84
3.2
3. Relativistic Quantum Mechanics
Multiply this equation from the left by ψ
∗ and multiply the complex
conjugate of this equation by ψ (assume V is real). Subtract the
two equations and show that your answer may be written in the
form of a continuity equation
∂ρ/∂t + ∇ · j = 0
where ρ = ψ
∗ ψ and j = i
−1 [ψ
∗ (∇ψ) − (∇ψ
∗ )ψ].
(b) Perform the same operations for the Klein–Gordon equation and
derive the corresponding ‘probability’ density current. Show also
that for a free-particle solution
−ip·x
φ = N e
with p
μ = (E, p), the probability current j
μ = (ρ, j) is proportional
to p
μ .
(a) Prove the following properties of the matrices α i and β:
(i) α i and β (i = 1, 2, 3) are all Hermitian [Hint : what is the
Hamiltonian?].
(ii) Trα i = Trβ = 0 where ‘Tr’ means the trace, i.e. the sum of
the diagonal elements [Hint : use Tr(AB) = Tr(BA) for any
matrices A and B – and prove this too!].
(iii) The eigenvalues of α i and β are ±1 [Hint : square α i and β].
(iv) The dimensionality of α i and β is even [Hint : the trace of a
matrix is equal to the sum of its eigenvalues].
(b) Verify explicitly that the matrices α and β of (3.31), and of (3.40),
satisfy the Dirac conditions (3.34) – (3.36).
3.3 For free-particle solutions of the Dirac equation
−ip·x
ψ = ωe
the four-component spinor ω may be written in terms of the two-component
spinors
( )
φ
ω =
.
χ
From the Dirac equation for ψ
i∂ψ/∂t = (−iα · ∇ + βm)ψ
using the explicit forms for the Dirac matrices
(
)
(
)
0 σ
1
0
α =
β =
σ 0
0 − 1