Mathematical Principles of Theoretical Physics

6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 367

In Section6.4.1we see that charged leptons and quarks are made up of three weaktons,
with masses caused by the deceleration of the constituent weaktons. Let the masses of the
constituent weaktons bem 1 ,m 2 ,m 3 , and the wave functions of these weaktons be given by

ψk=

(

ψ 1 k

ψ 2 k

)

fork= 1 , 2 , 3.

Hereψ 1 kandψ 2 krepresent the left-hand and right-hand states. The bound states are due to the
weak interaction, and the potential in (6.4.11) takes the form

gAμ= 2 gwWμ= ( 2 gwW 0 , 2 gwW~).

By (6.4.21)-(6.4.23), the spectral equations for charged leptons and quarks areas follows

−

̄h^2

2 mj

(∇+i

2 gw

hc ̄

W~)^2 ψj+ 2 (gww 0 +~μj·curl~W)ψj

(6.4.37) =λ ψj inρw<|x|<ρ, 1 ≤j≤ 3

ψ= (ψ^1 ,ψ^2 ,ψ^3 ) =0 at|x|=ρw,ρ,

whereρwis the weakton radius,ρthe attracting radius of weak interaction,W 0 is given by
(6.4.4) withN= 1 ,gw(ρ) =gw, and

~μj=

hg ̄ w

2 mj

~σ is the weak magnetic moment.

We are in position now to derive a few results on the energy levels for charged leptons
and quarks based on (6.4.37).

1.Bound states and energy levels. We know that the negative eigenvalues and eigen-
functions of (6.4.37) correspond to the bound energy and bound states. Let

−∞<λ 1 ≤ ··· ≤λN< 0

be all negative eigenvalues of (6.4.37) with eigenfunctions

ψ 1 ,···,ψN forψk= (ψk^1 ,ψ^2 k,ψk^3 )T.

Each bound stateψksatisfies

(6.4.38)

∫

Ω

|ψkj|^2 dx=1 for 1≤j≤ 3 , 1 ≤k≤N,

whereΩ={x∈R^3 |ρw<|x|<ρ}. Then, by (6.4.37) and (6.4.38) we get

λk =

h ̄

2 mj

∫

Ω

|(∇+i

gw

hc ̄

(6.4.39) ~W)ψkj|^2 dx

+ 2

∫

Ω

~μj·curlW~|ψkj|^2 dx+ 2

∫

Ω

gwW 0 |ψkj|^2 dx.