We willnormalize the statesso that there is one particle per unit volume.
ψ†ψ=1
V
N^2
(
1 +
p^2 c^2
(|E|+mc^2 )^2)
=
1
V
N^2
(
E^2 +m^2 c^4 + 2|E|mc^2 +p^2 c^2
(|E|+mc^2 )^2)
=
1
V
N^2
(
2 E^2 + 2|E|mc^2
(|E|+mc^2 )^2)
=
1
V
N^2
(
2 |E|
(|E|+mc^2 ))
=
1
V
N=
√
|E|+mc^2
2 |E|VWe have thefour solutions with for a free particle with momentum~p. For solutions 1 and
2,Eis a positive number. For solutions 3 and 4,Eis negative.
ψ(1)~p =√
|E|+mc^2
2 |E|V
1
0
pzc
E+mc^2
(px+ipy)c
E+mc^2
ei(~p·~x−Et)/ ̄h ψ(2)
~p =√
|E|+mc^2
2 |E|V
0
1
(px−ipy)c
E+mc^2
−pzc
E+mc^2
ei(~p·~x−Et)/ ̄hψ(3)~p =
√
|E|+mc^2
2 |E|V
−pzc
−E+mc^2
−(px+ipy)c
−E+mc^2
1
0
ei(~p·~x−Et)/ ̄h ψ(4)
~p =√
|E|+mc^2
2 |E|V
−(px−ipy)c
−Ep+zmcc^2
−E+mc^2
0
1
ei(~p·~x−Et)/ ̄hThe spinors areorthogonalfor states with the same momentum and the free particle waves are
orthogonal for different momenta. Note that the orthogonality condition is the same as for non-
relativistic spinors
ψ~p(r)†ψ(r′)
p~′ =δrr′δ(~p−~p′)It is useful to write the plane wave states as a spinoru
(r)
~p times an exponential. Sakurai picks a
normalization of the spinor so thatu†utransforms like the fourth component of a vector. We will
follow the same convention.
ψ~p(r)≡√
mc^2
|E|Vu(~pr)ei(~p·~x−Et)/ ̄hu(1)~p =√
E+mc^2
2 mc^2
1
0
pzc
E+mc^2
(px+ipy)c
E+mc^2
u(2)
~p =√
E+mc^2
2 mc^2
0
1
(px−ipy)c
E+mc^2
−pzc
E+mc^2
u
(3)
~p =√
−E+mc^2
2 mc^2
−pzc
−E+mc^2
−(px+ipy)c
−E+mc^2
1
0
u(4)
~p =√
−E+mc^2
2 mc^2
−(px−ipy)c
−Ep+zmcc^2
−E+mc^2
0
1