Introduction to Quantum Mechanics · Problem 9.13

The WKB Approximation — David J. Griffiths

Physics — Quantum Mechanics · wkb-power-law-potential-energies

Problem

Use the WKB approximation to find the allowed energies of the general power-law potential:

V(x) = \alpha |x|^\nu,

where

\nu

is a positive number. Check your result for the case

\nu = 2

. *Answer:*

E_n = \alpha \left[(n-1/2)\,\hbar\sqrt{\frac{\pi}{2m\alpha}}\frac{\Gamma\!\left(\frac{1}{\nu}+\frac{3}{2}\right)}{\Gamma\!\left(\frac{1}{\nu}+1\right)}\right]^{\!\left(\frac{2\nu}{\nu+2}\right)}.