The Heisenberg–Weyl Ring in Quantum Mechanics

Abstract

This chapter discusses the Heisenberg–Weyl ring in quantum mechanics. A program has been carried out that deals with quantum mechanics on a compact one-dimensional space. The system was defined as a quantum system whose momentum operator has a discrete, infinite spectrum of equally spaced eigenvalues. The importance of canonical transformations in quantum mechanics was recognized within a year of its original formulation. Point transformations have been used as transformation groups, the recent applications of which include many-body and scattering problems, while the role of linear canonical transformations has recently been appreciated. The transformation of a given physical system to a mathematically simpler one is a common technique in classical mechanics, usually by taking one of the new canonically conjugate observables to be a constant of motion as the Hamiltonian or the angular momentum. Linear transformations—in a higher dimensional space—become nonlinear when the radial part is isolated in a differential operator realization or when the requirement of the conservation of the commutator bracket is demanded between well-chosen states on a particular basis.