Schrödinger's operator - (ℏ/2m)[d2/dr2 + (2/r)d/dr] + V(r) is studied, and what happens when V(r) approaches - ∞ rapidly as r→∞ is shown. The cases in which V(r)∼ - βr δ (β>0, δ > 2) as r→∞are covered. If V(r) approaches - ∞ rapidly, then the above operator is not self-adjoint and in order to get a self-adjoint operator a boundary condition must be imposed. For such a self-adjoint operator there are states that belong to the discrete energy spectrum. To obtain the discrete energy spectrum, a quantization rule that corresponds to the quantization rule of Bohr and Sommerfeld in old quantum mechanics is considered.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics