Separate the time-independent Schrödinger equation (3.22) in spherical coordinates assuming that $\mathit{V}\mathbf{=}\mathit{V}\mathbf{\left(}\mathbf{r}\mathbf{\right)}$is independent of $\mathit{\theta }$and $\mathit{\varphi }$. (If V depends only on r , then we are dealing with central forces, for example, electrostatic or gravitational forces.) Hints: You may find it helpful to replace the mass m in the Schrödinger equation by M when you are working in spherical coordinates to avoid confusion with the letter m in the spherical harmonics (7.10). Follow the separation of (7.1) but with the extra term $\mathbf{[}\mathbf{V}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{-}\mathbf{E}\mathbf{]}\mathit{\Psi }$. Show that the $\mathit{\theta }\mathbf{,}\mathit{\varphi }$solutions are spherical harmonics as in (7.10) and Problem 16. Show that the r equation with $\mathit{k}\mathbf{=}\mathit{l}\mathbf{(}\mathbf{l}\mathbf{+}\mathbf{1}\mathbf{)}$is [compare (7.6)].

$\frac{\mathbf{1}}{\mathbf{R}}\frac{\mathbf{d}}{\mathbf{d}\mathbf{r}}\mathbf{\left(}{\mathbf{r}}^{\mathbf{2}}\frac{\mathbf{d}\mathbf{R}}{\mathbf{d}\mathbf{r}}\mathbf{\right)}\mathbf{-}\frac{\mathbf{2}\mathbf{M}{\mathbf{r}}^{\mathbf{2}}}{{\mathbf{h}}^{\mathbf{2}}}\mathbf{[}\mathbf{V}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{-}\mathbf{E}\mathbf{]}\mathbf{=}\mathit{l}\mathbf{(}\mathbf{l}\mathbf{+}\mathbf{1}\mathbf{)}$

The time independent Schrodinger equation is as below.

$\Delta \Psi -\frac{2M}{{\hslash }^2}(V\left(r\right)-E)\Psi =0$.

The Schrödinger equation is a partial differential equation that governs a quantum-mechanical system's wave function.

Write Laplace operator in spherical coordinates.

$\Delta =\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{r^2{\sin }^2\left(\theta \right)}\frac{{\partial }^2}{\partial {\varphi }^2}$

Use time independent Schrödinger equation.

$\Delta \Psi -\frac{2M}{{\hslash }^2}(V\left(r\right)-E)\Psi =0$

$\begin{array}{l}\frac{S\left(\theta \right)T\left(\varphi \right)}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial R}{\partial r}\right)+\frac{R\left(r\right)T\left(\varphi \right)}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial S}{\partial \theta }\right)+\frac{R\left(r\right)S\left(\theta \right)}{r^2{\sin }^2\left(\theta \right)}\frac{{\partial }^{2}T}{\partial {\varphi }^2}-\frac{2M}{{\hslash }^2}(V\left(r\right)-E)R\left(r\right)S\left(\theta \right)T\left(\varphi \right)=0\\ \frac{1}{r^{2}R}\frac{\partial }{\partial r}\left(r^2\frac{\partial R}{\partial r}\right)+\frac{1}{S{r}^2\sin \left(\theta \right)}\frac{\partial }{\partial \theta }\left(\sin \left(\theta \right)\frac{\partial S}{\partial \theta }\right)+\frac{1}{r^2{\sin }^2\left(\theta \right)T}\frac{{\partial }^{2}T}{\partial {\varphi }^2}-\frac{2M}{{\hslash }^2}(V\left(r\right)-E)=0\end{array}$

Check $\varphi$dependence.

$\begin{array}{l}\frac{1}{T}\frac{{\partial }^{2}T}{\partial {\varphi }^2}=-m^2\\ \frac{{\partial }^{2}T}{\partial {\varphi }^2}+m^{2}T=0\\ T\left(\varphi \right)=e^{\pm i\varphi }\end{array}$

$e^{\perp i\varphi }=\{\begin{array}{l}\mathrm{Re}\left(T\right)=\cos \left(m\varphi \right)\\ \mathrm{Im}\left(T\right)=\sin \left(m\varphi \right)\end{array}$

Check $\theta$dependence.

$\begin{array}{l}-\frac{1}{S}\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial S}{\partial \theta }\right)+\frac{m^2}{{\sin }^2\left(\theta \right)}=\alpha \\ \frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial S}{\partial \theta }\right)+(\alpha -\frac{m^2}{{\sin }^2\left(\theta \right)})S=0\end{array}$

Check r- independence

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

Hence $Y_{l,m}(\theta ,\varphi )=S\left(\theta \right)T\left(\varphi \right)$is spherical harmonics.