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Homework Help: How to apply the WKB approximation in this case?

  1. Dec 14, 2013 #1
    1. The problem statement, all variables and given/known data

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

    An electron, say, in the nuclear potential

    & -U_{0} \;\;\;\;\;\;\text{ if } r < r_{0} \\
    & k/r \;\;\;\;\;\;\;\;\text{ if } r > r_{0}

    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?

    2. Relevant equations

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

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

    3. 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?
  2. jcsd
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