Angular Momentum

For the common problem of central potentials $V(r)$, we use the obvious rotational symmetry to find that the angular momentum, $\vec{L}=\vec{x}\times \vec{p}$, operators commute with $H$,

$$\bgroup\color{black}[H,L_z]=[H,L_x]=[H,L_y]=0\egroup$$

but they do not commute with each other.

$$\bgroup\color{black}[L_x,L_y]\neq 0\egroup$$

We want to find two mutually commuting operators which commute with $H$, so we turn to $L^2=L_x^2+L_y^2+L_z^2$ which does commute with each component of $L$.

$$\bgroup\color{black}[L^2,L_z]=0\egroup$$

We chose our two operators to be $L^2$ and $L_z$.

Some computation reveals that we can write

$$\bgroup\color{black}p^2={1\over r^2}\left(L^2+(\vec{r}\cdot\vec{p})^2-i\hbar\vec{r}\cdot\vec{p}\right).\egroup$$

With this the kinetic energy part of our equation will only have derivatives in $r$ assuming that we have eigenstates of $L^2$.

$$\bgroup\color{black}{-\hbar^2\over 2\mu}\left[{1\over r^2}\le... ... r^2}\right] u_E(\vec{r})+V(r)u_E(\vec{r})=Eu_E(\vec{r})\egroup$$

The Schrödinger equation thus separates into an angular part (the $L^2$ term) and a radial part (the rest). With this separation we expect (anticipating the angular solution a bit)

$$\bgroup\color{black}u_E(\vec{r})=R_{E\ell}(r) Y_{\ell m}(\theta,\phi)\egroup$$

will be a solution. The $Y_{\ell m}(\theta,\phi)$ will be eigenfunctions of $L^2$

$$\bgroup\color{black}L^2 Y_{\ell m}(\theta,\phi)=\ell(\ell+1)\hbar^2 Y_{\ell m}(\theta,\phi)\egroup$$

so the radial equation becomes

$$\bgroup\color{black}{-\hbar^2\over 2\mu}\left[{1\over r^2}\le... ... r^2}\right] R_{E\ell}(r)+V(r)R_{E\ell}(r)=ER_{E\ell}(r)\egroup$$

We must come back to this equation for each $V(r)$ which we want to solve.

We solve the angular part of the problem in general using angular momentum operators. We find that angular momentum is quantized.

$$\bgroup\color{black}L_zY_{\ell m}(\theta,\phi)=m\hbar Y_{\ell m}(\theta,\phi) \egroup$$

$$\bgroup\color{black}L^2 Y_{\ell m}(\theta,\phi)=\ell(\ell+1)\hbar^2 Y_{\ell m}(\theta,\phi)\egroup$$

with $\ell$ and $m$ integers satisfying the condition $-\ell\leq m\leq\ell$. The operators that raise and lower the $z$ component of angular momentum are

$$\bgroup\color{black}L_\pm=L_x\pm iL_y\egroup$$

$$\bgroup\color{black} L_\pm Y_{\ell m}=\hbar\sqrt{\ell(\ell+1)-m(m\pm 1)}Y_{\ell(m\pm 1)}\egroup$$

We derive the functional form of the Spherical Harmonics $Y_{\ell m}(\theta,\phi)$ using the differential form of the angular momentum operators.