Chem 120A
Problem Set 7
Out: Tues. April 17, 2007; Due: Fri. April 27, 2007
1. One very simple model for magnetic behavior is the Ising model. This assumes that electron spins
in solids exist on separated sites and that the only interactions between spins are between nearest
neighbors. In that case, the Hamiltonian is
H
Ising
=
J
1
2
∑
i
6
=
j
s
(
i
)
z
s
(
j
)
z
,
(1)
where
J
is an exchangetype integral,
s
(
i
)
z
is the Zcomponent of the spin on site
i
, and the sum runs
only over nearest neighbors.
For the case of 2 sites on a line, ﬁnd the four possible eigenstates of
H
Ising
. Construct the energy level
diagram of these eigenstates for the two cases of i)
J
<
0 (ferromagnet) and ii)
J
>
0 (antiferromagnet).
Which of these cases has the greater total spin in the ground state?
2. This problem provides an exercise in using determinantal wavefunctions. As described in class, an
N
electron electronic wavefunction can be written as a determinant of spinspace orbitals with nor
malization
1
√
(
N
!
)
. Thus the wavefunction for the ground state of helium can be written as
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 Spring '07
 Whaley
 Physical chemistry, Electron, pH, ground state, spatial probability density, determinantal wavefunction

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