Quantization in Polar Coordinates and the Phase Operator

Abstract

We review some of the difficulties previously encountered in defining the phase operator for finite quantum systems. We then propose the Wigner–Weyl quantization of the angle function $\phi$ on phase space as the phase operator, and derive a closed expression for its matrix elements with respect to the Hermite functions. We also determine the quantization of $e^{i\phi }$, which turns out to be a weighted shift operator, its spectrum and that of its adjoint.

This is done in the framework of quantization of a certain symbol class of phase space distributions, specialized to those which depend on one variable only. After recalling some results for the position and momentum variables, we apply the scheme to functions of radius or angle. We give necessary and sufficient conditions for operators obtained by quantizing functions of the angle to be elements of ${\mathcal{L}}^{+}[\mathcal{S}(\mathbf{R})]$ and ${\mathcal{L}}^{+}[\mathcal{S}(\mathbf{R}),L^2(\mathbf{R})]$, and a sufficient condition for boundedness.

We then consider the associated questions of commutation relations and uncertainties for operators in ${\mathcal{L}}^{+}[\mathcal{S}(\mathbf{R}),L^2(\mathbf{R})]$, which we define as bilinear forms. As must be the case, the commutator between our phase operator and the number operator exhibits noncanonical terms. Not surprisingly, the Poisson bracket of their phase space symbols also exhibits a noncanonical term.