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| <math>\ \frac{d^{2}W}{ds^{2}}+\frac{2m}{\hbar^{2}}\left[E-V(e^{s})-\frac{\hbar^2}{2m}\left(l+\frac{1}{2}\right)^{2}e^{-2s}\right]e^{2s}=0.</math> | | <math>\ \frac{d^{2}W}{ds^{2}}+\frac{2m}{\hbar^{2}}\left[E-V(e^{s})-\frac{\hbar^2}{2m}\left(l+\frac{1}{2}\right)^{2}e^{-2s}\right]e^{2s}=0.</math> |
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| In this case the [[WKB Approximation#Bohr-Sommerfeld Quantization Rule|Bohr-Sommerfeld quantization rule]] becomes: | | In this case the [[WKB Approximation#Bound States Within the WKB Approximation|Bohr-Sommerfeld quantization rule]] becomes: |
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| <math>\ \int_{r_1}^{r_2}\sqrt{2m\left(E_n - V(r) - \frac{\hbar^2}{2m}\frac{(\ell+\frac{1}{2})^2}{r^2}\right)}dr = \left(n + \frac{1}{2}\right)\pi\hbar </math> | | <math>\ \int_{r_1}^{r_2}\sqrt{2m\left(E_n - V(r) - \frac{\hbar^2}{2m}\frac{(\ell+\frac{1}{2})^2}{r^2}\right)}dr = \left(n + \frac{1}{2}\right)\pi\hbar </math> |
Revision as of 02:15, 13 January 2014
It is possible to apply the WKB approximation to the radial equation using a method by R. E. Langer (1937).
Recall that
and that
satisfies the effective one-dimensional Schrödinger equation,
We now perform the following transformations:
Note that, for
The radial equation becomes
In this case the Bohr-Sommerfeld quantization rule becomes:
Worked Problem
Worked by team
WKB method for the Coulomb Potential
For the coulomb potential, the potential is given by:

Since the electron is bound to the nucleus, it can be veiwed as moving between two rigid walls at
and
with energy
. Thus, the energy of the electron is negative.
The energies of the s-state (
) can be obtained from:

Using the change of variable:

Where I have used the integral

Thus we have the expression:


Where
is the Bohr radius. Notice that this is the correct expression for the energy levels of a Coulomb potential.
Calculation of Gamow factor using WKB Aprroximation Method