Phy5645/Hydrogen Atom WKB

The WKB approximation is given by

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_{r_1}^{r_2} p(r)\,dr=(n+\tfrac{1}{2})\pi \hbar,}

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 p(r)=\sqrt {2m(E-V_{\text{eff}}(r))} =\sqrt {2m\left (E+\frac{e^{2}}{r}-{\frac{\hbar ^{2}(l+\tfrac{1}{2})^2}{2mr^{2}}}\right )}.}

We may rewrite the above as

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}\int_{r_1}^{r_2}\sqrt{1-\frac{\hbar^{2}(l+\tfrac{1}{2})^2}{2mEr^{2}}+\frac{e^{2}}{Er}}\,dr=(n+\tfrac{1}{2})\pi \hbar,}

or, making the substitution,

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 T=-\frac{\hbar^{2}(l+\tfrac{1}{2})}{2mE}} and 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 V=\frac{e^{2}}{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 \sqrt{2mE}\int_{r_1}^{r_2}\sqrt{1-\frac{V}{r}+\frac{T}{r^{2}}}\,dr=(n+\tfrac{1}{2})\pi \hbar.}

Using the fact that 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 r^{2}-Vr+T=(r_{1}-r)(r_{2}-r)\!} and that

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_{r_1}^{r_2}\sqrt{{\frac{(x-r_1)(x-r_2)}{x^{2}}}}\,dx=\frac{\pi }{2}(\sqrt {r_2} -\sqrt {r_1} )^{2},}

we obtain

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{\pi}{2}\sqrt {2mE}(\sqrt {r_{2}} -\sqrt {r_{1}} )^{2}=(n+\tfrac{1}{2})\pi \hbar.}

We now observe that 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 r^{2}-Vr+T=(r_{1}-r)(r_{2}-r)=r^{2}-(r_{1}+r_{2})r+r_{1}r_{2},\!} so that

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 V=r_{1}+r_{2}\!} and 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 T=r_{1}r_{2}.\!}

We thus obtain

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{\pi }{2}\sqrt {2mE}(V-2\sqrt {T} )=(n+\tfrac{1}{2})\pi \hbar,}

or

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+\tfrac{1}{2})^2}{2mE}} } \right )=(n+\tfrac{1}{2})\pi \hbar.}

This equation simplifies to

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\ell+1)\hbar =(2n+1)\hbar.}

We may now easily solve for 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,\!} obtaining

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=-\frac{me^{4}}{2\hbar ^{2}(n+\ell+1)^{2}}.}

Note that this is exactly the spectrum that we would obtain from an exact solution of the Coulomb problem.

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