36.3 The Dirac Equation
We canextend this concept to use the relativistic energy equation(for now with no EM
field). The idea is to replace~pwith~σ·~p.
(
E
c) 2
−p^2 = (mc)^2
(
E
c−~σ·~p)(
E
c+~σ·~p)
= (mc)^2
(
i ̄h
c∂
∂t+i ̄h~σ·∇~)(
i ̄h
c∂
∂t−i ̄h~σ·∇~)
φ= (mc)^2 φ
(
i ̄h∂
∂x 0+i ̄h~σ·~∇)(
i ̄h∂
∂x 0−i ̄h~σ·∇~)
φ= (mc)^2 φThis is again written in terms of a2 component spinorφ.
This equation is clearly headed toward being second order in the time derivative. As with Maxwell’s
equation, which is first order when written in terms of the field tensor, we can try to write a first
order equation in terms of a quantity derived fromφ. Define
φ(L) = φφ(R) =1
mc(
i ̄h∂
∂x 0−i ̄h~σ·∇~)
φ(L)Including the two components ofφ(L)and the two components ofφ(R), we now have four components
which satisfy the equations.
(
i ̄h∂
∂x 0+i ̄h~σ·∇~)(
i ̄h∂
∂x 0−i ̄h~σ·∇~)
φ(L) = (mc)^2 φ(L)
(
i ̄h∂
∂x 0+i ̄h~σ·∇~)
mcφ(R) = (mc)^2 φ(L)
(
i ̄h∂
∂x 0+i ̄h~σ·∇~)
φ(R) = mcφ(L)
(
i ̄h∂
∂x 0−i ̄h~σ·∇~)
φ(L) = mcφ(R)These (last) two equations couple the 4 components together unlessm= 0. Both of the above
equations are first order in the time derivative. We could continue with this set of coupled equations
but it is more reasonable towrite a single equation in terms of a 4 component wave function.
This will also be a first order equation. First rewrite the two equations together, putting all the
terms on one side.
(
i ̄h~σ·∇−~ i ̄h
∂
∂x 0)
φ(L)+mcφ(R)= 0
(
−i ̄h~σ·∇−~ i ̄h∂
∂x 0)
φ(R)+mcφ(L)= 0