ON THE QUANTUM-MECHANICAL REPRESENTATION IN PHASE-SPACE

The theory of the quantum mechanical representation in phase space recently proposed by Torres-Vega and Frederick is rigorously verified to be extended to the bound systems whose potential energy functions may contain terms of negative powers of coordinate and the quantum Liouville equation in the theory is also generalized to the same systems. It is suggested that the kernels of projection transformation take the forms e(ipq/2 ($) over bar h) and e(-(ipq/2 ($) over bar h)) instead of (1/root 4 pi ($) over bar h)e(ipq/2 ($) over bar h) and (1/root 4 pi ($) over bar h)e(-(ipq/2 ($) over bar h)) in order to keep the consistency for projections of the wave functions in all quantum problems. The validity of quantum average values and the virial theorem in the phase space representation are discussed in detail.