18 September 2000 Ž . Physics Letters A 274 2000 93–97 www.elsevier.nlrlocaterpla The formulation of Fermi’s golden rule in phase space D. Dragoman ) UniÕersity Bucharest, Physics Department, P.O. Box MG-11, 76900 Bucharest, Romania Received 19 November 1999; received in revised form 30 May 2000; accepted 13 August 2000 Communicated by P.R. Holland Abstract It is shown that the transition probability between two discrete energy levels, in the framework of time-dependent perturbation theory, can be expressed in terms of the Wigner distribution functions of these states. This phase space formulation of Fermi’s golden rule generalizes a previous result valid for Franck–Condon transitions. q 2000 Published by Elsevier Science B.V. PACS: 03.65 Keywords: Quantum theory; Quantum transitions 1. Introduction The phase space treatment of quantum mechanics and quantum optics has proven itself useful in ex- pressing averages of quantum operators in terms of distribution functions, defined on the quantum me- Ž . chanical phase space q, p . Here q and p are the position and momentum coordinates, which are real functions, not operators. There is a whole class of wx such phase space distribution functions 1 that sat- isfy the requirements of bilinearity in the wavefunc- tion and those of recovering the probability distribu- ) Correspondence address: P.O. Box 1-480, 70700 Bucharest, Romania, Tel.rFax: q 40-1-6473382. Ž . E-mail address: ddragoman@hotmail.com D. Dragoman . tions to find the quantum particle in the q or p space by a partial integration of the phase space distribu- tion function over p and q, respectively. Among these distributions one of the most extensively used Ž . wx is the Wigner distribution function WDF 2 . Apart from the above-mentioned properties it is also real but not positive defined. Moreover, the WDF is the only real phase space distribution function for which Ž . the transition probability between two states c q Ž . and f q can be expressed in terms of the corre- wx sponding distribution functions as 3 : 2 c ) q f q d q Ž . Ž . H s 2p " W q , pW q , p d q d p 1 Ž . Ž . Ž. H c f 0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. Ž . PII: S0375-9601 00 00530-2