How to apply the WKB approximation in this case?

[a1111]

Homework Statement

I'm trying to learn how to apply the WKB approximation. Given the following problem:

An electron, say, in the nuclear potential

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

1. What is the radial Schrödinger equation for the $\ell=0$ state?

2. Assuming the energy of the barrier (i.e. $k/r_{0}$) to be high, how do you use the WKB approximation to estimate the bound state energies inside the well?

Homework Equations

For the first question, I thought the radial part of the equation of motion was the following

$$\left \{ - {\hbar^2 \over 2m r^2} {d\over dr}\left(r^2{d\over dr}\right) +{\hbar^2 \ell(\ell+1)\over 2mr^2}+U(r) \right \} R(r)=ER(r)$$

The Attempt at a Solution

For the first part, do I simply just let $\ell=0$ and obtain the following? Which of the two potentials do I use?

$$\left \{ - {\hbar^2 \over 2m r^2} {d\over dr}\left(r^2{d\over dr}\right) +U(r) \right \} R(r)=ER(r)$$

For the other question, do I use $\int \sqrt{2m(E-U(r))}=(n+1/2)\hbar π$, where $n=0,1,2,...$ ? If so, what are the turning points? And again, which of the two potentials do I use?

 

[mark726]

For the first question, you are correct in setting $\ell=0$ to obtain the radial Schrödinger equation. However, you should use the potential given in the problem, which is a piecewise function. This means that the potential will be different for $r<r_0$ and $r>r_0$. Therefore, you will need to solve the equation separately for each region and then match the solutions at the boundary $r=r_0$.

For the second question, the WKB approximation can be used to estimate the bound state energies inside the well. The turning points refer to the points where the potential energy equals the total energy of the particle. In this case, the turning points will be at $r=r_0$ and at the point where $E=U(r)$. To use the WKB approximation, you will need to solve the Schrödinger equation for each region, just like in the first question, and then match the solutions at the turning points. This will give you an expression for the energy in terms of the potential and the turning points. You can then use this expression to estimate the bound state energies.