Physics 481
Final Exam
Spring 2004
The total number of points for this exam is 44. You have 2.0 hours to complete the exam.
1. A particle of mass m is in the potential well of the form
(
V (x) =
1
mω 2 (x2
2
0
− a2 ) if |x| < a,
if |x| > a.
(a) Sketch the potential. (3 points)
(b) Use the WKB approximation to estimate the energies of the bound states. (6
points)
(c) Find the condition under which there is only one bound state. (3 points)
2. An electron in the n = 4, l = 2, m = 1 state of hydrogen decays by a sequence of
electric dipole transitions.
(a) What decay sequences are possible? Specify them in the notation |421i →
|n0 l0 m0 i → |n00 l00 m00 i → · · · all the way until the lowest state reached by each
route. (6 points)
(b) If you had a bottle full of atoms in the |421i 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)
3. A free particle of mass m, traveling with momentum p parallel to the z-axis, scatters
elastically from the potential
V (~r) = V0 [δ(~r − ²ẑ) − δ(~r + ²ẑ)].
(a) Calculate the scattering amplitude f (θ) and the differential scattering cross
section dσ/dΩ in the Born approximation. Make the dependence on the scattering angle θ and on k = p/h̄ explicit in your answers. (6 points)
(b) Sketch the differential cross section as a function of the parameter ² for fixed
angle θ. At what values of the spacing between the δ-functions is the differential
cross section zero? (4 points)
The following two problems are ”short-answer” problems. Long explanations or calculations are not expected.
4. Consider an elastic scattering experiment A + B → A + B (A is the beam particle
and B is the target particle), where A and B both have spin 0 and B is much
heavier than A. The cross-section as a function of the momentum of A, h̄k exhibits
an enhancement (a resonance) near a particular value of h̄k at all scattering angles
except 90◦ . What can you deduce about the angular momentum J of the resonance?
(4 points)
1
5. Give a typical cross-section in units of barns (1 barn = 10−28 m) for each of the
following scattering processes:
(a) scattering of one atom by another atom (2 points),
(b) scattering of an electron by a proton (2 points).
Some useful Formulae
Z √
·
a2 − x2 dx =
Z ∞
0
Z ∞
0
Z ∞
Z0∞
0
Z ∞
0
Z ∞
0
µ ¶¸
1 √ 2
x
x a − x2 + a2 sin−1
2
a
r3 R42 (r)R32 (r) dr = −3.383a
r3 R42 (r)R31 (r) dr =
7.565a
r3 R42 (r)R30 (r) dr = −9.529a
r3 R42 (r)R21 (r) dr =
1.710a
r3 R42 (r)R20 (r) dr = −1.851a
r3 R42 (r)R10 (r) dr =
à !
sin
2
θ
2
0.2638a
1
= (1 − cos θ)
2
Good luck !
2