8
1 Basic Concepts
This shows that the Slater determinants are eigenfunctions of ˆ
S z and the corresponding eigenvalues M S are equal to 1 and 0, respectively. The result of applying ˆ
S 2 is
rather straightforward using Eq. 1.21 with the notion that (− ˆ
S z + ˆ
S 2
z ) gives zero when
applied on the spin functions αα, αβ and βα. Only remains to determine the action
of the two ladder operators to check whether the determinants are eigenfunctions
of ˆ
S 2
ˆ
S
+ ˆ
S
− |ϕ 1 ϕ 2 |=
ϕ 1 ϕ 2 − ϕ 2 ϕ 1
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ˆ s
− (1) +ˆ s
− (2))(αα)
=
ϕ 1 ϕ 2 − ϕ 2 ϕ 1
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(βα + αβ)
=
ϕ 1 ϕ 2 − ϕ 2 ϕ 1
√
2
(αα + αα) = 2 ·
ϕ 1 ϕ 2 − ϕ 2 ϕ 1
√
2
= 2|ϕ 1 ϕ 2 | (1.26)
ˆ
S
+ ˆ
S
− |ϕ 1 ϕ 2 |=
ϕ 1 ϕ 2
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ˆ s
− (1) +ˆ s
− (2))(αβ)
−
ϕ 2 ϕ 1
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ˆ s
− (1) +ˆ s
− (2))(βα)
=
ϕ 1 ϕ 2
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ββ) −
ϕ 2 ϕ 1
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ββ)
=
ϕ 1 ϕ 2
√
2
(αβ + βα) −
ϕ 2 ϕ 1
√
2
(αβ + βα) =|ϕ 1 ϕ 2 |+|ϕ 1 ϕ 2 |
= S(S + 1)|ϕ 1 ϕ 2 |
(1.27)
Hence, the single Slater determinant |ϕ 1 ϕ 2 | is a proper spin eigenfunction, while
|ϕ 1 ϕ 2 | is not. In general, linear combinations of Slater determinants are necessary to
ensure that the wave function is an eigenfunction of ˆ
S 2 .
In the following, three strategies will be illustrated to construct spin eigenfunctions
from scratch based on (i) projection techniques to eliminate the contributions of
unwanted spin eigenfunctions, (ii) diagonalization of the matrix representation of
ˆ
S 2 , and (iii) the genealogical construction of spin functions in which spins are added
one-by-one.
In the above demonstrations we have first developed the Slater determinants and
then applied the spin operators. This strategy becomes of course very laborious for
functions with more than two electrons. It should however be noted that it is not
necessary to work with the fully expanded determinants, one gets the same results
when working with the product of the diagonal elements.
1.5 Apply the total spin operator on Φ 1 =| ϕ 1 ϕ 1 | and Φ 2 ={ | ϕ 1 ϕ 2 |+
|ϕ 1 ϕ 2 |}/
√
2 and check that the same result is obtained when the determinants
are fully expanded.
1 Basic Concepts
This shows that the Slater determinants are eigenfunctions of ˆ
S z and the corresponding eigenvalues M S are equal to 1 and 0, respectively. The result of applying ˆ
S 2 is
rather straightforward using Eq. 1.21 with the notion that (− ˆ
S z + ˆ
S 2
z ) gives zero when
applied on the spin functions αα, αβ and βα. Only remains to determine the action
of the two ladder operators to check whether the determinants are eigenfunctions
of ˆ
S 2
ˆ
S
+ ˆ
S
− |ϕ 1 ϕ 2 |=
ϕ 1 ϕ 2 − ϕ 2 ϕ 1
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ˆ s
− (1) +ˆ s
− (2))(αα)
=
ϕ 1 ϕ 2 − ϕ 2 ϕ 1
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(βα + αβ)
=
ϕ 1 ϕ 2 − ϕ 2 ϕ 1
√
2
(αα + αα) = 2 ·
ϕ 1 ϕ 2 − ϕ 2 ϕ 1
√
2
= 2|ϕ 1 ϕ 2 | (1.26)
ˆ
S
+ ˆ
S
− |ϕ 1 ϕ 2 |=
ϕ 1 ϕ 2
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ˆ s
− (1) +ˆ s
− (2))(αβ)
−
ϕ 2 ϕ 1
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ˆ s
− (1) +ˆ s
− (2))(βα)
=
ϕ 1 ϕ 2
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ββ) −
ϕ 2 ϕ 1
√
2
(ˆ s
+ (1) +ˆ s
+ (2))(ββ)
=
ϕ 1 ϕ 2
√
2
(αβ + βα) −
ϕ 2 ϕ 1
√
2
(αβ + βα) =|ϕ 1 ϕ 2 |+|ϕ 1 ϕ 2 |
= S(S + 1)|ϕ 1 ϕ 2 |
(1.27)
Hence, the single Slater determinant |ϕ 1 ϕ 2 | is a proper spin eigenfunction, while
|ϕ 1 ϕ 2 | is not. In general, linear combinations of Slater determinants are necessary to
ensure that the wave function is an eigenfunction of ˆ
S 2 .
In the following, three strategies will be illustrated to construct spin eigenfunctions
from scratch based on (i) projection techniques to eliminate the contributions of
unwanted spin eigenfunctions, (ii) diagonalization of the matrix representation of
ˆ
S 2 , and (iii) the genealogical construction of spin functions in which spins are added
one-by-one.
In the above demonstrations we have first developed the Slater determinants and
then applied the spin operators. This strategy becomes of course very laborious for
functions with more than two electrons. It should however be noted that it is not
necessary to work with the fully expanded determinants, one gets the same results
when working with the product of the diagonal elements.
1.5 Apply the total spin operator on Φ 1 =| ϕ 1 ϕ 1 | and Φ 2 ={ | ϕ 1 ϕ 2 |+
|ϕ 1 ϕ 2 |}/
√
2 and check that the same result is obtained when the determinants
are fully expanded.