Homework Assignment 07: Quantum Dynamics
(Due: 03/24/2016)
Problem 1 (30%)
(a) [20%] Prove that
e ABe −A = B +[A,B]+
1
1
[A,[A,B]]+ [A,[A,[A,B]]]+ 
2!
3!
(1)
where A and B are two operators.
[Hint: Recall how we proved the Feynman-Glauber theorem in class… If one defines
F(x) ≡ e xABe −xA ,
then F(0) = B and F(1) = e ABe −A . Then what does Eq. (1) look like?]
(b) [10%] Let x̂(t) and p̂(t) be the coordinate and momentum operators for a free particle in one
dimension in the Heisenberg picture. Using Eq. (1) to evaluate p̂(t) , x̂(t) and [x̂(t), x̂(0)] .
Problem 2 (20%)
Read neutrino oscillations in textbook (Page 77-78), then (1) derive the neutrino oscillation
probability (Eq. 2.1.65 on Page 78) and (2) use this equation, along with the data in Figure 2.2
(Page 78), estimate the values of Δm 2c 4 (in unit of eV2) and the mixing angle θ .
Problem 3 (25%)
Consider a one-dimensional simple harmonic oscillator with charge e in a uniform static electric
field ε . Solving the Heisenberg equation of motion to get p̂(t) , x̂(t) and [x̂(t), x̂(t ′)] .
Problem 4 (25%)
Consider a particle with mass m in three dimensions whose Hamiltonian is given by
p̂2
H=
+V(r̂) .
2m
By calculating [r̂ ⋅ p̂, H ] using the fundamental commutator, show that
d
p̂2
r̂ ⋅ p̂ =
− r̂ ⋅ ∇V .
dt
m
[Notes: (1) This form is independent of picture. (2) This is the QM analogue of the classical
Virial theorem: if r ⋅ p is time-independent, then 2 T =
p2
= r̂ ⋅ ∇V .]
m