Hypervirial theorems and co-ordinate transformation in quantum mechanics

Abstract

The tensor virial theorem was derived in the framework of both classical and quantum mechanics (1.4). The virial theorem is a particular case of the hypervirial theorems for a special choice of the hypervirial operator and its validity is restricted to stationary states (s) as well as for scattering states (6). A method to derive the quantum virial theorem consists in using the cxtrcmum property of (/~ and a dilatation transformation (7). From this point of view, one possible application of the tensor virial theorem consists in the introduction of more than one scale factor in a trial wave function. This scaling method proposes different stretehings for the different spatial co-ordinates. Examples were given where the introduction of multiple scale factors and the imposition of the tensor virial theorem yield a better result than the usual procedure of subjecting the wave function to a single scale transformation and imposing the scalar virial theorem (3,s). But the most general co-ordinate transformation involves not only stretehings and rotations, but translations too. The purpose of the present communication is to analyse a general co-ordinate transformation through an optimization of the functional energy for a variational wave function, in which the variational parameters are associated to the geometric transformations of stretehings, rotations and translations. We shah follow the procedure of Pandres (2) and Einstein's convention of summation over repeated indices. Here, and henceforth, ~a~ is the i-th Cartesian co-ordinate of the ~-th particle of a N-particle system. We use the summation convention that repeated Greek subscripts are summed from 1 to N, while repeated Latin subscripts are summed from 1 to 3. Let the original set of Cartesian co-ordinates {~) be subject to the transformation