Physics 480
Final Exam
Fall 2004
The total number of points for this exam is 30. You have 120 minutes to complete the exam. Do all problems.
Show all your work so that I can see how you arrived at the answer. You may use Griffith ”Introduction to
Quantum Mechanics” and the lecture notes to look up any relevant formulas.
1. The electron in a hydrogen atom has the following wave function
µ ¯
À ¯
˦
¯ 3 −1
¯1 1
¯
¯
ψ = A R21 2 ¯
−¯
2 2
2 2
where the ket vectors are given in the |j mj i basis.
(a) Determine the normalization constant A. (2 points)
(b) If you measure the total angular momentum squared, J 2 , what values might you get and what is the
probability of each ? (3 points)
(c) If you measure the z component of the orbital angular momentum, ml , what values might you get
and what is the probability of each ? (3 points)
(d) If you measure the x component of the electron spin, Sx , what values might you get and what is the
probability of each ? (2 points)
2. The (unperturbed) Hamiltonian H 0 of a three state system is given
 



1
0
ψ1 =  0  ψ2 =  1  ψ3 = 
0
0
as


1 0 0
H 0 = V0  0 3 0  .
0 0 1
A perturbation of the form

0 0
H0 = ²  0 2
1 0
in the basis

0
0 
1

1
0 .
0
is introduced with ² ¿ V0 .
(a)
(b)
(c)
(d)
What are the energy levels in zeroth-order perturbation theory ? (1 points)
What are the energy levels in first-order perturbation theory ? (3 points)
What are the corresponding good states in zeroth-order perturbation theory ? (3 points)
If the system is in the state ψ1 at time t = 0, what is the (time-dependent) probability to find it in
state ψ3 at a later time t0 ? (3 points)
~1 · S
~2 . Assume that only the spin degrees of freedom
3. Two spin- 12 particles interact via the Hamiltonian S
are relevant for this problem.
Use the direct product basis |s1 sz,1 i|s¯2 sz,2
® ¯i when
® an explicit basis is required. You can use the ordered arrow notation in which e.g. ↑↓≡ ¯ 12 21 ¯ 21 −1
2 .
(a) Write down the operator Q = S1y S2y in matrix representation in this basis. (3 points)
(b) Write the wave function for this system such that if S1z is measured, the probability of obtaining
h̄/2 is 1, but if S2z is measured, the probability of obtaining h̄/2 is 50%. (2 points)
(c) Calculate the expectation value hQi for the state in part (b). (3 points)
(d) Show that the eigenstates of Q do not have definite energy. (2 points)
Good luck !
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