Phy5645/Hydrogen Atom WKB: Difference between revisions
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Use WKB approximation to estimate energy spectrum for Hydrogen atom. | Use WKB approximation to estimate energy spectrum for Hydrogen atom. | ||
Hints: | |||
<math>\text{use the relation r}^{2}-Vr+T=(r_{1}-r)(r_{2}-r)</math> | |||
and <math>\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} | |||
</math> | </math> | ||
---- | ---- | ||
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<math>\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 }</math> | <math>\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 }</math> | ||
<math>\text{ | <math>\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}</math> | ||
<math>\sqrt {2mE} *\frac{\pi }{2}*(\sqrt {r_{2}} -\sqrt {r_{1}} )^{2}=(n+\frac{1}{2})\pi \hbar </math> | <math>\sqrt {2mE} *\frac{\pi }{2}*(\sqrt {r_{2}} -\sqrt {r_{1}} )^{2}=(n+\frac{1}{2})\pi \hbar </math> | ||
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<math>\frac{4\hbar ^{2}\left ({n+\frac{1}{2}+\sqrt {l(l+1)} } \right )^{2}}{2me^{4}}=\frac{1}{-E}\text{ }</math> | <math>\frac{4\hbar ^{2}\left ({n+\frac{1}{2}+\sqrt {l(l+1)} } \right )^{2}}{2me^{4}}=\frac{1}{-E}\text{ }</math> | ||
Then if we finally pull out E, | |||
<math>\text{E=}\frac{-me^{4}}{2\hbar ^{2}\left ({n+\frac{1}{2}+\sqrt {l(l+1)} } \right )^{2}}</math> | <math>\text{E=}\frac{-me^{4}}{2\hbar ^{2}\left ({n+\frac{1}{2}+\sqrt {l(l+1)} } \right )^{2}}</math> | ||
Revision as of 20:32, 9 December 2009
Use WKB approximation to estimate energy spectrum for Hydrogen atom. Hints:
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 the relation r}^{2}-Vr+T=(r_{1}-r)(r_{2}-r)} 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 \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} }
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 )} )}
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\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.
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{\hbar ^{2}l(l+1)}{2mr^{2}E}+\frac{e^{2}}{Er})^{1/2}dr=(n+\frac{1}{2})\pi \hbar } if we do this 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 \text{let T=}-\frac{\hbar ^{2}l(l+1)}{2mE}\text{ and V=}-\frac{e^{2}}{r}\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 \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}}}