Phy5645/Hydrogen Atom WKB

Use WKB approximation to estimate energy spectrum for Hydrogen atom.

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The approximation is:

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 \int {P(r)} dr=(n+\frac{1}{2})\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 \text{P(r)=}\sqrt {2m(E-V(r))} =\sqrt {2m(E-\left ({\frac{\hbar ^{2}l(l+1)}{2mr^{2}}-\frac{e^{2}}{r}} \right )} )}

$\int \limits _{r1}^{r2}{\sqrt {2m(E-{\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}+{\frac {e^{2}}{r}})}}dr=(n+{\frac {1}{2}})\pi \hbar$ where r1 and r2 are turning points in this case.

${\sqrt {2mE}}\int \limits _{r1}^{r2}{(1-}{\frac {\hbar ^{2}l(l+1)}{2mr^{2}E}}+{\frac {e^{2}}{Er}})^{1/2}dr=(n+{\frac {1}{2}})\pi \hbar$

${\text{let T=}}-{\frac {\hbar ^{2}l(l+1)}{2mE}}{\text{ and V=}}-{\frac {e^{2}}{r}}{\text{ }}$

${\sqrt {2mE}}\int \limits _{r1}^{r2}{(1+{\frac {T}{r^{2}}}}+{\frac {V}{r}})^{1/2}dr=(n+{\frac {1}{2}})\pi \hbar {\text{ }}$

${\text{use the relation let r}}^{2}-Vr+T=(r_{1}-r)(r_{2}-r)$

${\sqrt {2mE}}\int \limits _{r1}^{r2}{\left({\frac {(r_{1}-r)(r_{2}-r)}{r^{2}}}\right)^{1/2}dr=(n+{\frac {1}{2}})\pi \hbar }$

${\text{ use definition let }}\int \limits _{r1}^{r2}{\left({\frac {(x-a)(x-b)}{x^{2}}}\right)^{1/2}dx}={\frac {\pi }{2}}({\sqrt {b}}-{\sqrt {a}})^{2}$

${\sqrt {2mE}}*{\frac {\pi }{2}}*({\sqrt {r_{2}}}-{\sqrt {r_{1}}})^{2}=(n+{\frac {1}{2}})\pi \hbar$

${\sqrt {2mE}}*{\frac {\pi }{2}}*(r_{2}+r_{1}-2{\sqrt {r_{1}r_{2}}})=(n+{\frac {1}{2}})\pi \hbar$

${\text{let r}}^{2}-Vr+T=(r_{1}-r)(r_{2}-r)=r^{2}-(r_{1}+r_{2})+r_{1}r_{2}$

${\text{so V=}}(r_{1}+r_{2}){\text{ and T=}}r_{1}r_{2}$

${\sqrt {2mE}}*{\frac {\pi }{2}}*(V-2{\sqrt {T}})=(n+{\frac {1}{2}})\pi \hbar$

${\sqrt {2mE}}\left({-{\frac {e^{2}}{E}}-2{\sqrt {-{\frac {\hbar ^{2}l(l+1)}{2mE}}}}}\right)=(n+{\frac {1}{2}})\pi \hbar$

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 -e^{2}\sqrt {\frac{2m}{E}} -2\sqrt {\hbar ^{2}l(l+1)} =2\hbar (n+\frac{1}{2})}

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 \text{ }2\hbar (n+\frac{1}{2})+2\hbar \sqrt {l(l+1)} =e^{2}\sqrt {\frac{2m}{-E}} }

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 \frac{4\hbar ^{2}\left ({n+\frac{1}{2}+\sqrt {l(l+1)} } \right )^{2}}{2me^{4}}=\frac{1}{-E}\text{ }}

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 \text{E=}\frac{-me^{4}}{2\hbar ^{2}\left ({n+\frac{1}{2}+\sqrt {l(l+1)} } \right )^{2}}}