Physics 481
Final Exam
Spring 2005
The total number of points for this exam is 30. You have 2.0 hours to complete the exam.
First do problems 1 – 3. If you have time left, do problem 4. 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. An electron in the n = 5, l = 3, m = 2 state of hydrogen decays by a sequence of
electric dipole transitions.
(a) What decay sequences are possible? Specify them in the notation |5 3 2i →
|n0 l0 m0 i → |n00 l00 m00 i → · · · all the way until the lowest state is reached by each
route. (4 points)
(b) If you had a bottle full of atoms in the |5 3 2i state, what fraction would
decay to which first intermediate state? (Hint: With careful reasoning you will
only need to do two angular integrals. All the possible radial integrals are listed
below. The fact that an integral is listed does not mean that you will need it
for this problem.) (8 points)
2. A free particle of mass m, traveling with momentum p parallel to the z-axis, scatters
elastically from the spherical δ-function shell potential
V (~r) = αδ(r − a)
where α and a are positive constants.
(a) the scattering amplitude f (θ) in the Born approximation. Make the dependence on the scattering angle θ and on k = p/h̄ explicit in your answers. (5
points)
(b) Calculate the differential scattering cross section dσ/dΩ and the total cross
section σ for low-energy scattering for the above potential. (5 points)
3. A four-state system has eigenstates |φ1 i, |φ2 i, |φ3 i and |φ4 i with equally spaced
energies E1 < E2 < E3 < E4 . Initially the system is in the state |φ1 i. Then, at
t = 0, a sinusoidal perturbation is turned on:


W (t) = λ 


0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0



 cos ωt.

For each state |φi i, i = 2, 3, 4, identify the lowest power of λ that will contribute
to the transition probability P1→i (t), when h̄ω is close to the energy level splitting
E4 − E3 = E3 − E2 = E2 − E1 . Do not do any explicit calculations of the transition
probability but justify your answer. (8 points)
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4. Extra credit: Do this problem only after you have done all other problems.
~ = V (~r), where
If a scattering potential is translation-invariant such that V (~r + R)
~ is a constant vector, show that in the Born approximation scattering occurs only
R
~ = 2πn, where ~κ = ~k 0 − ~k and n is an
in the directions defined by the condition ~κ · R
integer. (6 extra credit points)
Some useful integrals:
Z ∞
0
Z ∞
Z0∞
0
Z ∞
0
Z ∞
0
Z ∞
Z0∞
0
Z ∞
0
Z ∞
Z0∞
0
r3 R53 (r)R43 (r) dr = −5.316a
r3 R53 (r)R42 (r) dr = +14.065a
r3 R53 (r)R41 (r) dr = −18.880a
r3 R53 (r)R40 (r) dr = +20.897a
r3 R53 (r)R32 (r) dr = +3.319a
r3 R53 (r)R31 (r) dr = −2.6718a
r3 R53 (r)R30 (r) dr = +2.003a
r3 R53 (r)R21 (r) dr = +1.560a
r3 R53 (r)R20 (r) dr = −2.072a
r3 R53 (r)R10 (r) dr = +0.058a
Good luck !
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