Problems
147
where the constants a and b are to be chosen so that all the trajectories end
at the same point x(t 0 ).
5.3
(a) Use (5.57) and (5.63) to verify that
˙
p ˆ = mq ˆ
is consistent with the Heisenberg equation of motion for A ˆ = ˆ
q.
(b) By similar methods verify that
p ˆ ˙ = −mω
2 q. ˆ
5.4
(a) Rewrite the Hamiltonian H ˆ of (5.63) in terms of the operators ˆ
a
†
and ˆ
a .
(b) Evaluate the commutator between ˆ
a and ˆ
a
† and use this result
together with your expression for H ˆ from part (a) to verify equation (5.73).
(c) Verify that for |n> given by equation (5.81) the normalization condition

= 1
is satisfied.
(d) Verify (5.83) directly using the commutation relation (5.72).
∗
5.5 Treating ψ and ψ as independent classical fields, show that the Lagrangian density
∗
∗
L = iψ ψ ˙ − (1/2m)∇ψ · ∇ψ
∗
gives the Schr¨ odinger equation for ψ and ψ correctly.
5.6
(a) Verify that the commutation relations for a ˆ(k) and ˆ
a
† (k) (equations
(5.130)) are consistent with the equal time commutation relation
between φ ˆ and ˆ
π (equation (5.117)), and with (5.118).
(b) Consider the unequal time commutator D(x 1 , x 2 ) ≡ [φ ˆ (x 1 , t 1 ),
φ ˆ (x 2 , t 2 )], where φ ˆ is a massive KG field in three dimensions. Show
that
∫
d
3
k
ik·(x1−x2) ]
D(x 1 , x 2 ) =
[e
−ik·(x1−x2)
− e
(5.161)
(2π) 3 2E
where k · (x 1 − x 2 ) = E(t 1 − t 2 ) − k · (x 1 − x 2 ), and E = (k
2 +
m
2 )
1/2 . Note that D is not an operator, and that it depends only