6
1 Basic Concepts
The one-electron spin functions to be considered have the quantum numbers s =
1
2
and m s =±
1
2 and can be written in different formats:
|s, m s =[| 1 / 2 , 1 / 2 , | 1 / 2 , − 1 / 2 ] = [α, β] = [↑, ↓]
(1.14)
where [...] denotes the set of functions. When the spatial part of the wave function
is explicitly written, a similar notation can be used for spin orbitals:
|s, m s =
ϕ 1 , ϕ 2
(1.15)
where the barred orbital carries the electron with m s =− 1 / 2 . The notations by α, β
and ϕ 1 , ϕ 2 are most frequently used and will also be followed here. The corresponding
eigenvalues of the total spin operator ˆ
s 2 and the z-component of it (ˆ s z )are
ˆ
s
2 α = 1 / 2 ( 1 / 2 + 1) α = 3 / 4 α
ˆ
s z α = 1 / 2 α
(1.16a)
ˆ
s
2 β =− 1 / 2 (− 1 / 2 + 1) β = 3 / 4 β
ˆ
s z β =− 1 / 2 β
(1.16b)
The ladder operators ˆ
s ± =ˆ s x ± i ˆ
s y change the m s quantum number of the spin
functions by the following action
ˆ
s
+ |s, m s =
s(s + 1) − m s (m s + 1)|s, m s + 1
(1.17)
ˆ
s
− |s, m s =
s(s + 1) − m s (m s − 1)|s, m s − 1
This leads to the following simple relations when applied to the one-electron spin
functions α and β:
ˆ
s
+ α = 0
ˆ
s
− α = β
(1.18a)
ˆ
s
+ β = α
ˆ
s
− β = 0
(1.18b)
The substitution of ˆ
s x =
1
2 (ˆ s + +ˆ s − ) and ˆ
s y =
1
2i (ˆ s + −ˆ s − ) in the expression of the
total spin operator ˆ
s 2 =ˆ s 2
x +ˆ s 2
y +ˆ s 2
z gives a simple working equation to evaluate
the expectation value of ˆ
s 2 for spin functions:
ˆ
s
2 =ˆ s
+ ˆ
s
− −ˆ s z +ˆ s
2
z
(1.19)
For completeness, we also give the results of operating with ˆ
s x and ˆ
s y on α and β
ˆ
s x α =
1
2
β
ˆ
s y α =−
1
2i
β
(1.20a)
ˆ
s x β =
1
2
α
ˆ
s y β =
1
2i
α
(1.20b)