Talk:PlanetPhysics/Time Independent Schrodinger Equation in Spherical Coordinates

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: time independent Schr\"odinger equation in spherical coordinates %%% Primary Category Code: 03.65.-w %%% Filename: TimeIndependentSchrodingerEquationInSphericalCoordinates.tex %%% Version: 6 %%% Owner: bloftin %%% Author(s): bloftin, invisiblerhino %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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When writing the time independent Schr\"odinger equation in spherical coordinates, we need to plug the \htmladdnormallink{Laplacian in Spherical Coordinates}{http://planetphysics.us/encyclopedia/LaplacianInSphericalCoordinates.html} into the \htmladdnormallink{time independent Schr\"odinger equation}{http://planetphysics.us/encyclopedia/TimeIndependentSchrodingerEquation.html}. The \htmladdnormallink{Laplacian}{http://planetphysics.us/encyclopedia/LaplaceOperator.html} 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

\[ \hat{H} \psi(r,\theta, \phi) = -\frac{\hbar^2}{2m} \left [ \frac{1}{r^2} \frac{\partial }{\partial r}\left(r^2 \frac{\partial \psi(r,\theta, \phi)}{\partial r}\right) + \frac{1}{r^2 sin\theta} \frac{\partial}{\partial \theta} \left( sin \theta \frac{\partial \psi(r,\theta, \phi)}{\partial \theta}\right) + \frac{1}{r^2 sin^2 \theta} \frac{\partial^2 \psi(r,\theta, \phi)}{\partial \phi^2} \right ] + V(r,\theta, \phi) \psi(r,\theta, \phi) = E\psi(r, \theta, \phi) \] We can gain insight into this somewhat ugly equation by rewriting it using the \htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} of the \htmladdnormallink{angular momentum}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html} \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} in spherical polar coordinates: \[ \hat{L}^2 = {1 \over \sin\theta} {\partial\over\partial\theta}\left(\sin\theta{\partial\over\partial\theta}\right) +\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial \phi^2} \] This leads to \[ \left(-\frac{\hbar^2}{2m}\left({1 \over r^2} {\partial\over\partial r}\left(r^2{\partial\over\partial r}\right) \right)+\frac{1}{2m}\frac{\hat{L}^2}{r^2}+V(r, \theta, \phi)\right)\psi(r, \theta, \phi) = E\psi(r, \theta, \phi) \] \subsection{Spherically symmetric separable solution} This equation is only exactly solvable if $V=V(r)$, a \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} without angular dependence. We then write $\psi(r, \theta, \phi) = R(r)Y(\theta, \phi)$ leading to the following equation: \begin{align*} \left(-\frac{\hbar^2}{2m}\left({1 \over r^2} {\partial\over\partial r}\left(r^2 \frac{\partial}{\partial r}\right)\right)+\frac{1}{2m}\frac{\hat{L}^2}{r^2}+V(r, \theta, \phi)\right)\psi(r, \theta, \phi)R(r) Y(\theta, \phi) &= E R(r) Y(\theta, \phi)\\ -\frac{\hbar^2}{2m} \left( Y(\theta, \phi) \left( \frac{1}{r^2}{\partial \over \partial r}\left(r^2 \frac{\partial}{\partial r}\right) \right)R(r) \right) + \frac{R(r)}{2m}\frac{\hat{L}^2 \,Y(\theta, \phi)}{r^2} + V(r)R(r)Y(\theta, \phi) &= E R(r)(Y(\theta, \phi) \end{align*} To solve this equation we need to remove the angular dependence. This is simply done by substituting the eigenfunctions of $\hat{L}^2$ into the equation. These are known to be the spherical harmonics, $Y^m_l(\theta, \phi)$. We also know that these have eigenvalues $\hbar^2 l(l+1)$, i.e. \[ \hat{L}^2 \,Y_l^m(\theta, \phi) = \hbar^2 l(l+1) Y^m_l(\theta, \phi) \] We now substitute this result into the Schr\"odinger equation and divide through by a common factor of $Y_l^m(\theta, \phi)$ \[ \left( -\frac{\hbar^2}{2m} \left(\frac{1}{r^2}{\partial \over \partial r} \left(r^2 \frac{\partial}{\partial r}\right) +\frac{\hbar^2l(l+1)}{r^2}\right) + V(r) \right) R(r) = E R(r) \] This is the \htmladdnormallink{radial equation}{http://planetphysics.us/encyclopedia/RadialEquation.html}.\\ {\bf References}

[1] Griffiths, D. "Introduction to Quantum Mechanics" Prentice Hall, New Jersey, 1995.

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