time independent Schrödinger equation in spherical coordinatesTimeIndependentSchrodingerEquationInSphericalCoordinates2006-01-15 15:34:542006-01-15 15:34:54Definition% 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 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 version of the S.E. we then have
$$ C \psi(r,\theta, \phi) = -\frac{\hbar^2}{2m} \left [ \frac{1}{r^2} \frac{\partial }{\partial r}\left(r^2 \frac{\partial \psi(r,\theta, \phi)}{\partial r}\right) \frac{1}{r^2 sin\theta} \frac{\partial}{\partial \theta} \left( sin \theta \frac{\partial \psi(r,\theta, \phi)}{\partial \theta}\right) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2 \psi(r,\theta, \phi)}{\partial \phi^2} \right ] + V(r,\theta, \phi) \psi(r,\theta, \phi)
$$