PlanetPhysics/Time Independent Schrodinger Equation in Spherical Coordinates

When writing the time independent Schr\"odinger equation in spherical coordinates, we need to plug the [[../LaplacianInSphericalCoordinates/|Laplacian in Spherical Coordinates]] into the [[../TimeIndependentSchrodingerEquation/|time independent Schr\"odinger equation]]. The [[../LaplaceOperator/|Laplacian]] was found to be

$\underset{sph}{\overset{2}{\mathrm{\nabla }}}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{1}{r^{2}sin\theta }\frac{\partial }{\partial \theta }\left(sin\theta \frac{\partial }{\partial \theta }\right)+\frac{1}{r^{2}si{n}^2\theta }\frac{{\partial }^2}{\partial {\varphi }^2}$

Using the three dimensional Schr\"odinger equation we then have

$\hat{H}\psi (r,\theta ,\varphi )=-\frac{{\hslash }^2}{2m}[\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial \psi (r,\theta ,\varphi )}{\partial r}\right)+\frac{1}{r^{2}sin\theta }\frac{\partial }{\partial \theta }\left(sin\theta \frac{\partial \psi (r,\theta ,\varphi )}{\partial \theta }\right)+\frac{1}{r^{2}si{n}^2\theta }\frac{{\partial }^2\psi (r,\theta ,\varphi )}{\partial {\varphi }^2}]+V(r,\theta ,\varphi )\psi (r,\theta ,\varphi )=E\psi (r,\theta ,\varphi )$

We can gain insight into this somewhat ugly equation by rewriting it using the [[../PiecewiseLinear/|square]] of the [[../MolecularOrbitals/|angular momentum]] [[../QuantumOperatorAlgebra4/|operator]] in spherical polar coordinates:

${\hat{L}}^2=\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial }{\partial \theta })+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{\partial {\varphi }^2}$

This leads to

$(-\frac{{\hslash }^2}{2m}\left(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)\right)+\frac{1}{2m}\frac{{\hat{L}}^2}{r^2}+V(r,\theta ,\varphi ))\psi (r,\theta ,\varphi )=E\psi (r,\theta ,\varphi )$

Spherically symmetric separable solution

This equation is only exactly solvable if $V=V(r)$, a [[../Bijective/|function]] without angular dependence. We then write $\psi (r,\theta ,\varphi )=R(r)Y(\theta ,\varphi )$ leading to the following equation:

$\begin{array}{cc}(-\frac{{\hslash }^2}{2m}\left(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)\right)+\frac{1}{2m}\frac{{\hat{L}}^2}{r^2}+V(r,\theta ,\varphi ))\psi (r,\theta ,\varphi )R(r)Y(\theta ,\varphi )& =ER(r)Y(\theta ,\varphi )\\ -\frac{{\hslash }^2}{2m}(Y(\theta ,\varphi )\left(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)\right)R(r))+\frac{R(r)}{2m}\frac{{\hat{L}}^{2}Y(\theta ,\varphi )}{r^2}+V(r)R(r)Y(\theta ,\varphi )& =ER(r)(Y(\theta ,\varphi )\end{array}$

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, $Y_l^m(\theta ,\varphi )$. We also know that these have eigenvalues ${\hslash }^{2}l(l+1)$, i.e.

${\hat{L}}^2{Y}_l^m(\theta ,\varphi )={\hslash }^{2}l(l+1)Y_l^m(\theta ,\varphi )$

We now substitute this result into the Schr\"odinger equation and divide through by a common factor of $Y_l^m(\theta ,\varphi )$

$(-\frac{{\hslash }^2}{2m}(\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{{\hslash }^{2}l(l+1)}{r^2})+V(r))R(r)=ER(r)$

This is the [[../RadialEquation/|radial equation]].\\ {\mathbf References}

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