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PROFESSOR: OK, so
time for new subject.

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Let's introduce the subject and
pose the questions that we're

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going to try to answer.

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And I feel that with
identical particles,

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there's lots to think
about, and it makes it

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into an interesting way
to conclude the course.

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So identical particles.

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So there is the issue
of defining what do you

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mean by identical particles?

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And then the issue
of treating them.

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So when do we say that two
particles are identical?

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We say two particles
are identical if all

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their intrinsic properties--
like mass, spin, charge,

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magnetic moment--

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if all these things are the
same, these two particles--

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we have particle
1 and particle 2--

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are said to be identical.

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For example, all electrons
are said to be identical.

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And if you think about it,
well, what does that mean?

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You can have an electron
moving with some velocity

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and an electron standing here,
and they don't look identical.

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They have different states.

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Well, they're
identical in the sense

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that-- what we said-- the
intrinsic properties are

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the same.

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Those two particle
have the same mass.

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They have the same spin--

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in principle.

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They have the same charge.

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Have the same magnetic moment.

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Have all these same properties.

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Now, they can be in
different states.

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One electron can be
one momentum state,

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an electron can be in
another momentum state.

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One electron can be in
a spin state-- spin up.

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This can mean spin down.

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But what we would
mean is that if you

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would put, by saying these
particles are identical,

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we also mean they're
indistinguishable.

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What that would mean
is that if one of you

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gives me an electron with spin
up with some momentum state,

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and--

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yeah, let's say spin up
on some momentum state.

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And another one of you gives me
another electron with same spin

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up, same momentum state, and
I have those two electrons

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and I play with
them for a minute

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and I give them back
to you, you have

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no way of telling you
got the same electron

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or you got your
friend's electron.

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There's no possible
experiment that can

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tell which electron you got.

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So when we have identical
particles like electrons--

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elementary particles--
we understand

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what it means to be identical.

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Doesn't mean they're
in the same state.

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It means that this is a
particle that all the properties

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intrinsic of them are the same.

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If you have a more
complicated particle,

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you still can use the concept
of identical particles.

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So, for example,
you have a proton.

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A proton is a more
complicated particle.

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It is made of quarks.

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And if you have
two protons, they

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are identical in
that same sense.

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All the properties we
can give to the proton--

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the spin state of
the proton, mass

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of the proton, the dipole
moment of a proton,

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the magnetic moment of a--

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all those are the same.

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If you prepare those
protons in identical states,

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I cannot tell which is proton
1 and which is proton 2.

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Then you have the neutrons.

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Neutrons are the same thing.

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The neutrons can be
in several states,

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but two neutrons are
considered to be identical.

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We can complicate matters more.

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We can take hydrogen atoms.

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Are hydrogen atoms
identical particles?

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And in quantum mechanics,
we can think of them

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as identical particles--
or identical atoms,

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or identical molecules, or if
you write the wave function

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for a hydrogen atom-- a new
kind of entity of particle--

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we will use the axioms
of identical particles

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even for the hydrogen atom.

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But you could say, oh, no,
but they're not identical.

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A hydrogen atom can be
in the ground state,

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or it can be in
an excited state.

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But that's the same as
saying this electrode is

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going with little momentum,
and this is with high momentum.

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These are states of
the hydrogen atom.

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Just like an electron has
spin up and spin down,

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hydrogen atom has this state,
that state, that state.

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If you arrange them
in the same state,

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you cannot tell
they are different.

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So however clear these
comments can seem--

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or confusing, perhaps--
things can be a little subtle.

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In many ways, for
example, physicists

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used to think of protons and
neutrons as the same particle.

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Said what?

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Well, that's the way
they thought about it.

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It's a very nice thing.

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If you're working
with energy scales,

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the proton and neutron
mass difference

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is not that big, first of all.

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So at some scale for the
resolution of some experiments,

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or physicists that didn't
have that many tools,

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the proton and the neutron
were almost identical,

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and people invented this
term called isospin.

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And you might have heard of it.

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It's a very famous symmetry
of the strong interactions.

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In fact, for the
strong interactions,

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you have a nucleus.

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Whether you're a
proton or a neutron

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doesn't make that
much difference.

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So people used to think of
this thing as an isospin state.

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Just the spin one half--
you have spin up, spin down.

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Isospin means spin in some new
direction that is unimaginable.

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But the isospin up
would be the proton,

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the isospin down
would be the neutron.

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And you will have a doublet.

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So people used to think
of these two particles

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as different states
of a nucleon,

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and then they would say,
all nucleons are identical.

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Why do you complain?

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A proton is the
same as a neutron.

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It's just a different
state of the isospin,

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just like the spin electron
up or spin down is the same.

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So the power of the
[? formalism ?] in quantum

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mechanics is that it
allows you to treat

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these things as
identical particles,

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and this makes sense.

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For you-- for your experiments--
these are identical things

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that you can think of
different states of that.

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You may as well
treat them that way.

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So this is basically
what happens.

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Now, so we define
the identical--

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I didn't write anything here.

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I'm going to get notes
out today on scattering,

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and some of these
things as well.

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So in classical mechanics--

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classical mechanics-- identical
particles are distinguishable.

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And that's the main thing.

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How are they distinguishable?

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Well, you have two
particles, and I can follow--

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whenever they're moving, I can
say, OK, this is particle 1.

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This is particle 2.

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With quantum mechanics,
you can do the same thing

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when they are really far
away-- those particles--

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and they don't come
close together.

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There's some sense in which
classical mechanics sometimes

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applies, and that's
when they're far away.

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Now, when the particles
in quantum mechanics

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get close to each
other, then you

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lose track which one is which.

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They occupy the same position.

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But in classical mechanics,
they are distinguishable,

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because you can follow
their trajectories, which

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is very nice.

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You have an experiment.

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You follow the particle,
say, oh, this is particle 1.

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This is particle 2.

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They're here together.

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They're going around.

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And they split, you
follow the trajectories

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from the beginning to the end.

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In quantum mechanics, there's no
such thing as the trajectories.

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There's these waves.

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The waves mix together.

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They do things, then
they separate out,

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and you just can't
tell what they do.

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There's another technique that
we use in classical physics

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that it's probably
also relevant.

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We can tag the particles.

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That means, if you're doing an
experiment with billiard balls

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colliding, you could
take a little marker

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and put a red dot on one
of them and a black dot

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on the other one,
and that tagging

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doesn't affect the collisions.

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And you can tell, at the
end, where is the red ball,

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and which is the black ball.

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We do the same with
quantum mechanics.

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There's no way anyone
has figured out

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the tagging a particle
without changing drastically

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the way interactions happen.

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So it's a nice option
in classical physics,

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but doesn't work in
quantum mechanics.

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Even in classical physics, we
have something that survives.

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If you have a Hamiltonian
for identical particles--

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R2, P2-- that Hamiltonian is
symmetric under the exchange.

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00:11:21,810 --> 00:11:24,480
Whatever the
formula is, it's not

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00:11:24,480 --> 00:11:29,850
changed if you put
r2, p2 and r1, p1.

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It's a symmetric thing.

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The Hamiltonians
have that symmetry,

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and there's no way to do this.

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So let's get to the bottom--

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the real problem with
identical particles

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with quantum mechanics.

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We cannot tag them.

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00:12:01,230 --> 00:12:03,500
Once these particles
get together,

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you don't know what they did.

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You do an experiment
of scattering

202
00:12:07,910 --> 00:12:10,850
in the classical
mechanics, and you

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00:12:10,850 --> 00:12:16,000
put two particles coming
in, two detectors,

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00:12:16,000 --> 00:12:20,110
and you tag the particles
and you see what they do.

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00:12:20,110 --> 00:12:22,240
You do it in quantum
mechanics, and you

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00:12:22,240 --> 00:12:27,220
don't know if particle 1 did
that and particle 2 did this,

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00:12:27,220 --> 00:12:32,780
or if particle 1 did that
and particle 2 did that.

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00:12:32,780 --> 00:12:34,590
It's just not possible to tell.

209
00:12:38,710 --> 00:12:39,940
They're very different.

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00:12:39,940 --> 00:12:43,690
So how do we deal with this?

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00:12:43,690 --> 00:12:46,570
Well, that's the subject
of what we're going to do,

212
00:12:46,570 --> 00:12:53,350
but let's just conclude
today by stating the problem.

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00:12:53,350 --> 00:12:57,180
So the problem is that when we
had distinguishable particles

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00:12:57,180 --> 00:13:02,300
in quantum mechanics, we used
that tensor product to describe

215
00:13:02,300 --> 00:13:02,910
a state.

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00:13:02,910 --> 00:13:10,260
So for distinguishable
particles, this thing which

217
00:13:10,260 --> 00:13:19,220
all particles, say, 1 up to n,
we would use the tensor product

218
00:13:19,220 --> 00:13:25,200
and write PSI i1 for
the particle one,

219
00:13:25,200 --> 00:13:34,930
PSI i2 for the particle two,
PSI in for the particle n.

220
00:13:34,930 --> 00:13:39,270
And this says particle one
is in the state PSI i1,

221
00:13:39,270 --> 00:13:44,140
particle two is in the
state PSI i2, PSI in.

222
00:13:44,140 --> 00:13:47,730
And these states are one of
many states, for example.

223
00:13:47,730 --> 00:13:51,490
That's all good.

224
00:13:51,490 --> 00:13:55,250
And this is for
distinguishable particles.

225
00:13:55,250 --> 00:13:57,790
And that's all correct.

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00:13:57,790 --> 00:14:06,870
Now, suppose you have two
electrons, one up and one down.

227
00:14:09,550 --> 00:14:12,550
If they are
indistinguishable, how

228
00:14:12,550 --> 00:14:17,350
are we supposed to write the
state of the two electrons

229
00:14:17,350 --> 00:14:21,390
or to spin one-half particles,
or maybe, in some cases,

230
00:14:21,390 --> 00:14:23,560
maybe some other particles.

231
00:14:23,560 --> 00:14:35,450
Am I supposed to write that the
first particle is in state up

232
00:14:35,450 --> 00:14:37,510
and the second is in state down?

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00:14:37,510 --> 00:14:41,090
Or are I supposed to write
that the first particle is

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00:14:41,090 --> 00:14:46,960
in state down and the second
particle is in state up?

235
00:14:46,960 --> 00:14:56,920
How do I describe the state
with this one or with this one?

236
00:14:56,920 --> 00:15:00,780
They look equally plausible.

237
00:15:00,780 --> 00:15:07,790
So if you were in charge of
inventing quantum mechanics,

238
00:15:07,790 --> 00:15:11,140
one possibility that
it may occur to you

239
00:15:11,140 --> 00:15:16,780
is that if the particles
are defined to be identical,

240
00:15:16,780 --> 00:15:22,870
then those two states
should be identical.

241
00:15:22,870 --> 00:15:27,130
They should be
indistinguishable, identical,

242
00:15:27,130 --> 00:15:30,990
physically equivalent.

243
00:15:30,990 --> 00:15:36,740
And this might be a good
hypothesis to consider.

244
00:15:36,740 --> 00:15:39,420
Unfortunately,
that does not work.

245
00:15:39,420 --> 00:15:41,060
You say, why not?

246
00:15:41,060 --> 00:15:43,780
It seems so logical.

247
00:15:43,780 --> 00:15:46,240
I cannot tell the difference
between these two.

248
00:15:46,240 --> 00:15:48,950
Can I say that
they're equivalent?

249
00:15:48,950 --> 00:15:52,080
Well, no.

250
00:15:52,080 --> 00:16:00,520
If they would be equivalent,
you could form a state PSI alpha

251
00:16:00,520 --> 00:16:08,980
beta, which is alpha
times the first plus

252
00:16:08,980 --> 00:16:14,460
minus plus beta times
the second minus plus.

253
00:16:14,460 --> 00:16:17,830
And I'm not going to write all
the subscripts nor the tensors

254
00:16:17,830 --> 00:16:19,500
sometimes.

255
00:16:19,500 --> 00:16:27,520
With alpha and beta having
this for normalization.

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00:16:27,520 --> 00:16:29,410
That's a normalized state.

257
00:16:29,410 --> 00:16:33,410
Now, if all those
are equivalent,

258
00:16:33,410 --> 00:16:35,710
if those two are
equivalent, all these

259
00:16:35,710 --> 00:16:38,600
are equivalent for all
values in alpha and beta

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00:16:38,600 --> 00:16:41,490
because the superposition
of equivalence state

261
00:16:41,490 --> 00:16:43,990
is an equivalent state.

262
00:16:43,990 --> 00:16:48,370
But then let's ask for what is
the probability that we find

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00:16:48,370 --> 00:16:53,380
the two particles
in the state PSI 0,

264
00:16:53,380 --> 00:16:58,090
which is plus along
the x direction times

265
00:16:58,090 --> 00:17:01,480
plus along the x direction.

266
00:17:01,480 --> 00:17:06,130
You know, whatever I do
whether this hypothesis there

267
00:17:06,130 --> 00:17:09,540
that those two states are
equivalent is correct,

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00:17:09,540 --> 00:17:14,800
a state that is plus
and plus can only

269
00:17:14,800 --> 00:17:17,290
be described one
way-- plus plus.

270
00:17:17,290 --> 00:17:19,810
So I ask, what is
the probability

271
00:17:19,810 --> 00:17:22,359
that this state PSI
alpha beta in the plus

272
00:17:22,359 --> 00:17:25,690
along x and in the plus along x.

273
00:17:25,690 --> 00:17:30,400
Now, you'll remember those
pluses are states like plus,

274
00:17:30,400 --> 00:17:38,200
plus minus, 10 surplus
plus minus with a 1

275
00:17:38,200 --> 00:17:40,160
over square root of 2.

276
00:17:40,160 --> 00:17:41,510
And that becomes this.

277
00:17:41,510 --> 00:17:50,240
So this is 1/2 of plus
plus plus plus minus,

278
00:17:50,240 --> 00:17:55,330
plus minus plus
plus minus minus.

279
00:17:55,330 --> 00:17:57,510
That's the state.

280
00:17:57,510 --> 00:18:04,020
So what is the probability
that PSI alpha beta

281
00:18:04,020 --> 00:18:07,025
is found in the state PSI 0?

282
00:18:10,760 --> 00:18:12,890
It's this number.

283
00:18:12,890 --> 00:18:18,170
Now, if you do the inner
product of these two vectors,

284
00:18:18,170 --> 00:18:22,890
only the mixes ones
go with each other.

285
00:18:22,890 --> 00:18:27,995
And this gives you 1/2 of
alpha plus beta squared.

286
00:18:30,710 --> 00:18:33,200
And 1/2 of alpha
plus beta squared

287
00:18:33,200 --> 00:18:35,790
is the inner product
of these two states.

288
00:18:35,790 --> 00:18:40,550
And now you see that it depends
on the values of alpha and beta

289
00:18:40,550 --> 00:18:47,960
because this is in fact 1/2 of
alpha squared plus beta squared

290
00:18:47,960 --> 00:18:53,300
plus 2 real of alpha beta star.

291
00:18:53,300 --> 00:18:55,490
And since alpha and
beta are normalized,

292
00:18:55,490 --> 00:19:05,390
this is 1/2 plus real
of alpha beta star.

293
00:19:05,390 --> 00:19:10,700
So this hypothesis that these
two states are equivalent

294
00:19:10,700 --> 00:19:15,080
would mean that these states are
equivalent for all alpha beta

295
00:19:15,080 --> 00:19:16,980
that are normalized.

296
00:19:16,980 --> 00:19:20,690
And then you would have
that the probability

297
00:19:20,690 --> 00:19:23,240
to be found in
PSI 0 would depend

298
00:19:23,240 --> 00:19:26,310
on what the values of
alpha and beta you choose.

299
00:19:26,310 --> 00:19:28,460
So it's a contradiction.

300
00:19:28,460 --> 00:19:32,870
So we cannot solve the problem
of the degeneracy of identical

301
00:19:32,870 --> 00:19:37,010
particles by declaring that
all the states are the same.

302
00:19:37,010 --> 00:19:39,840
So we have to find a
different way to do it.

303
00:19:39,840 --> 00:19:42,610
And that's what we
will do next time.