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PROFESSOR: We've talked a
lot about de Broglie saying

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that the wavelength
is given by h over p.

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But we have not said much
yet about the frequency

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of the waves.

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So what is the frequency
of those matter waves?

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So what is the frequency--

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frequency--

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of the matter waves.

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So de Broglie did answer
that same question.

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And the answer was
obtained by analogy.

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We have p equal h bar k.

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And he said, well, just
like the wavelength

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is determined by the momentum,
we'll have e equal h bar omega.

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So the frequency--
so this equation

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is the one that now
completes the story.

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Omega is equal to e over h bar.

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Fixes omega in
terms of the energy.

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And we're going to
say a few things.

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In fact, this will be an
interesting digression

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into an important
subject about waves

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that illustrates why this
answer makes a lot of sense.

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And that's, really, all
you can do at this moment.

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This is a postulate
of quantum mechanics.

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That you do this thing,
and with this, you

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get quantum mechanics.

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So the best thing we
can do is to explain

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why it makes sense
in a number of ways,

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and then hope that the
theory that you built

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makes full sense.

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So I want to remind you
about velocities of waves.

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So if you have a wave now
that it has k and omega--

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you have this thing. k minus
omega, wave with a phase.

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kx minus omega t.

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Then there is something
called the phase velocity.

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And it's given by omega over k.

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It's the velocity in which the
nodes and maxima of this plane

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wave move.

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So let's see if this
makes some sense.

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Omega over k is the
same thing as e over p.

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We're nonrelativistic,
so let's continue.

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1/2 mv squared over mv.

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And this seems a little strange.

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1/2--

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v. So if I have a
particle, you see,

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this is matter waves of
energy e and momentum p.

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And e is 1/2 mv squared, the
velocity of the particle.

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p is equal to mv.

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And now, somehow
this wave seems to be

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moving with half the
speed of the particle.

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That looks pretty bad.

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What's going on?

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Well, this is the
usual story with waves.

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If the wave itself doesn't--

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a wave, a plane wave
carries no real information.

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It's not the signal.

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So many times when you try
to represent the particle--

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a little bit of information
traveling-- representing it

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with a plane wave is
actually quite wrong.

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You have to represent
it with a wave packet.

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And therefore, this
phase wave velocity

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being one half of the
velocity of the particles

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seems to just confirm the idea
that, first, these waves are

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a little strange.

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And second, phase velocity is
not very meaningful physically.

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The velocity that this more
meaningful is v group velocity.

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And it's d omega dk evaluated at
the value k that you're using.

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k is a proxy for momentum.

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So d omega dk may
depend on k and omega.

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So if it's d omega,
dk is a function.

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Which value should you use?

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Well, the value at the k
that you're propagating.

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And this would be the
same as d omega dk.

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Is because of the constant
separating-- the same as de vp.

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But what is the kinetic energy
in terms of the momentum?

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We wrote it last time.

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p squared over 2m.

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That's the kinetic energy
expressed in terms of momentum.

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So this is d dp of
p squared over 2m.

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Write p equal mv and you'll
recover the kinetic energy.

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And this is just--

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because of the 2--

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p over m, which is the
velocity of the particle.

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And this is the reason
people believe de Broglie.

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De Broglie made sense
because the group velocity

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of this [? package
?] would be correct.

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And that's a very
beautiful result.

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Actually, it's true
relativistically, as well.

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If you put the energy and
the momentum in relativity,

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this answer comes out exactly
the same, perfectly well.

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So to a large
degree, since it also

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works for energy and
momentum in relativity,

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there was a motivation
from relativity

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that I want to quote, although
not elaborate on it too much.

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So--

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the motivation--

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is that in special relativity--

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relativity--

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the components of the energy
divided by c and the momentum

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form a 4-vector.

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Just like position and time
forms a 4-vector and transform

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nicely about--

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with Lorenz transformation--
e and p form a 4-vector.

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Nevertheless, when
you consider phases--

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like this, and you have x
and t that form a 4-vector,

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the good behavior
of phases also imply

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that k and omega
form a 4-vector.

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In fact, omega-- in relativity,
omega over c and the k vector--

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form a 4-vector.

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You see, in all the
equations we've written--

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and de Broglie-- de Broglie
in three dimensions or more,

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really, is p vector
equal h bar k vector.

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And k is usually used for the
magnitude of this k vector.

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So this is also 4-vector
in special relativity.

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And therefore, vectors are
things that transform nicely.

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So it makes sense to
say that one 4-vector is

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equal to another 4-vector.

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Because if it's true
in one reference frame,

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it will be true in
every reference frame.

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So it's almost irresistible
to make them equal.

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And de Broglie, in some sense,
said this is equal to h bar.

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That's de Broglie.

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The interesting thing is that
this is true relativistically.

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But actually,
nonrelativistically, you

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can make sense of this and set
it equal to be the same things.

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And the phase velocities, group
velocities all makes sense.

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Certainly, we've now said
for [? minor ?] particles

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that e is equal to j bar omega.

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But another statement would
be that, yes, indeed, Einstein

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said that.

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That for photons, e was
equal to h bar omega or h nu.

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And therefore, yes,
whatever happens for photons

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happens for this matter waves.

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And so also, so this
is another argument.

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Group velocities is one.

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Special relativity
is another reason.

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And of, course photons.

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Einstein--

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said that e is equal h nu,
which is equal to h bar omega.