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ON A GENERALISED COMMUTATION RELATION

R. O'Raifeartaigh, Colm A. Ryan

Published Jan 1, 1963• 0 citations
OpenAlex

Abstract

The method of the complete set is used to study a generalized commutation relation. This commutation relation is of interest for two reasons. First, it is very similar to the commutation relations considered by various authors in connection with generalized statistics. Secondly, it allows us to investigate further a problem previously considered by Wigner, namely, Do the equations of motion determine the quantum mechanical commutation relations? '' (auth)

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The method of the complete set is used to study a generalized commutation relation. This commutation relation is of interest for two reasons. First, it is very similar to the commutation relations considered by various authors in connection with generalized statistics. Secondly, it allows us to investigate further a problem previously considered by Wigner, namely, Do the equations of motion determine the quantum mechanical commutation relations? '' (auth)
It has been shown by Wigner that the equations of motion of a quantum-mechanical system with a hamiltonian H=12p2+V(q)H=\frac{1}{2}{p}^{2}+V(q)H=21​p2+V(q) need not imply the commutation relation i[p,q]=1i[p,q]=1i[p,q]=1. Particular examples are furnished by the free particle and the harmonic oscillator. On the other hand, Wigner points out that the above commutation rule does follow from the equations of motion if V(q)V(q)V(q) is of the form V=aq+bV=aq+bV=aq+b or V=aq3V=a{q}^{3}V=aq3. In the present note, this latter result is generalized to the case of an arbitrary potential V=aqn+bV=a{q}^{n}+bV=aqn+b, where nnn is an odd integer; the last-named potentials are, in fact, particular cases of those satisfying a general criterion contained in the article.
The problem of whether or not the equations of motion of a quantum system determine the commutation relations was posed by E.P.Wigner in 1950. A similar problem (known as "The Inverse Problem in the Calculus of Variations") was posed in a classical setting as back as in 1887 by H.Helmoltz and has received great attention also in recent times. The aim of this paper is to discuss how these two apparently unrelated problems can actually be discussed in a somewhat unified framework. After reviewing briefly the Inverse Problem and the existence of alternative structures for classical systems, we discuss the geometric structures that are intrinsically present in Quantum Mechanics, starting from finite-level systems and then moving to a more general setting by using the Weyl-Wigner approach, showing how this approach can accomodate in an almost natural way the existence of alternative structures in Quantum Mechanics as well.