~vgroup=dω
dkˆk=dE
dppˆ=±pc^2
√
p^2 c^2 +m^2 c^4pˆClearly, we want waves that propagate in the right direction. Perhaps the momentum and energy
operators we developed in NR quantum mechanics are not the whole story.
For solutions 3 and 4, pick the solution for−~pto classify with solutions 1 and 2 with momentum~p
write everything in terms of the positive square rootE=
√
p^2 c^2 +m^2 c^4.ψ
(1)
~p =√
E+mc^2
2 EV
1
0
pzc
E+mc^2
(px+ipy)c
E+mc^2
ei(~p·~x−Et)/ ̄h ψ(2)
~p =√
E+mc^2
2 EV
0
1
(px−ipy)c
E+mc^2
−pzc
E+mc^2
ei(~p·~x−Et)/ ̄hψ(3)~p =√
E+mc^2
2 EV
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
E+mc^2
−pzc
E+mc^2
0
1
e−i(~p·~x−Et)/ ̄hWe have plane waves of the form
e±i(pμxμ)/ ̄h
which is not very surprising. In fact we picked the + sign somewhat randomly in the development
of NR quantum mechanics. For relativistic quantum mechanics, bothsolutions are needed. We have
no good reason to associate thee−i(pμxμ)solution with negative energy. Lets assume it also has
positive energy but happens to have the - sign on the whole exponent.
Consider the Dirac equation with the EM field term included. (While we are dealing with free
particle solutions, we can consider that nearly free particles will have a very similar exponential
term.)
∂
∂xμγμψ+
mc
̄hψ= 0
(
∂
∂xμ+
ie
̄hcAμ)
γμψ+
mc
̄hψ= 0The∂x∂μoperating on the exponential produces±ipμ/ ̄h. If we change the charge on the electron from
−eto +eand change the sign of the exponent, the equation remains the same. Thus, we can turn
the negative exponent solution (going backward in time) into the conventional positive exponent
solution if we change the charge to +e. Recall that the momentum has already been inverted (and
the spin also will be inverted).
Thenegative exponent electron solutions can be recast as conventional exponent positron
solutions.
36.7 “Negative Energy” Solutions: Hole Theory
Dirac’s goal had been to find a relativistic equation for electrons which was free of the negative
probabilities and the “negative energy” states of the Klein-Gordonequation. By developing and