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Orion1
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Hydrogen normalized position wavefunctions in spherical coordinates:
[tex]\Psi_{n \ell m}\left(r,\theta,\phi\right) = \sqrt{{\left( \frac{2}{n r_1} \right)}^3 \frac{\left(n - \ell - 1\right)!}{2n\left[\left(n + \ell\right)!\right]}} e^{-\frac{r}{n r_1}} \left({2r \over {n r_1}}\right)^{\ell} L_{n - \ell - 1}^{2 \ell + 1}\left(\rho\right) \cdot Y_{\ell}^{m}\left(\theta, \phi\right)[/tex]
Time independent Schrodinger equation:
[tex]i \hbar \frac{\partial \Psi\left(\mathbf{r},\,t\right)}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi\left(\mathbf{r},\,t\right) + V\left(\mathbf{r}\right)\Psi\left(\mathbf{r},\,t\right)
\end{equation*}[/tex]
Three dimensional Laplace operator:
[tex]\nabla^2 \Psi\left(\mathbf{r},\,t\right) = \frac{1}{r^2 \sin \theta} \left[\sin \theta \frac{d}{dr} \left(r^2 \frac{d\Psi}{dr}\right) + \frac{d}{d \theta}\left(\sin \theta \frac{d \Psi}{d\theta}\right) + \frac{1}{\sin \theta} \frac{d^2 \Psi}{d\phi^2}\right][/tex]
Differential time independent Hydrogen Schrodinger equation:
[tex]E\left(r\right) \Psi \left(r,\theta,\phi\right) = - \frac{\hbar^2}{2 \mu} \frac{1}{r^2 \sin \theta} \left[\sin \theta \frac{d}{dr} \left(r^2 \frac{d\Psi}{dr}\right) + \frac{d}{d \theta}\left(\sin \theta \frac{d \Psi}{d\theta}\right) + \frac{1}{\sin \theta} \frac{d^2 \Psi}{d\phi^2}\right] + U\left(r\right)\Psi \left(r,\theta,\phi\right)[/tex]
Relativistic Dirac equation:
[tex]i \hbar \frac{\partial \Psi\left(\mathbf{r},t\right)}{\partial t} = \left(\beta mc^2 + \sum_{k = 1}^3 \alpha_k p_k \, c\right) \Psi \left(\mathbf{r},t\right) + V\left(\mathbf{r}\right)\Psi\left(\mathbf{r},\,t\right)
\end{equation*}[/tex]
If the Schrodinger equation Laplace operator is describing a plane wave in spherical coordinates, what type of wave and coordinates does the relativistic Dirac equation describe?
What is the solution to the relativistic Hydrogen Dirac equation that is analogous to the differential time independent Hydrogen Schrodinger equation?
Reference:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html"
http://en.wikipedia.org/wiki/Hydrogen_atom#Mathematical_summary_of_eigenstates_of_hydrogen_atom"
http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation"
http://en.wikipedia.org/wiki/Dirac_equation"
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