WEBVTT
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PROFESSOR: With this, we
can phase the construction
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of the operators that
are going to help
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us build totally symmetric
states and totally
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anti-symmetric
states, and understand
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why we solve the problem
of degeneracy, exchange
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degeneracy.
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So let us look into that.
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So this is called
complete symmetrizers
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and anti-symmetrizers,
complete symmetrizers
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and anti-symmetrizers.
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Permutation operators
don't commute.
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So there's no hope ever of
simultaneously diagonalizing
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them.
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There's some of them
are not even Hermitian.
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So even worse, you
cannot find that, oh,
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I'm going to get a
complete basis of states,
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simultaneously diagonalizing,
some permutation operators are
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Hermitian.
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The majority are a unitary.
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The transposition operators
are the Hermitian ones.
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Those you could
try to diagonalize.
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But if you have the
whole permutation group,
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you cannot diagonalize it.
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It's just too many things
that don't commute.
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But while you cannot
diagonalize these things,
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you can find special states
that are eigenstates of all
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of the elements of
a permutation group.
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So you remember, when you say--
this is a very important point.
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Whenever you say you cannot
simultaneously diagonalize two
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operators, it means that you
cannot find a basis of states
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that are simultaneous
eigenstates.
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But it may happen that
you have one state that
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is an eigenstate of all
these other things that
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don't commute.
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It is possible to
have operators that
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don't commute and
have one state that's
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an eigenstate of all of them.
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You cannot have a basis that is
an eigenstate of all of them,
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because they don't commute.
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But one state is possible.
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So we can find special states.
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Special states that are
eigenstates of all permutation
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operators.
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So let's assume we have n
particles, each living on v,
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in v, so that the n
particles live on v tensor n.
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People write it like
that, v tensor n,
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which is supposed to
mean v tensor v with v
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appearing n times.
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So here is a claim
that we're going
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to postulate the existence
of symmetric states.
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Those are the states
that eventually will see
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are the ones physics ones.
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So postulate that there are the
existence of symmetric states,
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psi s, n v tensor n.
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In the whole big space,
there's symmetric states.
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And what is the characteristic
of a symmetric state?
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The p alpha, any permutation.
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Remember, alpha means all
these set of indices on psi
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s is equal to psi
s for all alpha.
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So the state is invariant.
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So we want to see that there
is such a thing, states that
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are invariant, under all
the permutation operators
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that we've constructed.
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This state would be eigenstates
of all the permutation
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operators with
eigenvalue equals to 1.
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So it's a simultaneous
eigenvector
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of all these operators.
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But since the permutation
operators don't commute,
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you cannot expect the basis.
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So if there are
symmetric states,
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they cannot form a basis
in the full Hilbert space.
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There must be some
smaller space.
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So we should be able to
reach them by a projector
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into a subspace of
symmetric states.
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How about defining now
postulate anti-symmetric states.
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Psi A. P alpha on psi A
should then be equal to--
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what should I put?
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Negative psi A.
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Well, yeah, that's the
first thing we would put,
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but that's pretty
problematic actually.
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So even when you
postulate things,
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you know, postulate means,
OK, we think they exist,
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then we'll try to build them.
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Here, we're trying to
postulate that there
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are states that do this.
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There's a little bit
of problems with this.
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First obvious problem,
you say, oh well, this
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is a mathematical technicality.
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The identity element is supposed
to be a permutation p 1, 2, 3.
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And the identity element is
not going to change this one.
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So that's not good.
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So suppose you have one
transposition, and changes
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the state, a transposition
should produce a minus sign,
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because it's anti-symmetric.
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But suppose you have
now two transpositions.
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You act on them with
two transpositions.
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One will change its sign.
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The other will change its sign.
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Now the total
double transposition
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is a permutation operator.
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Shouldn't change the
sign of the state.
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So in fact, this is untenable.
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We're not even-- so even
if we postulate something,
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we'd have to postulate
something that makes some sense.
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And so far, it
doesn't make sense.
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So what can we use?
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We can use the fact that
there's some even permutations
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and some odd permutations.
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So we'll put the
sign factor here,
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psi A. This is the only
way to solve this problem
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is to put the sign factor
epsilon sub alpha associated
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to the permutation.
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Sometimes it's
going to be a minus.
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Sometimes it's
going to be a plus.
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For example, for the identity
operator, it should be a plus.
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For a transposition,
it should be a minus.
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So what is this epsilon alpha?
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Epsilon alpha is
equal to 1 if p alpha
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is an even [? transpose ?] even
or minus 1 if p alpha is odd.
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So an odd permutation
is one that
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has an odd number
of transpositions.
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So that makes sense.
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This is a way to do
this consistently.
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If you have a single
transposition,
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and we'll put the minus, but
if you have two transposition,
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it will put a
plus, as it should.
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And the identity element
is an even permutation.
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Therefore, it works as well.
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So this is a nice thing.
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This is the only
way you can define
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this anti-symmetric states,
even before we construct them.
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So here are the names, and we'll
stop and build them next time.
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So the symmetric
state, symmetric states
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form a subspace of VN
called sym N V. Symmetric
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in N states of V. The
anti-symmetric states
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form a subspace of VN
called anti N of V.
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And our task for next time is
to construct the projectors that
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bring you down to
those spaces, analyze
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what are the properties
of these spaces,
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and show that it solves the
problem of exchange degeneracy,
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and that requires an
extra postulate in quantum
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mechanics, a postulate
for identical particles.