- 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 them=− 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. Theu-axis and the z-axis
are both perpendicular to the beam direction but have an angleθbetween them. Find the
relative intensities of the three beams separated in the second apparatus.
- Find the eigenstates of the harmonic oscillator lowering operatorA. They should satisfy the
equationA|α〉=α|α〉. Do this by finding the coefficients〈n|α〉where|n〉is thenthenergy
eigenstate. Make sure that the states|α〉are normalized so that〈α|α〉= 1. Suppose|α′〉is
another such state with a different eigenvalue. Compute〈α′|α〉. Would you expect these states
to be orthogonal?
- Find the matrix which represents thep^2 operator for a 1D harmonic oscillator. Write out the
upper left 5×5 part of the matrix.
- Let’s define the u axis to be in the x-z plane, between the positivex and z axes and at an
angle of 30 degrees to the x axis. Given an unpolarized spin^12 beam of intensityIgoing into
the following Stern-Gerlach apparati, what intensity comes out?
I→
{
+
−|
}
z
→
{
+
−|
}
x
→?
I→
{
+
−|
}
z
→
{
+|
−
}
u
→?
I→
{
+
−|
}
z
→
{
+|
−
}
u
→
{
+|
−
}
z
→?
I→
{
+
−|
}
z
→
{
+
−
}
u
→
{
+|
−
}
z
→?
I→
{
+
−|
}
z
→
{
+|
−
}
u
→
{
+|
−
}
x
→?
18.13Sample Test Problems
1.*We have shown that the Hermitian conjugate of a rotation operatorR(~θ) isR(−~θ). Use this
to prove that if theφiform an orthonormal complete set, then the setφ′i=R(~θ)φiare also
orthonormal and complete.
- Given thatunis thenthone dimensional harmonic oscillator energy eigenstate: a) Evaluate
the matrix element〈um|p^2 |un〉. b) Write the upper left 5 by 5 part of thep^2 matrix.
- A spin 1 system is in the following state in the usualLzbasis: χ=√^15
√
2
1 +i
−i
. What
is the probability that a measurement of thexcomponent of spin yields zero? What is the
probability that a measurement of theycomponent of spin yields + ̄h?
- In a three state system, the matrix elements are given as〈ψ 1 |H|ψ 1 〉=E 1 ,〈ψ 2 |H|ψ 2 〉=
〈ψ 3 |H|ψ 3 〉=E 2 ,〈ψ 1 |H|ψ 2 〉= 0,〈ψ 1 |H|ψ 3 〉= 0, and〈ψ 2 |H|ψ 3 〉=α. 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 stateψ 2. What isψ(t)?
