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Journal of Physics A Mathematical and General

Semiclassical mechanics in one dimension: II. Approximate matrix elements

Peter Kasperkovitz, Momtchil Peev

Published Mar 6, 1998• 2 citations
Publisher

Abstract

The semiclassical formula for matrix elements, sometimes called the Heisenberg correspondence principle, relates matrix elements (operators in energy representation) to phase space functions (Weyl representatives). The formula does not make sense for arbitrary operators; when it is valid it implicitly fixes the relative phases of the eigenfunctions of the Hamiltonian. The conventions, which have to be used for the wavefunctions in position or momentum representation, are given here in explicit form, and we present a class of operators related to coherent states of high energy for which the Heisenberg correspondence principle holds.

DOI
10.1088/0305-4470/31/9/011

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The semiclassical formula for matrix elements, sometimes called the Heisenberg correspondence principle, relates matrix elements (operators in energy representation) to phase space functions (Weyl representatives). The formula does not make sense for arbitrary operators; when it is valid it implicitly fixes the relative phases of the eigenfunctions of the Hamiltonian. The conventions, which have to be used for the wavefunctions in position or momentum representation, are given here in explicit form, and we present a class of operators related to coherent states of high energy for which the Heisenberg correspondence principle holds.
A semiclassical expression for matrix elements of an arbitrary operator with respect to the eigenstates of an integrable Hamiltonian is derived. This is essentially the Heisenberg correspondence principle, and it is shown via the Weyl correspondence that the approximation is valid through the lowest two orders in \ensuremath{\Elzxh}. The result is used to prove that an asymptotic form of the Clebsch-Gordan coefficients for two large and one small angular momenta is valid through two orders. \textcopyright{} 1996 The American Physical Society.
Heisenberg's form of the correspondence principle for non-relativistic matrix elements has been used to evaluate matrix elements for various potentials. These are compared with available quantum mechanical results to check the general validity of the use of classical mechanics and correspondence principles in the derivation of quantum mechanical expressions. Using a one dimensional harmonic oscillator potential matrix elements of the form (n mod qm mod n+s) and (n mod pm mod n+s) are worked out. A Morse potential is then considered and the matrix elements for position, momentum, and kinetic energy are evaluated. Using a Coulomb potential a similar procedure is used for functions of various position coordinates, from which dipole and quadrupole moments from transitions between n, l states are calculated. The agreement with quantum mechanics is found to be generally good, and in some cases identical results are obtained.
Semiclassical bound state theory is considered from the viewpoint of the phase space representations of quantum mechanics. In the small h/ limit, within a representation defined by the ordering chosen for the noncommuting position and momentum operators, we obtain: (1) equations of motion, which need not be classical and (2) correspondence principles in action-angle variables, regarded as those initial conditions on the equations of motion which dominate in the small h/ limit. The Weyl ordering is seen to generate the classical equations of motion and the normal correspondence result. Possible evaluation of semiclassical bound state energies is discussed by means of the eigenvalue approach to classical mechanics used by Prigogine and co-workers.
For physical systems with Hamiltonians analytic in the position and momentum operators the diagonal matrix elements of the density operator in the coherent state representation may be considered as a phase-space probability measure in the stochastic formulation of quantum mechanics. The correspondence rule relating quantum-mechanical observables to ordinary functions in phase space is obtained. It is shown that the dynamics of the time-dependent phase-space probability densities is again representable in the form of a stochastic process.
This paper presents a semiclassical theory for the computation of matrix elements of the type 〈u‖v〉 when either one or both ‖u〉 and ‖v〉 are coherent states (in different representations). Our results can be considered as an extension of Miller’s semiclassical theory [Adv. Chem. Phys. 25, 69 (1974)]. Such an extension has been presented also by Heller [J. Chem. Phys. 66, 5777 (1977)]. We were able to simplify considerably some of Heller’s results by exploiting the canonical properties of the classical coherent variables. This enabled us to relate the elements 〈u‖v〉 to certain generalized, complex generator functions in a manner that is very similar to the relations that appear in the original Miller’s theory. The advantages that are inherent in the coherent states representation are illustrated in a few elementary examples. We were able to derive an excellent approximation to the eigenstates of the harmonic oscillator which is valid even for the ground state. Furthermore, we have demonstrated that it is possible to describe semiclassically dynamical processes that are classically forbidden by real time trajectories in a certain generalized phase space.