Download Further Quantum Mechanics: Problem Set 2. Trinity term weeks 1 – 2

Further Quantum Mechanics: Problem Set 2.
Trinity term weeks 1 – 2
This problem set is taken mainly from the book by Binney and Skinner.
Qu 1. Problem 7.1 from Prof Blundell’s lecture course is repeated with some modifications below. If you have
already attempted it, explain how the approach you used fits within the more general understanding you should
now have of time-dependent problems in quantum mechanics. If you have not attempted it previously, you
should do so.
In the β decay H3 (1 proton + 2 neutrons in the nucleus) → (He3 )+ (2 protons + 1 neutron in the nucleus),
the emitted electron has a kinetic energy of 16 keV. We will consider the effects on the motion of the atomic
electron, i.e. the one orbiting the nucleus, which we assume is initially in the ground state of H3 .
Show by a brief justification that the perturbation is sudden, by considering the location of the emitted electron
at a time around τ = 5 × 10−17 s after emission. How does τ compare with the time-scale on which the
wavefunction changes?
Show that the probability for the electron to be left in the ground state of (He3 )+ is 23 (2/3)6 ' 0.7.
Qu 2. At early times (t ∼ −∞) a harmonic oscillator of mass m and natural angular frequency ω is in its
2
2
ground state. A perturbation δH = Exe−t /τ is then applied, where E and τ are constants.
What is the probability according to first-order theory that by late times the oscillator transitions to its second
excited state, |2i?
Show that to first order in δH the probability that the oscillator transitions to the first excited state, |1i, is
P =
πE 2 τ 2 −ω2 τ 2 /2
e
,
2m~ω
Plot P as a function of τ and comment on its behaviour as ωτ → 0 and ωτ → ∞.
Qu 3. Show that the number of states g(E) dE d2 Ω with energy in (E, E + dE) and momentum in the solid
angle d2 Ω around p = ~k of a particle of mass m that moves freely subject to periodic boundary conditions on
the walls of a cubical box of side length L is
3 3/2
L
m √
2
g(E) dE d Ω =
2E dE dΩ2 .
2π
~3
Hence show from Fermi’s golden rule that the cross-section for elastic scattering of such particles by a weak
potential V (x) from momentum ~k into the solid angle d2 Ω around momentum ~k0 is
2
Z
m2
2 3
i(k−k0 )·x
dσ =
d Ω d xe
V (x) .
2
4
(2π) ~
Explain in what sense the potential has to be ‘weak’ for this Born approximation to the scattering cross-section
to be valid.
Qu 4. Write down the selection rules for radiative transitions in the electric dipole approximation. Draw an
energy level diagram for hydrogen (use the vertical direction for energy, and separate the states horizontally by
angular momentum `). Show how the selection rules apply to hydrogen by marking allowed transitions on your
diagram.
Qu 5. Let |E, l, mi denote a stationary state of an atom with orbital angular-momentum quantum numbers
l, m, and let x± = x ± iy be complex position operators while L± = Lx ± iLy are the usual orbital angularmomentum ladder operators. Show that x± |E, l, mi is an eigenket of Lz with eigenvalue m ± 1. Show also
that
[L+ , x+ ] = [L− , x− ] = 0 and [L+ , x− ] = −[L− , x+ ] = 2z.
1
Hence show that
hE 0 , l0 , m|z|E, l, mi = α− (l0 , m)hE 0 , l0 , m − 1|x|E, l, mi − α+ (l, m)hE 0 , l0 , m|x|E, l, m + 1i.
p
where α± (l, m) ≡ l(l + 1) − m(m ± 1). [Hint: compute hE 0 , l0 , m|x|E, l, m + 1i]
Qu 6. A system has a Hamiltonian of the form
H=
p2
+ V (x) .
2me
Show that [x, [H, x]] = ~2 /me . By taking the expectation value of this expression in the state |ki, show that
X
|hn|x|ki|2 (En − Ek ) =
n6=k
~2
,
2me
where the sum runs over all the other stationary states.
The oscillator strength of a radiative transition |ki → |ni is defined to be
fkn ≡
Show that
of atoms?
P
n
2me
(En − Ek )|hn|x|ki|2 .
~2
fkn = 1. What is the significance of oscillator strengths for the allowed radiative transition rates
Qu 7. With |nlmi a stationary state of hydrogen, which of these matrix elements is non-zero?
h100|z|200i
h100|z|300i
h100|x|200i
h100|z|210i
h100|z|310i
h100|x|210i
h100|z|211i
h100|z|320i
h100|x|211i
Qu 8. With |nlmi a stationary state of hydrogen, and given that
r
r
r
6
3
3
iφ
Y10 (θ, φ) =
cos θ Y11 (θ, φ) = −
sin θ e
Y1−1 (θ, φ) =
sin θ e−iφ ,
8π
8π
8π
show that
√
h100|(x − iy)|211i = − 2h100|z|210i
h100(x − iy)|21 − 1i = 0.
Write down the values of h100|(x + iy)|21 − 1i and h100|(x + iy)|211i and hence show that with
1
|ψi ≡ √ (|211i − |21 − 1i),
2
h100|x|ψi = −h100|z|210i. Explain the physical significance of this result.
Qu 9. Derive the selection rules
hn0 l0 m0 |x+ |nlmi = 0 unless
hn0 l0 m0 |x− |nlmi = 0 unless
m0 = m + 1
m0 = m − 1.
where x± = x ± iy. From this selection rule one infers that when the atom sits in a magnetic field along the z
axis and the spectrometer looks along the z axis, the detected photons will be circularly polarised. Show that
linearly polarised photons can be detected from an atom that’s in a magnetic field.
From the above rules it might be argued that photons emitted along the z axis will be circularly polarised even
in the absence of a magnetic field. Why is this argument bogus?
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