Mathematical Principles of Theoretical Physics

364 CHAPTER 6. QUANTUM PHYSICS

with~D=∇+i ̄hcg~A, we derive that

~D×~D=ig

hc ̄

[

∇×~A+~A×∇

]

.

Note that as an operator we have

∇×~A=curl~A−~A×∇.

Hence we get
~D×~D=ig
̄hc

curl~A.

Thus, (6.4.19) is written as

(6.4.20) (~σ·~D)^2 =D^2 −
g
̄hc

~σ·curl~A.

By (6.4.20), the spectral equation (6.4.18) is in the form

(6.4.21)

[

−

̄h^2

2 mk

D^2 +gA 0

](

ψ 1 k

ψ 2 k

)

+~μk·curl~A

(

ψ 1 k

ψ 2 k

)

=λ

(

ψ 1 k

ψ 2 k

)

,

whereD= (D 1 ,D 2 ,D 3 ),(A 0 ,A 1 ,A 2 ,A 3 )is as in (6.4.11), and

(6.4.22) ~μk=

hg ̄

2 mk

~σ, Dk=∂k+i

g

hc ̄

Ak ( 1 ≤k≤ 3 ).

Since the fermions are bound in the interiorΩof subatomic particle,ψk( 1 ≤k≤N)are zero
outsideΩ. Therefore the equation (6.4.21) are supplemented with the Dirichlet boundary
conditions:

(6.4.23) (ψ 1 k,ψ 2 k)|∂Ω= 0 ( 1 ≤k≤ 3 ),

whereΩ⊂R^3 is a bounded domain.
The boundary value problem (6.4.21)-(6.4.23) is the model for the energy level theory of
massive subatomic particles.

Massless bound states

In order to obtain the spectral equations for massless subatomic particle, we have to de-
rive their wave equations, which are based on the basic quantum mechanics principle: the
Postulate 5.5.
We first recall the Weyl equation

(6.4.24)

∂ ψ

∂t

=c(~σ·∇)ψ,

which describes massless and free fermions. The Weyl equation (6.4.24) is derived from the
de Broglie relation

(6.4.25) E=cp (see (6.2.12))