Sample Test Problems

1. Two identical spin ${3\over 2}$ particles are bound together into a state with total angular momentum $l$. a) What are the allowed states of total spin for $l=0$ and for $l=1$? b) List the allowed states using spectroscopic notation for $l=0$ and 1. $(^{2s+1}L_j)$

2. A hydrogen atom is in the state $\psi=R_{43}Y_{30}\chi_+$. A combined measurement of of $J^2$ and of $J_z$ is made. What are the possible outcomes of this combined measurement and what are the probabilities of each? You may ignore nuclear spin in this problem.

3. We want to find the eigenstates of total $S^2$ and $S_z$ for two spin 1 particles which have an $S_1\cdot S_2$ interaction. ( ${\bf S}={\bf S_1}+{\bf S_2}$)
   1. What are the allowed values of $s$, the total spin quantum number.
   2. Write down the states of maximum $m_s$ for the maximum $s$ state. Use $\vert sm_s\rangle$ notation and $\vert s_1m_1\rangle\vert s_2m_2\rangle$ for the product states.
   3. Now apply the lowering operator to get the other $m_s$ states. You only need to go down to $m_s=0$ because of the obvious symmetry.
   4. Now find the states with the other values of $s$ in a similar way.

4. Two (identical) electrons are bound in a Helium atom. What are the allowed states $\vert jlsl_1 l_2\rangle$ if both electrons have principal quantum number $n=1$? What are the states if one has $n=1$ and the other $n=2$?

5. A hydrogen atom is in an eigenstate $(\psi)$ of $J^2$, $L^2$, and of $J_z$ such that $J^2 \psi = {15\over4} \hbar^2 \psi$, $L^2 \psi = 6 \hbar^2 \psi$, $J_z \psi = - {1\over 2} \hbar \psi$, and of course the electron's spin is ${1\over 2}$. Determine the quantum numbers of this state as well as you can. If a measurement of $L_z$ is made, what are the possible outcomes and what are the probabilities of each.

6. A hydrogen atom is in the state $\psi = R_{32}Y_{21}\chi_-$. If a measurement of $J^2$ and of $J_z$ is made, what are the possible outcomes of this measurement and what are the probabilities for each outcome? If a measurement of the energy of the state is made, what are the possible energies and the probabilities of each? You may ignore the nuclear spin in this problem.

7. Two identical spin 1 particles are bound together into a state with orbital angular momentum $l$. What are the allowed states of total spin (s) for $l=2$, for $l=1$, and for $l=0$? List all the allowed states giving, for each state, the values of the quantum numbers for total angular momentum $(j)$, orbital angular momentum $(l)$ and spin angular momentum $(s)$ if $l$ is 2 or less. You need not list all the different $m_j$ values.

8. List all the allowed states of total spin and total z-component of spin for 2 identical spin 1 particles. What $\ell$ values are allowed for each of these states? Explicitly write down the $(2s+1)$ states for the highest $s$ in terms of $\chi^{(1)}_+ ,\chi^{(2)}_+ ,\chi^{(1)}_0 ,\chi^{(2)}_0 ,\chi^{(1)}_-$, and $\chi^{(2)}_-$.

9. Two different spin ${1\over 2}$ particles have a Hamiltonian given by $H=E_0+{A\over\hbar^2}\vec{S_1}\cdot\vec{S_2}+{B\over\hbar}(S_{1z}+S_{2z})$. Find the allowed energies and the energy eigenstates in terms of the four basis states $\vert++\rangle$, $\vert+-\rangle$, $\vert-+\rangle$, and $\vert-\rangle$.

10. A spin 1 particle is in an $\ell =2$ state. Find the allowed values of the total angular momentum quantum number, $j$. Write out the $\vert j,m_j\rangle$ states for the largest allowed $j$ value, in terms of the $\vert m_l,m_s\rangle$ basis. (That is give one state for every $m_j$ value.) If the particle is prepared in the state $\vert m_l=0,m_s=0\rangle$, what is the probability to measure $J^2=12\hbar^2$?

11. Two different spin ${1\over 2}$ particles have a Hamiltonian given by $H=E_0+A\vec{S_1}\cdot\vec{S_2}+B(S_{1z}+S_{2z})$. Find the allowed energies and the energy eigenstates in terms of the four product states $\chi^{(1)}_+\chi^{(2)}_+$, $\chi^{(1)}_+\chi^{(2)}_-$, $\chi^{(1)}_-\chi^{(2)}_+$, and $\chi^{(1)}_-\chi^{(2)}_-$.