Quantization as an Eigenvalue Problem

Erwin Schrödinger

Harold V. McIntosh

Erwin Schrödinger

Akira Yoshioka, Kiyotaka Ii

In the presence of a magnetic field, the Maslov quantization condition is not available in the original form. An alternative quantization condition is proposed with the aid of a principal U(1) bundle over a phase space and a connection whose curvature form is the charged symplectic form. By means of this quantization condition, quasiclassical eigenvalues of the Kepler problem with a centrifugal potential and Dirac’s monopole field are calculated, which turn out to coincide with the eigenvalues of the quantized problem.

М. Н. Сергеенко

The eigenvalue problem in quantum mechanics is reduced to quantization of the classical action of the physical system. State function of the system, $ψ_0(ϕ)$, is written in the form of superposition of two plane waves in the phase space. Quantization condition is derived from the basic requirements of continuity and finiteness for $ψ_0(ϕ)$ in the whole region.

Bruno Cordani

In Chapter 4 we have quantized the Kepler Problem using the Schrödinger method, i.e., we have found the eigenvalues and the eigenfunctions of the Hermitian operator $$\begin{array}{*{20}{c}} {\hat{H} = \frac{1}{2}{{{\hat{p}}}^{2}} - \frac{1}{q},} & {where} & {{{{\hat{p}}}_{k}} = - i\frac{\partial }{{\partial {{q}^{k}}}}.} \\ \end{array}$$