WEBVTT
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PROFESSOR: It's a statement
about the time dependence
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of the expectation values.
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It's a pretty
fundamental theorem.
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So here it goes.
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You have d dt of the
expectation value
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of Q. This is what
we want to evaluate.
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We Now this would be d dt of
integral psi star of x and t,
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Q psi of x and t.
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And the d dt acts
on the two of them.
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So it gives you integral
partial psi star dt Q
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psi of x and t plus psi
star Q partial psi dt.
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And this is the
integral over the x.
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You've seen that kind of stuff.
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And what is it?
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Well, integral dx, this is
this Schrodinger equation,
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d psi star dt is i
over h bar, H psi star.
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From the Schrodinger equation.
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Then you have the
Q psi of x and t.
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On this term, you will
have a very similar thing.
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Minus i over H bar this time,
psi star QH psi of x and t.
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So we use the
Schrodinger equation
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in the form, i d psi dt--
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i H bar d psi dt--
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equal H psi.
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I used it twice.
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So then, it's actually
convenient to multiply here
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by i H bar d dt of Q. So
I multiplied by i H bar,
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and I will cancel the i
and the H bar in this term,
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minus them this term.
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So we'll have d cube x psi star
Q H hat psi minus H hat psi
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star cube psi.
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OK.
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Things have simplified
very nicely.
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And there's just one
more thing we can do.
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Look, this is the
product of Q and H.
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But by hermiticity,
H in here can
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be brought to the other side
to act on this wave function.
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So this is actually equal to
the integral d cube dx psi star
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QH hat psi minus--
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the H can go to the other side--
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psi star H cube psi.
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But then, what do we see there?
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We recognize a commutator.
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This commutator is just
like we did for x and p,
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and we started practicing
how to compute them.
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They show up here.
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And this is maybe one of the
reasons commutators are so
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important in quantum mechanics.
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So what do we have here?
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i H bar d dt of the
expectation value of Q
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is equal to the integral dx
of psi star, QH minus HQ psi.
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This is all of x and t.
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Well, this is nothing else
but the commutator of Q and H.
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So our final result is that
iH bar d dt of the expectation
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value of Q is equal to--
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look.
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It's the expectation
value of the commutator.
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Remember, expectation
value for an operator--
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the operator is the
thing here-- so this
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is nothing else than the
expectation value of Q with H.
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This is actually a
pretty important result.
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It has all the dynamics of the
physics in the observables.
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Look, the wave functions
used to change in time.
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Due to their change in
time, the expectation values
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of the operators change in time.
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Because this integral
can't depend on time.
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But here what you
have succeeded is
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to represent the change in
time of the expectation value--
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the change in time
of the position
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that you expect you
find your particle--
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in terms of the expectation
value of a commutator
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with a Hamiltonian.
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So if some quantity
commutes with a Hamiltonian,
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its expectation value
will not change in time.
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If you have a Hamiltonian,
say with a free particle,
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well, the momentum
commutes with this.
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Therefore the expected
value of the momentum,
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you already know, since
the momentum commutes
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with H. This is 0.
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The expected value of this is 0.
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And the expected value of
the momentum will not change,
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will be conserved.
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So conservation laws
in quantum mechanics
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have to do with things that
commute with the Hamiltonian.
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And it's an idea we're
going to develop on and on.