174 CHAPTER 3. MATHEMATICAL FOUNDATIONS
Example 3.40(Number of Atomic Energy Levels).As an application of Theorem3.38, we
consider here the estimates for the number of atomic energy levels. Let the atom number be
Z. If ignoring interactions between orbital electrons, thenthe spectral equation is as follows
−∆ψ+V(x)ψ=λ ψ forx∈Br 0 ,
ψ= 0 forx∈∂Br 0 ,andr 0 is the atom radius,Vis the potential energy, given by
V(x) =−
2 Zme^2
h ̄^21
r, mthe mass of electron.The parameters in (3.6.28) for this system are
(3.6.36) α=− 1 , n= 3 , r=r 0 = 10 −^8 cm.
In addition,V 0 andΩin (3.6.23) and (3.6.24) are as
V 0 =−
2 Zme^2
̄h^2, Ω=B 1.
Therefore, the parameterθis given by
(3.6.37) θ=
2 Zme^2
h ̄^2.
According to physical parameters, it is known that
(3.6.38) r 0 ×
mc
̄h=
1
4
× 103 ,
e^2
hc ̄=
1
137
.
Hence, by (3.6.36)-(3.6.38) the formulas (3.6.28) becomes
(3.6.39) N=
(
2 Z
λ 1e^2
hc ̄mcr 0
h ̄) 3 / 2
=
(
103 Z
274 λ 1) 3 / 2
,
whereλ 1 is the first eigenvalue of−δonB 1.
The formulas is derived in the ideal situation ignoring interactions between orbital elec-
trons, and only holds for a bigger atom numberZ.
3.6.5 Spectrum of Weyl operators
In Section6.4.2, we shall deduce from Basic Postulates of Quantum Mechanicsthat the
spectral equations for the massless subatomic particles are in the following form
(3.6.40)
− ̄hc(~σ·~D)^2 φ+ig
2{(~σ·~D),A 0 }φ=iλ(~σ·~D)φ,φ|∂Ω= 0 ,