Sample Test Problems

1. * We have shown that the Hermitian conjugate of a rotation operator $R(\vec{\theta})$ is $R(-\vec{\theta})$. Use this to prove that if the $\phi_i$ form an orthonormal complete set, then the set $\phi'_i=R(\vec{\theta})\phi_i$ are also orthonormal and complete.

2. Given that $u_n$ is the $n^{th}$ one dimensional harmonic oscillator energy eigenstate: a) Evaluate the matrix element $\langle u_m \vert p^2 \vert u_n \rangle$. b) Write the upper left 5 by 5 part of the $p^2$ matrix.

3. A spin 1 system is in the following state in the usual $Lz$ basis: $\chi={1\over \sqrt{5}}\left(\matrix{\sqrt{2}\cr 1+i\cr -i}\right)$. What is the probability that a measurement of the $x$ component of spin yields zero? What is the probability that a measurement of the $y$ component of spin yields $+\hbar$?

4. In a three state system, the matrix elements are given as $\langle\psi_1\vert H\vert\psi_1\rangle =E_1$, $\langle\psi_2\vert H\vert\psi_2\rangle =\langle\psi_3\vert H\vert\psi_3\rangle =E_2$, $\langle\psi_1\vert H\vert\psi_2\rangle =0$, $\langle\psi_1\vert H\vert\psi_3\rangle =0$, and $\langle \psi_2\vert H\vert\psi_3\rangle =\alpha$. Assume all of the matrix elements are real. What are the energy eigenvalues and eigenstates of the system? At $t=0$ the system is in the state $\psi_2$. What is $\psi(t)$?

5. Find the (normalized) eigenvectors and eigenvalues of the $S_x$ (matrix) operator for $s=1$ in the usual ($S_z$) basis.

6. * A spin ${1\over 2}$ particle is in a magnetic field in the $x$ direction giving a Hamiltonian $H=\mu_B B \sigma_x$. Find the time development (matrix) operator $e^{-iHt/\hbar}$ in the usual basis. If $\chi(t=0)=\left(\matrix{1\cr 0}\right)$, find $\chi(t)$.

7. A spin ${1\over 2}$ system is in the following state in the usual $S_z$ basis: $\chi={1\over \sqrt{5}}\left(\matrix{\sqrt{3}\cr 1+i\cr}\right)$. What is the probability that a measurement of the $x$ component of spin yields $+{1\over 2}$?

8. A spin ${1\over 2}$ system is in the state $\chi = {1\over \sqrt{5}} \left(\matrix{i\cr 2\cr}\right)$ (in the usual $S_z$ eigenstate basis). What is the probability that a measurement of $S_x$ yields ${-\hbar\over 2}$? What is the probability that a measurement of $S_y$ yields ${-\hbar\over 2}$?

9. A spin ${1\over 2}$ object is in an eigenstate of $S_y$ with eigenvalue $\hbar \over 2$ at t=0. The particle is in a magnetic field $\bf {B}$ = $(0,0,B)$ which makes the Hamiltonian for the system $H = \mu_B B \sigma_z$. Find the probability to measure $S_y = {\hbar \over 2}$ as a function of time.

10. Two degenerate eigenfunctions of the Hamiltonian are properly normalized and have the following properties.
    $$\begin{eqnarray*} H\psi_1 = E_0 \psi_1 & H\psi_2 = E_0 \psi_2 \\ P\psi_1 = -\psi_2 & P\psi_2 = -\psi_1 \end{eqnarray*}$$
    
    
    
    
    What are the properly normalized states that are eigenfunctions of H and P? What are their energies?

11. What are the eigenvectors and eigenvalues for the spin ${1\over 2}$ operator $S_x+S_z$?

12. A spin ${1\over 2}$ object is in an eigenstate of $S_y$ with eigenvalue $\hbar \over 2$ at t=0. The particle is in a magnetic field $\bf {B}$ = $(0,0,B)$ which makes the Hamiltonian for the system $H = \mu_B B \sigma_z$. Find the probability to measure $S_y = {\hbar \over 2}$ as a function of time.

13. A spin 1 system is in the following state, (in the usual $L_z$ eigenstate basis):
    
    $$\chi = {1\over \sqrt{5}} \left(\matrix{i\cr\sqrt{2}\cr 1+i\cr}\right).$$
    
    
    What is the probability that a measurement of $L_x$ yields 0? What is the probability that a measurement of $L_y$ yields $-\hbar$?

14. A spin ${1\over 2}$ object is in an eigenstate of $S_z$ with eigenvalue $\hbar \over 2$ at t=0. The particle is in a magnetic field $\bf {B}$ = $(0,B,0)$ which makes the Hamiltonian for the system $H = \mu_B B \sigma_y$. Find the probability to measure $S_z = {\hbar \over 2}$ as a function of time.

15. A spin 1 particle is placed in an external field in the $u$ direction such that the Hamiltonian is given by
    
    $$H=\alpha \left({\sqrt{3}\over 2}S_x+{1\over 2} S_y\right)$$
    
    
    Find the energy eigenstates and eigenvalues.

16. A (spin ${1\over 2}$) electron is in an eigenstate of $S_y$ with eigenvalue $-{\hbar\over 2}$ at $t=0$. The particle is in a magnetic field $\vec{B}$ = $(0,0,B)$ which makes the Hamiltonian for the system $H = \mu_B B \sigma_z$. Find the spin state of the particle as a function of time. Find the probability to measure $S_y=+{\hbar\over 2}$ as a function of time.