Quantization in generalized coordinates—II

Abstract

In a previous article (Gruber, 1971), the author considered what the operator form of the generalized canonical momenta was is quantum mechanics. As noted in the article, Pauli (1950), through a different method, found what the generalized momentum operator was also, and both results (the author's and Pauli's) were in agreement. However, the prescription of incorporating the momentum operator into the Hamiltonian, in some cases does not give the correct form of the Hamiltonian operator. In the present article, the author finds exactly how to incorporate the total momentum operatorp _q1 =iħ∂/∂q _i into the generalized classical Hamiltonian to get the correct quantum mechanical Hamiltonian operator for all cases. The author also shows a clear-cut way of making the transition from classical observable functions of the canonical momenta to their quantum mechanical operator analogs, in generalized spaces.