Problem 1:
A system has a wave function y(x,y,z) = N*(x + y + z)*exp(-r2/a2)
with a real. If Lz and L2 are
measured, what are the probabilities of finding 0 and 2h2?
Problem 2:
A measurement of L2 and Lz for a free particle yields the
values l = 1 and m = 1. Later a measurement of Ly is made.
(a) What are the possible values of Ly?
(b) Calculate the probabilities for each of the possible values in part (a).
Problem 3:
The quantum numbers l1 and l2 of the orbital momenta of particle A and particle B are 1 and 2, respectively. Find the 15 possible ‘kets’ in the coupled representation (notation |l1,l2;L,ML>) where L represents the quantum number of the total orbital momentum.
Problem 4:
Consider two particles with angular momenta:
J = J1 + J2, Jx = J1x + J2x,
Jy = J1y + J2y, Jz = J1z
+ J2z.
J1 and J2 are the angular momentum operators of particle 1 and 2.
Show that the commutators [J2,J12] and [J1z,J2] are zero and nonzero, respectively.
What does it mean in terms of measurements and Heisenberg’s uncertainty
principle?
Problem 5:
For any two quantum-mechanical operators A and B, the uncertainty principle says that
<(DA)2><(DB)2>
³ (1/4)|<[A,B]>|2. Consider a
spin ½ particle. Show that for the spin
operators Sx and Sy the eigenstate |+> of the Sz
operator is a minimum uncertainty state.
Problem 6:
Let Si, i = 1, 2 denote the spin vectors of two spin-1/2 particles. The interaction is given by
H = V0 (S1 · S2 − 3 S1zS2z).
Find the energy eigenstates
and eigenvalues.
Problem 7:
Some organic molecules have a triplet (S = 1) excited state
that is located at an energy D above the
singlet (S = 0) ground state. Consider an ensemble of N such molecules where N
is of the order of Avogadro’s number
(a) Find the average magnetic moment <m>
per molecule in the presence of a magnetic field B. Assume Boltzmann
statistics. You may also assume that D
is large compared to the field-induced level splittings.
(b) Show that the magnetic susceptibility c
= N d<m>/dB is approximately
independent of D when kBT >>
D.