86
3. Relativistic Quantum Mechanics
(b) Construct two two-component spinors φ + and φ − which are eigenstates of σ·u ˆ belonging to eigenvalues ±1, and normalized to φ
† φ s =
r
δ rs for (r, s) = (+, −), for the case u ˆ = (sin θ cos φ, sin θ sin φ, cos θ)
( )
1
[Hint : take the arbitrary φ =
].
0
3.8 Positive-energy spinors u(p, s) are defined by
(
)
φ
s
u(p, s) = (E + m)
1/2 ( σ · p
)
s = 1, 2
φ
s
E + m
†
with φ
s† φ
s = 1. Verify that these satisfy u u = 2E.
In a similar way, negative-energy spinors v(p, s) are defined by
(
)
σ · p χ
s
m)
1/2 ( E + m )
v(p, s) = (E +
s = 1, 2
χ
s
†
with χ
s† χ
s = 1. Verify that v v = 2E.
3.9 Using the KG equation together with the replacement ∂
μ
→ ∂
μ + iqA
μ ,
find the form of the potential V ˆ KG in the corresponding equation
(❗ + m
2 )φ = −V ˆ KG φ
in terms of A
μ .
3.10 Evaluate
{σ · (−i∇ − qA)}
2 ψ
by following the subsequent steps (or doing it your own way):
(a) Multiply the operator by itself to get
{(σ · −i∇)
2 + iq(σ · ∇)(σ · A) + iq(σ · A)(σ · ∇) + q
2 (σ · A)
2
}ψ.
2
A
2
The first and last terms are, respectively, −∇
2 and q
where the
2 × 2 unit matrix 1 is understood. The second and third terms are
iq(σ · ∇)(σ · Aψ) and iq(σ · A)(σ · ∇ψ). These may be simplified
using the identity of problem 4.4(b), but we must be careful to treat
∇ correctly as a differential operator.
(b) Show that (σ · ∇)(σ · A)ψ = ∇ · (Aψ) + iσ · {∇ × (Aψ)}. Now use
∇ × (Aψ) = (∇ × A)ψ − A × ∇ψ to simplify the last term.
(c) Similarly, show that (σ · A)(σ · ∇)ψ = A · ∇ψ + iσ · (A × ∇ψ).
(d) Hence verify (3.116).