| Abstract: | It has been suggested by several authors [1, 2] that quantum mechanical canonical transformations may be generalized by admitting partially isometric operators instead of unitary transformations used so far [3, 4]. It is known that it is possible to transform a Heisenberg couple into a corresponding one in a different Hilbert space. We shall show that the operators Q = VqV+ and P = VpV+ obtained in this way — which are unitarily equivalent to EqE and EpE, respectively, in the initial domain M of V onto which E projects — though symmetric in general will not be selfadjoint, and also present an example of this. Although it does not seem to be possible to settle the question of the existence of self-adjoint extensions definitely in the general case, the example of operators generated from the Schrödinger couple q and p shows the existence of such extensions having the spectrum of angle and z-component of angular momentum. Transducing the argument further we shall show that by choosing a different subspace N ? M ?? H it is well possible to generate the action and phase operator of the quantum mechanical harmonic oscillator with the correct spectrum. |
