Abstract
The set of all pairs of polynomials X(x,p) and P(x,p) in the position x and momentum p are found such that X and P have the same commutator as x and p. Several other types of representations of the commutation relations are studied. The reducibility, time, and space inversion properties of these are discussed. The unitary transformations connecting different representations are exhibited. The use of different representations to solve scattering and bound state problems is indicated. The application of one of the transformations to a partial diagonalization of a perturbed harmonic oscillator Hamiltonian is made.
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