Mathematical Principles of Theoretical Physics

392 CHAPTER 6. QUANTUM PHYSICS

B. FIELD EQUATION OF SYSTEMSnl. The precise model of atomic spectrum should take

(6.5.56) as anN-particle system. Also,Sn=

n− 1

∑

l= 0

Snlis again divided intonsub-systems

Sn:Sn 0 ,···,Snn− 1.

Hence, the systemSnlhas more sub-systems thanSn, i.e. ifSnhasNsub-systems, thenSnl
has^12 N(N+ 1 )sub-systems.
LetSnlhaveKnlelectrons with wave functions:

(6.5.63) Snl:Ψnl= (ψnl^1 ,···,ψnlKnl), 1 ≤n≤N, 0 ≤l≤n− 1 ,

andKnl≤ 2 ( 2 l+ 1 ). Then the action of (6.5.63) takes as

(6.5.64) L=

∫n− 1

∑

l= 0

N

∑

n= 1

(LSU(Knl)+LDnl)dx,

whereLSU(Knl)andLDnlare similar to that of (6.5.59). Thus, the field equation of the system
(6.5.63) is determined by (6.5.64).

Remark 6.32.The reason why atomic spectrum can be divided into two systems (6.5.57)
and (6.5.63) to be considered is that in the system (6.5.63) the electrons in eachSnlhave the
same energy, and in (6.5.57) the electons in eachSnhave the same energy if we ignore the
interaction energy between differentl-orbital electrons ofSnl. Hence, the system ofSnlis
precise and the system ofSnis approximative.

3.Atomic spectrum equations. For simplicity, we only consider the systemSn, and for
Snlthe case is similar. Since the electrons in eachSnhave the same energyλn, the wave
functions in (6.5.57) can take as

(6.5.65) ψnj=φnj(x)e−iλnt/h ̄ for 1≤j≤Kn.

It is known that theEMfieldsAaμin atomic shells are independent of timet,i.e.∂tAaμ=0.
Therefore, inserting (6.5.65) into (6.5.61) and (6.5.62) we derive the spectrum equation in the
form

(6.5.66) λnΦn=ic ̄h(~α·D)Φn−eVΦn+mec^2 α 0 Φn+eAa 0 nτanΦn for 1≤n≤N,

∆Aa 0 n−

e

hc ̄

λbanncn~Abn·(∇Ac 0 n+

e

hc ̄

λdcnnfnAdn~Afn)−eΦ†nτanΦn=

e

̄hck∑ 6 =n

(6.5.67) A( 0 k)φan,

∆~Aan−∇(div~Aan)+

e

̄hc

λbanncngα α~AαbnAcαn+eΦ)n~γ τanΦn= (∇+

e

̄hck∑ 6 =n

(6.5.68) A~(k))φan,

whereΦn= (φn^1 ,···,φKnn)T,Aaμn= (Aa 0 n,~Aan),~α= (α 1 ,α 2 ,α 3 )andα 0 are as in (3.1.15),~γ=
(γ^1 ,γ^2 ,γ^3 )is as in (6.2.8),V=ze/ris the Coulomb potential of the nuclear,~A= (A 1 ,A 2 ,A 3 )
is the magnetic potential of the nuclear, and

DΦn= (∇−

ie

hc ̄

~A−ie

hc ̄

~Aanτan)Φn,

~Abαn=∂α~Abn−∇Abαn−e

̄hc

λcbnndnAcαn~Adn.