This was an important test of the Dirac equation and it passed with flying colors. TheDirac equa-
tion naturally incorporatesrelativistic corrections, the interaction with electron spin, and gives
an additional correction for s states that is found to be correct experimentally. When the Dirac
equation is used to make a quantum field theory of electrons and photons, Quantum ElectroDy-
namics, we can calculate effects to very high order and compare thecalculations with experiment,
finding good agreement.
36.6 Solution of Dirac Equation for a Free Particle
As with the Schr ̈odinger equation, the simplest solutions of the Dirac equation are those for a free
particle. They are also quite important to understand. We will find thateach component of the
Dirac spinor represents a state of a free particle at restthat we can interpret fairly easily.
We can show that a free particle solution can be written as aconstant spinor times the usual
free particle exponential. Start from the Dirac equation and attempt to develop an equationto
show that each component has the free particle exponential. We willdo this by making a second
order differential equation, which turns out to be the Klein-Gordonequation.
(
γμ
∂
∂xμ+
mc
̄h)
ψ= 0γν∂
∂xν(
γμ∂
∂xμ+
mc
̄h)
ψ= 0γν∂
∂xνγμ∂
∂xμψ+γν∂
∂xνmc
̄hψ= 0γνγμ∂
∂xν∂
∂xμψ+mc
̄hγν∂
∂xνψ= 0γνγμ∂
∂xν∂
∂xμψ−(mc
̄h) 2
ψ= 0γμγν∂
∂xμ∂
∂xν
ψ−(mc
̄h) 2
ψ= 0(γνγμ+γμγν)∂
∂xμ∂
∂xνψ− 2(mc
̄h) 2
ψ= 02 δνμ∂
∂xμ∂
∂xνψ− 2(mc
̄h) 2
ψ= 02
∂
∂xμ∂
∂xμ
ψ− 2(mc
̄h) 2
ψ= 0Thefree electron solutions all satisfy the wave equation.
(
✷−
(mc
̄h) 2 )
ψ= 0Because we have eliminated theγmatrices from the equation, this is an equation for each component
of the Dirac spinorψ.Each component satisfies the wave (Klein-Gordon) equationand a
solution can be written as a constant spinor times the usual exponential representing a wave.