91影视

The time independent Schrodinger equation is as below.

$螖唯鈭�\frac{2M}{{\mathrm{鈩�}}^2}(V\left(r\right)鈭�E)唯=0$.

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

Write Laplace operator in spherical coordinates.

$螖=\frac{1}{r^2}\frac{鈭�}{鈭�r}\left(r^2\frac{鈭�}{鈭�r}\right)+\frac{1}{r^2\sin 胃}\frac{鈭�}{鈭�胃}\left(\sin 胃\frac{鈭�}{鈭�胃}\right)+\frac{1}{r^2{\sin }^2\left(胃\right)}\frac{{鈭�}^2}{鈭�{蠒}^2}$

Use time independent Schr枚dinger equation.

$螖唯鈭�\frac{2M}{{\mathrm{鈩�}}^2}(V\left(r\right)鈭�E)唯=0$

$\begin{array}{l}\frac{S\left(胃\right)T\left(蠒\right)}{r^2}\frac{鈭�}{鈭�r}\left(r^2\frac{鈭�R}{鈭�r}\right)+\frac{R\left(r\right)T\left(蠒\right)}{r^2\sin 胃}\frac{鈭�}{鈭�胃}\left(\sin 胃\frac{鈭�S}{鈭�胃}\right)+\frac{R\left(r\right)S\left(胃\right)}{r^2{\sin }^2\left(胃\right)}\frac{{鈭�}^{2}T}{鈭�{蠒}^2}鈭�\frac{2M}{{\mathrm{鈩�}}^2}(V\left(r\right)鈭�E)R\left(r\right)S\left(胃\right)T\left(蠒\right)=0\\ \frac{1}{r^{2}R}\frac{鈭�}{鈭�r}\left(r^2\frac{鈭�R}{鈭�r}\right)+\frac{1}{S{r}^2\sin \left(胃\right)}\frac{鈭�}{鈭�胃}\left(\sin \left(胃\right)\frac{鈭�S}{鈭�胃}\right)+\frac{1}{r^2{\sin }^2\left(胃\right)T}\frac{{鈭�}^{2}T}{鈭�{蠒}^2}鈭�\frac{2M}{{\mathrm{鈩�}}^2}(V\left(r\right)鈭�E)=0\end{array}$

Check $蠒$dependence.

$\begin{array}{l}\frac{1}{T}\frac{{鈭�}^{2}T}{鈭�{蠒}^2}=鈭�m^2\\ \frac{{鈭�}^{2}T}{鈭�{蠒}^2}+m^{2}T=0\\ T\left(蠒\right)=e^{卤i蠒}\end{array}$

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

Check $胃$dependence.

$\begin{array}{l}鈭�\frac{1}{S}\frac{1}{\sin 胃}\frac{鈭�}{鈭�胃}\left(\sin 胃\frac{鈭�S}{鈭�胃}\right)+\frac{m^2}{{\sin }^2\left(胃\right)}=伪\\ \frac{1}{\sin 胃}\frac{鈭�}{鈭�胃}\left(\sin 胃\frac{鈭�S}{鈭�胃}\right)+(伪鈭�\frac{m^2}{{\sin }^2\left(胃\right)})S=0\end{array}$

Check r- independence

$\frac{1}{R}\frac{鈭�}{鈭�r}\left(r^2\frac{鈭�R}{鈭�r}\right)鈭�\frac{2M{r}^2}{{\mathrm{鈩�}}^2}(V\left(r\right)鈭�E)=伪=l(l+1)$

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