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BARTON ZWIEBACH: Now
that we've introduce h,

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h is a very important
quantity in quantum mechanics.

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So let's talk a little
more about h, its units,

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and we already put one
number that I really

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wish you will remember.

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Now let's talk
about the units of h

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and some other things
you can do with h.

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So units of h.

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So if you have a quantity that
appears for the first time

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and as it appears
here, E equal h nu,

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this is a good place to
understand the units of h

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because the units of h
would be units of energy

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divided by units of frequency.

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And I put this square
brackets to denote units.

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Now what are the
units of energy?

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We're going to work with
units that are characterized

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by M, L and T--

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mass, length, and time.

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So energy, you think kinetic
energy and you say, MV squared,

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so that's a mass
and velocity squared

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is L squared over T squared.

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So that's units of energy.

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Frequency is cycles
per unit time.

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Cycles have a number of units,
so it's 1 over time here.

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So then you say that
it's ML squared over T.

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So that's the first
answer and that's

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a nice answer,
although it's never

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quite that useful in this way,
so we try to rearrange it.

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And I will rearrange it the
following way to think--

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you see, it's nice to think of
what physical quantity that we

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are familiar, hats
units of h-bar.

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We know these units of h-bar
are energy over frequency,

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but that's not a single physical
quantity, so let's look at it

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and separate this as L times
MLT. That's the same thing.

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And then I see an
interesting thing--

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this is the units of
position, or length.

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Length or radius.

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Distance.

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And this has the
units of momentum p.

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Momentum.

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So this product has the
units of angular momentum.

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And perhaps that's the
most important quantity

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that has the units of h-bar.

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It's something that
you should remember.

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Of h-bar has units
of angular momentum,

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that's why when people talk
about a particle of spin 1/2,

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they say the angular
momentum is 1/2 of h-bar,

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and that has the right units.

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So spin 1/2 particle--

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1/2 particle-- means
that the magnitude

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of the intrinsic angular
momentum is 1/2 of h-bar.

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h or h-bar have the same units,
they just differ by a 2 pi

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that--

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unfortunately, we have to
be careful about that 2 pi,

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it affects numbers
and some formulas

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are nicer without the bar,
some formulas are less nice.

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So OK.

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So another thing
that you could say

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is that this h allows
you to construct

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all kinds of new quantities.

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And that's a nice thing to do.

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Whenever you have a new constant
of nature that comes up,

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and we have the speed of
light, Planck's constant,

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Newton's constant-- seem to
be the three fundamental units

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of nature--

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you can do some things.

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And you can look
at this quantity--

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h is proportional to rp
and get an inspiration.

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So you can think h has
units of r times p.

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And you can say, look--

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if I have any
particle with mass M,

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I can now associate
a length to it.

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I can invent a length
associated to that particle.

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And how do I do it?

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Well, this has units of
length, so all I have to do

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is take h and divide by p.

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Well, that will be one
way to get the length

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where p is the momentum, and it
will be called the de Broglie

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wavelength.

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But there is another way.

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Suppose this particle
is just not moving

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and you have the momentum
and you say, wow,

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momentum is not moving,
so what's going on here?

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So think of it at
rest and then you say,

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well, you still can
construct a length.

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You can put h and divide by
the mass times the velocity

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of light, why not?

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That's a velocity, it
is a constant of nature.

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So that way, you
associate a length

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to any particle of a given mass.

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You don't have to tell
me what is the momentum.

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You can just know
the mass and it

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has a length associated to it.

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So it's called the Compton--

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Compton-- wavelength
of a particle.

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And I want to make
sure you don't confuse,

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it's not the same as
de Broglie wavelength

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that we will see later.

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It's not the same as the
de Broglie wavelength.

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This is the Compton
wavelength of the particle.

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And you can say,
all right, good,

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you give me a
particle of some mass,

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I can tell you what a
length associated to it--

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why would it be important?

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It will be important
in two different ways--

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through an experiment and
through a thought experiment,

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which I want to do right now.

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You see, I could ask
the following question--

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I have this particle, has a mass
M. I use the speed of light,

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so with that mass M, I could
associate out to this particle

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a rest energy.

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MC squared.

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That's the rest energy.

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And then I could ask,
what is the wavelength

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of a photon that has the
same energy as the rest

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energy of this particle?

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So you translate the question
into a question of a length.

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Once you have some energy,
there is a natural length,

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which is the wavelength of
a photon with that energy.

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So let's ask this question
independently of what we did.

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So what is the wavelength--

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wavelength-- of a photon whose
energy is the rest mass--

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rest mass-- of a particle?

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So the rest mass is
MC squared, and that's

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the energy of this photon.

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And we know that energy of a
photon h nu or hC over lambda,

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and there, we can
calculate the lambda.

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Lambda is hC over MC
squared, and no surprise,

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it gives us h over
MC and that thing

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is the Compton wavelength.

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So it's sometimes
called l-Compton

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of the particle of mass M.

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So this is a way that you
can think of this particle.

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You think of a particle, you
have a Compton wavelength,

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and that Compton wavelength
is the wavelength of light

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that has that rest energy.

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And that actually has
experimental implications

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in high energy particle physics.

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Because if you have an
electron and it has a Compton

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wavelength, and you shine a
photon that has that size,

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that photon is
carrying as much energy

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as the rest energy
of the electron.

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And in particle theory
and quantum field theory,

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particles can be created and
destroyed, so this photon maybe

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can do some things and
create more particles out

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of this electron, particle
equation could start.

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happening.

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So it will be difficult to
isolate a particle to a size

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smaller than its
Compton wavelength,

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because the photons could do
such damage to the particle

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by creating new particles
or doing other things to it.

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So for an electron, let's
calculate the Compton

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wavelength.

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So l-Compton of an electron
would be h over MeC,

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and you would do h--

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you would do 2 pi h-bar
C over and MeC squared.

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And you've got 2 pi
197.33 MeV fermi,

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and here you would
have 0.511 MeV.

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So this gives you 2,426 fermi,
or about 2.426 picometers.

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Picometers is kind
of a natural length.

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Picometer is 10 to
the minus 12 meters.

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The Bohr radius is
about 50 picometers,

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so that's how big this thing is.

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Is it still much bigger than
the size of the nucleus?

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The nucleus is a few fermis.

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A single proton is
about a fermi big.

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And nucleus grow
slowly, so you can

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have a big nucleus with
200 particles maybe

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of a radius of 7 or 8 fermi.

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So this is still a
lot bigger and it's

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a very interesting quantity
that will show up very soon.