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Homework Help: How to find the wave function using WKB?

  1. Dec 15, 2013 #1
    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
  2. jcsd
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