- #1
Juqon
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Homework Statement
Calculate the zero-point energy of the hydrogen atom using the Ritz variaton principle and the approach [itex]\psi_{\alpha}[/itex].
Hint: The stationary Schroedinger equation in spherical coordinates is ...
Homework Equations
[tex] \left\{ -\frac{\hbar^{2}}{2\mu}\left[\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial}{\partial r}\right]+\frac{\hat{\vec{L}}^{2}}{2\mu r^{2}}+V(r)\right\} \psi=E\psi [/tex][tex]
\psi_{\alpha}(\vec{r})=C\exp\left\{ -\alpha r\right\} ;C>0;\alpha>0; C,\alpha=const. [/tex] [tex]
\int_{0}^{\infty}dr\cdot r\cdot\exp\left\{ -2\alpha r\right\} =\frac{1}{4}\alpha^{2} [/tex][tex]
\int_{0}^{\infty}dr\cdot r^{2}\cdot\exp\left\{ -2\alpha r\right\} =\frac{1}{4}\alpha^{3} [/tex]
The Attempt at a Solution
[tex]
\left\{ -\frac{\hbar^{2}}{2\mu}\left[\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial}{\partial r}\right]+\frac{\hat{\vec{L}}^{2}}{2\mu r^{2}}+V(r)\right\} \psi=E\psi [/tex][tex]
\Leftrightarrow \left\{ -\frac{ \hbar^{2} }{2\mu} \left [\frac{\partial^{2}} {\partial r^{2}}+ \frac{2}{r} \frac {\partial} {\partial r} \right]+ \frac {\left(\hbar^{2} l(l+1) \right)^{2}} {2\mu r^{2}}+\frac{1} {4\pi\epsilon_{0} } \frac{qQ}{r} \cdot C \cdot \exp -\alpha r \right\} \cdot C\cdot\exp\left\{ -\alpha r\right\} = E \cdot C\cdot\exp\left\{ -\alpha r\right\} [/tex][tex]
\Leftrightarrow \left\{ -\frac{\hbar^{2}}{2\mu}\left [\alpha^{2}-\frac{2}{r}\alpha\right] +\frac{\hbar^{4}l^{2} (l^{2}+2l+1)}{2\mu r^{2}} +\frac{1}{4\pi\epsilon_{0}}\frac{qQ} {r}\right\} \cdot C\cdot\exp\left\{ -\alpha r\right\} = E \cdot C\cdot\ exp\left\{ -\alpha r\right\} [/tex][tex]
\Leftrightarrow E=\left\{ -\frac{\hbar^{2}}{2\mu}\left[\alpha^{2}-\frac{2}{r}\alpha \right]+\frac{\hbar^{4}l^{2}(l^{2}+2l+1)}{2\mu r^{2}}+\frac{1}{4\pi\epsilon_{0}} \frac{qQ}{r}\right\} [/tex]
Where do I need to integrate at all?
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