## Homework Problems

1. An angular momentum 1 system is in the state . What is the probability that a measurement of yields a value of 0?

2. A spin particle is in an eigenstate of with eigenvalue at time . At that time it is placed in a constant magnetic field in the direction. The spin is allowed to precess for a time . At that instant, the magnetic field is very quickly switched to the direction. After another time interval , a measurement of the component of the spin is made. What is the probability that the value will be found?

3. Consider a system of spin . What are the eigenstates and eigenvalues of the operator ? 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 yields ?

4. The Hamiltonian matrix is given to be

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 for a spin 1 system?

6. Calculate the operator for arbitrary rotations about the x-axis. Use the usual eigenstates as a basis.

7. An electron is in an eigenstate of with eigenvalue . What are the amplitudes to find the electron with a) , b) , , , where the -axis is assumed to be in the plane rotated by and angle from the -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 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 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 . They should satisfy the equation . Do this by finding the coefficients where is the energy eigenstate. Make sure that the states are normalized so that . Suppose is another such state with a different eigenvalue. Compute . Would you expect these states to be orthogonal?

10. Find the matrix which represents the operator for a 1D harmonic oscillator. Write out the upper left 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 beam of intensity going into the following Stern-Gerlach apparati, what intensity comes out?

Jim Branson 2013-04-22