PHY4605–Introduction to Quantum Mechanics II
Spring 2004
Test 3
April 16, 2004
No materials allowed for exam. If you can’t do one part of a problem, solve subsequent
parts in terms of unknown answer–define clearly. All parts 10 pts., max=70. Problem
1 required, attempt 1 of remaining 3 problems; circle which ones you want graded.
Possibly helpful formulae and constants
1Zt
dthm|V̂ (t)|niei(Em −En )t/h̄
ih̄ 0
|hf |V̂0 |ii|2 sin2 [(ω − ω0 )t/2]
Pf =
(ω − ω0 )2
h̄2
|eiθ − 1|2 = 4 sin2 θ/2
me4 1
En =
2h̄2 n2
ˆ·B
H = −~µ
ˆ = − ge S
~µ
2mc
Hψ = Eψ
∂ψ
Hψ = ih̄
∂t
h̄2 2
∇ + V (r)
H =−
2m
δE (1) = hn|V̂ |ni
δE (2) =
Rf =
Ã
Z r0
En − Em
0
X hm|V̂ |ni|mi
m6=n
s
=∓
1
e−ikr
eikr
ψ =
−
+ η0
2i
kr
kr
!
−2
En − Em
3
sin θe±iφ
8π
1
+
X
`m
dr r sin kr = k (sin kr0 − kr0 cos kr0 )
dσ
= |f (θ, φ)|2
dΩ
dσ
sin2 δ0
|0 =
dΩ
k2
eikr
ψ → eik·r + f
r
η0 − 1
f0 =
η0 = e2iδ0
2ik
1
Y00 = √
4π
s
3
Y10 =
cos θ
4π
Y1±1
Pf
π|hf |V̂0 |ii|2
dPf
'
=
ρ(ω0 )
dt
t
2h̄2
hφ|H|φi
≥ E0
hφ|φi
Ã
!
−ikr
ikr
X
1
e
e
eik·r =
−
+
+
g`m Y`m
2i
kr
kr
`m
X |hn|V̂ |mi|2
m6=n
δ|ni =
cm '
0
g`m
Y`m
1. Short Answer. Must attempt (only) 4 of 6. Circle answers to be graded.
(a) Using the figure sketching the wave
function of a particle with the scattering potential a) turned off and
b) turned on, state whether the potential was most likely attractive or
repulsive, and estimate (quantitatively!) the cross-section of the scattering process in the s-wave approximation in terms of the wave vector
k of the scattering particle.
(b) Explain qualitatively the difference between classical hard sphere cross
section and quantum mechanical low-energy cross section for hard spheres.
(c) Optical theorem. State the optical theorem and discuss.
2
(d) Fermi Golden rule. A beam of protons is scatters off a target containing
neutrons. The scattering amplitude for p − n scattering is modelled in a
certain energy range by
f (θ) = χ†f (A + Bσp · σn )χi
(1)
where χf and χi are the final and initial spin states of the n − p system.
Assume the initial (probe) proton spin is up, the inital (target) neutron
spin is unknown, and only the final proton spin is measured. Use the
Fermi Golden rule to find the ratio of the rates of spin up and spin down
measurements.
(e) Spontaneous emission. A hydrogen atom in an excited state in free space
decays after a certain time of order 10−9 sec. typically. How can this be?
After all, the atom is in an eigenstate, which is a stationary state of the
Hamiltonian, i.e. the time evolution of the state does not mix in any other
eigenstates. Discuss this paradox.
(f) Electric dipole radiation selection rule. State the selection rule for ` derived
in the homework for an electric dipole transition in a spherically symmetric
atom, and give a hand-waving explanation.
3
2. Spin exchange. The Hamiltonian which describes the interaction of a proton
and a neutron interacting by their dipolar moments (assume equal), is (for this
problem!)
H0 = J S~p · S~n
where J is a constant.
(a) What are the eigenfunctions and eigenvalues of H0 ? Give your answer for
the eigenfunctions explicitly in terms of single particle Ŝz eigenstates like
| ↑↓i, where the 1st spin is the proton and the 2nd the neutron.
At t = 0 the system is in its ground state, and an external magnetic field in the
ẑ direction, B(t) = (B0 cos ωt)ẑ is turned on.
(b) Write a Hamiltonian which includes the magnetic field coupling to the
spins. Assume the magnetic moment operators for the particles are ~µˆp =
gp S and ~µˆn = gn S, where gp and gn are unequal constants.
4
(c) Calculate perturbatively the probability for the system to make a transition
to the upper state after a time T .
(d) Sketch the probability of a transition as a function of time for fixed ω.
Discuss with this picture the criterion for the breakdown of perturbation
theory.
5
3. Hard sphere. Particles are scattered by a potential
V (r)



 V0 r < r0


 0 r > r0
with V0 > 0.
(a) Find the differential cross section in Born approximation.
(b) What is the limit of your result when the particles are very slow?
6
(2)
(c) For this part, take dσ/dΩ = 10−12 cm2 . If 1016 particles per cm2 per sec are
incident on this target at slow speeds, how many per second are detected
in a 1 cm2 detector located at 90◦ from the incident direction 1m (À r0 )
from the target?
(d) Let V0 → ∞. Write down and solve the radial Schrödinger equation for
the radial wave fctn. u0 = rψ|`=0 in the s-wave scattering approximation.
Find the s-wave scattering phase shift δ0 (k), and use it to calculate the
differential scattering cross section. Compare with your answer in b) and
explain the difference.
7
4. Scattering from molecular hydrogen. A light, neutral spinless probe particle of wave vector k scatters weakly off protons. The asymptotic (r → ∞)
wave function has the form
ψ(r) = e−k·r + a
eikr
,
r
(3)
where a is a constant.
(a) Find the differential scattering cross section and the total scattering cross
section for a single hydrogen atom in this approximation.
(b) A hydrogen molecule now consists of two protons separated by a displacement d, which is unaltered by the scattering event. For small k, the scattering is not exactly isotropic. Find the differential cross section for an
angle α between k and d, as a function of the scattering angle θ, assuming
that the scattering plane contains d.
8
(c) Find the ` = 0 and ` = 1 phase shifts δ0 and δ1 in the low energy limit
k → 0. If you could not do part b), discuss how you would go about this
calculation, beginning from an arbitrary scattering amplitude f (θ, φ).
(d) Find the total cross section for fixed α, either in terms of α, if you did part
b) or in terms of the δ` .
9