2.3 Further Removal of the Degeneracy of the N-electron States
51
Fig. 2.2 stabilization of the
p z orbital by E with respect
to the degenerate p x and p y
orbitals due to an external
potential
ψ
(0)
0 = p z α = p 0 α
ψ
(0)
0 = p z β = p 0 β
(2.60)
ψ
(0)
i
={p x α, p x β, p y α, p y β}={p + α, p + β, p − α, p − β}
(2.61)
Now we apply Eq. 2.50 to obtain the expression of the first-order corrected wave
functions for the lowest two levels
ψ
(1)
0 = p 0 α +
1
2
ζ
p + | ˆ
L z |p 0
E
p + α +
p − | ˆ
L z |p 0
E
p − α
+
1
2
ζ
p + | ˆ
L + |p 0
E
p + β +
p − | ˆ
L + |p 0
E
p − β
= p 0 α +
1
2
√
2ζ p + β
(2.62)
ψ
(1)
0 = p 0 β −
1
2
ζ
p + | ˆ
L z |p 0
E
p + β +
p − | ˆ
L z |p 0
E
p − β
+
1
2
ζ
p + | ˆ
L − |p 0
E
p + α +
p − | ˆ
L − |p 0
E
p − α
= p 0 β +
1
2
√
2ζ p − α
(2.63)
The values of g zz and g xx = g yy are determined from Eqs. 2.55 and 2.59 and lead to
g zz = g e + 2ζ
p + | ˆ
L z |p 0 p 0 | ˆ
L z |p +
E
+
p − | ˆ
L z |p 0 p 0 | ˆ
L z |p −
E
= g e (2.64)
g xx = g e + 2ζ
p + | ˆ
L x |p 0 p 0 | ˆ
L x |p +
E
+
p − | ˆ
L x |p 0 p 0 | ˆ
L x |p −
E
= g e + 2ζ
1
2
√
2 ·
1
2
√
2
E
+
1
2
√
2 ·
1
2
√
2
E
= g e +
2ζ
E
(2.65)