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arXiv:quant-ph/9909077

Legitimacy of wave-function expansion

Chen, Changye

Published Sep 25, 1999• 1 citation
arXivHTMLPDF

Abstract

In this letter we investigate the common procedure in which any wave function is expanded into a series of eigenfunctions. It is shown that as far as dynamical systems are concerned the expanding procedure involves various mathematical and physical difficulties. With or without introducing phase factors, such expansions do not represent dynamical wave functions.

DOI
10.48550/arxiv.quant-ph/9909077

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In this letter we investigate the common procedure in which any wave function is expanded into a series of eigenfunctions. It is shown that as far as dynamical systems are concerned the expanding procedure involves various mathematical and physical difficulties. With or without introducing phase factors, such expansions do not represent dynamical wave functions.
Any problem is concerned with the mind, but what do minds make a decision on? Here we show that there are three conditions for the mind to make a certain answer. We found that some difficulties in physics and mathematics are in fact introduced by infinity, which can not be rightly expressed by minds. Based on this point, we suggest a general observation system, where we use region (a type of infinity) to substitute for infinitesimal (another type of infinity) and thus get a consistent image with the mind. Furthermore, we declare that without world pictures we can never have ideas to any expressive events, which is the primary condition for a wave function like mind to collapse to a series of numbers. A following observation by expanding algorithm brings the final collapse: classifying the numbers and coming up with a certain yes or no answer.
Quantum dynamical properties of a general time-dependent coupled oscillator are investigated based on the theory of two-dimensional (2D) dynamical invariants. The quantum dynamical invariant of the system satisfies the Liouville–von Neumann equation and it coincides with its classical counterpart. The mathematical formula of this invariant involves a cross term which couples the two oscillators mutually. However, we show that, by introducing two pairs of annihilation and creation operators, it is possible to uncouple the original invariant operator so that it becomes the one that describes two independent subsystems. The eigenvalue problem of this decoupled quantum invariant can be solved by using a unitary transformation approach. Through this procedure, we eventually obtain the eigenfunctions of the invariant operator and the wave functions of the system in the Fock state. The wave functions that we have developed are necessary in studying the basic quantum characteristics of the system. In order to show the validity of our theory, we apply our consequences to the derivation of the fluctuations of canonical variables and the uncertainty products for a particular 2D oscillatory system whose masses are exponentially increasing.