I'm trying to learn how to apply WKB. I asked a similar question already, but that question was related to finding the energies. Here, I would like to understand how to find the wave functions using WKB.
An electron, say, in the nuclear potential
$$U(r)=\begin{cases}
& -U_{0} \;\;\;\;\;\;\text{ if } r < r_{0} \\
& k/r \;\;\;\;\;\;\;\;\text{ if } r > r_{0}
\end{cases}$$
What is the wave function inside the barrier region ($r_{0} < r < k/E$)?
Shouldn't the wave function have the following form?
$$\psi(r)=\frac{A}{\sqrt{2m(E-U(r))}}e^{\phi(r)}+\frac{B}{\sqrt{2m(E-U(r))}}e^{-\phi(r)}$$
where
$$\phi(r)=\frac{1}{\hbar}\int_{0}^{r} \sqrt{2m(E-U(r))} dr'$$
Best Answer
Inside vs outside, there is a sign change inside the square root, so that changes the nature of the "phase" $\phi(r)$.
Normally, when you match wave functions you require that $\psi_\mathrm{left}(x) = \psi_\mathrm{right}(x)$ (continuity) and that the derivative changes according to what you get when you integrate the Schrodinger equation: $\int_\mathrm{left}^\mathrm{right}\!dx\, \left( -\frac{\hbar^2}{2m} \frac{d ^2\psi(x)}{dx^2} + V(x) \psi(x) \right) = 0$ but in fact your WKB solutions are only approximations and they get worse as $V(x) \approx E$. So your textbook (Griffiths) devotes a few pages to deriving "matching conditions" instead. I suggest you start there and come back with specific questions if you are unsure about how to use those conditions.