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PROFESSOR: Here is
where the power of this

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comes when you
decide that you're

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going to invent all possible
Hamiltonians at this moment.

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You've reduced the infinite
dimensional space of functions

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in the line to
two points, so you

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have a two-dimensional vector
space, dramatic reduction.

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So here we decide, OK,
here is the Hamiltonian.

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And it's going to be
a two-by-two matrix,

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and it better be Hermitian.

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So what options do I have?

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Well, Hermitian means transpose
complex conjugated gives you

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back the same matrix.

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So let's try to
parametrize such a matrix.

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I could put a0, a real quantity
here, and another real quantity

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in the bottom size.

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And if they are real, the
transpose complex conjugate

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will remain the same.

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That's OK.

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So I could put a0 and a1 here.

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I'll do it in a
little different way.

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I'll put a0 plus a3,
and a0 minus a3 here.

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Now, the thing is that a0
and a3 have to be real.

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So I'll use a0, a1, a2, and a3.

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And they all should be real.

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So here, transpose
complex conjugate

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doesn't affect these things.

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They are the same.

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That's good.

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Here, we can a1 minus ia2.

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This is a complex number.

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And the only thing
that must happen

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is that, when I transpose
a complex conjugate,

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I must get the same thing.

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So I should put
here a1 plus ia2.

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Because if I transpose this,
I will have it on this side.

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And then I complex conjugate
it, and it becomes this term.

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Similarly, if I transpose
this term, it goes here.

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But then complex
conjugated, it becomes that.

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So actually, I claim the most
general two-by-two Hermitian

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matrix.

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Time independent-- you see,
all our quantum mechanics

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this semester has been time
independent potentials.

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So here it's time independent.

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And now, this is the
most general Hamiltonian

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you could have.

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That's it.

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So when you see
something like that,

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you realize that in an hour
or two or after some thinking,

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you will have solved the
most general dynamical system

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with two degrees of freedom
in quantum mechanics.

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So I will write this as
a0 times this matrix,

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plus a1 times this matrix,
plus a2 times this matrix,

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plus a3 times this matrix.

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That's exactly what
you have in there.

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Multiply in these constants
and add these matrices,

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and they give you
all what we have.

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So actually, these are the basic
Hermitian two-by-two matrices.

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And if you multiply
them by real numbers,

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you still are Hermitian.

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And if you add them,
you still are Hermitian.

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So the most general Hermitian
matrix has four parameters.

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And it is a space of matrices
spanned by these four matrices.

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They are so famous,
these matrices,

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that this is called sigma
1, this is called sigma 2,

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and this is called sigma 3.

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And they're called
the Pauli matrices.

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Well, but let's put
units to these things.

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We want to write Hamiltonians.

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So let's make sure we have
units that do the job.

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The Hamiltonian must
have units of energy.

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So we could do a Hamiltonian
that has units of energy.

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So I'll write h omega, which has
units of energy, omega 1, sigma

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1, plus h--

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I'll put it even over 2--

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h omega 2, over 2, sigma 2,
plus h omega 3, over 2, sigma 3.

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Now you would say,
well, why didn't you

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use the first matrix.

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I could have used
the first matrix,

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but the first matrix is
proportional to the identity.

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We already learned in
our course that if you

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have an extra constant
operator in the Hamiltonian,

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it doesn't change your
calculations in any way.

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You had the Hamiltonian for
the harmonic oscillator.

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It was h omega N plus 1/2.

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And the 1/2 was an
additive constant

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that never played
any important role.

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So this would be an additive
constant to the energy.

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It would tell you
how you're measuring

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the energy from what level.

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So it's not very interesting.

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You can use it sometimes,
but it's definitely not

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all that interesting.

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So I'll do a little
variation of this

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by writing omega 1, h over
2, sigma 1, plus omega 2,

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h over 2, sigma 2, plus
omega 3, h over 2, sigma 3.

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And then you say, look,
that's interesting, OK,

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I have an omega on this thing.

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But omega is fine.

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We know what it is.

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It's a frequency,
1 over time unit.

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But this has units
of angular momentum.

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h bar has units of
angular momentum.

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And the thing that is a
little mysterious here

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is that we seem to
have three of them.

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So maybe somehow this has
to do with angular momentum.

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So let's investigate
it a little bit.

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Well, they have units
of angular momentum.

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So maybe I can call some first
component of angular momentum,

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h bar over 2 sigma 1, second
component of angular momentum,

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h 1 over 2 sigma 2, and the
third component of angular

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momentum, h bar over 2 sigma 3.

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Well, those are just names.

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But we can try to do a
computation with them.

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We can try to see what is
the commutator of Sx with Sy.

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And happily, these are matrices,
so it's a natural thing

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to do commutators.

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So you would have h
bar over 2, sigma 1,

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with h bar over 2,
sigma 2, commutator.

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And it's equal to h bar
over 2 times h bar over 2,

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sigma 1, sigma 2,
minus sigma 2, sigma 1.

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So it's h bar over 2
times h bar over 2.

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And let's do this.

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Sigma 1 is 0, 1, 1, 0.

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Sigma 2 is 0, minus i, i, 0,
minus 0, minus i, i, 0, 0, 1,

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1, 0.

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OK, I have to do
all that arithmetic.

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Happily, this is not that bad.

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Let's see if I
don't make mistakes.

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OK, here I get two terms,
an i from the first,

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a 0 here, a 0, and
a minus i here--

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minus-- and minus i,
a 0, a 0, and an i,

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which is h bar over 2,
times h bar over 2, times--

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oh, they don't cancel.

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They seem to cancel,
but there's some minus--

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it's actually twice--

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of those, so 2i minus 2i, 0, 0.

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And here we get a 2 cancels
this and then i goes out.

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So I'll have with
this factor and i out

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is i h bar times h bar over
2, times the matrix 1 minus 1,

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0, 0.

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Somehow, it gave that.

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h bar over 2, 1 minus 1--

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1 minus 1 is sigma 3.

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And h bar over 2 sigma 3
is Sz, so this is all Sz.

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So it's i h bar Sz.

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So this stuff, Sx, Sy,
is giving you i h bar Sz.

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And that was exactly
like angular momentum.

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So not only it has the
units of angular momentum,

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it has the commutation
relations of angular momentum.

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Hermitian operators,
two-by-two matrices,

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they used to be r cross
p, all these derivatives,

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complicated stuff.

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Here it is-- with
two-by-two matrices,

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you've constructed
angular momentum.

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What we've constructed at
this moment is spin 1/2.

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A whole spin 1/2 system is
nothing else than that--

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angular momentum and the freedom
of having two discrete degrees

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of freedom.

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The interpretation that
what they have to do

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is spin up and spin
down is something

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that physicists came up with.

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But the mathematics was there
waiting as the simplest quantum

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mechanical problem.

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Considering [? who wrote ?]
the Schrodinger equation,

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maybe, if he had been more
mathematically inclined,

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he could have discovered,
five minutes later, spin.

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But he wanted to figure out the
wave function of the hydrogen

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atom and scattering and all
these very complicated things.

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So needless to say, the other
commutation relations work out.

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So if you check that Sy with
Sz, you will get i h bar Sx.

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And if you do finally Sz with
Sx, you will get i h bar Sy.

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So these two-by-two matrices
satisfy this property.

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And there is a little
more to be said.

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I want to say a few
more things about it

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because it's counter-intuitive
and therefore very nice.

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Half of the semester in
805 is devoted to spin 1/2.

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It takes a while
to understand it.

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So I wanted you to
see it, at least once.

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And the problem is the
physical interpretation

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takes time to get accustomed.

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So on the other hand, we
did write the Hamiltonian.

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So the Hamiltonian was
omega 1 Sx, plus omega 2 Sy,

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plus omega 3 Sz.

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And it's there--

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S1, S2, S3, second line.

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And this is the Hamiltonian.

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So people write it sometimes as
omega dotted with an S vector,

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as it's saying it has
three components, as omega

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has three components as well.

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And there's a lot of
physics in this Hamiltonian.

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It's the simplest Hamiltonian,
but it actually represents

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a spin in a magnetic field.

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And what it will make
it do, this Hamiltonian,

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if we solve the differential,
this two-by-two matrix

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equation, we will find that
the spin starts to precess.

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That's the origin of
nuclear magnetic resonance,

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00:15:31,030 --> 00:15:36,850
spinning, precessing spins, that
the machine makes them precess.

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00:15:36,850 --> 00:15:39,820
And they send a
signal and you detect

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00:15:39,820 --> 00:15:43,530
the density of different
fluids in the body.