Quantum theory of rotation angles

Abstract

The formulation of the quantum description of the rotation angle of the plane rotator has been beset by many of the long-standing problems associated with harmonic-oscillator phases. We apply methods recently developed for oscillator phases to the problem of describing a rotation angle by a Hermitian operator. These methods involve use of a finite, but arbitrarily large, state space of dimension 2l+1 that is used to calculate physically measurable quantum properties, such as expectation values, as a function l. Physical results are then recovered in the limit as l tends to infinity. This approach removes the indeterminacies caused by working directly with an infinite-dimensional state space. Our results show that the classical rotation angle observable does have a corresponding Hermitian operator with well-determined and reasonable properties. The existence of this operator provides deeper insight into the quantum-mechanical nature of rotating systems.