A Precaution Needed in Using the Phase-space Formulation of Quantum Mechanics

Recently there has been renewed interest in applications of the so-called phase-space formulation of quantum mechanics’) for both equilibrium2) and non-equilibrium phenomena3). Its basic feature is to provide a framework for the treatment of quantum mechanical problems in terms of classical phasespace language. Following the appearance of the well-known Wigner distribution function’), many other distribution functions have been considered. For instance, there is anti-normal ordered (smoothed Wigner distribution4) and the normal-ordered5) (P-)distribution functions, the anti-standard-ordered6) and the standard-ordered’) distribution functions. Each of those distribution functions was created for a particular purpose. In most cases, the Wigner distribution function is the simplest one to use8-1o ). But it can assume negative values. The smoothed Wigner distribution function (SWDF) is obtained by convolution of the Wigner distribution function with a phase-space Gaussian function and is always non-negative. It has been reported recently”) that employment of the SWDF in calculations for nonlinear systems, such as an anharmonic oscillator and the hydrogen atom, causes severe discrepancy with the results obtained by standard (Schrodinger) quantum mechanics. We are able to show that this disagreement may be attributed to an incorrect use of the SWDF. It is the purpose of the present paper to clarify this point and illustrate how to obtain correct results by use of the SWDF.