Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 273
for the interaction energy between two magnetic moments. Because the electrons responsible for
magnetic behavior are localized near the atoms of a regular lattice in most magnetic materials,
we consider the simple case of two localized electrons. Each electron has spin 1/2 and are aligned
up or down along the axis specified by the applied magnetic field. The electrons interact with
each other and with nearby atoms and are described in part by the spatial wavefunction ψ(r1,r2).
This wavefunction must be multiplied by the spin eigenstates to obtain the actual state of the two
electron system. We denote the basis for the spin eigenstates as
|↑↑〉,|↓↓〉,|↑↓〉,|↓↑〉, (5.139)
where the arrows correspond to the spin of the electrons. These states are eigenstates of the
z-component of the total spin angular momentum 15 ˆ Sz such that ˆ Sz operating on any of the
states in (5.139) has an eigenvalue equal to the sum of the spins in the z direction. For example,
ˆSz|↑↑〉 = 1|↑↑〉 and ˆ Sz|↑↓〉 = 0|↑↓〉. Similarly, ˆ Sx or ˆ Sy give zero if either operator acts on these
eigenstates.
Because electrons are fermions, the basis states in (5.139) are not acceptable because if two
electrons areinterchanged, the wavefunction must be antisymmetric. Thus, ψ(r1,r2) = +ψ(r2,r1)
if the spin state is antisymmetric, and ψ(r1,r2) = −ψ(r2,r1) if the spin state is symmetric. The
simplest normalized linear combinations of the eigenstates in (5.139) that satisfy this condition are
1
√ 2 [|↑↓〉−|↓↑〉], (5.140a)
|↑↑〉, (5.140b)
1
√ [|↑↓〉+|↓↑〉],
2
(5.140c)
|↓↓〉. (5.140d)
The state in (5.140a) is antisymmetric, because interchanging the two electrons leads to minus the
original state. This state has a total spin S = 0 and is called the singlet state. The collection of the
last three states is called the triplet state and has S = 1. If the spins are in the triplet state, then
ψ(r1,r2) = −ψ(r2,r1). Similarly, if the spins are in the singlet state, then ψ(r1,r2) = +ψ(r2,r1).
Hence, when r1 = r2, ψ is zero for the triplet state, and thus the electrons stay further apart, and
their electrostatic energy is smaller. For the singlet state at r1 = r2, ψ is nonzero, and thus the
electrons can be closer to each other, with a larger electrostatic energy. To find a relation between
the energy and the spin operators we note that
ˆS · ˆ S = ( ˆ S1 + ˆ S2) 2 = ˆ S 2 1 + ˆ S 2 2 +2 ˆ S1· ˆ S2, (5.141)
where the operator ˆ S is the total spin. Because both electrons have spin 1/2, the eigenvalues of ˆ S 2 1
and ˆ S 2 2 are equal and are given by (1/2)(1+1/2) = 3/4. We see that the eigenvalue S of ˆ S is zero
for the singlet state and is one for the triplet state. Hence, the eigenvalue of ˆ S 2 is S(S +1) = 0
for the singlet state and (1(1+1) = 2 for the triplet state. Similarly, the eigenvalue S12 of ˆ S1 · ˆ S2
equals −3/4 for the singlet state and 1/4 for the triplet state. These considerations allows us to
write
E = c−JS12, (5.142)
15 We will denote operators with a caret symbol in this section.