Problem set 11
Question 1 Consider a one dimensional harmonic oscillator (mass m, frequency ω) in its
ground state perturbed by
H 0 (t) = αx2 sin(f t)
(1)
for 0 ≤ t ≤ π/f , where α is small compared to mω 2 . What is the probability of measuring
the oscillator in its first excited state after time t = π/f ? What about for the second excited
state? For the second excited state, plot this probability as a function of f /ω0 , where
ω0 = (E2 − E0 )/~. For this problem, use the exact formula for the transition probability in
first order perturbation theory. You can use a computer to graph the results for you.
Question 2 Consider a particle of mass m in the ground state of an infinite square well
potential of width L (with V = 0 for x ∈ [0, L]) at t = −∞. During the times −∞ ≤ t ≤ ∞,
the system is perturbed by
0
H (t) = Aδ(x − L/2)e
−
t
t0
2
(2)
If we measure the energy at t = ∞, what are the possible values we might obtain and what
are the probabilities of obtaining them (assuming first order perturbation theory is valid)?
Note: you may wish to use that
r
Z ∞
π β2
2
e 4α
e−αt +βt dt =
(3)
α
−∞
b) The parameter t0 controls how quickly the perturbation turns on and off. Describe
qualitatively how the transition probability depends on t0 as t0 ranges from 0 to infinity.
c) There is a theorem in quantum mechanics called the adiabatic theorem. It states that if
we are initially in the nth eigenstate of some Hamiltonian H0 , and the Hamiltonian changes
via some time dependent process to a final Hamiltonian H1 (which could be the same or
different than H0 ), then in a limit where the Hamiltonian is changing slowly at all times,
the final state will be the nth eigenstate of H1 . Explain how your result is consistent with
this theorem.
PS: If you are interested, you can read more about the adiabatic theorem in section 10.1 of
Griffiths. PPS: the “proof ” in 10.1.2 isn’t really a proof.
Question 3 An electron in a system with H0 = E0 /~Sz is initially in a spin up (Sz = ~/2)
state. At t = 0, we turn on a magnetic field
~
B(t)
= B0 x̂ cos(ωt) ,
(4)
Using first order perturbation theory, calculate the probability that a measurement of Sz at
time t > 0 will give −~/2. You can use the approximate result for sinusoidal perturbations.
1
Question 4 Read the handout on charged particles in electromagnetic fields. For a charged
~ = B ẑ, what is a possible Hamiltonian
particle in a uniform, constant magnetic field B
(ignoring spin)?
2