- An angular momentum 1 system is in the state
.
What is the probability that a measurement of yields a value of 0?
- 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?
- 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
?
- The Hamiltonian matrix is given to be

What are the eigen-energies and corresponding eigenstates of the system? (This isn't too messy.) - What are the eigenfunctions and eigenvalues of the operator for a spin 1
system?
- Calculate the operator for arbitrary rotations about the x-axis. Use the usual
eigenstates as a basis.
- 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.
- 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.
- 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?
- Find the matrix which represents the operator for a 1D harmonic oscillator.
Write out the upper left part of the matrix.
- 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