Semiclassical approximations in phase space with coherent states

We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

[1]  J. H. Van Vleck,et al.  The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics , 1928 .

[2]  E. Wigner.  On the quantum correction for thermodynamic equilibrium , 1932 .

[3]  J. Mayer,et al.  On the Quantum Correction for Thermodynamic Equilibrium , 1947 .

[4]  William H. Miller,et al.  Classical S Matrix: Numerical Application to Inelastic Collisions , 1970 .

[5]  M. Gutzwiller,et al.  Periodic Orbits and Classical Quantization Conditions , 1971 .

[6]  E. Heller.  Time‐dependent approach to semiclassical dynamics , 1975 .

[7]  U. Smilansky,et al.  The hamiltonian path integrals and the uniform semiclassical approximations for the propagator , 1977 .

[8]  Asymptotic $\hbar $-expansions of stationary quantum states , 1977 .

[9]  J. Klauder.  CONTINUOUS REPRESENTATIONS AND PATH INTEGRALS, REVISITED , 1978 .

[10]  John R. Klauder,et al.  Path integrals and stationary-phase approximations , 1979 .

[11]  M. S. Zubairy,et al.  Path-integral approach to problems in quantum optics , 1982 .

[12]  Y. Weissman.  Semiclassical approximation in the coherent states representation , 1982 .

[13]  Y. Weissman.  On the stationary phase evaluation of path integrals in the coherent states representation , 1983 .

[14]  Eric J. Heller,et al.  Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits , 1984 .

[15]  M. Scully,et al.  Distribution functions in physics: Fundamentals , 1984 .

[16]  E. Kluk,et al.  A semiclasical justification for the use of non-spreading wavepackets in dynamics calculations , 1984 .

[17]  J. Klauder,et al.  COHERENT STATES: APPLICATIONS IN PHYSICS AND MATHEMATICAL PHYSICS , 1985 .

[18]  Michael F. Herman.  Time reversal and unitarity in the frozen Gaussian approximation for semiclassical scattering , 1986 .

[19]  E. Kluk,et al.  Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator , 1986 .

[20]  J. Klauder.  Global, uniform, asymptotic wave-equation solutions for large wavenumbers , 1987 .

[21]  J. Klauder.  Some Recent Results on Wave Equations, Path Integrals, and Semiclassical Approximations , 1987 .

[22]  E. Bogomolny.  Smoothed wave functions of chaotic quantum systems , 1988 .

[23]  Voros.  Wentzel-Kramers-Brillouin method in the Bargmann representation. , 1989, Physical review. A, General physics.

[24]  Leboeuf,et al.  Semiclassical approximations in the coherent-state representation. , 1989, Physical review. A, General physics.

[25]  M. Berry,et al.  Quantum scars of classical closed orbits in phase space , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[26]  J. Robbins.  Maslov indices in the Gutzwiller trace formula , 1991 .

[27]  Quantum and semiclassical Husimi distributions for a one-dimensional resonant system R. Egydio de Carvalho , 1992 .

[28]  Quantum and semiclassical Husimi distributions for a one-dimensional resonant system. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[29]  P. Brumer,et al.  Semiclassical collision theory in the initial value representation: Efficient numerics and reactive formalism , 1992 .

[30]  Walter Dittrich,et al.  Classical and Quantum Dynamics , 1992 .

[31]  K. Kay,et al.  Integral expressions for the semiclassical time‐dependent propagator , 1994 .

[32]  K. Kay,et al.  Numerical study of semiclassical initial value methods for dynamics , 1994 .

[33]  D. Manolopoulos,et al.  APPLICATION OF THE FROZEN GAUSSIAN APPROXIMATION TO THE PHOTODISSOCIATION OF CO2 , 1995 .

[34]  Uniform semiclassical wave-packet propagation and eigenstate extraction in a smooth chaotic system. , 1994, Physical review letters.

[35]  S. Garashchuk,et al.  Wave packet correlation function approach to H2(ν)+H→H+H2(ν′): semiclassical implementation , 1996 .

[36]  Marcus A. M. de Aguiar,et al.  Complex Trajectories in the Quartic Oscillator and Its Semiclassical Coherent-State Propagator , 1996 .

[37]  Semiclassical approximations to the coherent-state propagator for a particle in a box , 1996 .

[38]  Semiclassical approximations to the coherent-state propagator for a particle in a box. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[39]  A. Vourdas.  THE GROWTH OF BARGMANN FUNCTIONS AND THE COMPLETENESS OF SEQUENCES OF COHERENT STATES , 1997 .

[40]  K. Kay.  SEMICLASSICAL TUNNELING IN THE INITIAL VALUE REPRESENTATION , 1997 .

[41]  William H. Miller,et al.  Mixed semiclassical-classical approaches to the dynamics of complex molecular systems , 1997 .

[42]  F. Grossmann,et al.  Harmonic inversion of semiclassical short time signals , 1997 .

[43]  P. Brumer,et al.  SEMICLASSICAL INITIAL VALUE THEORY FOR DISSOCIATION DYNAMICS , 1997 .

[44]  Michael F. Herman.  IMPROVING THE ACCURACY OF SEMICLASSICAL WAVEPACKET PROPAGATION USING INTEGRAL CONDITIONING TECHNIQUES , 1997 .

[45]  A. L. Xavier,et al.  Phase-Space Approach to the Tunnel Effect: A New Semiclassical Traversal Time , 1997 .

[46]  F. Grossmann,et al.  From the coherent state path integral to a semiclassical initial value representation of the quantum mechanical propagator , 1998 .

[47]  Michael F. Herman,et al.  A numerical test of different integral conditioning approximations for a semiclassical initial value representation for wavepacket propagation , 1998 .

[48]  A. M. Almeida.  The Weyl representation in classical and quantum mechanics , 1998 .

[49]  Semiclassical coherent-state path integrals for scattering , 1998 .

[50]  Galactic halo models and particle dark-matter detection , 1997, hep-ph/9710337.

[51]  Semiclassical quantization of a system with mixed regular/chaotic dynamics , 1998 .

[52]  D. McCormack.  An evaluation of the semiclassical Herman–Kluk (HK) propagator for molecule–surface reactive scattering , 2000 .

[53]  Sean X. Sun.  Semiclassical approximations to real-time quantum-mechanical effects in correlation functions of complex molecular systems , 2000 .

[54]  N. Maitra.  Semiclassical maps: A study of classically forbidden transitions, sub-h structure, and dynamical localization , 2000 .

[55]  M. Thoss,et al.  Semiclassical description of nonadiabatic quantum dynamics: Application to the S1–S2 conical intersection in pyrazine , 2000 .

[56]  Exact and asymptotic solutions of the mixed quantum-classical Liouville equation , 2000 .

[57]  Semiclassical real-time tunneling by multiple spawning of classical trajectories , 2000, Physical review letters.

[58]  William H. Miller,et al.  The Semiclassical Initial Value Representation: A Potentially Practical Way for Adding Quantum Effects to Classical Molecular Dynamics Simulations , 2001 .

[59]  M. Baranger,et al.  Reply to ‘Comment on ‘‘Semiclassical approximations in phase space with coherent states’’’ , 2002 .

[60]  W. Miller.  An alternate derivation of the Herman—Kluk (coherent state) semiclassical initial value representation of the time evolution operator , 2002 .

[61]  Comment on ‘Semiclassical approximations in phase space with coherent states’ , 2003 .

[62]  J. H. Van Vleck.  The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics. , 2022, Proceedings of the National Academy of Sciences of the United States of America.