Mathematical Principles of Theoretical Physics

376 CHAPTER 6. QUANTUM PHYSICS

Hence, it is natural to think that

Bwργ

ρw

=

ργ

ρ

= 102 ∼ 104.

Thus, by (6.4.71) and (6.4.6) we get hat

(6.4.72) N∼

(

ρn

ρw

) 18

≥ 1090.

Remark 6.22.In the estimates (6.4.66), (6.4.69) and (6.4.71), the energy level numbersN
for various subatomic particles are counting the multiplicities of eigenvalues. However, the
numbersNhave the same order as the real energy level numbers. It is because that the
multiple eigenvalues are unstable, under the perturbationof electromagnetism together with
weak and strong magnetism, most multiple eigenvalues become the simple eigenvalues.

4.Energy level gradient of photons.Each energy levelEk( 1 ≤k≤N)of photons can be
written as
Ek=E 0 +λk ( 1 ≤k≤N).

It is clear that the largest and smallest energy levels are given by

Emax=E 0 +λN, Emin=E 0 +λ 1.

The total energy level difference is

Emax−Emin=λN−λ 1.

Since|λ 1 | ≫ |λN|, the average energy level gradient (for two adjacent energylevels) is ap-
proximatively given by

(6.4.73) ∆E=

Emax−Emin

N

≃

|λ 1 |

N

.

The first eigenvalueλ 1 of (6.4.70) is

λ 1 ≃ −K

(

the unit is

hc ̄

ργ

)

.

Hence, by (6.4.71) and (6.4.73)

∆E=β 13 K−^2

̄hc

ργ

, K=

Bwργ

ρw

g^2 w

̄hc

.

Then we get the estimates

(6.4.74) ∆E=

(

ρw

ρn

) 12

̄hc

ργ

.

As we take ρ
w
ρn

= 10 −^5 , ργ= 10 −^20 cm,