Derivation of the Husimi symbols without antinormal ordering, scale transformation and uncertainty relations

Abstract

We propose a new method for the derivation of Husimi symbols, for operators that are given in the form of products of an arbitrary number of coordinates, and momentum operators, in an arbitrary order. For such an operator, in the standard approach, one expresses coordinate and momentum operators as a linear combination of the creation and annihilation operators, and then uses the antinormal ordering to obtain the final form of the symbol. In our method, one obtains the Husimi symbol in a much more straightforward fashion, departing directly from operator explicit form without transforming it through creation and annihilation operators. With this method the mean values of some operators are found. It is shown how the Heisenberg and the Schrodinger-Robertson uncertainty relations, for position and momentum, are transformed under scale transformation (q; p) - GT (lambda q; lambda p). The physical sense of some states which can be constructed with this transformation is also discussed.