Separations of Variables (20 min)

Show by using the definition of the Laplace operator in polar coordinates and the definition of the angular momentum square,

$${\bf\hat{L}}^2= -\hbar^2 \left[\frac{1}{\sin \theta}\frac{\partia... ...ta}\right) +\frac{1}{\sin^2 \theta}\frac{\partial^2}{\partial \varphi^2}\right]$$ (43)

that the stationary Schrödinger equation for energy $E$ for the motion of a particle with mass $m$ in a central potential $U(r)$ can be separated with the Ansatz for the wave function

$$\Psi(r,\theta,\phi)=R(r)Y_{lm}(\theta,\phi).$$ (44)

In order to do so, define the radial function $\chi(r):=rR(r)$ and show

$$\frac{d^2\chi(r)}{dr^2}+\left[\frac{2m}{\hbar^2}(E-U(r))-\frac{l(l+1)}{r^2}\right]\chi(r)=0.$$ (45)

Which values are possible for $l$ (without proof)?