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I would like to understand how to find wave functions using WKB.
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?
See below.
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?
Homework Statement
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?
Homework Equations
See below.
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?
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