Center-of-Mass Motion in Many-Particle Systems. II. Critique of the Gartenhaus-Schwartz Transformation

Abstract

We examine the Gartenhaus and Schwartz (G-S) transformation, which generates a translationally invariant wave function from one which does not possess this property. A clarification of the G-S procedure for the calculation of matrix elements is achieved if one takes two independent limits of the parameters $\ensuremath{\Lambda}$, ${\ensuremath{\Lambda}}^{\ensuremath{'}}$ which appear in the expression for a transformed operator ${\stackrel{\ensuremath{\rightarrow}}{\mathfrak{a}}}^{\ensuremath{'}}=({\mathrm{lim}}_{{\ensuremath{\Lambda}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{{\ensuremath{\Lambda}}^{\ensuremath{'}}}^{\ifmmode\dagger\else\textdagger\fi{}})\stackrel{\ensuremath{\rightarrow}}{\mathfrak{a}}({\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{\ensuremath{\Lambda}})$. It is shown that the G-S operator, $U={\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{\ensuremath{\Lambda}}={\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}\mathrm{exp}[\ensuremath{-}\frac{1}{2}i\ensuremath{\Lambda}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}}+\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}})]$, is equivalent to the operator $U={(2\ensuremath{\pi})}^{3}\ensuremath{\delta}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}})\ensuremath{\delta}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}})$. This operator is seen to be a specific example of the more general class of operators presented earlier.