Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras
Affiliation(s)
1 Ciclo Básico Común-Cátedra de Física, Universidad de Buenos Aires, Buenos Aires, Argentina. 2 Laboratorio de Sistemas Complejos, Departamento de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires, Buenos Aires, Argentina. 3 Instituto de Física La Plata-CCT-CONICET, Universidad Nacional de La Plata, La Plata, Argentina. 4 Physics Department and IFISC-CSIC, University of the Balearic Islands, Palma de Mallorca, Spain.
1 Ciclo Básico Común-Cátedra de Física, Universidad de Buenos Aires, Buenos Aires, Argentina. 2 Laboratorio de Sistemas Complejos, Departamento de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires, Buenos Aires, Argentina. 3 Instituto de Física La Plata-CCT-CONICET, Universidad Nacional de La Plata, La Plata, Argentina. 4 Physics Department and IFISC-CSIC, University of the Balearic Islands, Palma de Mallorca, Spain.
ABSTRACT
We show that the non-linear semi-quantum Hamiltonians which may be expressed as
(where
is the set of generators of some Lie algebra and are the classical conjugated canonical variables) always close a partial semi Lie algebra under commutation and
, because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion:
(where
is the Maximum Entropy Principle density operator) and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.
We show that the non-linear semi-quantum Hamiltonians which may be expressed as
(whereis the set of generators of some Lie algebra and are the classical conjugated canonical variables) always close a partial semi Lie algebra under commutation and, because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion:(whereis the Maximum Entropy Principle density operator) and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.
Cite this paper
Sarris, C. , Plastino, A. , (2014) Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras. Applied Mathematics, 5, 3277-3296. doi: 10.4236/am.2014.520306.
Sarris, C. , Plastino, A. , (2014) Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras. Applied Mathematics, 5, 3277-3296. doi: 10.4236/am.2014.520306.
References
[1] Kowalski, A.M., Plastino, A. and Proto, A.N. (1995) Semiclassical Model for Quantum Dissipation. Physical Review E, 52, 165-177.
http://dx.doi.org/10.1103/PhysRevE.52.165 [2] Kowalski, A.M., Martin, M.T., Nuñez, J., Plastino, A. and Proto, A.N. (1988) Quantitative Indicator for Semiquantum Chaos. Physical Review A, 58, 2596-2599.
http://dx.doi.org/10.1103/PhysRevA.58.2596 [3] Sarris, C.M. and Proto, A.N. (2009) Information Entropy and Nonlinear Semiquantum Dynamics. International Journal of Bifurcation and Chaos, 19, 3473-3484. [4] Alhassid, Y. and Levine, R.D. (1977) Entropy and Chemical Change. III. The Maximal Entropy (Subject to Constraints) Procedure as a Dynamical Theory. The Journal of Chemical Physics, 67, 4321-4339.
http://dx.doi.org/10.1063/1.434578 [5] Kowalski, A.M., Plastino, A. and Proto, A.N. (2002) Classical Limits. Physics Letters A, 297, 162-172.
http://dx.doi.org/10.1016/S0375-9601(02)00034-8 [6] Kowalski, A.M., Martin, M.T., Nuñez, J., Plastino, A. and Proto, A.N. (2000) Semiquantum Chaos and the Uncertainty Principle. Physica A: Statistical Mechanics and Its Applications, 276, 95-108.
http://dx.doi.org/10.1016/S0378-4371(99)00280-0 [7] Von Neumann, J. (1955) Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton. [8] Fano, U. (1957) Description of States in Quantum Mechanics by Density Matrix and Operator Techniques. Reviews of Modern Physics, 29, 74-93.
http://dx.doi.org/10.1103/RevModPhys.29.74 [9] Jaynes, E.T. (1957) Information Theory and Statistical Mechanics. Physical Review, 106, 620-630.
http://dx.doi.org/10.1103/PhysRev.106.620 [10] Jaynes, E.T. (1957) Information Theory and Statistical Mechanics II. Physical Review, 108, 171-190.
http://dx.doi.org/10.1103/PhysRev.108.171 [11] Otero, D., Plastino, A., Proto, A.N. and Zannoli, G. (1982) Ehrenfest Theorem and Information Theory. Physical Review A, 26, 1209-1217.
http://dx.doi.org/10.1103/PhysRevA.26.1209 [12] Ballentine, L.E. (2001) Is Semiquantum Chaos Real? Physical Review E, 63, Article ID: 056204.
http://dx.doi.org/10.1103/PhysRevE.63.056204 [13] Blum, T.C. and Elze, H.T. (1996) Semiquantum Chaos in the Double Well. Physical Review E, 53, 3123-3133.
http://dx.doi.org/10.1103/PhysRevE.53.3123 [14] Bonilla, L.L. and Guinea, F. (1992) Collapse of the Wave Packet and Chaos in a Model with Classical and Quantum Degrees of Freedom. Physical Review A, 45, 7718-7728.
http://dx.doi.org/10.1103/PhysRevA.45.7718 [15] Cukier, R.I. and Morillo, M. (2000) Comparison between Quantum and Approximate Semiclassical Dynamics of an Externally Driven Spin-Harmonic Oscillator System. Physical Review A, 61, Article ID: 024103.
http://dx.doi.org/10.1103/PhysRevA.61.024103 [16] Pattayanak, A.K. and Schieve, W.C. (1994) Semiquantal Dynamics of Fluctuations: Ostensible Quantum Chaos. Physical Review Letters, 72, 2855-2858.
http://dx.doi.org/10.1103/PhysRevLett.72.2855 [17] Kowalski, A.M., Martin, M.T., Plastino, A., Proto, A.N. and Rosso, O.A. (2003) Wavelet Statistical Complexity Analysis of the Classical Limit. Physical Review A, 311, 189-191.
http://dx.doi.org/10.1016/S0375-9601(03)00470-5 [18] Cooper, F., Dawson, J.F., Meredith, D. and Shepard, H. (1994) Semiquantum Chaos. Physics Letters, 72, 1337-1340.
http://dx.doi.org/10.1016/S0375-9601(03)00470-5 [19] Kowalski, A.M., Plastino, A. and Proto, A.N. (2003) Classical Limit and Chaotic Regime in a Semi-Quantum Hamiltonian. International Journal of Bifurcation and Chaos, 13, 2315-2325.
http://dx.doi.org/10.1142/S0218127403007977 [20] Porter, M.A. (2001) Nonadiabatic Dynamics in Semiquantal Physics. Reports on Progress in Physics, 64, 1165-1189.
http://dx.doi.org/10.1088/0034-4885/64/9/203 [21] Kowalski, A.M., Plastino, A. and Proto, A.N. (1997) A Semiclassical Model for Quantum Dissipation. Physica A: Statistical Mechanics and Its Applications, 236, 429-447.
http://dx.doi.org/10.1016/S0378-4371(96)00379-2 [22] Porter, A.M. and Liboff, R.L. (2001) Vibrating Quantum Billiards on Riemannian Manifolds. International Journal of Bifurcation and Chaos, 11, 2305-2315. [23] Plastino, A. and Sarris, C. (2014) Information Theory and Semi-Quantum MaxEnt: Semiquantum Physics. LAP Lambert Academic Press, Saarbrücken. [24] Blumel, R. and Esser, B. (1994) Quantum Chaos in the Born-Oppenheimer Approximation. Physical Review Letters, 72, 3658-3661.
http://dx.doi.org/10.1103/PhysRevLett.72.3658 [25] Schanz, H. and Esser, B. (1997) Mixed Quantum-Classical versus Full Quantum Dynamics: Coupled Quasiparticle-Oscillator System. Physical Review A, 55, 3375-3387.
http://dx.doi.org/10.1103/PhysRevA.55.3375 [26] Ma, J. and Yuan, R.K. (1997) Semiquantum Chaos. Journal of the Physical Society of Japan, 66, 2302-2307.
http://dx.doi.org/10.1143/JPSJ.66.2302 [27] Sarris, C.M., Plastino, A. and Sassano, M.P. (2014) Peculiar Dynamics of Phase Space Embedded SU(2) Hamiltonians. International Journal of Sciences, 3, 32-44.
http://www.ijsciences.com/pub/article/379 [28] Cohen-Tannouudji, C., Diu, B. and Laloë, F. (1977) Quantum Mechanics. Wiley, New York. [29] Aliaga, J., Otero, D., Plastino, A. and Proto, A.N. (1987) Constants of Motion, Accessible States and Information Theory. Physical Review A, 35, 2304-2311.
http://dx.doi.org/10.1103/PhysRevA.35.2304 [30] Merzbacher, E. (1963) Quantum Mechanics. Wiley, New York. [31] Düering, E., Otero, D., Plastino, A. and Proto, A.N. (1987) General Dynamical Invariants for Time-Dependent Hamiltonians. Physical Review A, 35, 2314-2320.
http://dx.doi.org/10.1103/PhysRevA.35.2304 [32] Sarris, C.M., Caram, F. and Proto, A.N. (2004) Entropy Invariants of Motion. Physica A: Statistical Mechanics and Its Applications, 331, 125-139.
http://dx.doi.org/10.1016/j.physa.2003.07.008 [33] Sarris, C.M., Caram, F. and Proto, A.N. (2004) The Uncertainty Principle as Invariant of Motion for Time-Dependent Hamiltonians. Physics Letters A, 324, 1-8.
http://dx.doi.org/10.1016/j.physleta.2004.02.036 [34] Sarris, C.M. and Proto, A.N. (2005) Time-Dependent Invariants of Motion for Complete Sets of Non-Commuting Observables. Physica A: Statistical Mechanics and Its Applications, 348, 97-109.
http://dx.doi.org/10.1016/j.physa.2004.09.038 [35] Sarris, C.M. and Proto, A.N. (2007) Generalized Metric Phase Space for Quantum Systems and the Uncertainty Principle. Physica A: Statistical Mechanics and Its Applications, 377, 33-42.
http://dx.doi.org/10.1016/j.physa.2006.10.093 [36] Tung, W.K. (1985) Group Theory in Physics. World Scientific Publishing, Singapore.
http://dx.doi.org/10.1142/0097 [37] Sarris, C.M., Plastino, A. and Proto, A.N. (2013) Difficulties in Evaluating Lyapunov Exponents for Lie Governed Dynamics. Journal of Chaos, 2013, Article ID: 587548, 7 p.
http://dx.doi.org/10.1155/2013/587548 [38] Louisell, W. (1973) Quantum Statistical Properties of Radiation. Wiley, New York. [39] Cooper, F., Dawson, J., Habib, S. and Ryne, R.D. (1998) Chaos in Time-Dependent Variational Approximation to Quantum. Physical Review E, 57, 1489-1498.
http://dx.doi.org/10.1103/PhysRevE.57.1489 [40] Cooper, F., Habib, S., Kluger, Y. and Mottola, E. (1997) Nonequilibrium Dynamics of Symmetry Breaking in λΦ? Theory. Physical Review D, 55, 6471-6503.
http://dx.doi.org/10.1103/PhysRevD.55.6471 [41] Aliaga, J., Crespo, G. and Proto, A.N. (1990) Thermodynamics of Squeezed States for the Kanai-Caldirola Hamiltonian. Physical Review A, 42, 4325-4335.
http://dx.doi.org/10.1103/PhysRevD.55.6471 [42] Aliaga, J., Crespo, G. and Proto, A.N. (1990) Non-Zero Temperature Coherent and Squeezed States for the Harmonic-Oscillator: The Time-Dependent Frequency Case. Physical Review A, 42, 618-626.
http://dx.doi.org/10.1103/PhysRevA.42.618 [43] Hirayama, M. (1991) SO(2,1) Structure of the Generalized Harmonic Oscillator. Progress of Theoretical Physics, 86, 343-354.
http://dx.doi.org/10.1143/ptp/86.2.343 [44] Cerveró, J.M. and Lejarreta, J.D. (1989) SO(2,1) Invariant Systems and the Berry Phase. Journal of Physics A: Mathematical and General, 22, L633-L666.
http://dx.doi.org/10.1088/0305-4470/22/14/001 [45] Aliaga, J., Otero, D., Plastino, A. and Proto, A.N. (1988) Quantum Thermodynamics and Information Theory. Physical Review A, 38, 918-929.
http://dx.doi.org/10.1103/PhysRevA.38.918 [46] Dattoli, G., Dipace, A. and Torre, A. (1986) Dynamics of the SU(1,1) Bloch Vector. Physical Review A, 33, 4387-4389.
http://dx.doi.org/10.1103/PhysRevA.33.4387
[1] Kowalski, A.M., Plastino, A. and Proto, A.N. (1995) Semiclassical Model for Quantum Dissipation. Physical Review E, 52, 165-177.
http://dx.doi.org/10.1103/PhysRevE.52.165 [2] Kowalski, A.M., Martin, M.T., Nuñez, J., Plastino, A. and Proto, A.N. (1988) Quantitative Indicator for Semiquantum Chaos. Physical Review A, 58, 2596-2599.
http://dx.doi.org/10.1103/PhysRevA.58.2596 [3] Sarris, C.M. and Proto, A.N. (2009) Information Entropy and Nonlinear Semiquantum Dynamics. International Journal of Bifurcation and Chaos, 19, 3473-3484. [4] Alhassid, Y. and Levine, R.D. (1977) Entropy and Chemical Change. III. The Maximal Entropy (Subject to Constraints) Procedure as a Dynamical Theory. The Journal of Chemical Physics, 67, 4321-4339.
http://dx.doi.org/10.1063/1.434578 [5] Kowalski, A.M., Plastino, A. and Proto, A.N. (2002) Classical Limits. Physics Letters A, 297, 162-172.
http://dx.doi.org/10.1016/S0375-9601(02)00034-8 [6] Kowalski, A.M., Martin, M.T., Nuñez, J., Plastino, A. and Proto, A.N. (2000) Semiquantum Chaos and the Uncertainty Principle. Physica A: Statistical Mechanics and Its Applications, 276, 95-108.
http://dx.doi.org/10.1016/S0378-4371(99)00280-0 [7] Von Neumann, J. (1955) Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton. [8] Fano, U. (1957) Description of States in Quantum Mechanics by Density Matrix and Operator Techniques. Reviews of Modern Physics, 29, 74-93.
http://dx.doi.org/10.1103/RevModPhys.29.74 [9] Jaynes, E.T. (1957) Information Theory and Statistical Mechanics. Physical Review, 106, 620-630.
http://dx.doi.org/10.1103/PhysRev.106.620 [10] Jaynes, E.T. (1957) Information Theory and Statistical Mechanics II. Physical Review, 108, 171-190.
http://dx.doi.org/10.1103/PhysRev.108.171 [11] Otero, D., Plastino, A., Proto, A.N. and Zannoli, G. (1982) Ehrenfest Theorem and Information Theory. Physical Review A, 26, 1209-1217.
http://dx.doi.org/10.1103/PhysRevA.26.1209 [12] Ballentine, L.E. (2001) Is Semiquantum Chaos Real? Physical Review E, 63, Article ID: 056204.
http://dx.doi.org/10.1103/PhysRevE.63.056204 [13] Blum, T.C. and Elze, H.T. (1996) Semiquantum Chaos in the Double Well. Physical Review E, 53, 3123-3133.
http://dx.doi.org/10.1103/PhysRevE.53.3123 [14] Bonilla, L.L. and Guinea, F. (1992) Collapse of the Wave Packet and Chaos in a Model with Classical and Quantum Degrees of Freedom. Physical Review A, 45, 7718-7728.
http://dx.doi.org/10.1103/PhysRevA.45.7718 [15] Cukier, R.I. and Morillo, M. (2000) Comparison between Quantum and Approximate Semiclassical Dynamics of an Externally Driven Spin-Harmonic Oscillator System. Physical Review A, 61, Article ID: 024103.
http://dx.doi.org/10.1103/PhysRevA.61.024103 [16] Pattayanak, A.K. and Schieve, W.C. (1994) Semiquantal Dynamics of Fluctuations: Ostensible Quantum Chaos. Physical Review Letters, 72, 2855-2858.
http://dx.doi.org/10.1103/PhysRevLett.72.2855 [17] Kowalski, A.M., Martin, M.T., Plastino, A., Proto, A.N. and Rosso, O.A. (2003) Wavelet Statistical Complexity Analysis of the Classical Limit. Physical Review A, 311, 189-191.
http://dx.doi.org/10.1016/S0375-9601(03)00470-5 [18] Cooper, F., Dawson, J.F., Meredith, D. and Shepard, H. (1994) Semiquantum Chaos. Physics Letters, 72, 1337-1340.
http://dx.doi.org/10.1016/S0375-9601(03)00470-5 [19] Kowalski, A.M., Plastino, A. and Proto, A.N. (2003) Classical Limit and Chaotic Regime in a Semi-Quantum Hamiltonian. International Journal of Bifurcation and Chaos, 13, 2315-2325.
http://dx.doi.org/10.1142/S0218127403007977 [20] Porter, M.A. (2001) Nonadiabatic Dynamics in Semiquantal Physics. Reports on Progress in Physics, 64, 1165-1189.
http://dx.doi.org/10.1088/0034-4885/64/9/203 [21] Kowalski, A.M., Plastino, A. and Proto, A.N. (1997) A Semiclassical Model for Quantum Dissipation. Physica A: Statistical Mechanics and Its Applications, 236, 429-447.
http://dx.doi.org/10.1016/S0378-4371(96)00379-2 [22] Porter, A.M. and Liboff, R.L. (2001) Vibrating Quantum Billiards on Riemannian Manifolds. International Journal of Bifurcation and Chaos, 11, 2305-2315. [23] Plastino, A. and Sarris, C. (2014) Information Theory and Semi-Quantum MaxEnt: Semiquantum Physics. LAP Lambert Academic Press, Saarbrücken. [24] Blumel, R. and Esser, B. (1994) Quantum Chaos in the Born-Oppenheimer Approximation. Physical Review Letters, 72, 3658-3661.
http://dx.doi.org/10.1103/PhysRevLett.72.3658 [25] Schanz, H. and Esser, B. (1997) Mixed Quantum-Classical versus Full Quantum Dynamics: Coupled Quasiparticle-Oscillator System. Physical Review A, 55, 3375-3387.
http://dx.doi.org/10.1103/PhysRevA.55.3375 [26] Ma, J. and Yuan, R.K. (1997) Semiquantum Chaos. Journal of the Physical Society of Japan, 66, 2302-2307.
http://dx.doi.org/10.1143/JPSJ.66.2302 [27] Sarris, C.M., Plastino, A. and Sassano, M.P. (2014) Peculiar Dynamics of Phase Space Embedded SU(2) Hamiltonians. International Journal of Sciences, 3, 32-44.
http://www.ijsciences.com/pub/article/379 [28] Cohen-Tannouudji, C., Diu, B. and Laloë, F. (1977) Quantum Mechanics. Wiley, New York. [29] Aliaga, J., Otero, D., Plastino, A. and Proto, A.N. (1987) Constants of Motion, Accessible States and Information Theory. Physical Review A, 35, 2304-2311.
http://dx.doi.org/10.1103/PhysRevA.35.2304 [30] Merzbacher, E. (1963) Quantum Mechanics. Wiley, New York. [31] Düering, E., Otero, D., Plastino, A. and Proto, A.N. (1987) General Dynamical Invariants for Time-Dependent Hamiltonians. Physical Review A, 35, 2314-2320.
http://dx.doi.org/10.1103/PhysRevA.35.2304 [32] Sarris, C.M., Caram, F. and Proto, A.N. (2004) Entropy Invariants of Motion. Physica A: Statistical Mechanics and Its Applications, 331, 125-139.
http://dx.doi.org/10.1016/j.physa.2003.07.008 [33] Sarris, C.M., Caram, F. and Proto, A.N. (2004) The Uncertainty Principle as Invariant of Motion for Time-Dependent Hamiltonians. Physics Letters A, 324, 1-8.
http://dx.doi.org/10.1016/j.physleta.2004.02.036 [34] Sarris, C.M. and Proto, A.N. (2005) Time-Dependent Invariants of Motion for Complete Sets of Non-Commuting Observables. Physica A: Statistical Mechanics and Its Applications, 348, 97-109.
http://dx.doi.org/10.1016/j.physa.2004.09.038 [35] Sarris, C.M. and Proto, A.N. (2007) Generalized Metric Phase Space for Quantum Systems and the Uncertainty Principle. Physica A: Statistical Mechanics and Its Applications, 377, 33-42.
http://dx.doi.org/10.1016/j.physa.2006.10.093 [36] Tung, W.K. (1985) Group Theory in Physics. World Scientific Publishing, Singapore.
http://dx.doi.org/10.1142/0097 [37] Sarris, C.M., Plastino, A. and Proto, A.N. (2013) Difficulties in Evaluating Lyapunov Exponents for Lie Governed Dynamics. Journal of Chaos, 2013, Article ID: 587548, 7 p.
http://dx.doi.org/10.1155/2013/587548 [38] Louisell, W. (1973) Quantum Statistical Properties of Radiation. Wiley, New York. [39] Cooper, F., Dawson, J., Habib, S. and Ryne, R.D. (1998) Chaos in Time-Dependent Variational Approximation to Quantum. Physical Review E, 57, 1489-1498.
http://dx.doi.org/10.1103/PhysRevE.57.1489 [40] Cooper, F., Habib, S., Kluger, Y. and Mottola, E. (1997) Nonequilibrium Dynamics of Symmetry Breaking in λΦ? Theory. Physical Review D, 55, 6471-6503.
http://dx.doi.org/10.1103/PhysRevD.55.6471 [41] Aliaga, J., Crespo, G. and Proto, A.N. (1990) Thermodynamics of Squeezed States for the Kanai-Caldirola Hamiltonian. Physical Review A, 42, 4325-4335.
http://dx.doi.org/10.1103/PhysRevD.55.6471 [42] Aliaga, J., Crespo, G. and Proto, A.N. (1990) Non-Zero Temperature Coherent and Squeezed States for the Harmonic-Oscillator: The Time-Dependent Frequency Case. Physical Review A, 42, 618-626.
http://dx.doi.org/10.1103/PhysRevA.42.618 [43] Hirayama, M. (1991) SO(2,1) Structure of the Generalized Harmonic Oscillator. Progress of Theoretical Physics, 86, 343-354.
http://dx.doi.org/10.1143/ptp/86.2.343 [44] Cerveró, J.M. and Lejarreta, J.D. (1989) SO(2,1) Invariant Systems and the Berry Phase. Journal of Physics A: Mathematical and General, 22, L633-L666.
http://dx.doi.org/10.1088/0305-4470/22/14/001 [45] Aliaga, J., Otero, D., Plastino, A. and Proto, A.N. (1988) Quantum Thermodynamics and Information Theory. Physical Review A, 38, 918-929.
http://dx.doi.org/10.1103/PhysRevA.38.918 [46] Dattoli, G., Dipace, A. and Torre, A. (1986) Dynamics of the SU(1,1) Bloch Vector. Physical Review A, 33, 4387-4389.
http://dx.doi.org/10.1103/PhysRevA.33.4387