Operators and Expectation Values

Bruce Cameron Reed

Donald H. Kobe

Because of domain considerations for the z component of the angular momentum operator, the time rate of change of the expectation value of the angular displacement φ is not equal to the expectation value of the angular velocity operator. The angular velocity operator is the angular momentum operator divided by the moment of inertia. Ehrenfest’s theorem is obtained for the time rate of change of the expectation value of cos φ and sin φ.

J. Ben-Benjamin, William George Unruh

Abstract We present a method for calculating expectation values of operators in terms of a corresponding c-function formalism which is not the Wigner–Weyl position-momentum phase-space, but another space. Here, the quantity representing the quantum system is the expectation value of the displacement operator, parametrized by the position and momentum displacements, and expectation values are evaluated as classical integrals over these parameters. The displacement operator is found to offer a complete orthogonal basis for operators, and some of its other properties are investigated. Connection to the Wigner distribution and Weyl procedure are discussed and examples are given.

Juan Pablo Paz, Augusto J. Roncaglia

A programmable gate array is a circuit whose action is controlled by input data. In this paper we describe a special-purpose quantum circuit that can be programmed to evaluate the expectation value of any operator O acting on a space of states of N dimensions. The circuit has a program register whose state $|\ensuremath{\Psi}(O){〉}_{P}$ encodes the operator O whose expectation value is to be evaluated. The method requires knowledge of the expansion of O in a basis of the space of operators. We discuss some applications of this circuit and its relation to known instances of quantum state tomography.