1. (a) [5 points] Describe the similarities and di↵erences between spin and o
ital
angular momentum. Include in your discussion at least the following concepts:
classical equivalence of each, eigenvalues and eigenstates, Hamiltonians. Use
words, equations, and diagrams as needed.
(b) [5 points] Give a complete explanation of the meaning of “CSCO,” and
what it implies for the states of the quantum systems it represents. Justify you
discussion with appropriate mathematics, and cite examples from class that are
CSCOs, and also some that are not (and why they are or aren’t).
2. An electron in the hydrogen atom is in the superposition state of the 1s, 2s,
and 2p o
itals,
(r, ✓,�) =
1p
2
R10(r)Y00(✓,�) +
1
2
R20(r)Y00(✓,�) +
1
2
R21(r)Y1,�1(✓,�)
where Ylm(✓,�) are the usual spherical harmonics, and Rnl is the radial function
and
1s �! R10(r) = 2a�3/20 e�
a0
2s �! R20(r) = (2a0)�3/2
✓
2�
a0
◆
e�
(2a0)
2p �! R21(r) =
1p
3
(2a0)
�3/2
✓
a0
◆
e�
(2a0)
.
(a)[3 points] If the Lz angular momentum is measured, what results can be
found and with what probabilities? Explain your reasoning.
(b) [3 points] Show how you would calculate the probability of finding the elec-
tron between a = a0 and a = 5a0, where a0 is the Bohr radius (but don’t do the
calculation!). Be sure to explain all terms in your calculation and show all steps!
(c) [4 points] How you would calculate the average radial position of the elec-
tron in this state? Be sure to explain all terms in your calculation and show all
steps.
3. For this problem, recall that
|+iu = cos
✓
2
!
e�i�/2|+i+ sin
✓
2
!
e+i�/2|�i
|�iu = � sin
✓
2
!
e�i�/2|+i+ cos
✓
2
!
e+i�/2|�i
(a) [3 points] For what angles ✓,� are the states |±ix and |±iy recovered?
Show your work.
(b) [3 points] A particle is prepared in the state |�iu oriented at an unknown
angle ✓ (with � = 0). If there is a 0.35 probability of measuring the state |+ix,
determine the preperation angle ✓ of the original state.
(c) [4 points] A particle is prepared in the state |+i. It passes through a
Stern-Gerlach apparatus oriented in the û�direction (at angles ✓ and �), but
whose action is unknown. That is, the interim state could be |+iu or |�iu with
equal probability.
Show that the probability of measuring the state |�i after the particle emerges
from the SG device is P = 14 cos
2 ✓.
[Hint: Recall your trig identities....]