On the Wigner distribution function for an oscillator

✍ Scribed by R.W Davies; K.T.R Davies

• Publisher: Elsevier Science
• Year: 1975
• Tongue: English
• Weight: 579 KB
• Volume: 89
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(75)90182-7

✦ Synopsis

We present two new derivations of the Wigner distribution function for a simple harmonic oscillator Hamiltonian. Both methods are facilitated using a formula which expresses the Wigner function as a simple trace. The first method of derivation then utilizes a modification of a theorem due to Messiah. An alternative procedure makes use of the coherent state representation of an oscillator. The Wigner distribution function gives a semiclassical joint probability for finding the system with given coordinates and momenta, and the joint probability is factorable for the special case of an oscillator. An important application of this result occurs in the theory of nuclear fission for calculating the probability distributions for the masses, kinetic energies, and vibrational energies of the fission fragments at infinite separation.

1. Introduction

In classical statistical mechanics the canonical distribution function,f(X, ..-X, , Pl *.. PN), for the joint probabilities of the coordinates and momenta of a dynamical system is factorable for any Hamiltonian of the form Yf = 2 (g + V(X,)).