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Journal de physique

Semiclassical matrix mechanics. I. The harmonic oscillator

Richard M. More

Published Jan 1, 1990• 2 citations
Publisher

Abstract

Matrices of dynamical variables of the one-dimensional harmonic oscillator are calculated by a new semiclassical method formulated by More and Warren. The results are semiclassical matrices Xn, m, Pn, m, etc., which i) exactly obey selection rules for zero matrix-elements, ii) are very accurate, circa 1 %, for nonzero matrix-elements, iii) which diagonalize the Hamiltonian H, and iv) which exactly satisfy the Heisenberg commutation relations and equations of motion. These statements are true for matrix-elements Fn, m with m ≽ n. The results show that most of the quantum physics is contained in the extended semiclassical theory.

DOI
10.1051/jphys:0199000510104700

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Matrices of dynamical variables of the one-dimensional harmonic oscillator are calculated by a new semiclassical method formulated by More and Warren. The results are semiclassical matrices Xn, m, Pn, m, etc., which i) exactly obey selection rules for zero matrix-elements, ii) are very accurate, circa 1 %, for nonzero matrix-elements, iii) which diagonalize the Hamiltonian H, and iv) which exactly satisfy the Heisenberg commutation relations and equations of motion. These statements are true for matrix-elements Fn, m with m ≽ n. The results show that most of the quantum physics is contained in the extended semiclassical theory.
Semiclassical angular momentum matrices are calculated using a new contour-integral formula for matrix-elements. WKB spherical harmonic functions are found to be exaclty orthonormal with the contour-integral inner-product. Matrix-elements obtained from these wave-functions are accurate to about 2 % and the matrices obey the commutation relations expected of quantum angular momentum operators. The semiclassical wave-functions are related to a superposition of allowed classical orbits and this illustrates the connection to the Feynman pathsummation formulation of quantum mechanics for electrons in a spherically symmetric potential.
The original version of matrix mechanics is little used in applications nowadays. In that early theory position, momentum and other quantities were represented by infinite matrices, whereas many modern applications of quantum mechanics represent such observables as operators. Infinite matrices are unwieldy to use and have some disturbing properties, e.g. they do not obey the associative rule. By far the most commonly appearing matrix in quantum mechanics is the matrix of the energy operator as set up in some finite basis of trial functions. The procedure has been used for perturbed oscillator problems; the recurrence relation approach is particularly simple when the matrix has only one non-zero element beyond the diagonal element in each row. It seems to the author that there are still many hybrid methods which need investigation.
For an anharmonic oscillator, it follows from the number of zeros of the exact wave functions as functions of the quantum number nnn that the matrix elements $〈n|x|{n}^{\ensuremath{'}}〉$ and $〈n|p|{n}^{\ensuremath{'}}〉$ should be rapidly decreasing functions of $|n\ensuremath{-}{n}^{\ensuremath{'}}|$. Matrix elements of polynomials in xxx and ppp should therefore be well approximated by a finite number of terms in their sum-rule decomposition. From the matrix elements of the equations of motion for xxx and for ppp and of the commutator ∣x,p∣|x,p|∣x,p∣, one thereby obtains closed sets of nonlinear algebraic equations to characterize subspaces of the Hilbert space of exact eigenfunctions. The approximations are also derived from a novel variational principle, and numerous variant approximation schemes are suggested. Essentially exact numerical results are obtained and compared with previous work. The broad applicability of the techniques is emphasized.
Abstract In both classical and quantum physics the harmonic oscillator is one of the most important physical systems [1–4]. Furthermore, because of the simplicity and symmetry of the Hamiltonian operator, the quantum mechanical oscillator is particularly suited to analysis by abstract operator methods. For this reason the study of the one dimensional oscillator is a useful introduction to the application of operator methods in quantum mechanics. In Sections 2.1–2.4 we use these methods to determine energy eigenvalues and eigenkets, and matrix elements; the analysis is extended to isotropic multidimensional oscillators in Section 2.10. Quasiclassical states of the oscillator are discussed in Section 2.5. In Section 2.6 shift operators are used to calculate wave functions. In Sections 2.7–2.11 we discuss some applications of the theory of the oscillator.
Matrix mechanic methods are used to find approximate equations and solutions for quantum anharmonic oscillator problems. A series of hypotheses are introduced that truncate and partially decouple the infinite set of coupled equations that specify the problem in the matrix mechanics formulation. The dependent variables or unknowns in these equations are the matrix elements of the coordinate and momentum operators. The independent variables are the matrix indices and coupling strengths. The equations themselves specify that the off diagonal matrix elements of the Hamiltonian and the commutators expressed in terms of the unknowns vanish and that the diagonal commutator matrix elements vanish except for canonical pairs in which case they are equal to −ih/. The truncation and decoupling hypotheses offer an orderly procedure for dealing approximately with the vast array of equations of the exact problems. Only the leading behavior of the coordinate and momentum operator matrix elements is found in terms of the matrix indices and coupling parameters. Although general techniques are presented to find the equations, the solutions discussed and the applications are brief extensions of problems that have already been treated.
After a brief review of the extensive work done on the theory of invariant tori and their quantization, we show that nevertheless an important connection between the quantum and classical theories remains to be exploited. This is the relationship between matrix elements of operators in the energy diagonal representation and Fourier components of the corresponding classical dynamical variables that was the basis for Heisenberg's invention of quantum mechanics. We describe a number of previously unknown or little-known aspects of this relationship, with special emphasis on variational principles and the connection between commutation relations and quantization of action variables. As a single illustration of the utility of these ideas we show that it is possible to obtain approximate solutions to the quantum scheme that are more accurate than the semiclassical approximation with little additional effort compared to the latter.