Physics 315, Problem Set 8
1. Using
φ(~x) =
Z
1
d3 p
√
(ap~ ei~p·~x + ap†~ e−i~p·~x )
3
(2π) 2ωp~
as the solution to the free scalar Lagrange density, find π(~x).
2. From the commutation relation [ap~ , ap†~0 ] = (2π)3 δ (3) (~p − p~0 ), find the commutation
relations satisfied by φ(~x) and π(~x), using your results from the previous problem.
3. Find the Hamiltonian density, H, corresponding to
1
1
L = (∂µ φ)2 − m2 φ2 .
2
2
Plug in for φ and π above to find the Hamiltonian H =
and annihilation operators and their commutators.
R
dx H in terms of creation
4. We discussed the action of the translation operator on a real scalar field φ(x):
~
~
0
0
eiP ·~x φ(t, ~x)e−iP ·~x = φ(t, ~x + ~x0 ) .
What does the translation operator do to creation and annihilation operators? With
this information, and the fact that the vacuum is translation invariant, show that the
particle state |~ki transforms like a plane wave state.
5. Evaluate h0|φ(~x)|~pi. From this result, interpret the significance of φ(~x)|0i. (φ(~x) is a
real scalar field.)
6. Filling in the blanks for the two real scalar fields ↔ single complex scalar field case.
1
1
1
1
L = ∂µ φ(1) ∂ µ φ(1) + ∂µ φ(2) ∂ µ φ(2) − m2 (φ(1) )2 − m2 (φ(2) )2 .
2
2
2
2
(a) Find the conserved “charge” Q associated with the symmetry L possesses as the
fields φ(1) and φ(2) are scrambled.
(b) Plug in a modal expansion for both φ(1) and φ(2) and show that you obtain
Q = −i
Z
d3 k[a(2) (~k)† a(1) (~k) − a(1) (~k)† a(2) (~k)] .
(c) Show that
H=
Z
d3 k ωk [a(1) (~k)† a(1) (~k) + a(2) (~k)† a(2) (~k)] + infinite piece .
(d) Show that Q commutes with H.
(e) Define b(~k) = √12 [a(1) (~k) + ia(2) (~k)] and c(~k) = √12 [a(1) (~k) − ia(2) (~k)], along with
their hermitian conjugates, and find all the commutation relations between and
among the b, b† , c, and c† .
1
(f) Show that
H=
Z
d3 k ωk [b(~k)† b(~k) + c(~k)† c(~k)] + infinite piece .
and
Q=
Z
d3 k[b(~k)† b(~k) − c(~k)† c(~k)] .
(g) Re-express L above in terms of the field
1
ψ = √ (φ(1) + iφ(2) )
2
and its conjugate.
7. Review of Time Reversal.
(a) Determine how time, position, momentum, orbital angular momentum, and spin
angular momentum transform under a time reversal operation.
(b) Consider a spin-1/2 particle under the influence of a scalar potential and a spinorbit force. The Schroedinger equation is:
i
dψ
1 2
~ 2 (r))ψ ,
= (−
∇ + V1 (r) + ~σ · LV
dt
2m
where ~σ are the vector of Pauli matrices, and |~r| = r. We desire this equation to
be invariant under a time reversal transformation, T . Show that
T ψ(t, ~r) = iσ2 ψ ∗ (−t, ~r)
and your results from part (a) will accomplish this.
(c) If
ik·x
ψ=e
1
0
!
,
where k and x are four-vectors, find T ψ. Does this result make physical sense to
you? Why or why not?
(d) Show that T conserves probabilities even though it is not unitary. Show that T
is anti-unitary.
8. Warm up for fermions. Employ {a, a† } = 1 to find the spectrum of the harmonic
oscillator hamiltonian under fermion anti-commutation conditions.
2