Mathematical Principles of Theoretical Physics

368 CHAPTER 6. QUANTUM PHYSICS

λkis independent ofj( 1 ≤j≤ 3 ), i.e. each weakton has the same bound energyλkbut in
different bound stateψkj.
In the right-hand side of (6.4.39), the first term stands for the kinetic energy, the second
term for the weak magnetic energy, and the third term for the weak potential energy, the
potential energy in (6.4.39) is negative. Hence, the bound energy can be written as

λk=kinetic energy + magnetic energy + potential energy.

In addition, the energy distributions of charged leptons and quarks are discrete and finite:

(6.4.40) 0 <E 1 <···<EN 0 (N 0 ≤N),

andNis the number of negative eigenvalues. Each energy levelEkcan be expressed as

Ek= 3 (E 0 +λk), E 0 =g^2 w/ρwis the intrinsic energy.

2.Masses.At an energy levelEkof (6.4.40), the massMkof a lepton or a quark satisfies
the relation

Mk=

3

∑

j= 1

mj+Ek/c^2 =

3

∑

j= 1

mj+

3 g^2 w

ρwc^2

+

3 λk

c^2

.

3.Parameters of electrons.In all charged leptons and quarks, only the electrons are long
life-time and observable. Hence the physical parameters ofelectrons are important. By the
spectral equation (6.4.37) we can derive some information for electronic parameters.
To this end, we recall the weak interaction potential for theweaktons, which is written as

(6.4.41) W 0 =gw

[

1

r

−

Bw

ρw

( 1 +

2 r

r 0

)e−r/r^0

]

e−r/r^0 ,

whereBwis the constant for weaktons.
Assume that the masses of three weaktons are the same. We ignore the magnetism, i.e.
let~W=0. Then (6.4.37) is reduced in the form

(6.4.42) −

h ̄^2

2 m∆ψ+^2 gwW^0 ψ=λ ψ forρw<|x|<ρ,

ψ= 0 , for|x|=ρw,ρ.

We shall apply (6.4.41) and (6.4.42) to derive some basic parameters and their relations
for the electrons.
Letλeandψebe the spectrum and bound state of an electron, which satisfy(6.4.42). It
follows from (6.4.41) and (6.4.42) that

(6.4.43) λe=

1

2

mv^2 + 2 g^2 w

(

1

re

−

κe

ρw

)