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Journal of Chuxiong Normal University

The deductive method for the operator theory of the uncertainty relation of one-dimensional harmonic oscillator

Luo Ling-xiao

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

The uncertainty relation of one-dimensional harmonic oscillator is deduced with the operator theory.

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A quantum state corresponds to a specific wave function. We adopt a new mathematical method [1] to improve Dirac's ladder operator method. A set of orthonormal wave functions will be used to associate the operator with the square matrix corresponding to it. These allow us to determine the matrix elements by using the operator relations without having to know the specific wave functions. As a result, we can get the direct results of matrix mechanics and wave mechanics on one-dimensional Harmonic oscillator and their descriptions will be also unified.
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.