arXiv:1906.01709 · Physica Scripta

The ambiguity function and the displacement operator basis in quantum mechanics

J. Ben-Benjamin, William George Unruh

Published Jul 18, 2019• 2 citations

Abstract

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.

DOI: 10.1088/1402-4896/ab3376

Related papers

Quantum operators in Weyl quantization procedure via Wigner representation of quantum mechanics ‐ quantum phase operator as a special case

M. Davidović

Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.

Fortschritte der PhysikFeb 26, 20035 citations

Weyl Transformation and the Classical Limit of Quantum Mechanics

Boris Leaf

The Weyl correspondence for obtaining quantum operators from functions of classical coordinates and momenta is known to be incorrect. To calculate quantum-mechanical expectation values as phase-space averages with the Wigner density function, one cannot use classical functions but must use Weyl transforms. These transforms are defined and their properties derived from quantum mechanics. Their properties are expressed in terms of a Hermitian operator Δ(Q, K) whose Weyl transform is a δ function. The Wigner function is the transform of the density operator. Every Weyl transform is exhibited as a difference of two functions which are nonnegative on the phase spece. Weyl transforms do not obey the algebra of classical functions. In the classical limit ℏ → 0, Weyl transforms become classical functions, the Wigner function becomes nonnegative throughout the phase space, and the Hilbert space is spanned by an orthonormal set of vectors which are simultaneous eigenkets of the commuting coordinate and momentum operators.

Journal of Mathematical PhysicsJan 1, 196898 citations

Wigner distribution for angle coordinates in quantum mechanics

Narasimhaiengar Mukunda

The method of Wigner distribution functions, and the Weyl correspondence between quantum and classical variables, are extended from the usual kind of canonically conjugate position and momentum operators to the case of an angle and angular momentum operator pair. The sense in which one has a description of quantum mechanics using classical phase-space language is much clarified by this extension.

American Journal of PhysicsFeb 1, 197974 citations

Quantum-mechanical phase space: A generalization of Wigner phase-space formulation to arbitrary coordinate systems

Ashok V. Pimpale, Mohsen Razavy

This paper deals with certain aspects of the phase-space formulation of quantum mechanics. Noting the ambiguities involved in quantizing the classical phase-space dynamics, a generalized quantum-mechanical phase space is introduced in terms of eigenvalues of two mutually incompatible complete sets of operators. General definitions are given for the phase-space distribution function corresponding to a given abstract state of the system and the phase-space function for an arbitrary operator. Our formulation goes over to the Wigner formulation when the complete sets correspond to position and momentum operators in Cartesian coordinates. Explicit examples are discussed with use of action-angle and polar coordinates for simple one-, two-, and three-dimensional systems. Symmetry changes in phase space are discussed by applying Gel'fand-Levitan transformations and it is shown that, in general, the degeneracies of motion in phase space are not reflected in the energy eigenvalues.

Physical Review ADec 1, 198810 citations

General approach to quantum mechanics as a statistical theory

R. P. Rundle, Todd Tilma, John H. Samson +3 more

Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 (2016) and Phys. Rev. A 96, 022117 (2017)] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function---the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics---extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. First we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space---putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of ``parity'' and ``displacement'' in a natural broadening of previously developed phase-space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

arXiv:1708.03814Physical Review AJan 16, 201942 citations

Quantum mechanics in phase space

Philip Watson

The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.

Jan 1, 19910 citations