Introduction to Quantum Mechanics · Problem 9.15

The WKB Approximation — David J. Griffiths

Physics — Quantum Mechanics · wkb-logarithmic-potential-spherical

Problem

For spherically symmetrical potentials we can apply the WKB approximation to the radial part (Equation 4.37). In the case

l = 0

it is reasonable to use Equation 9.48 in the form

\int_0^{r_0} p(r)\,dr = (n - 1/4)\,\pi\hbar,

where

r_0

is the turning point (in effect, we treat

r = 0

as an infinite wall). Exploit this formula to estimate the allowed energies of a particle in the logarithmic potential

V(r) = V_0 \ln(r/a)

(for constants

V_0

and

a

). Treat only the case

l = 0

. Show that the spacing between the levels is independent of mass. *Partial answer:*

E_{n+1} - E_n = V_0 \ln\!\left(\frac{n+3/4}{n-1/4}\right).