New Perspective on Inner Product Quantization

Abstract

We devise a new and highly accurate quantization procedure for the inner product representation, both in configuration and momentum space. Utilizing the representation $\ensuremath{\Psi}(\ensuremath{\xi})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{i}^{}{a}_{i}[E]{\ensuremath{\xi}}^{i}{R}_{\ensuremath{\beta}}(\ensuremath{\xi})$, for an appropriate reference function, ${R}_{\ensuremath{\beta}}(\ensuremath{\xi})$, we demonstrate that the (convergent) zeros of the coefficient functions, ${a}_{i}[E]\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$, approximate the exact bound state energies with increasing accuracy as $i\ensuremath{\rightarrow}\ensuremath{\infty}$. The validity of the approach is shown to be based on an approximation to the Hill determinant quantization procedure. Our method has been applied, with remarkable success, to various quantum mechanical problems in one and two space dimensions.