Description
1 Dirac algebra
a) Based on the properties of the Dirac algebra only (which do not depend on the representation), prove that β and α
i matrices must have even dimensionality.
b) Show explicitly that the Dirac algebra cannot be satisfied in a representation with
dimension d = 2. Hint: assume α
i = σ
i as Pauli matrices and show that one cannot find a
matrix for β that satisfies the Dirac algebra.
c) Find the explicit representation for (β, αi
), where
α
3 =
1 0
0 −1
,
with 1 the 2 × 2 identity matrix.
2 Free Dirac particle
The normalization of the solutions of the Dirac equation with positive and negative energy
satisfy
ψψ¯ = ±1,
where + corresponds to the positive energy states and − to the negative energy ones, with
ψ¯ = ψ
†
γ
0
the Dirac adjoint.
a) Find the eigenenergies and the normalized eigenfunctions that solve the Dirac equation
for a free particle,
(iγµ
∂µ − m)Ψ(x, t) = 0.
b) Show that the orbital angular momentum L of a free Dirac particle is not a constant
of the motion. Use the fact that
[Li
, pj ] = i~ijkpk,
where ijk is the Levi-Civita tensor.
Defining the spin operator as Σ ≡ 1 ⊗ σ,where 1 is the
2 × 2 identity matrix and σ = (σx, σy, σz), show that the total angular momentum
J = L +
~
2
Σ
1
is conserved.
c) Now show that the operators p · Σ and p · L are each one constants of the motion. The
operator
p · Σ
|p|
is called helicity.
d) Calculate the equation of motion for the position operator x of a free Dirac particle.
Show that the velocity operator v ≡
d
dt
x is not a constant of the motion, unlike the momentum
p.
3 Central potential
a) Show that the Dirac equation for a central potential V (r) can be written in the form
χ =
c
E − V (r) + mc2
(σ · p)ϕ
where the total wavefunction is a four component spinor
Ψ =
ϕ
χ
,
in the bi-spinor representation.
b) Assume that ϕ describes an s-wave orbital with spin ↓ of the form
ϕ(r, t) = R(r)exp
−
iEt
~
0
1
.
Calculate χ explicitly and show that it describes a p-wave function with spin s = 1/2 and
orbital angular momentum ` = 1. Hint: express χ in terms of spherical harmonics and
spinors.
c) Using your previous result, show that χ(r, t) describes a j = 1/2 state with m = −1/2,
where j and m are the total angular momentum J = L + S quantum number.
Hint: use a
table of Clebsh-Gordan coefficients.
