Define the4 by 4 matricesγμare by.
γi =(
0 −iσi
iσi 0)
γ 4 =(
1 0
0 − 1
)
With this definition, the relativistic equation can be simplified a great deal
(
γμ
∂
∂xμ+
mc
̄h)
ψ= 0where the gamma matrices are given by
γ 1 =
0 0 0 −i
0 0 −i 0
0 i 0 0
i 0 0 0
γ 2 =
0 0 0 − 1
0 0 1 0
0 1 0 0
−1 0 0 0
γ 3 =
0 0 −i 0
0 0 0 i
i 0 0 0
0 −i 0 0
γ 4 =
1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 − 1
and they satisfy anti-commutation relations.{γμ,γν}= 2δμνIn fact any set of matrices that satisfy the anti-commutation relations would yield equivalent physics
results, however, we will work in the above explicit representation of the gamma matrices.
Definingψ ̄=ψ†γ 4 ,
jμ=icψγ ̄μψ
satisfies the equation of a conserved 4-vector current
∂
∂xμjμ= 0and also transforms like a 4-vector. The fourth component of thevector shows that the probability
density isψ†ψ. This indicates that the normalization of the state includes all four components of
the Dirac spinors.
For non-relativistic electrons, the first two components of the Dirac spinor are large while the last
two are small.
ψ=(
ψA
ψB)
ψB≈c
2 mc^2~σ·(
~p+e
cA~
)
ψA≈pc
2 mc^2ψAWe use this fact to write an approximate two-component equation derived from the Dirac equation
in the non-relativistic limit.
(
p^2
2 m
−
Ze^2
4 πr−
p^4
8 m^3 c^2+
Ze^2 L~·S~
8 πm^2 c^2 r^3+
Ze^2 ̄h^2
8 m^2 c^2δ^3 (~r))
ψ = E(NR)ψ