WKB in Spherical Coordinates: Difference between revisions
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It is possible to apply the [[WKB Approximation|WKB approximation]] to the radial equation using a method by R. E. Langer (1937). | It is possible to apply the [[WKB Approximation|WKB approximation]] to the radial equation using a method by R. E. Langer (1937). | ||
Recall | Recall that | ||
<math>u(r)=rR(r),\!</math> | |||
and that <math>u(r)\!</math> satisfies the effective one-dimensional [[Schrödinger Equation|Schrödinger equation]], | |||
<math>\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.</math> | |||
We now perform the following transformations: | |||
<math>\ | <math>\ r = e^{s};</math> <math>\ u(r) = W(s)e^{\frac{1}{2}s}</math> | ||
Note that, for <math>0<r<\infty,\!</math> <math>-\infty<s<\infty.\!</math> The radial equation becomes | |||
<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> | |||
In this case the [[WKB Approximation#Bohr-Sommerfeld Quantization Rule|Bohr-Sommerfeld quantization rule]] becomes: | In this case the [[WKB Approximation#Bohr-Sommerfeld Quantization Rule|Bohr-Sommerfeld quantization rule]] becomes: |
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:
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:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -Ze^2\pi\sqrt{-\frac{2m}{E}} = n\pi\hbar }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow E_n = -\frac{mZ^2e^4}{\hbar^2} = -\frac{Z^2e^2}{2a_0}}
Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_0\!} is the Bohr radius. Notice that this is the correct expression for the energy levels of a Coulomb potential.