# Angular Momentum Commutation Relations

## Prerequisites

Students should be able to:

## In-class Content

- Spherical Harmonics and the $H$, $L^2$, and $L_z$ Operators (Lecture, 60 minutes)

## Homework for Spherical Harmonics

- (AngMomCommute)
Show that the components of the angular momentum operator $\vec L$, written in differential operator form in rectangular components, satisfy the commutation relations:

$$ \left[L_x,L_y\right]=+i\hbar L_z \qquad\qquad\hbox{(and cyclic permutations)} $$

- (HarmonicsLookUpP)
Find a list of the first few spherical harmonics in a quantum mechanics textbook. Remember where you found it! Look at the functional forms of the spherical harmonics to get familiar with them. What patterns do you notice? Calculate a few from the formulas given in class.

- (HarmonicsVerifyP)
In spherical coordinates, the square of the angular momentum vector $L^2$ and the $z$-component of the angular momentum vector $L_z$ are given by:

$$L^2=\Vec L \cdot \Vec L= -\hbar^2\left({1\over \sin\theta}{\partial\over\partial\theta}\left( \sin\theta{\partial\over\partial\theta}\right)+{1\over\sin^2\theta} {\partial^2\over \partial \phi^2}\right)$$

$$L_z=-i\hbar{\partial\over \partial\phi}$$

Show explicitly that the spherical harmonics are eigenvectors of $L^2$. What are the eigenvalues?

Show explicitly that the spherical harmonics are eigenvectors of $L_z$. What are the eigenvalues?

- (Sphere)
Consider the normalized function:

$$f(\theta,\phi)= \begin{cases} N\left({\pi^2\over 4}-\theta^2\right)&0<\theta<\frac{\pi}{2}\\ 0&{\pi\over 2}<\theta<\pi \end{cases} $$

where

$$N=\frac{1}{\sqrt{\frac{\pi^5}{8} +2\pi^3-24\pi^2+48\pi}}$$

Find the $\left|\ell,m\right\rangle=\left|0,0\right\rangle$, $\left|1,-1\right\rangle$, $\left|1,0\right\rangle$, and $\left|1,1\right\rangle$ terms in a spherical harmonics expansion of $f(\theta,\phi)$.

If a quantum particle, confined to the surface of a sphere, is in the state above, what is the probability that a measurement of the square of the total angular momentum will yield $2\hbar^2$? $4\hbar^2$?

If a quantum particle, confined to the surface of a sphere, is in the state above, what is the probability that the particle can be found in the region $0<\theta<{\pi\over 6}$ and $0<\phi<{\pi\over 6}$? Repeat the question for the region ${5\pi\over 6}<\theta<{\pi}$ and $0<\phi<{\pi\over 6}$. Plot your approximation from part (a) above and check to see if your answers seem reasonable.

- (SphereQuestions)
Consider the following normalized state for the rigid rotor given by:

$$\left|\psi\right\rangle={1\over\sqrt{2}}\left\vert 1, -1\right\rangle + {1\over\sqrt{3}}\left\vert 1, 0\right\rangle + {i\over\sqrt{6}}\left\vert 0, 0\right\rangle $$

What is the probability that a measurement of $L_z$ will yield $2\hbar$? $-\hbar$? $0\hbar$?

If you measured the z-component of angular momentum to be $-\hbar$, what would the state of the particle be immediately after the measurement is made? $0\hbar$?

What is the expectation value of $L_z$ in this state?

What is the expectation value of $L^2$ in this state?

What is the expectation value of the energy in this state?

- (SphereTable) Complete the table (EigenTableRigidRotorEmpty.pdf) which summarizes much of what you've learned about the eigenstates for the rigid rotor problem by filling in all the empty boxes.