A displacement parameter such as the angle of rotation or the position of a quantum system, or the phase of a harmonic oscillation, is to be estimated by observing the system with an apparatus that applies to it an operator-valued measure (o.v.m.). The o.v.m. minimizing the average cost of errors in the estimate is determined by quantum estimation theory for a system in a pure state. The best estimate of the parameter is found to be the more accurate, the greater the uncertainty of the dynamical variable serving as the infinitesimal generator of the displacement group. The relation of this result to such uncertainty principles as those between angle and angular momentum, and between oscillator phase and photon number, is discussed. A lower bound to the variance of an unbiased estimate of the time of occurrence of an event in the evolution of a system is derived from the quantum-mechanical Cramer-Rao inequality. It is inversely proportional to the square of the uncertainty of the energy of the system.

Estimation of a Displacement Parameter of a Quantum System

The formalism developed in Part I of the present paper for the conjugate momentum of label operators is applied to standard physical examples, such as the regular one-dimensional lattice, the one-dimensional box, and the angular momentum of the plane rotator. A non-standard application is also considered, i.e., the timeoperator, conjugate to the one-dimensional harmonic Hamiltonian.

Canonical conjugate momentum of discrete label operators in quantum mechanics II: formalism

The concept of an azimuthal-angle operator canonically conjugate to the z component of angular-momentum operator is known to present certain difficulties in quantum mechanics. These difficulties are resolved rigorously for the first time, and their implications are discussed.

G. J. Troup

Papers

Is there an Azimuthal Angle Observable