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Quantized discrete space oscillators

Charles A. Uzes, E. Kapuścik

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

A quasi-canonical sequence of finite dimensional quantizations was found which has canonical quantization as its limit. In order to demonstrate its practical utility and its numerical convergence, this formalism is applied to the eigenvalue and 'eigenfunction' problem of several harmonic and anharmonic oscillators.

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In quantum mechanics the linear harmonic oscillator representatives of the canonical observables q and p are certain well-known infinite square matrices, qmn and pmn, with denumerable sets of rows and columns. It is pedagogically useful to investigate various problems which arise when one considers the finite matrices Qmn and Pmn which result when qmn and pmn are truncated after the Nth row and column, where N( 2≥2) is some preassigned positive integer. So that the enquiry may be more instructive, it begins with an abstract formulation of the problem of the “truncated harmonic oscillator.” Formally, the latter has the usual Hamiltonian H = 12( Q2+P2 ) , but Q and P are not canonical—their commutator [Q,P] now differing from i by a certain projection operator. The spectrum of H and the (finite) basis { |n〉} generated by H are found by standard methods, though there are now some complications which are absent from the theory of the linear harmonic oscillator. The diagonalization of Q and P presents no difficulties if full use is made of the classical theory of Hermite polynomials. The spectrum of Q is the set of N zeros hs of the Nth Hermite polynomial. Although the wavefunctions 〈 Q′|n〉 are therefore functions of the discrete variable hs, one can formally contemplate them as functions of a continuous real variable h; and some curious intimations of Schrödinger representation appear. The analog of a component of orbital angular momentum is considered, and its spectrum obtained. Finally, the asymptotic limit N → ∞ is briefly discussed, in order thus explicitly to dispel the impression that this limit might somehow represent the “usual” linear harmonic oscillator.
Assume a solved quantum-mechanical problem for the one-dimensional Schr\"odinger equation, which has a discrete spectrum. An algorithm is presented to calculate exactly the change in the original potential and eigenfunctions brought about by arbitrary changes in the positions of the original eigenvalues and/or the normalizations of the corresponding eigenfunctions. As a first example, we consider the modification of the harmonic oscillator and thereby obtain potentials for anharmonic oscillators with exact eigenvalues and eigenfunctions. Next, we introduce potentials in the one-dimensional box with rigid walls which alter the usual spectrum in a predetermined way. We also sketch the process of obtaining a change in the Coulomb potential which deletes the lowest eigenvalue for the zero-angular-momentum equation.