336 CHAPTER 6. QUANTUM PHYSICS
wherep 0 is a scalar momentum. It follows that
(6.2.12) E=cp 0 ,
By (6.2.5), corresponding top 0 the Hermitian operator reads
pˆ 0 =i ̄h(~σ·∇).Thus, the relation (6.2.12) leads to the following Weyl equation:
(6.2.13)
∂ ψ
∂t=c(~σ·∇)ψ,whereψ= (ψ 1 ,ψ 2 )Tis a two-component Weyl spinor. The equation (6.2.13) describes free
neutrinos.
4.Dirac equations.For a massive particle, the de Broglie matter-wave relation(6.2.12)
should be rewritten in the form
(6.2.14) E=cp 1 ±mc^2 ,
wherep 1 is a scalar momentum for massive fermions, and by (6.2.5) the Hermitian operators
forp 1 andmare given by
pˆ 1 =−ih ̄(~α·∇), mˆ=mα 0 ,
where~α= (α 1 ,α 2 ,α 3 ),
α 0 =(
I 0
0 −I
)
, αk=(
0 σk
σk 0)
, k= 1 , 2 , 3 ,andσk( 1 ≤k≤ 3 )are the Pauli matrices as in (3.5.36).
Thus, the de Broglie matter-wave relation (6.2.14) for massive fermions leads to the fol-
lowing Dirac equations:
(6.2.15) ih ̄
∂ ψ
∂t
=−ihc ̄ (~α·∇)ψ+mc^2 α 0 ψ,whereψ= (ψ 1 ,ψ 2 ,ψ 3 ,ψ 4 )Tis the Dirac spinor.
Multiplying both sides of (6.2.15) by the matrixα 0 , then the Dirac equation is rewritten
in the usual form
(6.2.16)
(
iγμ∂μ−mc
̄h)
ψ= 0 ,whereγμ= (γ^0 ,γ^1 ,γ^2 ,γ^3 )is the Dirac matrices as in (6.2.8), withγ^0 =α 0 ,γk=α 0 αk( 1 ≤
k≤ 3 ).
Remark 6.7.Based on the spinor theory, the Weyl equation (6.2.13) and the Dirac equations
(6.2.16) are Lorentz invariant (see Section2.2.6) and space rotation invariant. In addition, the
Dirac equations are invariant under the space reflection
(6.2.17) ̃x=−x, ̃t=t.