Phy5645/Hydrogen Atom WKB: Difference between revisions

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The approximation is:
The WKB approximation is given by


<math>\int {P(r)} dr=(n+\frac{1}{2})\pi \hbar</math>
<math>\int_{r_1}^{r_2} p(r)\,dr=(n+\tfrac{1}{2})\pi \hbar,</math>


<math>\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 )} )</math>
where


<math>\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 </math> where r1 and r2 are turning points in this case.
<math>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 )}.</math>


<math>\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 </math>
We may rewrite the above as
if we do this substitution:
 
<math>\text{let T=}-\frac{\hbar ^{2}l(l+1)}{2mE}\text{  and V=}-\frac{e^{2}}{r}\text{    }</math>
<math>\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,</math>
 
or, making the substitution,
 
<math>T=-\frac{\hbar^{2}(l+\tfrac{1}{2})}{2mE}</math> and <math>V=\frac{e^{2}}{E},</math>


<math>\sqrt {2mE} \int\limits_{r1}^{r2} {(1+\frac{T}{r^{2}}}+\frac{V}{r})^{1/2}dr=(n+\frac{1}{2})\pi \hbar \text{ }</math>
<math>\sqrt {2mE} \int\limits_{r1}^{r2} {(1+\frac{T}{r^{2}}}+\frac{V}{r})^{1/2}dr=(n+\frac{1}{2})\pi \hbar \text{ }</math>

Revision as of 02:43, 13 January 2014

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\limits_{r1}^{r2} {(1+\frac{T}{r^{2}}}+\frac{V}{r})^{1/2}dr=(n+\frac{1}{2})\pi \hbar \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{the relation r}^{2}-Vr+T=(r_{1}-r)(r_{2}-r)}

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\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{ the definition }\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}} }

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{ }}

Then if we finally pull out 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 \text{E=}\frac{-me^{4}}{2\hbar ^{2}\left ({n+\frac{1}{2}+\sqrt {l(l+1)} } \right )^{2}}}

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