Analysis of classical solutions in Schroedinger equations based on the idea of non-standard analysis and harmonic-oscillator motions

Abstract

Classical limits of quantum mechanics are studied on the basis of non-standard analysis. Following the idea of ultra eigenpairs in quantum mechanics extended to non-standard numbers, the exact equality in the Schroedinger equation is replaced by the approximate equality by taking Ħ as an infinitesimal (Ħ ≈0) and the classical limit is introduced as the operation represented by taking the standard part with respect to Ħ defined in the non-standard analysis. We find out that the ultra eigenfunctions which result from the extention of the exact equality to the approximate equality in the equation represent classical stationary states in the classical limit. As an explicit example we study harmonic-oscillator motions and derive two types of solutions: one is given by the solutions for stationary states described by ultra eigenfunctions on the extended Hilbert space* þ and the other by the solutions for non-stationary states represented by coherent states which are expressed as the superposition of eigenfunctions given by the non-standard extension of Hermite polynomials. We see that the former represents the statistical nature of quantum wave functions, while the latter their particle nature. Both solutions are shown to reproduce classical motions exactly in the classical limit. All interferences among different classical solutions disappear in the limit. We also see that the ultra eigenfunctions representing the stationary states cover not only the extensions of the eigenfunctions for the original potential, but also those for potentals which differ from the original one by an order smaller than Ħ. That is, they are spanned over many Hilbert spaces which differ one another in quantum-mechanical worlds whereĦ = a non-infinitesimal number is required. In the argument for measurements on the stationary states we see that the crucial difference between quantum mechanics and classical mechanics in expressed as follows; quantum mechanics does not have any infinitesimal physical constants in the theory, while classical mechanics does,i.e. Ħ ≈ 0 in this approach. Relations between detectors in physics and filters in non-standard analysis is also pointed out. The Schroedinger equation with the approximate equality based on the idea of the non-standard analysis is proposed as the fundamental equation for the unified theory describing quantum and classical phenomena simultaneously.