Q$1$: For spherically symmetrical potentials we can apply the WKB approximation to the radial part. Using the following quantization condition:

${\displaystyle \underset{0}{\overset{r_0}{}}}p(r)dr=(n-\frac{1}{4}),$

to estimate the allowed energies of a particle in the logarithmic potential of:

$V(r)=V_{0}ln(\frac{r}{a})$

where $r_0$ is the turning point and Show that the spacing between the levels is independent of mass and it is equal to:

$E_{n+1}-E_n=V_{0}ln(\frac{n+\frac{3}{4}}{n-\frac{1}{4}}).$