Prof. Saurabh Basu: Let us now discuss how
one gets a magnetic Hamiltonian involving
only spins. We are particularly talking about
Ising kind of Hamiltonians or Heisenberg kind
of Hamiltonians. Mainly, we would be talking
about Ising Hamiltonians where the spin can
only have either pointing up or down. These
are the two possible orientations, and let
us see that how we derive Hamiltonian, which
we have introduced or rather we have talked
about when we spoke on magnetism during our
lectures.
So we want to study magnetic Hamiltonian 
and a derivation of a magnetic Hamiltonian
would require that we know the addition of
spins. So, say, addition of two spins and
these are spin half particles. So we have
spin half particles and we would see that
how one actually add the spin vectors.
So let us consider two particles with spin
vectors S1 and S2. The total angular momentum
-- 
I mean what I mean by angular momentum is
that the total spin angular momentum is S
= S1 + S2, where S1 and S2 are the spin vectors
for the two particles that we are considering.
So just to remind you that both are spin half.
Now the direct product space that consists
of -- it's of four dimensions. So the direct
product space is 4-dimensional and we can
use the basis use (S, ms) basis for each.
So what I mean by that is that the eigenvalue
for the spin operator, S has eigenvalue S
and Sz has eigenvalue ms. So we can form the
basis of each of the particles by this S,
ms and the total space will be produce of
two such S, ms that is S1. So total space
is (S1, ms1) x (S2, ms2).
Let the -- since we have for each one of them,
so ms = ą 1/2 h, so let's represent the states
by ? and ?, so each of these 1/2 h will correspond
to say a ? and this minus half will correspond
to minus half h, so +h/2 and this is -h/2,
and hence we'll have -- we can write it in
two ways. So the spin space or the direct
product space is either you call it a(1),
so maybe this is called as a and this is called
as a ß. So it's a(1) a(2), which means both
are in |?>, a(1) ß(2) means one of them in
|?> and the other in |?>, and a(2) ß(1),
the first one is in the down and the second
is in up; or both of them are in the down.
This is one option 
whereas the other option is that we can write
it as |??> as a states, |??> and |??>, and
|??>, okay. So this is other option. We can
simply choose one of them, but let us choose
this option 
in order to write the wave function and -- I
mean to discuss this problem of two spins.
So what is the total value of ms, which is
ms1 +ms2, which can take value 1, 0, 0, -1;
1 when they both add up 1/2 + 1/2 and this
is when 1/2 - 1/2, this is - 1/2 1/2, and
this -1/2 -1/2, and the total spin quantum
number S, which is equal to S1 + S1, which
can take value 0 and 1, okay. So for S = 0,
we have just one eigen function and that eigen
function, let's write it with a form which
is |?00>, which is 1/v2 and I have a |??-??>. So
this is called as a singlet wave function,
and this is antisymmetric.
What I mean by antisymmetric is the following:
that you have two particles, so the first
one is in upstate, the second one is downstate.
Here, the first one is in downstate and the
second one is in upstate, and now if you interchange
? to ? one gets a negative sign. So that's
why it's called as a antisymmetric, and for
S=1, we would need -- so for S=1, we'll have
three combinations, because we'll have to
take care of |?11>, which will be simply |??> state,
|?10> which will simply be combination of
|??+??>, and |?1-1>, which is equal to a |??> state.
Now all these are called as triplets and triplet
states, because they are three in number and
one can easily check that they are symmetric,
because if the first particle is swapped with
the second particle, the wave function remains
the same.
So these are the state or the wave functions
for the two particles, both spin half, and
there's a -- for a system consisting or comprising
of two spin half particles and all possible
combinations have been taken. We get four
states and those four states are one singlet
and three triplet states. The singlet state
is antisymmetric with respect to the change
in the position of the particle, and the triplet
states are symmetric with respect to the change
in the position of the particle.
So these are the state what about the eigenvalues,
because in order to solve a full quantum mechanical
problem, we need both the information and
the eigenvalues and the eigen functions.
So let us see that, so all these three states,
so now we'll talk about the eigenvalues. These
states are |??>, |??>, |??>, 
and |??>, so this forms the basis of the problem
of a two-particle spin half system.
So they are eigenstates of S12, S22, S1z and
S2z. So the total spin S can be 0 or 1, okay.
So now we can see how these total spin operators
act on each of these states. So the total
spin operator, which Sz = (S1z + S2z), that
action, the state acting on -- or let us write
is here as well. So Sz acting on the |??> state,
this will give me S1z will only act on the
first spin on the left and S2z will act on
the spin on the right. So this will give me
h/2 for each one of them and h/2 and a |??>. So
as we have said that these are eigenstates
of these operators, so I get an eigenvalue
equation, which is Sz acting on a |??> state
gives me h/2 as the eigenvalue and returns
me the |??> state as well.
Similarly, for Sz acting on |??> would give
me 0, because S1z will give me a h/2 and S2z
will give me -h/2, and similarly we'll also
have Sz acting on the state |??> state should
also give me 0, and Sz now acting on the |??> state
will give me a -h -- sorry there is a -h/2
for each so this should be simply h. So for
each one of them there's an h/2, so there
are two h/2 which makes this h. So Sz on |??> will
give me a -h and so on. So these are the eigenvalues
of this Sz operator.
So furthermore, we have S2 = (S1 + S2)2 = S12
+ S22 + 2S1.S2, S1 and S2 will commute with
each other, because they pertain to different
particles. So S12 will be h -- so it's 1/2
(1/2 + 1), this acting on. So S2 acting on
any of these states, so say we talk about
|??>, say for example, so this is equal to
1/2 (1/2 +1) -- it's S (S + 1), so that is
this. Then again, for the S22 this will be
1/2 (1/2 +1).
Now we of course don't know what is 2S1 +
S2, so we'll leave it for the moment, and
let us see that what we can do for the S1.S2.
so S1.S2 if you see it is equal to S1x S2x
+ S1y S2y + S1z S2z. Now if you introduce
these ladder operators for the spins, so S+
can be written as Sx + iSy and S- can be written
as Sx = iSy. Now this will give me (S1+ S1-
+ S1- S2+) and then there'll be a factor of
1/2 there and + S1z S2z.
So this is S1.S2, and hence what we can do
is we can see that S2 acting on a |??>, which
we have already saw that the first term gives
3/4 h2, second terms gives 3/4 h2 as well.
Now we have a 2S1.S2. Now for the |??> state,
this will raise the sin and hence it will
be 0, because up is the maximally aligned
state and though S2- can give you a non-zero
contribution, but S1+ will give 0, and similarly
S2+ will give 0 and that's why these two terms
do no contribute, and that simplifies the
problems and then we are left with S1z S2z,
for which we know the operation. So that's
why we have done this, and this is 2(h/2)
and this whole thing or rather acted upon
by this. So it is a eigenvalue equation and
this is, if you simplify it, it becomes equal
to 2h2 |??> and so on
Similarly, for the |??> as well, one gets
the same answer by doing the same technique.
One gets this as a -- so on these, acting
on the |??> state will give us 2h2 and a |??>. So
they have -- so these state |??> and |??> have
total spin S=1 and ms = ąh, okay.
Of course, S=-1 should have three states,
which are equal to ms = ąh and 0. So the
third state, so ms = ąh is there, so ms = 0
state is obtained by a particular operation,
so by the application of S- on |??> state.
Let's see how one gets it; or you can also
consider -- or S+ on the |??> states. So S-
on the |??> state gives me S1- + S2- on the
|??> state, which gives me -- so S1- will
lower this spin and now this is something
that you should have done in quantum mechanics.
This gives me an eigenvalue which is -- these
are not eigenstates of |??>, but it will operate
on this and give me, this S1 will give me
a h and will give me a |??> -- sorry it will
be a |??>. 
The first one will lower, so it's a |??+??>.
S2 will lower the other one with an eigenvalue
which is given by h. So 1/h S- |??> is nothing
but 1/v2, which comes as a normalization factor,
|??+??>, it doesn't matter we have written
down the second term ahead of the first term,
and this will correspond to Sz = 0. So these
three will be called as the triplet states.
So the singlet states are of course which
corresponds to -- 
so these are the triplet states. So the 2
that's coming over here with spin S = 1 and
ms = 1, which is here and the other one comes
from here. So these are the three states.
Now we'll just look at the single state, which
corresponds to S = 0, ms = 0. Let's just call
it as, we can call it as |?00> or we can also
use a notation, which is like |00>, which
is equal to 1/2 (|??-??>). So why is it a
singlet state? So Sz acting on this |00> will
give me a 3/2 -- it's S1z + S2z which will
act on this, it will be a 3/2 h2 - 2(h/2)2
- h2, acting on |00> and it will give me a
0|00>, which means that ms value of this equal
to 0, and this has S = 0. So we have found
out all the four eigenstates of this 2-particle
problem.
So let us now look at the spin Hamiltonian
consisting of these -- if you want to construct
a Hamiltonian only consisting of these two
spins, which is like, as I said, like a Ising
Hamilton or Heisenberg Hamiltonian if h has
a full rotational symmetry. So let's just
discuss the construction of a magnetic Hamiltonian.
So we have S2 = S12 + S22 + 2S1.S2, now 
the eigenvalue of S2 = 3/2 h2, as we have
discussed that 3/2 comes from two terms of
3/4 h2, each of S1 and S2, and plus a 2S1.S2.
So for the singlet state, that is S=0, we'll
have to put S=0, the S1.S2 has an eigenvalue,
which is equal 2 -1/2 -3/4 h2, because this
is equal to 0. If you put the right hand side
equal to 0, the S1. S2 will have an eigenvalue
which is half of or minus of half of 3/2 h2,
which is -3/4 h2...
Whereas for the triplet state, which corresponds
to S=1, so that will have 1(1 + 1) h2 for
the left hand side, which is equal to 3/2
h2 + 2S1.S2, so this is equal to 2, so 2h2
- 3/2 h2/2 is the eigenvalue for S1.S2 for
the triplet state. So this is equal to 2 - 3/2
is just 1/2, so this is equal to 1/4 h2. So
1/4 h2 is the eigenvalue, in short e-value
I am writing, for the operator S1.S2 for a
2-particle problem. So let's just summarize
this quick result. So for singlet states S1.S2
-- so this is singlet and triplet. So this
singlet one has -3/4 h2 and this is 1/4 h2.
So this is the eigenvalue of S1.S2.
Now if we write down a Hamiltonian, which
is H = 1/4 (Es + 3Et), I'll tell you what
these are, (Es - Et) S1.S2, we have written
it in a particular way of this term where
Es is the energy of the singlet state 
and Et is the energy of the triplet state.
Why have we written it in this fashion, is
that H acting on the single state which is
|00> will be simply equal to this 1/4 (Es
+ 3Et) and (Es - Et) S1.S2 acting on |00>
-- we can skip the comma in between -- so
that's a singlet states. So with Es = -3/4
h2 and Et = 1/4 h2, one can simply check that
H|00> will give me a -3/4 h2 |00>, and similarly
H acting on either of these |??> states or
|??> states or |??+??> states, all those multitude
of |??> or |??+??> up state with a normalization
will give me a 1/4 h2 and these states that
we have written such as |??>, |??>, |??+??>.
So that says that. We have arrived at a Hamiltonian
which gives us -- for a 2-particle problem,
which gives us the correct energy eigenvalues
for a two spin half particles, for a system
of two spin half particles, and this is that
Hamiltonian.
Now we can see that if you redefine the zero
of the energy, we may omit the constant (Es
+ 3Et/4), which is common to all the states,
all the four states. Then we can write down
a 
spin Hamiltonian as H = J S1.S2, where J is
nothing but the difference between the singlet
and the triplet energies. Here of course we
have the singlet energy to be lower, which
is equal to - 3/4 h2 and Et being 1/4 h2.
So J will be negative.
Now if we say that such Hamiltonians can be
written for N particles with a pair wise interaction
between the particles, then we can write a
generic Hamiltonian for a magnetic system
or spin half system. We can extend it to spin
having any values. It should be then -- it's
a J and then there is a Si.Sj, it's i and
j. It's between the neighboring sites and
this is of Heisenberg Hamiltonian, if S has
a full rotational symmetry, and it's just
the Ising Hamiltonian if S is taken as ą1/2,
but however it gives magnetic properties of
the magnetic system such as antiferromagnet
or ferromagnet, and of course if J is positive,
now we are not restricting ourselves to only
two particles where we know that J is negative,
but we also go ahead and consider J to be
positive as well.
So if J is positive in this particular model,
in this Hamiltonian given by (1), (1) favors
-- we can write it with a minus sign, putting
a minus sign from outside, then this favors
parallel arrangements of spins, which are
essential for ferromagnetism. And if J is
negative, then (1) favors antiparallel arrangement
and it is antiferromagnetism. We have seen
this phenomena from a purely electronic model,
which is Hubbard model, but however, we have
also gotten exposed to this kind of spin only
models, which are there.
So if J is positive, then the energy is lowered.
If the Si.Sj that is the Si and Sj, they point,
spin point vectors point in the same direction,
which are in a sense we talk about ferromagnetism,
whereas if J is negative, then that means
that whole energy would be negative if Si
and Sj are antiparallelly aligned which are
the features of antiferromagnetism.
So this can be actually compared with the
magnetic dipolar interaction, like this which
is 1/r3 and it's m1.m2, so these are the two
magnetic moments and these are related -- this
you are familiar in the context of classical
electromagnetic theory, and the relative distance
between m1 and m2 are involved, but here,
we have a purely spin Hamiltonian, which neglects
all special symmetries.
Now this is H written as J [Si.Sj] has -- there
are a large number of approximations that
are going on, namely i, j are nearest neighbors.
One doesn't have to be, on can include longer
than nearest neighbor, that is next to next
nearest neighbor interactions as well, and
we can also write this inside, the J to be
inside and it doesn't have to be constant,
and it can be depend from one bond to another.
So these are possible Hamiltonians and they
have all been explored in the context of spin
systems.
