p0 ¼ 0) for the amplitude of the driving force g ¼ 10.0. The classical dynamics for the initial phase point (x0 ¼ 2; p0 ¼ 0) is integrable. A spear-head like structure is observed in the phase space. The classical dynamics corresponds to a regular island for the phase point (x0 ¼ 2; p0 ¼ 0). The phase plot for the Bohmian trajectory reflects the underlying classical dynamics, i.e., a regular dynamics for the initial phase point (x0 ¼ 2; p0 ¼ 0). It appears as a cantorus-like structure which is the quantum equivalent of the classical torus. However, the adjusted Bohmian trajectory while more localised in phase space is also less toroidal in nature. For the phase point (x0 ¼ þ2;
Chaos 24, 043123 (2014)
p0 ¼ 0) corresponding to classically chaotic regime, the Bohmian quantum trajectory as well as the adjusted Bohmian trajectory exhibit non integrable features. The classical trajectory shows a chaotic sea in the phase space. The Bohmian trajectory mimicks the classical behavior and also shows a chaotic sea for the phase point (x0 ¼ þ2; p0 ¼ 0). However, the adjusted Bohmian trajectory is restricted to its original potential well and does not cross back and forth between the two wells like the Bohmian or the classical trajectory. The major difference between “classical” and “adjusted Bohmian” trajectories might stem from the fact that momentum is not an independent variable in the latter case and the density conservation is constrained to configuration space and not to phase space. It is interesting to note that the trajectories generated from the “classical” Schr€odinger equation akin to a dissipative dynamical system seem to fall down and move around one of the potential wells. The Kolmogorov Sinai entropy plots are shown in Figure 2, where in the classically integrable regime, the Bohmian quantum trajectory and the classical trajectory exhibit similar lower values of the KS entropy, while the KS entropy value for the adjusted Bohmian trajectory is high. For the classically chaotic domain, the KS entropy plots depict a consistently high value for the adjusted Bohmian
FIG. 3. Comparison between phase plots for classical, Bohmian, and adjusted Bohmian trajectories for a driven quartic oscillator with other initial conditions. For details, see caption of Fig. 1.
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Chaos 24, 043123 (2014)
FIG. 4. Comparison between Kolmogorov-Sinai entropy for classical, Bohmian, and adjusted Bohmian trajectories for a driven quartic oscillator with other initial conditions. For details, see caption of Fig. 2.
trajectory compared to the other two cases. The KS entropy, h, measures the rate of the loss of information for predicting the future course of the trajectory. A dynamical system is said to be chaotic if KS entropy is positive definite. It may, however, be noted that the orders of magnitude of two types of Bohmian trajectories are same, which is different from that of the respective classical trajectory. Figures 3 and 4 present the plots of corresponding diagnostics for different initial conditions, viz., (x0 ¼ 1, p0 ¼ 0), (x0 ¼ 0, p0 ¼ 0), and (x0 ¼ þ1, p0 ¼ 0) for classical, Bohmian, and adjusted Bohmian dynamics encompassing regular and chaotic behavior. These results mimick the same obtained in the above mentioned cases. For the initial phase point (x0 ¼ 1, p0 ¼ 0), the classical trajectory shows integrable features just like that for the initial phase point (x0 ¼ 2, p0 ¼ 0). The Bohmian trajectory shows a cantorus-like structure, which is slightly space filling for the initial phase point (x0 ¼ 1, p0 ¼ 0). For the initial phase points (x0 ¼ 0, p0 ¼ 0) and (x0 ¼ þ1, p0 ¼ 0), the classical as well as the Bohmian trajectory mimick the chaotic dynamics as seen before for the initial phase point (x0 ¼ þ2, p0 ¼ 0). However, the adjusted Bohmian trajectory is more localized in the phase space for the initial phase points (x0 ¼ 1, p0 ¼ 0), (x0 ¼ 0, p0 ¼ 0), and (x0 ¼ þ1, p0 ¼ 0). The KS entropy plots are similar to that observed earlier. Since all initial conditions studied give rise to similar trends in the adjusted Bohmian trajectories, the observed phenomena call for a generalization. IV. CONCLUSION
It appears that the “classical” trajectories obtained here from the “classical” Schr€odinger equation do not exhibit any qualitative similarity with the classical mechanical trajectories and in fact the Bohmian quantum trajectories appear to reflect the dynamical aspects of the classical hamiltonian better albeit with a different order of magnitude. One possible reason might be the fact that the classical Schr€ odinger equation is nonlinear and the solution obtained is a nonlinear superposition of classical solutions as alluded to by Rosen.24 ACKNOWLEDGMENTS
M.K. thanks CSIR, New Delhi for financial assistance. P.K.C. thanks DST, New Delhi for the J. C. Bose National Fellowship. 1
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