Homework Problems

  1. An angular momentum 1 system is in the state $\chi={1\over\sqrt{26}}\left(\matrix{1\cr 3\cr 4}\right)$. What is the probability that a measurement of $L_x$ yields a value of 0?

  2. A spin ${1\over 2}$ particle is in an eigenstate of $S_y$ with eigenvalue $+{\hbar\over 2}$ at time $t=0$. At that time it is placed in a constant magnetic field $B$ in the $z$ direction. The spin is allowed to precess for a time $T$. At that instant, the magnetic field is very quickly switched to the $x$ direction. After another time interval $T$, a measurement of the $y$ component of the spin is made. What is the probability that the value $-{\hbar\over 2}$ will be found?

  3. Consider a system of spin ${1\over 2}$. What are the eigenstates and eigenvalues of the operator $S_x+S_y$? Suppose a measurement of this quantity is made, and the system is found to be in the eigenstate with the larger eigenvalue. What is the probability that a subsequent measurement of $S_y$ yields ${\hbar\over 2}$?

  4. The Hamiltonian matrix is given to be

    \begin{displaymath}H=\hbar\omega\left(\matrix{8&4&6\cr 4&10&4\cr 6&4&8}\right).\end{displaymath}

    What are the eigen-energies and corresponding eigenstates of the system? (This isn't too messy.)

  5. What are the eigenfunctions and eigenvalues of the operator $L_xL_y+L_yL_x$ for a spin 1 system?

  6. Calculate the $\ell =1$ operator for arbitrary rotations about the x-axis. Use the usual $L_z$ eigenstates as a basis.

  7. An electron is in an eigenstate of $S_x$ with eigenvalue ${\hbar\over 2}$. What are the amplitudes to find the electron with a) $S_z=+{\hbar\over 2}$, b) $S_z=-{\hbar\over 2}$, $S_y=+{\hbar\over 2}$, $S_u=+{\hbar\over 2}$, where the $u$-axis is assumed to be in the $x-y$ plane rotated by and angle $\theta$ from the $x$-axis.

  8. Particles with angular momentum 1 are passed through a Stern-Gerlach apparatus which separates them according to the z-component of their angular momentum. Only the $m=-1$ component is allowed to pass through the apparatus. A second apparatus separates the beam according to its angular momentum component along the u-axis. The u-axis and the z-axis are both perpendicular to the beam direction but have an angle $\theta$ between them. Find the relative intensities of the three beams separated in the second apparatus.

  9. Find the eigenstates of the harmonic oscillator lowering operator $A$. They should satisfy the equation $A\vert\alpha\rangle =\alpha\vert\alpha\rangle $. Do this by finding the coefficients $\langle n\vert\alpha\rangle $ where $\vert n\rangle $ is the $n^{th}$ energy eigenstate. Make sure that the states $\vert\alpha\rangle $ are normalized so that $\langle\alpha\vert\alpha\rangle =1$. Suppose $\vert\alpha'\rangle $ is another such state with a different eigenvalue. Compute $\langle\alpha'\vert\alpha\rangle $. Would you expect these states to be orthogonal?

  10. Find the matrix which represents the $p^2$ operator for a 1D harmonic oscillator. Write out the upper left $5\times 5$ part of the matrix.

  11. Let's define the u axis to be in the x-z plane, between the positive x and z axes and at an angle of 30 degrees to the x axis. Given an unpolarized spin ${1\over 2}$ beam of intensity $I$ going into the following Stern-Gerlach apparati, what intensity comes out?

    \begin{displaymath}I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarrow
\left\{\matrix{+ \cr - \vert}\right\}_x\rightarrow ?\end{displaymath}


    \begin{displaymath}I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarrow
\left\{\matrix{+ \vert\cr - }\right\}_u\rightarrow ?\end{displaymath}


    \begin{displaymath}I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarr...
...ghtarrow
\left\{\matrix{+ \vert\cr - }\right\}_z\rightarrow ?\end{displaymath}


    \begin{displaymath}I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarr...
...ghtarrow
\left\{\matrix{+ \vert\cr - }\right\}_z\rightarrow ?\end{displaymath}


    \begin{displaymath}I\rightarrow \left\{\matrix{+ \cr - \vert}\right\}_z\rightarr...
...ghtarrow
\left\{\matrix{+ \vert\cr - }\right\}_x\rightarrow ?\end{displaymath}

Jim Branson 2013-04-22