Angular Momentum Commutation Relations

Students should be able to:

1. (AngMomCommute)

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

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

2. (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.

3. (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:

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

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

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

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

4. (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

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

   1. 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 $f(\theta,\phi)$.

   2. 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 $2\hbar^2$? $4\hbar^2$?

   3. 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.

5. (SphereQuestions)

   Consider the following normalized state for the rigid rotor given by:

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

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

   2. 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? $0\hbar$?

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

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

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

6. (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.