For some calculations it is useful to write the Klein-Gordon equation in the two-component form
We see that and .
The charge density for these state is (setting )
which is rather simple and somewhat similar to the nonrelativistic case.
obey the coupled equation
If we define the 2-component spinor
we can combine equations 4.87 and 4.88 to be
Using the Pauli matrices, ,
and we can write
This is a first order Schrödinger equation (cf. ). The quantity in square brackets is .
Also in this notation, the charge density can be written as
The normalization condition becomes
It will be shown later that the sign is determined by whether we start with particles () or antiparticles ().
The Klein-Gordon Hamiltonian is
The Hamiltonian appears to be non-hermitian, since
We notice that
Because of the normalization condition, the Hamiltonian is effectively hermitian.
Consider the free particle solutions
A positive-energy plane-wave solution normalized to unit density (equation 4.41) is
We can write
A corresponding negative-energy plane-wave solution is
and it can also be shown that
In the nonrelativistic limit we have
The components of equation 4.108 are
Equation 4.108 in the nonrelativistic limit is
which holds to second order in the velocity. Similarly
By completeness, any wavepacket can be expanded in terms of a linear combination of positive- and negative-energy solutions.
where only depends on the magnitude of , and is a function of time and the magnitude of .
If the wave function is normalized to unity we have