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Lecture notes in physics

Operator relations, the eigenvalue problem, and representability for quantum phase space distributions

Leon W. Cohen

Published Jan 1, 1987• 1 citation
Publisher
DOI
10.1007/3-540-17894-5_360

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We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c-function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi-representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc.
In a previous paper a certain new probability distribution function q(z) relating to blackbody radiation was introduced. In the present paper the properties of this function for a general radiation field are studied. Unlike the phase-space distribution function of Sudarshan (1963), this function is nonnegative and is an ordinary function. A series expansion for q(z) is given, and it is shown that the series is absolutely convergent for all eigenvalues z of the destruction operator. It is also shown that the density matrix in the Fock representation can be uniquely determined from this probability distribution function, and vice versa. The relation between q(z) and the Sudarshan's phase-space distribution function is discussed.
The problem of quantization in the phase-space formulation of quantum mechanics is considered. An integral equation for the phase-space eigenfunctions is derived which is equivalent to the standard eigenvalue equation for a quantum mechanical operator. A differential form is also given. A variational principle is derived for quasiprobability distributions. It is shown that the expected value of the classical Hamiltonian calculated with a trial quasiprobability distribution will be greater than the ground state energy only if the distribution is chosen from a certain class of functions. The notion of ψ-representability is introduced to classify these functions. They represent distributions which correspond to possible quantum mechanical states. Also, a general relation is given between different distribution functions.