Mathematical Principles of Theoretical Physics

6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 369

where^12 mv^2 is the kinetic energy of each weakton in electronrethe radius of the naked
electron,κeis the bound parameter of electron, and they are expressed as

re=

∫

Bρ

1

r

e−r/r^0 |ψe|^2 dx,

κe=Bw

∫

Bρ

( 1 +

2 r

r 0

)e−^2 r/r^0 |ψe|^2 dx,

(

mwv

̄h

)^2 =

∫

Bρ

|∇ψe|^2 dx.

These three parameters are related with the energy levels ofan electron, i.e. with theλkand
ψk. However, the most important case is the lowest energy levelstate. We shall use the
spherical coordinate to disscuss the first eigenvalueλ 1 of (6.4.42). Let the first eigenfunction
ψebe in the form
ψe=φ 0 (r)Y(θ,φ).

Thenφ 0 andYsatisfy

(6.4.44) −

̄h^2

2 m

1

r^2

d

dr(r

2 d

dr)φ^0 +^2 gsW^0 φ^0 +

βk

r^2 φ^0 =λeφ^0 ,

φ 0 (ρw) =φ 0 (ρ) = 0 ,

and

(6.4.45)

[

1

sinθ

∂

∂ θ

(

sinθ

∂

∂ θ

)

+

1

sin^2 θ

∂^2

∂ φ^2

]

Yk=βkYk,

whereβk=k(k+ 1 ),k= 0 , 1 ,···.
Becauseλeis the minimal eigenvalue, it implies thatβk=β 0 =0 in (6.4.44). The eigen-
functionY 0 of (6.4.45) is given by

Y 0 =

1

√

4 π

.

Thusψeis as follows

ψe=

1

√

4 π

φ 0 (r),

andλe,φ 0 are the first eigenvalue and eigenfucntion of the following equation

(6.4.46) −

h ̄^2

2 m

1

r^2

d

dr(r

2 d

dr)φ^0 +^2 gwW^0 φ^0 =λ^1 φ^0

φ 0 (ρw) =φ 0 (ρ) = 0.

In this case, the parameters in (6.4.43) are simplified as

re=

∫ρ

ρw

rφ^20 (r)e−r/r^0 dr,

κe=Bw

∫ρ

ρw

r^2 ( 1 +

2 r
r 0
(6.4.47) )e−^2 r/r^0 φ 02 (r)dr,

(mv

h ̄

) 2

=

∫ρ

ρw

r^2

(

dφ 0

dr

) 2

dr,

whereφ 0 satisfies (6.4.46).