time independent Schr\"odinger equation in spherical coordinatesTimeIndependentSchrodingerEquationInSphericalCoordinates2006-01-15 15:34:542006-10-30 21:50:45Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereWhen writing the time independent Schr\"odinger equation in spherical coordinates, we need to plug the Laplacian in spherical coordinates into the time independent Schr\"odinger equation. The Laplacian was found to be
$$
\nabla _{sph}^{2} = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r}\right) + \frac{1}{r^2 sin\theta} \frac{\partial}{\partial \theta} \left( sin \theta \frac{\partial}{\partial \theta}\right) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2}{\partial \phi^2}
$$
Using the three dimensional Schr\"odinger equation we then have
$$ C \psi(r,\theta, \phi) = -\frac{\hbar^2}{2m} \left [ \frac{1}{r^2} \frac{\partial }{\partial r}\left(r^2 \frac{\partial \psi(r,\theta, \phi)}{\partial r}\right) \frac{1}{r^2 sin\theta} \frac{\partial}{\partial \theta} \left( sin \theta \frac{\partial \psi(r,\theta, \phi)}{\partial \theta}\right) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2 \psi(r,\theta, \phi)}{\partial \phi^2} \right ] + V(r,\theta, \phi) \psi(r,\theta, \phi)
$$
{\bf References}
[1] Griffiths, D. "Introduction to Quantum Mechanics" Prentice Hall, New Jersey, 1995.