On the quantization of the potential amplitude in the Schrödinger equation

Abstract

A consistent scheme is proposed for quantizing the potential amplitude in the Schrödinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ${\mathit{\phi }}_{\mathit{\nu }}\mathrm{(}\mathit{r}\mathrm{)}$ and eigenvalues ${\mathit{\alpha }}_{\mathit{\nu }}$ corresponding to zero, small, and large absolute values of energy $\mathit{E}\mathrm{<}0$ are analyzed. Expansion in the set ${\mathit{\phi }}_{\mathit{\nu }}\mathrm{(}\mathit{r}\mathrm{)}$ is used to develop a regular perturbation theory (for $\mathit{E}\mathrm{<}0$ ), and a general expression is found for the Green function associated with the time-independent Schrödinger equation. A similar method is used to solve several physical problems: the polarizability of a bound quantum-mechanical system, the two-center problem, and the elastic scattering of slow particles. The proposed approach is advantageous in that it does not require the use of continuum states (for $\mathit{E}\mathrm{>}0$ ).