Harmonic oscillations and rotations in quantum theory

Similarly to the classical connection between simple harmonic motion and rotation about an axis there exists the possibility of a unified quantum treatment of angle and harmonic phase in the case of the electromagnetic field mode. This can be accomplished within the framework of a single mathematical construction based on the tensor product of the Hilbert spaces of two harmonic oscillators. The construction can be used to obtain PV extensions of the harmonic oscillator phase POV measure and define relative phase measurements.; We have examined the limits placed by quantum mechanics on the variance of an ideal phase measurement, along with the improvement that can be achieved with the use of a collapsible relative phase measurement. While the optimizing input states were determined and some of their properties studied, no suggestions have been made about experimental generation of such states. The similarity of the quantum angle measurement to that of the relative phase measurement was exploited to find optimum input states that give the least variance in the angle variable of axial rotation. For sufficiently small values of J the optimizing states were shown to be close to the states of maximum angular momentum projection along the direction that is perpendicular to the rotation axis and lies in the plane of the most probable angle value. These two types of states become essentially different for higher values of J.; The description of the simultaneous measurement of two spin 1/2 components of angular momentum was also accomplished. Different methods for the derivation of appropriate overcomplete sets of vectors were presented for the case of components at right angle and the more general case of components at an arbitrary angle. The results were applied to exploring how the violation of a Bell's inequality depends on the ideal nature of the quantum measurements involved, showing how the violation of the inequality brakes down as we move away from the ideal quantum measurements. This result could be of substantial interest in quantum information processing and quantum encryption, where the restrictions on the statistics of measurement outcomes pertaining to widely separated parts of entangled systems are essential.