Appendix E
Solutions
Exercises and Problems of Chap. 1
Exercise 1.1 Ψ(1, 2, 3) =|φ a (1)φ b (2)φ c (3)|=
1
√
6
φ a (1)φ b (2)φ c (3) − φ a (1)φ c (2)
φ b (3)−φ b (1)φ a (2)φ c (3)+φ b (1)φ c (2)φ a (3)+φ c (1)φ a (2)φ b (3)−φ c (1)φ b (2)φ a (3)
Ψ(2, 1, 3) =
1
√
6
φ b (1)φ a (2)φ c (3)−φ c (1)φ a (2)φ b (3)− φ a (1)φ b (2)φ c (3)+φ c (1)φ b
(2)φ a (3)+φ a (1)φ c (2)φ b (3)−φ b (1)φ c (2)φ a (3)
=
1
√
6
(−φ a (1)φ b (2)φ c (3)+φ a (1)φ c
(2)φ b (3) + φ b (1)φ a (2)φ c (3) − φ b (1)φ c (2)φ a (3)−φ c (1)φ a (2)φ b (3) + φ c (1)φ b
(2)φ a (3)) =− Ψ(1, 2, 3) Assume φ a = φ b , then Ψ(1, 2, 3) =
1
√
6
φ a (1)φ a (2)φ c
(3) − φ a (1)φ c (2)φ a (3) − φ a (1)φ a (2)φ c (3) + φ a (1)φ c (2)φ a (3) + φ c (1)φ a (2)φ a (3) −
φ c (1)φ a (2)φ a (3)
= 0.
Exercise 1.2 ˆ
A(1, 2) =
1
√
2
(1− ˆ
P 12 ); ˆ
Aϕ 1 ϕ 2 =
1
√
2
(ϕ 1 ϕ 2 −ϕ 2 ϕ 1 ) ⇒
√
N! ˆ
Aϕ 1 ϕ 2 =
ϕ 1 ϕ 2 − ϕ 2 ϕ 1 ; ˆ
A ˆ
Aϕ 1 ϕ 2 =
1
√
2
(1 − ˆ
P 12 )
1
√
2
(ϕ 1 ϕ 2 − ϕ 2 ϕ 1 ) =
1
2 (ϕ 1 ϕ 2 − ϕ 2 ϕ 1 − ϕ 2 ϕ 1 +
ϕ 1 ϕ 2 ) = ϕ 1 ϕ 2 − ϕ 2 ϕ 1 .
Exercise 1.3 For a given S, M S runs from S to −S in steps of 1. Hence, the degeneracy
is 2S + 1.
Exercise 1.4 Substituting s = 1/2 and m s =±1/2 in the normalization factor of ˆ
s +
gives
√
1/2(1/2 + 1) − 1/2(1/2 + 1) = 0 for α and
√
1/2(1/2 + 1) −−1/2(−1/2 + 1)
=
√
3/4 + 1/4 = 1forβ. (b) ˆ
s 2 = 1/4(ˆ s + +ˆ s − )(ˆ s + +ˆ s − ) − 1/4(ˆ s + +ˆ s − )(ˆ s + +
ˆ
s − )+ˆ s 2
z = 1/4(ˆ s + ˆ
s + +ˆ s + ˆ
s − +ˆ s − ˆ
s + +ˆ s − ˆ
s − )−1/4(ˆ s + ˆ
s + −ˆ s + ˆ
s − −ˆ s − ˆ
s + +ˆ s − ˆ
s − )+ˆ s 2
z
(remember ˆ
s + and ˆ
s − do no commute) = 1/2(ˆ s + ˆ
s − +ˆ s − ˆ
s + ) +ˆ s 2
z = 1/2(ˆ s + ˆ
s − +
ˆ
s − ˆ
s + ) + (1/2)ˆ s + ˆ
s − − (1/2)ˆ s + ˆ
s − +ˆ s 2
z =ˆ s + ˆ
s − − 1/2[ˆ s + , ˆ
s − ]+ˆ s 2
z =ˆ s + ˆ
s − −ˆ s z +ˆ s 2
z .
(c) ˆ
s + ˆ
s − α = α, −ˆ s z α =− 1/2α, ˆ
s 2
z α = 1/4α. Combining the three terms gives
(1−1/2+1/4)α; the expectation value is 3/4. ˆ
s + ˆ
s − β = 0, −ˆ s z β = 1/2, ˆ
s 2
z β = 1/4β.
Combining the terms, gives the expectation value (0 + 1/2 + 1/4) = 3/4forβ.
© Springer International Publishing Switzerland 2016
C. Graaf and R. Broer, Magnetic Interactions in Molecules and Solids,
Theoretical Chemistry and Computational Modelling,
DOI 10.1007/978-3-319-22951-5
223