# How to find the wave function using WKB?

1. Dec 15, 2013

### a1111

I would like to understand how to find wave functions using WKB.

1. The problem statement, all variables and given/known data

Given an electron, say, in the nuclear potential

$$U(r)=\begin{cases} & -U_{0} \;\;\;\;\;\;\text{ if } r < r_{0} \\ & k/r \;\;\;\;\;\;\;\;\text{ if } r > r_{0} \end{cases}$$

With the barrier region given by:

$$r_{0} < r < k/E$$

What is the wave function inside the barrier region?

2. Relevant equations

See below.

3. The attempt at a solution

For E > U(r):

$$\psi(r)=\frac{Ae^{i\phi(x)}}{\sqrt{2m(E-U(r)}}+\frac{Be^{-\phi(x)}}{\sqrt{2m(E-U(r)}}$$

$$\phi(r)=\frac{1}{\hbar}\int^{r}_{0}\sqrt{2m(E-U(r')}=\frac{1}{\hbar}\int^{r}_{0}\sqrt{2m(E+U_{0})}dr'=\frac{r\sqrt{2m(E+U_{0})}}{\hbar}$$

Is the above integral correct? Is that simply all that I needed to do?

Last edited: Dec 15, 2013