Phy5645/Hydrogen Atom WKB

Use WKB approximation to estimate energy spectrum for Hydrogen atom.

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The approximation is:

$\int {P(r)}dr=(n+{\frac {1}{2}})\pi \hbar$

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

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

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

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

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

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

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

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{ 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}}

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 \sqrt {2mE} *\frac{\pi }{2}*(\sqrt {r_{2}} -\sqrt {r_{1}} )^{2}=(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 \sqrt {2mE} *\frac{\pi }{2}*(r_{2}+r_{1}-2\sqrt {r_{1}r_{2}} )=(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{let r}^{2}-Vr+T=(r_{1}-r)(r_{2}-r)=r^{2}-(r_{1}+r_{2})+r_{1}r_{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{so V=}(r_{1}+r_{2})\text{ and T=}r_{1}r_{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 \sqrt {2mE} *\frac{\pi }{2}*(V-2\sqrt {T} )=(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 \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}} }

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

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