The rotational wigner function

Abstract

The quantum mechanics of angular momentum is studied using the Wigner phase-space distribution function. To accomplish this in a way which allows one to retain the bilinear form of the distribution function originally conceived by Wigner, a new set of generalized variables is introduced, viz. the Cayley-Klein (C.K.) coordinates and momenta. The equations relating the four C.K. coordinates to the three Euler angles give rise to a constraint which is relaxed in a natural way by introducing a fourth “radial” variable ϱ whose conjugate momentum is p ϱ . Physical quantities in the classical mechanics of rotational systems are re-expressed in the C.K. representation; and the canonical transformation between the C.K. and Eulerian variables is established. In a similar manner, after introducing the basic quantum mechanical operators for the C.K. coordinates and momenta, the wavefunctions and operators for rotational systems are re-expressed in these variables. With the aid of the Weyl correspondence to link the operators of quantum mechanics with the dynamical functions of classical mechanics, the phase-space formulation is established. Rotational Wigner distribution functions are constructed, their properties investigated, and their form is explicitly given for several quantum states. Expectation values are then calculated according to the procedure identical to the one in classical statistical mechanics for calculating phase space averages. The equation of motion for the distribution function is derived. It turns out to be the sine expansion of the Poisson-bracket operator expressed in the C.K. variables and operating on the Hamiltonian and distribution function. This equation reduces to Liouville's equation in the classical limit. The Heisenberg picture of the phase-space formulation is introduced, and the equations of motion for the body-fixed components of the angular momentum are studied. In the classical limit Euler's equations of motion are recovered. Finally, the thermodynamic properties of a system in equilibrium are investigated. The temperature dependence of the ensemble Wigner function is shown to be given by a cosine expansion of the Poisson bracket operator expressed in the C.K. variables, and operating on the Hamiltonian and distribution function. An ansatz is then made on the form of the distribution function, specifically, a power series in Planck's constant with coefficients which are functions of the coordinates and momenta. Substitution of this postulated form into the phase-space Bloch equation allows an explicit determination of these functions. Integration of the Wigner function over all momenta gives the configurational distribution function. A further integration over the coordinates gives the partition function. This procedure is carried out in detail for the asymmetric top, and the first quantum corrections to both the configurational distribution function and partition function are obtained. Upon setting appropriate moments of inertia equal to each other, these quantum corrections reduce to those of the symmetrical and spherical tops.