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Journal of Experimental and Theoretical Physics

Squeezed states in the semiclassical limit

Pavel S. Alekseev, Fedor V. Moroseev

Published Apr 1, 2009• 3 citations
Publisher

Abstract

A symplectically covariant quantum-mechanical uncertainty relation more accurate than previously known ones is derived for multidimensional systems. It is shown that the quantum-mechanical description of a linear Hamiltonian system in terms of squeezed states is completely equivalent to its description in terms of a phase-space distribution function. A new approach to the semiclassical limit is proposed, based on the use of squeezed states. By analyzing explicit formulas for squeezed states, a semiclassical asymptotic form of the solution to the Cauchy problem for a multidimensional Schrödinger equation is found in the limit of ħ → 0. The behavior of the semiclassical solution in the neighborhood of a caustic is analyzed in the one-dimensional case, and the phase shift across the caustic is determined. General properties and examples of squeezed states are discussed that point to the fundamental importance of squeezed states for developing a nonrelativistic quantum-mechanical description of a system of charged particles in an electromagnetic field in the dipole approximation.

DOI
10.1134/s1063776109040037

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A symplectically covariant quantum-mechanical uncertainty relation more accurate than previously known ones is derived for multidimensional systems. It is shown that the quantum-mechanical description of a linear Hamiltonian system in terms of squeezed states is completely equivalent to its description in terms of a phase-space distribution function. A new approach to the semiclassical limit is proposed, based on the use of squeezed states. By analyzing explicit formulas for squeezed states, a semiclassical asymptotic form of the solution to the Cauchy problem for a multidimensional Schrödinger equation is found in the limit of ħ → 0. The behavior of the semiclassical solution in the neighborhood of a caustic is analyzed in the one-dimensional case, and the phase shift across the caustic is determined. General properties and examples of squeezed states are discussed that point to the fundamental importance of squeezed states for developing a nonrelativistic quantum-mechanical description of a system of charged particles in an electromagnetic field in the dipole approximation.
It is by now well recognized how the quantum time evolution for time-dependent Hamiltonians that are quadratic both in position and momentum variables generates ‘‘squeezing.’’ On the other hand, ‘‘squeezed states’’ and their ‘‘high-n generalizations’’ are known to provide a simple classical mechanical description of general quantum time evolution. In this paper, these ideas are combined to show the relevance of ‘‘generalized squeezed states’’ in the semiclassical limit of the evolution of time-dependent wave packets for general time-dependent Hamiltonians, including the multidimensional case.
In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant ( E,H )-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.
Semiclassical limits for the dynamics of the Wigner function, and for the dynamics of the low-order moments of physical interest, are discussed in fluid-dynamical terms. The harmonic-oscillator solution yields local Maxwellian (ideal) fluid-dynamical behavior, and this is extended to locally quadratic problems in a straightforward manner. Coherent states are shown to have a particularly simple fluid-dynamical interpretation, and coherence may be easily demonstrated for mixed states having an arbitrary degree of uncertainty in the field variables. The semiclassical behavior of pure and mixed states is conveniently discussed in terms of coherent and incoherent limits, and the origin of quantum-mechanical behavior in the simple low-order approximations proposed is simply explained. The relation to the pure-state theory of Madelung is made explicit. A recently proposed method for the solution of a certain hierarchy of quantum-fluid-dynamical equations is analyzed in detail, and is demonstrated to be inconsistent with these semiclassical limits.
Little is known about dark matter particles save that their most important interactions with ordinary matter are gravitational and that, if they exist, they are stable, slow moving and relatively massive. Based on these assumptions, a semiclassical approximation to the Schrödinger equation under the action of a Coulomb potential should be relevant for modelling their behaviour. We investigate the semiclassical limit of the Schrödinger equation for a particle of mass M under a Coulomb potential in the context of Nelson’s stochastic mechanics. This is done using a Freidlin-Wentzell asymptotic series expansion in the parameter ϵ=ħ/M for the Nelson diffusion. It is shown that for wave functions ψ ∼ exp((R + iS)/ϵ2) where R and S are real valued, the ϵ = 0 behaviour is governed by a constrained Hamiltonian system with Hamiltonian Hr and constraint Hi = 0 where the superscripts r and i denote the real and imaginary parts of the Bohr correspondence limit of the quantum mechanical Hamiltonian, independent of Nelson’s ideas. Nelson’s stochastic mechanics is restored in dealing with the nodal surface singularities and by computing (correct to first order in ϵ) the relevant diffusion process in terms of Jacobi fields thereby revealing Kepler’s laws in a new light. The key here is that the constrained Hamiltonian system has just two solutions corresponding to the forward and backward drifts in Nelson’s stochastic mechanics. We discuss the application of this theory to modelling dark matter particles under the influence of a large gravitating point mass.
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.
We describe a multi-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of the corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions and the corresponding photon statistics are discussed. Some applications to quantum optics, cavity quantum electrodynamics and superfocusing in channelling scattering are mentioned. Explicit solutions of the Heisenberg equations for radiation field operators with squeezing are found.