130_notes.dvi

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|V

We 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







e

i(~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







e

i(~p·~x−Et)/ ̄h

ψ(3)~p =

√

|E|+mc^2

2 |E|V









−pzc

−E+mc^2

−(px+ipy)c

−E+mc^2

1

0







e

i(~p·~x−Et)/ ̄h ψ(4)

~p =

√

|E|+mc^2

2 |E|V









−(px−ipy)c

−Ep+zmcc^2

−E+mc^2

0

1







e

i(~p·~x−Et)/ ̄h

The 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|V

u(~pr)ei(~p·~x−Et)/ ̄h

u(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







