1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

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

    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?

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: How to apply the WKB approximation in this case?
Loading...