General quantum mechanical canonical point transformations

Abstract

Problems related to the operator form of the generalized canonical momenta in quantum mechanics are resolved by use of the general quantum mechanical canonical point transformation method. This method can be applied to any general canonical point transformation irrespective of the relationship between the domains of the original and transformed variables. The differential representation of the original canonical momenta pi in the original coordinate space is −i\(\begin{array}{*{20}c} / \\ h \\ \end{array}\) ∂/∂x i and of the transformed canonical momentap i ′ in the transformed coordinate space is −i\(\begin{array}{*{20}c} / \\ h \\ \end{array}\) ∂/∂x i ′. Relationships are derived between the eigenvalues of the original and transformed momenta in either space. The ordering problem for general point transformations is discussed and is shown to be solved. As an example of the generality of the method, it is demonstrated on the point transformation in three dimensions from Cartesian rectilinear to spherical rectilinear coordinates.