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| ==WKB method for the Coulomb Potential ==
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| For the coulomb potential, the potential is given by:
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| :<math> V(r) = -\frac{-Ze^2}{r} </math>
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| Since the electron is bound to the nucleus, it can be veiwed as moving between two rigid walls at <math> r = 0 \!</math> and <math> r = a \!</math> with energy <math> E = V(a), a = -\frac{-Ze^2}{E}\!</math>. Thus, the energy of the electron is negative.
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| The energies of the s-state (<math> \ell = 0 \!</math>) can be obtained from:
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| :<math> \int_0^a \sqrt{2m\left(E+\frac{Ze^2}{r}\right)}dr = n\pi\hbar </math>
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| Using the change of variable: <math> x = \frac{a}{r} </math>
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| :<math>
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| \begin{align}
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| \int_0^a \sqrt{2m\left(E+\frac{Ze^2}{r}\right)}dr &= \sqrt{-2mE} \int_0^a dr \sqrt{\frac{a}{r} - 1} \\
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| &= a\sqrt{-2mE} \int_0^1 dx\sqrt{\frac{1}{x} - 1} \\
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| &= \frac{\pi}{2}a\sqrt{-2mE} \\
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| &= -Ze^2\pi\sqrt{-\frac{2m}{E}}
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| \end{align}
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| </math>
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| Where I have used the integral
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| :<math> \int_0^1\sqrt{\frac{1}{x} -1} = \frac{\pi}{2} </math>
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| Thus we have the expression:
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| :<math>-Ze^2\pi\sqrt{-\frac{2m}{E}} = n\pi\hbar </math>
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| :<math>\Rightarrow E_n = -\frac{mZ^2e^4}{\hbar^2} = -\frac{Z^2e^2}{2a_0}</math>
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| Where <math> a_0\!</math> is the Bohr radius. Notice that this is the correct expression for the energy levels of a Coulomb potential.
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| [[Phy5645/Gamowfactor|Calculation of Gamow factor using WKB Aprroximation Method]] | | [[Phy5645/Gamowfactor|Calculation of Gamow factor using WKB Aprroximation Method]] |
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,
![{\displaystyle \left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}+V(r)-E\right]u(r)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29583ed7b829c27f7bfc7be11741f5e5444e42eb)
We now perform the following transformations:
Note that, for 
The radial equation becomes
![{\displaystyle \ {\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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41c418bf1db3ab3f0a6b285a08d1ca1accfc1d18)
In this case the Bohr-Sommerfeld quantization rule becomes:
Worked Problem
Worked by team
Calculation of Gamow factor using WKB Aprroximation Method
