=
√
E+mc^2
2 EV′
1
0
0
βγmc^2
γmc^2 +mc^2
ei(~p·~x−Et)/ ̄h=
√
E+mc^2
2 EV′
1
0
0
βγ
γ+1
ei(~p·~x−Et)/ ̄h=
√
E+mc^2
2 EV′
1
0
0
sinhχ
coshχ+1
ei(~p·~x−Et)/ ̄h=
√
E+mc^2
2 EV′
1
0
0
2 sinhχ 2 coshχ 2
cosh^2 χ 2 +sinh^2 χ 2 +cosh^2 χ 2 −sinh^2 χ 2
ei(~p·~x−Et)/ ̄h=
√
E+mc^2
2 EV′
1
0
0
2 sinhχ 2 coshχ 2
2 cosh^2 χ 2
ei(~p·~x−Et)/ ̄h=
√
E+mc^2
2 EV′
1
0
0
tanhχ 2
ei(~p·~x−Et)/ ̄hwhere the normalization factor is now set to be√^1 V′, defining this as the primed system.
We can also find the same state by boosting the at rest solution. Recall that we are boosting in the
x direction with−β, implyingχ→−χ.
Sboost = coshχ
2−iγ 1 γ 4 sinhχ
2= coshχ
2−i
0 0 0 −i
0 0 −i 0
0 i 0 0
i 0 0 0
1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 − 1
sinhχ
2= coshχ
2−i
0 0 0 i
0 0 i 0
0 i 0 0
i 0 0 0
sinhχ
2= coshχ
2+
0 0 0 1
0 0 1 0
0 1 0 0
1 0 0 0
sinhχ
2=
coshχ 2 0 0 sinhχ 2
0 coshχ 2 sinhχ 2 0
0 sinhχ 2 coshχ 2 0
sinhχ 2 0 0 coshχ 2