2 Concepts in Magnetism
49
We have considered the eigenvalues of ˆ
S a · ˆ
S b , but what about the eigenstates? The
most straightforward basis to consider is
| ↑↑↑ , | ↑↓↓ , | ↓↑↑ , | ↓↓↓ .
(2.28)
The first arrow refers to the z component of the spin labelled a and the second arrow
refers to the z component of the spin labelled b. The eigenstates of ˆ
S a · ˆ
S b are linear
combinations of these basis states and are listed in Table 2.1. The value of m s is equal
to the sum of the z components of the individual spins. Since the eigenstates are a
mixture of states in the original basis, we cannot know both the z components of the
original spins and the total spin of the resultant entity. This is a general feature which
will become more important in more complicated situations.
The basis in (2.28) also fails to satisfy the condition that the overall wave function
must be antisymmetric with respect to exchange of the two electrons. Since the wave
function is a product of a spatial function ψ space (r 1 , r 2 ) and the spin function χ ,
the spatial wave function can be either symmetric or antisymmetric with respect to
exchange of electrons. For example, the spatial wave function
ψ space (r 1 , r 2 ) =
φ(r 1 )ξ(r 2 ) ± φ(r 2 )ξ(r 1 )
√
2
(2.29)
is symmetric (+) or antisymmetric (−) with respect to exchange of electrons depending on the ±. This type of symmetry is known as exchange symmetry. In (2.29),
φ(r i ) and ξ(r i ) are single-particle wave functions for the ith electron. Whatever is
the exchange symmetry of the spatial wave function, the spin-wave function χ must
have the opposite exchange symmetry. Hence χ must be antisymmetric when the
spatial wave function is symmetric and vice versa. This is in order that the product
ψ space (r 1 , r 2 ) × χ is antisymmetric overall.
States such as | ↑↑↑ and | ↓↓↓ are clearly symmetric under exchange of electrons,
but exchanging the two electrons in | ↑↓↓ yields | ↓↑↑ which is not a multiple of
| ↑↓↓. Thus | ↑↓↓, and also by an identical argument | ↓↑↑, are both neither symmetric
nor antisymmetric under exchange of the two electrons. The true eigenstates must,
therefore, be linear combinations of these two states (see Table 2.1). The state (| ↑↓↓ +
| ↓↑↑)/
√
2 is symmetric under exchange of electrons (in common with the other two
s = 1 states) while the state (| ↑↓↓ − | ↓↑↑)/
√
2 (the s = 0 state) is antisymmetric
under exchange of electrons.
The energy levels are shown in Fig. 2.5. Without the flip-flop term
1
2
( ˆ
S
+
a
ˆ
S
−
b +
ˆ
S
−
a
ˆ
S
+
b ) in (2.22), the Hamiltonian is simply ˆ
H = A ˆ
S
z
a
ˆ
S
z
b and this leads to two degenerate doublets as shown. The upper doublet (consisting of the states | ↑↑↑ and | ↓↓↓)
is unchanged when the flip-flop terms are switched on. The lower doublet (consisting of the states | ↑↓↓ and | ↓↑↑, although strictly speaking it should of course be
the symmetric and antisymmetric combinations of these two states) splits with the
addition of the flip-flop terms to make ˆ
H = A ˆ
S a · ˆ
S b and the symmetric combination
49
We have considered the eigenvalues of ˆ
S a · ˆ
S b , but what about the eigenstates? The
most straightforward basis to consider is
| ↑↑↑ , | ↑↓↓ , | ↓↑↑ , | ↓↓↓ .
(2.28)
The first arrow refers to the z component of the spin labelled a and the second arrow
refers to the z component of the spin labelled b. The eigenstates of ˆ
S a · ˆ
S b are linear
combinations of these basis states and are listed in Table 2.1. The value of m s is equal
to the sum of the z components of the individual spins. Since the eigenstates are a
mixture of states in the original basis, we cannot know both the z components of the
original spins and the total spin of the resultant entity. This is a general feature which
will become more important in more complicated situations.
The basis in (2.28) also fails to satisfy the condition that the overall wave function
must be antisymmetric with respect to exchange of the two electrons. Since the wave
function is a product of a spatial function ψ space (r 1 , r 2 ) and the spin function χ ,
the spatial wave function can be either symmetric or antisymmetric with respect to
exchange of electrons. For example, the spatial wave function
ψ space (r 1 , r 2 ) =
φ(r 1 )ξ(r 2 ) ± φ(r 2 )ξ(r 1 )
√
2
(2.29)
is symmetric (+) or antisymmetric (−) with respect to exchange of electrons depending on the ±. This type of symmetry is known as exchange symmetry. In (2.29),
φ(r i ) and ξ(r i ) are single-particle wave functions for the ith electron. Whatever is
the exchange symmetry of the spatial wave function, the spin-wave function χ must
have the opposite exchange symmetry. Hence χ must be antisymmetric when the
spatial wave function is symmetric and vice versa. This is in order that the product
ψ space (r 1 , r 2 ) × χ is antisymmetric overall.
States such as | ↑↑↑ and | ↓↓↓ are clearly symmetric under exchange of electrons,
but exchanging the two electrons in | ↑↓↓ yields | ↓↑↑ which is not a multiple of
| ↑↓↓. Thus | ↑↓↓, and also by an identical argument | ↓↑↑, are both neither symmetric
nor antisymmetric under exchange of the two electrons. The true eigenstates must,
therefore, be linear combinations of these two states (see Table 2.1). The state (| ↑↓↓ +
| ↓↑↑)/
√
2 is symmetric under exchange of electrons (in common with the other two
s = 1 states) while the state (| ↑↓↓ − | ↓↑↑)/
√
2 (the s = 0 state) is antisymmetric
under exchange of electrons.
The energy levels are shown in Fig. 2.5. Without the flip-flop term
1
2
( ˆ
S
+
a
ˆ
S
−
b +
ˆ
S
−
a
ˆ
S
+
b ) in (2.22), the Hamiltonian is simply ˆ
H = A ˆ
S
z
a
ˆ
S
z
b and this leads to two degenerate doublets as shown. The upper doublet (consisting of the states | ↑↑↑ and | ↓↓↓)
is unchanged when the flip-flop terms are switched on. The lower doublet (consisting of the states | ↑↓↓ and | ↓↑↑, although strictly speaking it should of course be
the symmetric and antisymmetric combinations of these two states) splits with the
addition of the flip-flop terms to make ˆ
H = A ˆ
S a · ˆ
S b and the symmetric combination