Mathematical Principles of Theoretical Physics

374 CHAPTER 6. QUANTUM PHYSICS

the spectral equations for charged leptons and quarks are given by (6.4.42), and which are
written as

(6.4.64)

−h ̄

2

2 mw∆ψ+^2 gwW^0 ψ=λ ψ inρw<|x|<ρ,

ψ= 0 , on|x|=ρw,ρ,

wheremwis the masses of weaktons in leptons and quarks,ρis the attracting radius of weak
interaction.
For the weak interaction potentialW 0 , we approximatively take

W 0 =−

gwBw

ρw

.

Then take the dimensional transformation

x→ρx (ρ as in (6.4.64)).

Note that the weakton radiusρwfor smaller thanρ,

ρw≪ρ.

Hence, the problem (6.4.64) can be approximatively expressed as

(6.4.65)

−∆ψ−^4 mwBwρ

2

̄h^2 ρw g

2

wψ=λ ψ for|x|<^1 ,

ψ= 0 on|x|= 1.

It is clear that for the equation (6.4.65), the parametersr,α,θas in (3.6.28) of Theorem
2.37 are as follows

r= 1 , α= 0 , θ=

4 mwBwρ^2

̄h^2 ρw

g^2 w

Thus the numberNof the energy levels of charged leptons and quarks is approximatively
given by

(6.4.66) N=

[

4

λ 1

Bwρ^2

ρw

mwc

h ̄

g^2 w

hc ̄

]^32

,

whereλ 1 is the first eigenvalue of−∆in unit ballB 1 ⊂R^3.
By the de Broglie relation,
mwc
h ̄

∼

1

6 λ

,

where 6 λis the wave length of weaktons, i.e. 6 λ=nrm(n= 1 , 2 ,···),rmthe mediator radius.
We take 6 λ=rm≃ρ, and

4

λ 1

≃ 1 ,

Bwρ

ρw

≃ 102 ,

mwc

h ̄

ρ≃ 1.

Then, by (6.4.6) the numberNin (6.4.66) has the estimate

(6.4.67) N∼

(

ρn

ρw

) 9

≥ 1045.