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Bohmian quantum trajectories in a square billiard in the bouncing ball regime
Physica A 290 (2001) 101–106
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Bohmian quantum trajectories in a square billiard in the bouncing ball regime J.A. de Sales, J. Florencio ∗ Departamento de FÃsica, Universidade Federal de Minas Gerais, 30.123-970 Belo Horizonte, MG, Brazil Received 20 March 2000; received in revised form 24 July 2000
Abstract We have studied the behavior of the Gaussian wave packet in the 2-D square billiard using Bohmian quantum mechanics. We found non-chaotic ballistic trajectories for particles with high energies. The most important aspect of these trajectories is that upon collision they obey c 2001 Elsevier Science B.V. the laws of re ection in spite of curved paths near the walls. All rights reserved. PACS: 05.45.+b; 03.65.Ge; 03.40.Kf Keywords: Bohmian quantum mechanics; Quantum chaos; Billiards
The 2-D square billiard is one of the simplest billiards. It is completely integrable in classical mechanics: there are two constants of motion, energy and square of momentum. In conventional quantum mechanics, however, one does not have trajectories like in classical mechanics and the dynamic behavior in billiards is frequently analyzed by using spectral statistics [1]. Random matrix theory [2,3] and other auxiliary theories, like periodic orbits theories in the semiclassical regime [4] have been used to study chaos in conventional quantum mechanics. In square billiards, the energy levels are uncorrelated and the energy spacing distribution is Poisson-like, which indicates regular motion, according to random matrix theory. The famous Bohigas–Gianinni–Schmit conjecture [5] is very often employed to characterize chaos, but there are some examples in the literature which show its breakdown, namely, the hydrogen atom in a magnetic eld and a two-dimensional quartic oscillator [6]. The fact is that there is no general consensus about the signature of chaos in quantum mechanics [4]. ∗ Corresponding author. E-mail address: jfj@ sica.ufmg.br (J. Florencio).
c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 4 9 3 - 3
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J.A. de Sales, J. Florencio / Physica A 290 (2001) 101–106
Within the framework of Bohm’s quantum mechanics [7–9], the square billiard can show chaos [10] depending on the initial position of the particle when an initial wave packet is built from a nite number of Hamiltonian eigenfunctions. Chaotic behavior can be expected in that case because the particle is scattered at the nodal lines in x–y plane. In instances where the motion is quasiperiodic, the trajectories are Lissajous-like pictures. On the other hand, when the motion is chaotic the trajectories are quite irregular. Thus, both chaotic and semi-periodic regimes can be found in that billiard. According to Bohr’s correspondence principle, the classical limit of a quantum system should be achieved when the quantum numbers go to in nity. However, in a square box that is questionable, because there is always a great number of nodal lines when the energy of the system is large. On the other hand, in Bohm’s theory those lines are regions where the velocity is not well-de ned. In order to study the semiclassical regime of Bohmian trajectories in the square billiard, we shall consider the case of an initial Gaussian wave packet, since it does not have any nodes within the billiard. To perform this, we use a sharp packet localized far from the scatterer. This case is similar to those studied by Dewdney and Hiley [11] where a square barrier was used like a scatterer for a Gaussian packet. Another reason to use Gaussian packets is that we are interested in ballistic trajectories in the billiard domain, then we need a particle that is initially free, i.e., having negligible contact with the walls. The plane wave description is not good for this because the particle is initially not free, its wave function carries all information of the external scattering potential throughout space [9]. This is the principal reason to use the Gaussian wave packet instead of plane waves when we want to describe scattering phenomena, like tunnel e ect [11] and double-slit experiments [12]. As we shall see, such a choice seems to be the best representation for the particle state at high energies. We nd that away from the boundaries the trajectories are straight lines, as expected for motions governed by Gaussian wave packets. However, near the walls the quantum potential modi es the particle’s trajectories which become curved. Since the in uence of the quantum potential diminishes as the particle moves away from the collision region, the trajectories become straight lines. The most striking feature of these trajectories is that the straight segments before and after collision obey the re ection laws of classical billiards. The dynamics of the system is described in Bohm’s theory by writing down the wave function (x; y; t) in polar coordinates, i.e., = R exp(−iS=˝) and inserting it in the Schrodinger equation. The amplitude R and the phase S are separated into two equations, which after some simple algebraic manipulations become ∇S @ =0; (1) +∇ @t M (∇S)2 @S + +V +Q=0; (2) @t 2M where = R 2 is the probability density, V is the ordinary quantized potential, M is the mass of the particle, and Q = −˝2 =2M ∇2 R=R is the quantum potential. The rst equation is simply the conservation of probability ux, whereas the second equation is
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a generalized Hamilton–Jacobi equation, where the conventional potential is replaced by an e ective potential Ve = V + Q. In Bohm’s theory, an additional postulate is introduced, which de nes ∇S as the momentum of the particle. The velocity of the particle can then be given in terms of the wave function as ˝ (3) ( ∗ ∇ − ∇ ∗ )=( ∗ ) : v(x; y; t) = 2M i The position and velocity of the particle are well-de ned at all times provided the initial position is not lying on a node. The particle now moves governed by a modi ed Newton’s second law d2 r (4) M 2 = −∇(V + Q)|r=r(t) : dt In our study, this equation is integrated by using a fourth-order Runge–Kutta method [13] with the initial conditions x(t = 0) and v(t = 0) = v0 , where the velocity v0 is pre-de ned in Bohm’s equation and depends on the initial position and initial wave packet of the particle in the billiard’s domain. We also need to know the wave function (x; y; t) at all times. We use a suitably sharp Gaussian packet to prevent any appreciable amplitude at the borders of the billiard. It is given by Y 1 2 exp(ik0 − | − 0 |2 =20 ); (5) (x; y; 0) = 2 (0 )1=4 =x; y
for which the mean energy is 2 2 Z 1 1 −˝ ∇ ∗ 2 dr = k0x2 + k0y + 2 + 2 hEi = 2M 20x 20y
(6)
in a system of units where ˝2 = 2M = L = 1. Here 0x and 0y are the initial widths of the wave packets along the x and y directions, respectively. The quantities k0x and k0y are the average momenta along those directions, and x0 and y0 are the coordinates of the center of the initial wave packet. The evaluation of the time-dependent wave function in this system is not trivial and was calculated using the unconditionally stable algorithm of De Raedt [14,15]. In this method, the wave function norm is conserved with good precision; this fact is very important because of the two simultaneous integration processes needed to obtain the Bohmian trajectories. The Bohm’s trajectories are obtained from numerical integration of Eq. (3) using the Runge–Kutta method, where the derivative of the wave function is replaced by a ve-point nite-di erence approximation. The billiard area was discretized by a 500×500 lattice. We performed extensive calculations to determine the Bohmian trajectories for several initial Gaussian wave packets and initial positions. The most important features of our results can be seen in Figs. 1– 4, which are discussed in the following. 2 2 + 0y = Fig. 1 shows three quantum trajectories for Gaussian packets with 02 = 0x −4 9×10 starting at distinct positions with the particle localized at the center of the
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Fig. 1. Quantum trajectories corresponding to an initial Gaussian wave packet. The wave packet parameters are k0x = 80; k0y = 100; and 02 = 9×10−4 . We used a 500×500 integration mesh. The initial positions are (x0 ; y0 ) = (0:8; 0:7), (0:6; 0:8), and (0:1; 0:8).
Fig. 2. Same as in Fig. 1, but with 02 = 6:4×10−6 . Notice the sharper bounces o the walls, as compared with those of Fig. 1.
wave packet. We can note that away from the borders, the trajectories are ballistic. Near the walls, however, they curve, and then emerge following the laws of re ection. Note that these are results straight from numerical integration, and no bias was used. The particle never hits the billiard walls, where the quantum potential is very strong. Fig. 2 uses the same parameters as in Fig. 1, except that now the initial packet is narrower, whose, 02 = 6:4×10−6 , that is, a packet with higher mean energy. In this case, the trajectories resemble the classical regime even more. The trajectories now
J.A. de Sales, J. Florencio / Physica A 290 (2001) 101–106
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Fig. 3. Several quantum trajectories corresponding to initial Gaussian wave packets starting from initial ◦ position (x0 ; y0 ) = (0:6; 0:8). The wave vector is k = 100, and the angle of incidence is 60 , with 0 = 0:2; 0:1; 0:08; 0:06; 0:03. We used a 500×500 integration mesh. The wiggled curves are an artifact of the numerical integration: smooth curves could be obtained with more re ned (time-consuming) integration steps.
Fig. 4. Minimum distance of the trajectory to the wall rmin versus the width of the Gaussian wave packet. Notice the linear relationship between rmin and . Here, rmin ( = 0) = 0:001 ± 0:001 and slope = 0:36 ± 0:01. We used a 500×500 integration mesh.
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have a smaller radius of curvature in the collision region. We expect that this e ect will be most pronounced when the width 0 → 0; that is, when the wave packet is a delta-function. One should keep in mind that the system is still in a quantum regime. The Schrodinger equation and all quantum formalisms still apply; however, the trajectories show prominent classical features. The classical features are more evident when the Gaussian wave packet becomes sharper. In Fig. 3, we show several trajectories starting in the same position with di erent widths. We can note that the trajectories with higher mean energy approach the wall more closely, with smaller radius of curvature. The behavior of the radius of curvature with the width of the Gaussian packet is shown in Fig. 4. One can infer from this gure that a true classical regime can only be reached when the width of the Gaussian packet is much smaller than the sides of the wall, with the quantum trajectory hitting the wall. The minimum distance rmin to the wall turns out to be a linear function of the width of the packet and goes to zero when goes also to zero. In summary, the Bohmian trajectories in a square billiard do not exhibit chaos when we use Gaussian wave packets. In the cases studied, we found a bouncing ball regime for high energies. The particle’s quantum trajectories are very close to those in the classical regime when the energy of Gaussian wave packet is large. The angles of re ection and incidence for the particle were obtained as outcomes of the numerical integration of the equation of motion of the Bohmian particle. We would like to thank O.F. de Alcantara Bon m and F.C. Sa Barreto for fruitful discussions. This work was partially supported by CNPq, FINEP, FAPEMIG and MCT (Brazilian agencies). References [1] S.W. McDonald, A.N. Kaufman, Phys. Rev. Lett. 42 (1979) 1189. [2] M.L. Mehta, Random Matrix Theory, Academic Press, New York, 1991. [3] T. Guhr, A. Muller-Groeling, H.A. Weidenmuller, Random-Matrix Theories in Quantum Physics: Common Concepts, Phys. Rep. 299 (4 – 6) (1998) 189–428. [4] M.C. Gutzwiller, Chaos in Classical and Quantum Systems, Springer, Berlin, 1990. [5] O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984) 1. [6] J. Zakrzewski, K. Dupret, D. Delande, Phys. Rev. Lett. 74 (1995) 522. [7] D. Bohm, Phys. Rev. 85 (1952) 166, 180. [8] D. Bohm, B.J. Hiley, The Undivided Universe, Routledge, London, 1993. [9] P.R. Holland, The Quantum Theory of Motion, Cambridge University Press, Cambridge, 1993. [10] O.F. de Alcantara Bon m, J. Florencio, F.C. de Sa Barreto, Phys. Rev. E 58 (1998) 2693. [11] C. Dewdney, B.J. Hiley, Found. Phys. 12 (1982) 27. [12] C. Philippidis, C. Dewdney, B.J. Hiley, Nuovo Cimento B 52 (1979) 15. [13] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, 1992. [14] H. De Raedt, Comput. Phys. Rep. 7 (1987) 1. [15] H. De Raedt, K. Michielsen, Comput. Phys. 8 (1994) 600.
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Bohmian quantum trajectories in a square billiard in the bouncing ball regime J.A. de Sales, J. Florencio ∗ Departamento de FÃsica, Universidade Federal de Minas Gerais, 30.123-970 Belo Horizonte, MG, Brazil Received 20 March 2000; received in revised form 24 July 2000
Abstract We have studied the behavior of the Gaussian wave packet in the 2-D square billiard using Bohmian quantum mechanics. We found non-chaotic ballistic trajectories for particles with high energies. The most important aspect of these trajectories is that upon collision they obey c 2001 Elsevier Science B.V. the laws of re ection in spite of curved paths near the walls. All rights reserved. PACS: 05.45.+b; 03.65.Ge; 03.40.Kf Keywords: Bohmian quantum mechanics; Quantum chaos; Billiards
The 2-D square billiard is one of the simplest billiards. It is completely integrable in classical mechanics: there are two constants of motion, energy and square of momentum. In conventional quantum mechanics, however, one does not have trajectories like in classical mechanics and the dynamic behavior in billiards is frequently analyzed by using spectral statistics [1]. Random matrix theory [2,3] and other auxiliary theories, like periodic orbits theories in the semiclassical regime [4] have been used to study chaos in conventional quantum mechanics. In square billiards, the energy levels are uncorrelated and the energy spacing distribution is Poisson-like, which indicates regular motion, according to random matrix theory. The famous Bohigas–Gianinni–Schmit conjecture [5] is very often employed to characterize chaos, but there are some examples in the literature which show its breakdown, namely, the hydrogen atom in a magnetic eld and a two-dimensional quartic oscillator [6]. The fact is that there is no general consensus about the signature of chaos in quantum mechanics [4]. ∗ Corresponding author. E-mail address: jfj@ sica.ufmg.br (J. Florencio).
c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 4 9 3 - 3
102
J.A. de Sales, J. Florencio / Physica A 290 (2001) 101–106
Within the framework of Bohm’s quantum mechanics [7–9], the square billiard can show chaos [10] depending on the initial position of the particle when an initial wave packet is built from a nite number of Hamiltonian eigenfunctions. Chaotic behavior can be expected in that case because the particle is scattered at the nodal lines in x–y plane. In instances where the motion is quasiperiodic, the trajectories are Lissajous-like pictures. On the other hand, when the motion is chaotic the trajectories are quite irregular. Thus, both chaotic and semi-periodic regimes can be found in that billiard. According to Bohr’s correspondence principle, the classical limit of a quantum system should be achieved when the quantum numbers go to in nity. However, in a square box that is questionable, because there is always a great number of nodal lines when the energy of the system is large. On the other hand, in Bohm’s theory those lines are regions where the velocity is not well-de ned. In order to study the semiclassical regime of Bohmian trajectories in the square billiard, we shall consider the case of an initial Gaussian wave packet, since it does not have any nodes within the billiard. To perform this, we use a sharp packet localized far from the scatterer. This case is similar to those studied by Dewdney and Hiley [11] where a square barrier was used like a scatterer for a Gaussian packet. Another reason to use Gaussian packets is that we are interested in ballistic trajectories in the billiard domain, then we need a particle that is initially free, i.e., having negligible contact with the walls. The plane wave description is not good for this because the particle is initially not free, its wave function carries all information of the external scattering potential throughout space [9]. This is the principal reason to use the Gaussian wave packet instead of plane waves when we want to describe scattering phenomena, like tunnel e ect [11] and double-slit experiments [12]. As we shall see, such a choice seems to be the best representation for the particle state at high energies. We nd that away from the boundaries the trajectories are straight lines, as expected for motions governed by Gaussian wave packets. However, near the walls the quantum potential modi es the particle’s trajectories which become curved. Since the in uence of the quantum potential diminishes as the particle moves away from the collision region, the trajectories become straight lines. The most striking feature of these trajectories is that the straight segments before and after collision obey the re ection laws of classical billiards. The dynamics of the system is described in Bohm’s theory by writing down the wave function (x; y; t) in polar coordinates, i.e., = R exp(−iS=˝) and inserting it in the Schrodinger equation. The amplitude R and the phase S are separated into two equations, which after some simple algebraic manipulations become ∇S @ =0; (1) +∇ @t M (∇S)2 @S + +V +Q=0; (2) @t 2M where = R 2 is the probability density, V is the ordinary quantized potential, M is the mass of the particle, and Q = −˝2 =2M ∇2 R=R is the quantum potential. The rst equation is simply the conservation of probability ux, whereas the second equation is
J.A. de Sales, J. Florencio / Physica A 290 (2001) 101–106
103
a generalized Hamilton–Jacobi equation, where the conventional potential is replaced by an e ective potential Ve = V + Q. In Bohm’s theory, an additional postulate is introduced, which de nes ∇S as the momentum of the particle. The velocity of the particle can then be given in terms of the wave function as ˝ (3) ( ∗ ∇ − ∇ ∗ )=( ∗ ) : v(x; y; t) = 2M i The position and velocity of the particle are well-de ned at all times provided the initial position is not lying on a node. The particle now moves governed by a modi ed Newton’s second law d2 r (4) M 2 = −∇(V + Q)|r=r(t) : dt In our study, this equation is integrated by using a fourth-order Runge–Kutta method [13] with the initial conditions x(t = 0) and v(t = 0) = v0 , where the velocity v0 is pre-de ned in Bohm’s equation and depends on the initial position and initial wave packet of the particle in the billiard’s domain. We also need to know the wave function (x; y; t) at all times. We use a suitably sharp Gaussian packet to prevent any appreciable amplitude at the borders of the billiard. It is given by Y 1 2 exp(ik0 − | − 0 |2 =20 ); (5) (x; y; 0) = 2 (0 )1=4 =x; y
for which the mean energy is 2 2 Z 1 1 −˝ ∇ ∗ 2 dr = k0x2 + k0y + 2 + 2 hEi = 2M 20x 20y
(6)
in a system of units where ˝2 = 2M = L = 1. Here 0x and 0y are the initial widths of the wave packets along the x and y directions, respectively. The quantities k0x and k0y are the average momenta along those directions, and x0 and y0 are the coordinates of the center of the initial wave packet. The evaluation of the time-dependent wave function in this system is not trivial and was calculated using the unconditionally stable algorithm of De Raedt [14,15]. In this method, the wave function norm is conserved with good precision; this fact is very important because of the two simultaneous integration processes needed to obtain the Bohmian trajectories. The Bohm’s trajectories are obtained from numerical integration of Eq. (3) using the Runge–Kutta method, where the derivative of the wave function is replaced by a ve-point nite-di erence approximation. The billiard area was discretized by a 500×500 lattice. We performed extensive calculations to determine the Bohmian trajectories for several initial Gaussian wave packets and initial positions. The most important features of our results can be seen in Figs. 1– 4, which are discussed in the following. 2 2 + 0y = Fig. 1 shows three quantum trajectories for Gaussian packets with 02 = 0x −4 9×10 starting at distinct positions with the particle localized at the center of the
104
J.A. de Sales, J. Florencio / Physica A 290 (2001) 101–106
Fig. 1. Quantum trajectories corresponding to an initial Gaussian wave packet. The wave packet parameters are k0x = 80; k0y = 100; and 02 = 9×10−4 . We used a 500×500 integration mesh. The initial positions are (x0 ; y0 ) = (0:8; 0:7), (0:6; 0:8), and (0:1; 0:8).
Fig. 2. Same as in Fig. 1, but with 02 = 6:4×10−6 . Notice the sharper bounces o the walls, as compared with those of Fig. 1.
wave packet. We can note that away from the borders, the trajectories are ballistic. Near the walls, however, they curve, and then emerge following the laws of re ection. Note that these are results straight from numerical integration, and no bias was used. The particle never hits the billiard walls, where the quantum potential is very strong. Fig. 2 uses the same parameters as in Fig. 1, except that now the initial packet is narrower, whose, 02 = 6:4×10−6 , that is, a packet with higher mean energy. In this case, the trajectories resemble the classical regime even more. The trajectories now
J.A. de Sales, J. Florencio / Physica A 290 (2001) 101–106
105
Fig. 3. Several quantum trajectories corresponding to initial Gaussian wave packets starting from initial ◦ position (x0 ; y0 ) = (0:6; 0:8). The wave vector is k = 100, and the angle of incidence is 60 , with 0 = 0:2; 0:1; 0:08; 0:06; 0:03. We used a 500×500 integration mesh. The wiggled curves are an artifact of the numerical integration: smooth curves could be obtained with more re ned (time-consuming) integration steps.
Fig. 4. Minimum distance of the trajectory to the wall rmin versus the width of the Gaussian wave packet. Notice the linear relationship between rmin and . Here, rmin ( = 0) = 0:001 ± 0:001 and slope = 0:36 ± 0:01. We used a 500×500 integration mesh.
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J.A. de Sales, J. Florencio / Physica A 290 (2001) 101–106
have a smaller radius of curvature in the collision region. We expect that this e ect will be most pronounced when the width 0 → 0; that is, when the wave packet is a delta-function. One should keep in mind that the system is still in a quantum regime. The Schrodinger equation and all quantum formalisms still apply; however, the trajectories show prominent classical features. The classical features are more evident when the Gaussian wave packet becomes sharper. In Fig. 3, we show several trajectories starting in the same position with di erent widths. We can note that the trajectories with higher mean energy approach the wall more closely, with smaller radius of curvature. The behavior of the radius of curvature with the width of the Gaussian packet is shown in Fig. 4. One can infer from this gure that a true classical regime can only be reached when the width of the Gaussian packet is much smaller than the sides of the wall, with the quantum trajectory hitting the wall. The minimum distance rmin to the wall turns out to be a linear function of the width of the packet and goes to zero when goes also to zero. In summary, the Bohmian trajectories in a square billiard do not exhibit chaos when we use Gaussian wave packets. In the cases studied, we found a bouncing ball regime for high energies. The particle’s quantum trajectories are very close to those in the classical regime when the energy of Gaussian wave packet is large. The angles of re ection and incidence for the particle were obtained as outcomes of the numerical integration of the equation of motion of the Bohmian particle. We would like to thank O.F. de Alcantara Bon m and F.C. Sa Barreto for fruitful discussions. This work was partially supported by CNPq, FINEP, FAPEMIG and MCT (Brazilian agencies). References [1] S.W. McDonald, A.N. Kaufman, Phys. Rev. Lett. 42 (1979) 1189. [2] M.L. Mehta, Random Matrix Theory, Academic Press, New York, 1991. [3] T. Guhr, A. Muller-Groeling, H.A. Weidenmuller, Random-Matrix Theories in Quantum Physics: Common Concepts, Phys. Rep. 299 (4 – 6) (1998) 189–428. [4] M.C. Gutzwiller, Chaos in Classical and Quantum Systems, Springer, Berlin, 1990. [5] O. Bohigas, M.J. Giannoni, C. Schmit, Phys. Rev. Lett. 52 (1984) 1. [6] J. Zakrzewski, K. Dupret, D. Delande, Phys. Rev. Lett. 74 (1995) 522. [7] D. Bohm, Phys. Rev. 85 (1952) 166, 180. [8] D. Bohm, B.J. Hiley, The Undivided Universe, Routledge, London, 1993. [9] P.R. Holland, The Quantum Theory of Motion, Cambridge University Press, Cambridge, 1993. [10] O.F. de Alcantara Bon m, J. Florencio, F.C. de Sa Barreto, Phys. Rev. E 58 (1998) 2693. [11] C. Dewdney, B.J. Hiley, Found. Phys. 12 (1982) 27. [12] C. Philippidis, C. Dewdney, B.J. Hiley, Nuovo Cimento B 52 (1979) 15. [13] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, 1992. [14] H. De Raedt, Comput. Phys. Rep. 7 (1987) 1. [15] H. De Raedt, K. Michielsen, Comput. Phys. 8 (1994) 600.