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Let's pick up from where we
were on Friday.

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We had discovered the nucleus.
Now we were faced with the

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problem, as all the scientific
community was in 1911,

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in trying to understand the
structure of the atom.

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Where was the nucleus in the
atom?

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Where was the electron?
How were they bound?

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How did they hang together?
And we talked about the fact

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that the electron in the
nucleus, the force of

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interaction is the Coulomb
force.

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And we started talking about
how, at that time,

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the only equation of motion
that was going to allow us to

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figure out how the electron and
nucleus moved under influence of

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this Coulomb force was Newton's
equations of motion,

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in particular the Second Law,
F equals ma.

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And so, in order to apply that
equation of motion,

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we needed a model for the atom.
And what was the simplest and

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most obvious thing to do was to
suggest the planetary model.

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After all, that is how the
astronomical bodies moved around

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the sun.
And so the model that is set up

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is one where this electron has a
uniform circular motion around

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the nucleus with a well-defined
radius, which we called r star.

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We said that given this,

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the acceleration was a
constant.

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It was given by v squared over.

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The linear velocity over r.
We plugged that into F equals,

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put in the Coulomb force,

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and from that we were able to
calculate the kinetic energy of

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that electron going around the
nucleus.

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Well, the reason I want to
calculate the kinetic energy

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from this model is because I
want to ultimately calculate the

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total energy.
And why I want to calculate the

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total energy is going to be
obvious in just a few minutes.

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My goal is to get the total
energy.

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Actually, I am using my notes
from Friday because I didn't

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finish them.
You may need to get them out.

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This will probably often be the
case, is that I won't quite

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finish the notes from the other
lecture.

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I will start out the next
lecture where I left off,

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so you should bring your
previous day's notes to class

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if, in fact, you use them during
class.

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I want the kinetic energy plus
the potential energy because I

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want both of them to add them up
to get the total energy.

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I know the kinetic energy.
Now, we need the potential

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energy.
What is the potential energy?

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Well, the potential energy is
the integral over the operating

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force over the appropriate
limits.

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In this case,
if our force of interaction is

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the Coulomb force,
which I will just represent as

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F of r,
I am going to integrate this

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from r star out,
and this is going to be minus

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the integral of the force.
Now, some of you may have seen

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this before.
This is a general case,

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the potential energy of
anything is minus the integral

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of the operating force over the
appropriate coordinates.

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If you have seen it before,
that is fine,

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you are happy.
If you have not seen this

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before, you are panicked.
Don't panic.

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I do not hold you responsible
for this.

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You will see it in 8.01 later
on this semester.

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When you see it later on,
you can come back here and say,

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okay, now I know what is going
on.

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But I just need it right now to
make a point about the total

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energy of the system.
And that is what is going to

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lead me to the conundrum.
I need the potential energy.

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It is the integral of the
force.

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Let me plug in,
here, my force that is

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operating, e squared 4 pi
epsilon nought r squared.

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I do that integral and put in

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the appropriate limits.
It is minus e squared over 4 pi

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epsilon nought r star.

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Now I have kinetic energy plus
potential energy.

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Let me add them up.
The kinetic is one-half e

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squared 4 pi epsilon nought r
star.

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Potential, minus e squared over

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4 pi epsilon nought r star.

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The result is minus one-half e
squared 4 pi epsilon nought r

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star.

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That is the total energy here

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of this particular system.
Well, why I wanted this total

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energy is to show you that this
total energy is negative.

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What that negative means to us
is that the system is bound.

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The electron and the nucleus
are stuck together.

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And I can show you that maybe a
little more clearly if I draw an

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energy level diagram.
Let me plot here the total

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energy.
And I am plotting it as a

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function of r,
the distance between the

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electron and the nucleus.
Well, what you can see is that

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for very large r,
the energy here is going to be

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zero.
Way out here,

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for very large r,
where we have the electron and

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the nucleus separated infinitely
apart, the energy is zero.

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And, of course,
as you bring them closer

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together, the energy goes down.
And when you are exactly,

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and we calculated this,
at r star here,

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well, then the total energy is
minus one-half e squared over 4

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pi epsilon nought times r star.

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If you brought the electron and

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the nucleus into this value here
of r star,

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the energy would change like
that.

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But the big point is this
energy is negative,

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or it is lower than the
electron and the nucleus

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separated.
That means that the electron

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and nucleus are stuck together.
You are going to have to put

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this much energy into the system
in order to pull them apart.

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That is the big point here,
is that this model so far looks

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like everything is hunky-dory.
Everything is working.

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The electron stuck to the
nucleus.

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It is not going anywhere.
It looks terrific.

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But here comes the conundrum.
The conundrum is that classical

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electromagnetism,
which was pretty well

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understood by this time,
1911, 1912.

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Maxwell's equations,
that was down pat.

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But what classical
electromagnetism says is that

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when you have a charge,
and this electron is a charge,

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that is accelerating,
that charge has to be emitting

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radiation.
It has to be giving off energy.

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After all, that is actually how
an antenna works.

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In an antenna,
what you are doing is taking

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charge and sloshing it,
accelerating it.

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When it accelerates,
it emits radiation.

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That is how you broadcast.
That is true,

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it was known in 1911.
Synchrotron radiation works the

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same way.
When you have a synchrotron,

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the way you get synchrotron
radiation is essentially by

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accelerating charge.
That is a given and is

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actually, again,
something you will talk about

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in much more detail in 8.02.
But the point here is if this

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charge is being accelerated,
and it is, then it must be

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giving off radiation.
It must be giving off energy.

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Well, if it is giving off
energy, we look at our energy

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expression here.
That must mean that the energy

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in the system is going down
because it is losing the energy.

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It is giving it off to
radiation.

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If E is going down,
it is getting more negative

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here.
The only way for E to get more

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negative is for this r star
right here to be changing.

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Is for r star to be getting
smaller and smaller and smaller.

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Well, we could set up another
set of equations using what we

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know from classical
electromagnetism and from what

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we have already done here.
What we would find is that this

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value here of r star would go to
zero in t equal 10^-10 seconds

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if r was originally on the order
of an angstrom to begin with.

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Here is the problem.
Classical equations of motion

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coupled with classical
electromagnetism,

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they are making a prediction
that my atom is not going to

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live more than 10^-10 seconds.
Because in 10^-10 seconds,

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that electron is on top of the
nucleus.

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We no longer have an atom that
was already known to have a

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volume associated with a
diameter that is about an

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angstrom.
The classical way of thinking

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is making a prediction that is
not consistent with the

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observations at that time.
And even now,

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it is predicted that the atom
essentially kind of annihilates,

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collapses in 10^-10 seconds.
And that is the problem that

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the scientific community had in

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That is the problem we have
right now.

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And they had it for 10,
12 years.

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Now you can say what is wrong
here?

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Well, it is possible,
and they were thinking about

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this, too.
It is possible that maybe this

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force is wrong,
this Coulomb force.

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That is a possibility.
Or, of course,

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maybe it is the equations of
motion that are wrong.

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That is possible.
Or, maybe it is classical

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electromagnetism that is wrong.
Well, of course what it is

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going to turn out to be is the
equations of motion,

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F equals ma.
Bottom line is that you cannot

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use classical mechanics to
explain the motion of this

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microscopic particle,
the atom, in the constrained

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environment of an atom.
That is the bottom line.

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We need different mechanics.
We cannot use classical

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mechanics to describe how that
electron hangs on that nucleus,

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how they are bound.
And so that was the problem.

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This signaled something was
really amiss in the scientific

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community in the world at that
time.

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That is our problem now,
too.

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What is the next step?
Well, historically the clues

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about why the electron did not
actually collapse into the

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nucleus, like classical physics
predicted, came from a

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completely different area of
discussion.

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It came from the discussion of
the wave-particle duality of

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light and matter.
It was long believed that

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matter, with its particle-like
behavior, was distinct from

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light, which was this
transmission of energy through

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space.
But, in the last 1800s and

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00:14:24,000 --> 00:14:29,000
early 1900s, there were a few
experiments that appeared on the

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horizon that began to suggest
that maybe this boundary between

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matter with its particle-like
behavior and radiation with its

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wave-like behavior was not as
rigid as thought.

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And, in fact,
what we are going to see is

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that radiation has both
wave-like properties and

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particle-like properties.
It depends on the particular

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experiment that you do which one
of those behaviors you see.

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00:15:00,000 --> 00:15:04,000
And, consequently,
matter behaves both as a

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particle and a wave.
Again, it depends on exactly

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what experiment you do,
which one of those properties

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you observe.
What we are going to do right

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now is put aside the discussion
of the structure of the atom.

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We are going to put it aside
until next Monday.

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We have to do that because we
need some more information in

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order to take a big leap to get
us out of this constraint of

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classical mechanics.
And those clues,

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as I said, came from this
discussion of the wave-particle

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duality of light and matter.
And that is what we are going

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to be talking about for the next
three lectures.

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Then we are going to come back
and tie in those results to the

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structure of the atom.
Of course, where that is going

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to lead us is a new equation of
motion called quantum mechanics.

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00:16:05,000 --> 00:16:10,000
That is where we are going.
Let's start off by talking

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00:16:10,000 --> 00:16:14,000
about radiation or light.
We are going to talk about its

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00:16:14,000 --> 00:16:19,000
wave-like properties,
then Wednesday we are going to

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00:16:19,000 --> 00:16:23,000
talk about the particle-like
properties of light,

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00:16:23,000 --> 00:16:28,000
and Friday we are going to talk
about the wave-like properties

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00:16:28,000 --> 00:16:32,000
of matter.
That is where we are going.

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00:16:32,000 --> 00:16:38,000
Let's talk about waves here.
You all know that waves are

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some periodic variation of a
quantity.

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00:16:42,000 --> 00:16:45,000
A water wave,
for example,

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00:16:45,000 --> 00:16:49,000
is a periodic variation of the
level of water.

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00:16:49,000 --> 00:16:54,000
At some points in space,
the water level is high.

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00:16:54,000 --> 00:17:00,000
At other points,
the water level is low.

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00:17:00,000 --> 00:17:03,000
Sound wave.
Well, a sound wave is the

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00:17:03,000 --> 00:17:07,000
periodic variation of the
density of air.

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00:17:07,000 --> 00:17:11,000
At some points in space,
the air is very dense.

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00:17:11,000 --> 00:17:16,000
At other points in space,
the air is not dense.

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00:17:16,000 --> 00:17:22,000
Well, light or radiation is a
period variation of an electric

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00:17:22,000 --> 00:17:25,000
field, as I depict here on this
slide.

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00:17:25,000 --> 00:17:32,000
Electric field versus position.
There is a periodic variation

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00:17:32,000 --> 00:17:37,000
of the electric field.
Now, exactly what is an

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00:17:37,000 --> 00:17:40,000
electric field?
Some of you know this,

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00:17:40,000 --> 00:17:44,000
some of you don't,
but an electric field is

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literally the space through
which the Coulomb force

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00:17:49,000 --> 00:17:51,000
operates.
For example,

237
00:17:51,000 --> 00:17:57,000
if we have a negatively charged
plate and a positively charged

238
00:17:57,000 --> 00:18:02,000
plate here.
The space through which the

239
00:18:02,000 --> 00:18:06,000
Coulomb force is operating here,
and the Coulomb force is

240
00:18:06,000 --> 00:18:11,000
operating because we have two
plates here that are oppositely

241
00:18:11,000 --> 00:18:14,000
charged.
The electric field is the space

242
00:18:14,000 --> 00:18:17,000
through which the Coulomb force
operates.

243
00:18:17,000 --> 00:18:22,000
If we put a positive charge in
that space, you know what is

244
00:18:22,000 --> 00:18:25,000
going to happen.
In this coordinated system,

245
00:18:25,000 --> 00:18:30,000
the positive charge is going to
float up.

246
00:18:30,000 --> 00:18:34,000
Because the negatively charged
plate is up above.

247
00:18:34,000 --> 00:18:40,000
If we reversed the potential
difference and put a positively

248
00:18:40,000 --> 00:18:44,000
charged particle in this
electric field,

249
00:18:44,000 --> 00:18:48,000
in this space,
it is going to move down

250
00:18:48,000 --> 00:18:53,000
because now the negatively
charged plate is lower.

251
00:18:53,000 --> 00:18:59,000
This electric field here has
not only magnitude --

252
00:18:59,000 --> 00:19:02,000
You can imagine here the
magnitude is given by the

253
00:19:02,000 --> 00:19:05,000
difference in the potentials of
these plates.

254
00:19:05,000 --> 00:19:09,000
The larger the difference,
the larger the magnitude.

255
00:19:09,000 --> 00:19:12,000
But it also has direction.
In one case,

256
00:19:12,000 --> 00:19:15,000
it is pointed this way.
In the other case,

257
00:19:15,000 --> 00:19:19,000
it is pointed that way.
And that is reflected here on

258
00:19:19,000 --> 00:19:22,000
this plot of the electric field
here.

259
00:19:22,000 --> 00:19:26,000
What you see is that right
here, the magnitude of the

260
00:19:26,000 --> 00:19:31,000
electric field is small.
As you move along in x,

261
00:19:31,000 --> 00:19:34,000
that magnitude increases,
goes to a maximum,

262
00:19:34,000 --> 00:19:39,000
then turns around and at some
point literally is zero.

263
00:19:39,000 --> 00:19:44,000
And then the electric field
changes direction and its

264
00:19:44,000 --> 00:19:48,000
magnitude increases in the
opposite direction.

265
00:19:48,000 --> 00:19:51,000
Increases, increases,
gets to a point,

266
00:19:51,000 --> 00:19:55,000
then turns around and becomes
zero again.

267
00:19:55,000 --> 00:20:00,000
If you have a charge in a
radiation field and you put it

268
00:20:00,000 --> 00:20:05,000
right here --
Well, it would be pulled in one

269
00:20:05,000 --> 00:20:08,000
direction.
If you put it over here,

270
00:20:08,000 --> 00:20:12,000
it would be pulled in the other
direction.

271
00:20:12,000 --> 00:20:16,000
We have a magnitude and we have
a direction.

272
00:20:16,000 --> 00:20:21,000
Now, not only is light a
periodic variation of the

273
00:20:21,000 --> 00:20:26,000
electric field in space,
it is also a periodic variation

274
00:20:26,000 --> 00:20:32,000
of the electric field in time.
That is, this is a picture of

275
00:20:32,000 --> 00:20:35,000
that field, that one instant in
time.

276
00:20:35,000 --> 00:20:39,000
We will call it t equals 0.

277
00:20:39,000 --> 00:20:42,000
However, that electric field
moves.

278
00:20:42,000 --> 00:20:45,000
It propagates.
And the distance,

279
00:20:45,000 --> 00:20:50,000
or the time it takes for the
electric field here to move over

280
00:20:50,000 --> 00:20:54,000
one wavelength,
I have shown this as a star.

281
00:20:54,000 --> 00:20:59,000
The time it takes for this
maximum to go from here to here,

282
00:20:59,000 --> 00:21:05,000
one wavelength,
is defined as one period.

283
00:21:05,000 --> 00:21:09,000
And a period is given by one
over nu,

284
00:21:09,000 --> 00:21:12,000
where nu is the frequency of
the radiation.

285
00:21:12,000 --> 00:21:16,000
It is the number of cycles per
second.

286
00:21:16,000 --> 00:21:20,000
In other words,
if you were sitting here at x

287
00:21:20,000 --> 00:21:24,000
equals 0,
you were tied at x equals 0,

288
00:21:24,000 --> 00:21:30,000
and you were just watching this
electric field come by.

289
00:21:30,000 --> 00:21:34,000
You would see a maximum in that
electric field,

290
00:21:34,000 --> 00:21:40,000
one maximum every second if the
frequency is one Hertz.

291
00:21:40,000 --> 00:21:44,000
In other words,
the frequency is the number of

292
00:21:44,000 --> 00:21:49,000
maxima you would see pass by you
per second.

293
00:21:49,000 --> 00:21:54,000
Well, we have a unit to
characterize frequency.

294
00:21:54,000 --> 00:22:00,000
I call it cycles per second.
It is cycles per second,

295
00:22:00,000 --> 00:22:06,000
but the formal unit is Hertz.
Hertz is inverse seconds.

296
00:22:06,000 --> 00:22:10,000
We leave out the number of
cycles.

297
00:22:10,000 --> 00:22:16,000
The number of cycles is implied
in the unit of Hertz.

298
00:22:16,000 --> 00:22:22,000
To give another example,
here, suppose we had some

299
00:22:22,000 --> 00:22:30,000
radiation and the frequency of
that radiation was one Hertz.

300
00:22:30,000 --> 00:22:34,000
Suppose we had an electron,
an electron is charged,

301
00:22:34,000 --> 00:22:40,000
and we put here at x equals 0
and we tie it at x equals 0.

302
00:22:40,000 --> 00:22:44,000
What is going to happen to this
electron?

303
00:22:44,000 --> 00:22:50,000
Well, what is going to happen
is that this electron is going

304
00:22:50,000 --> 00:22:56,000
to be pulled down and then it is
going to be pushed back up once

305
00:22:56,000 --> 00:23:00,000
every second because the
frequency, here,

306
00:23:00,000 --> 00:23:04,000
is one Hertz.
It is charged,

307
00:23:04,000 --> 00:00:00,000
and we are tying it at x equals

308
00:00:00,000 --> 00:23:07,000
That is our proof that the

309
00:23:07,000 --> 00:23:13,000
It is going to go like this.
It is going to oscillate once

310
00:23:13,000 --> 00:23:17,000
every second if this frequency
is one hertz.

311
00:23:17,000 --> 00:23:22,000
Here it goes.
An electron is pulled down and

312
00:23:22,000 --> 00:23:26,000
then pushed back up once every
second.

313
00:23:26,000 --> 00:23:30,000
Now, we, of course,
can write an equation to

314
00:23:30,000 --> 00:23:36,000
describe this oscillation of the
electric field in both space and

315
00:23:36,000 --> 00:23:41,000
time.
x is the position variable,

316
00:23:41,000 --> 00:23:45,000
t, the time variable,
and I have written it down

317
00:23:45,000 --> 00:23:49,000
here.
I will explain this more in

318
00:23:49,000 --> 00:23:53,000
just a moment,
but what I also want to point

319
00:23:53,000 --> 00:23:57,000
out is that an oscillating
electric field always,

320
00:23:57,000 --> 00:24:01,000
always, always has
perpendicular to it an

321
00:24:01,000 --> 00:24:06,000
oscillating magnetic field.
That is well described by

322
00:24:06,000 --> 00:24:10,000
Maxwell's equations.
Again, you are going to see

323
00:24:10,000 --> 00:24:13,000
that in 8.02.
And the magnetic field here has

324
00:24:13,000 --> 00:24:18,000
the same essentially function
form and characteristics as this

325
00:24:18,000 --> 00:24:20,000
electric field.
And, because it does,

326
00:24:20,000 --> 00:24:24,000
I am just going to talk about
the electric field.

327
00:24:24,000 --> 00:24:27,000
Here is the expression for the
magnetic field.

328
00:24:27,000 --> 00:24:32,000
I just call it H.
But, again, it is a function of

329
00:24:32,000 --> 00:24:36,000
position and time.
Here is an illustration,

330
00:24:36,000 --> 00:24:40,000
just the variation of the
electric field.

331
00:24:40,000 --> 00:24:45,000
Light, radiation is actually a
variation in space and time of

332
00:24:45,000 --> 00:24:49,000
both the electric and a magnetic
field.

333
00:24:49,000 --> 00:24:53,000
That is why it is
electromagnetic radiation.

334
00:24:53,000 --> 00:24:56,000
Now, let me show you on the
8.02 website.

335
00:24:56,000 --> 00:25:01,000
Let me get that rolling.
There it is.

336
00:25:01,000 --> 00:25:04,000
Now we have to start it.
All right.

337
00:25:04,000 --> 00:25:07,000
One of these is the electric
field.

338
00:25:07,000 --> 00:25:10,000
The other one is the magnetic
field.

339
00:25:10,000 --> 00:25:14,000
This is a simulation that 8.02
has made for you.

340
00:25:14,000 --> 00:25:18,000
You can go and look at it on
the 8.02 website,

341
00:25:18,000 --> 00:25:22,000
but you can see it propagating
here in time,

342
00:25:22,000 --> 00:25:28,000
and you can see its variation
in space of this electromagnetic

343
00:25:28,000 --> 00:25:32,000
field.
Let's look at this functional

344
00:25:32,000 --> 00:25:35,000
form just a little more
carefully, just to make sure

345
00:25:35,000 --> 00:25:39,000
everybody is on the same page.
I think many of you have seen

346
00:25:39,000 --> 00:25:42,000
this before.
What we are going to do,

347
00:25:42,000 --> 00:25:46,000
because we have two variables,
is we are going to hold one

348
00:25:46,000 --> 00:25:50,000
variable constant and plot it as
a function of the other

349
00:25:50,000 --> 00:25:53,000
variable, just to explain the
parameters that go into this

350
00:25:53,000 --> 00:25:57,000
functional form.
At time t equals 0,

351
00:25:57,000 --> 00:26:01,000
if in this equation here I
stick in t equals 0,

352
00:26:01,000 --> 00:26:03,000
I have a form that looks like
this.

353
00:26:03,000 --> 00:26:07,000
It is just the cosine function
in x.

354
00:26:07,000 --> 00:26:11,000
And you can see that the
amplitude goes from positive A

355
00:26:11,000 --> 00:26:14,000
to minus A.
And so what you see is that

356
00:26:14,000 --> 00:26:19,000
this A in front of the cosine,
the physical meaning of it is

357
00:26:19,000 --> 00:26:23,000
just the maximum amplitude.
If you were given a functional

358
00:26:23,000 --> 00:26:27,000
form with a number in front of a
cosine, well,

359
00:26:27,000 --> 00:26:32,000
you could read off the
amplitude immediately.

360
00:26:32,000 --> 00:26:36,000
The other parameter that
characterizes this wave is the

361
00:26:36,000 --> 00:26:40,000
wavelength.
It is the distance between two

362
00:26:40,000 --> 00:26:44,000
successive maxima or two
successive minima.

363
00:26:44,000 --> 00:26:50,000
And you can also see here that
the field is going to be at its

364
00:26:50,000 --> 00:26:55,000
maximum amplitude whenever this
x is an integral multiple of the

365
00:26:55,000 --> 00:26:59,000
wavelength, lambda,
2 lambda, 3 lambda,

366
00:26:59,000 --> 00:27:04,000
or minus lambda,
minus 2 lambda or zero.

367
00:27:04,000 --> 00:27:08,000
If you were given a waveform
and there was a number in front

368
00:27:08,000 --> 00:27:12,000
of the x, you can almost,
by inspection,

369
00:27:12,000 --> 00:27:17,000
tell what the wavelength is.
That number would be equal to 2

370
00:27:17,000 --> 00:27:20,000
pi over lambda.
Now what we are going to do is

371
00:27:20,000 --> 00:27:25,000
hold x constant and set it equal
to zero, and then plot this

372
00:27:25,000 --> 00:27:30,000
functional form as a function of
time.

373
00:27:30,000 --> 00:27:36,000
Again, we have the cosine
function, oscillates from plus A

374
00:27:36,000 --> 00:27:39,000
to minus A.
Now the time between two

375
00:27:39,000 --> 00:27:46,000
successive maxima or minima is
what we spoke earlier of as the

376
00:27:46,000 --> 00:27:50,000
period.
It is the time for one cycle.

377
00:27:50,000 --> 00:27:56,000
In other words,
is it one over the frequency.

378
00:27:56,000 --> 00:28:01,000
And you get the maxima then
whenever the time is an integral

379
00:28:01,000 --> 00:28:06,000
multiple of the period,
whenever time is 1 over nu,

380
00:28:06,000 --> 00:28:09,000
2 over nu, 3 over nu,
or minus 1 over nu,

381
00:28:09,000 --> 00:28:11,000
minus 2 over nu,
or zero.

382
00:28:14,000 --> 00:28:18,000
These are the
characteristics of the

383
00:28:18,000 --> 00:28:22,000
functional form,
amplitudes, wavelengths,

384
00:28:22,000 --> 00:28:26,000
frequencies.
Now, I told you that the period

385
00:28:26,000 --> 00:28:31,000
was given by 1 over nu.

386
00:28:31,000 --> 00:28:36,000
Let's just do a quick proof
that the period is actually 1

387
00:28:36,000 --> 00:28:42,000
over nu, one over frequency.
How are we going to do that?

388
00:28:42,000 --> 00:28:48,000
Well, what I said was the
definition for a period was the

389
00:28:48,000 --> 00:28:53,000
time it takes the wave to move
one wavelength.

390
00:28:53,000 --> 00:28:59,000
If this is the wave at t equals
0, this then coming up here

391
00:28:59,000 --> 00:29:05,000
should be the wave at one period
later.

392
00:29:05,000 --> 00:29:09,000
And so, if we moved over
exactly one cycle,

393
00:29:09,000 --> 00:29:16,000
what this means is that at one
period later the functional form

394
00:29:16,000 --> 00:29:21,000
ought to look exactly like it
did at t equals 0.

395
00:29:21,000 --> 00:29:27,000
If I take my general expression
for the waveform and plug in t

396
00:29:27,000 --> 00:29:33,000
equals 1 over nu,
I get this.

397
00:29:33,000 --> 00:29:36,000
What you can see at first
glance is that it doesn't really

398
00:29:36,000 --> 00:29:39,000
look like this,
or at least not just yet,

399
00:29:39,000 --> 00:29:43,000
but we are going to make it
look like this and we are going

400
00:29:43,000 --> 00:29:46,000
to do so legally.
What are we going do?

401
00:29:46,000 --> 00:29:48,000
This just repeats that
equation.

402
00:29:48,000 --> 00:29:51,000
You can already see we have
some cancellation here.

403
00:29:51,000 --> 00:29:55,000
These two nu's go away,
so I just have cosine ((2 pi x)

404
00:29:55,000 --> 00:30:00,000
over lambda minus 2 pi).

405
00:30:00,000 --> 00:30:03,000
In order to simplify this,
I am going to need a

406
00:30:03,000 --> 00:30:06,000
trigonometric identity,
which you may or may not

407
00:30:06,000 --> 00:30:11,000
remember, cosine (alpha minus
beta) is the cosine alpha times

408
00:30:11,000 --> 00:30:15,000
cosine beta plus the sine of the
alpha sine beta.

409
00:30:19,000 --> 00:30:23,000
I am going to let 2(pi)x be
alpha, and beta will be 2pi.

410
00:30:23,000 --> 00:30:27,000
I am going to plug that in.
Here, we can see some nice

411
00:30:27,000 --> 00:30:31,000
simplification.
This cosine 2pi,

412
00:30:31,000 --> 00:30:35,000
of course, is one.
The sign of

413
00:30:35,000 --> 00:30:40,000
2pi is zero, this term goes
away, and what I have left is A

414
00:30:40,000 --> 00:30:45,000
cosine 2 pi x over lambda at t
equals 1 over nu.

415
00:30:45,000 --> 00:30:48,000
And, indeed,

416
00:30:48,000 --> 00:30:52,000
that is the same functional
field as the field at t equals

417
00:30:55,000 --> 00:31:00,000
period is equal to 1 over nu.

418
00:31:00,000 --> 00:31:03,000
Now, this wave also propagates
in space.

419
00:31:03,000 --> 00:31:06,000
It moves.
It goes from here to here.

420
00:31:06,000 --> 00:31:10,000
And another important
characteristic of

421
00:31:10,000 --> 00:31:15,000
electromagnetic reaction is the
speed with which it propagates.

422
00:31:15,000 --> 00:31:20,000
Let's just quickly calculate
what that speed is.

423
00:31:20,000 --> 00:31:23,000
We have enough information to
do that.

424
00:31:23,000 --> 00:31:27,000
Speed is always distance
traveled divided by time

425
00:31:27,000 --> 00:31:32,000
elapsed.
And we said that at t equals 0,

426
00:31:32,000 --> 00:31:36,000
this is what our waveform
looked like.

427
00:31:36,000 --> 00:31:41,000
We also said that one period
later, this is what our waveform

428
00:31:41,000 --> 00:31:45,000
looked like.
We know at one period that the

429
00:31:45,000 --> 00:31:48,000
waveform moved over one
wavelength.

430
00:31:48,000 --> 00:31:53,000
The speed is the distance
traveled, which is a wavelength,

431
00:31:53,000 --> 00:31:57,000
divided by the time elapsed,
which is 1 over nu,

432
00:31:57,000 --> 00:32:03,000
the period.
Therefore, the speed is lambda

433
00:32:03,000 --> 00:32:07,000
times nu.
That is the speed with which

434
00:32:07,000 --> 00:32:11,000
this wave propagates.
And, of course,

435
00:32:11,000 --> 00:32:17,000
you already know that all
electromagnetic radiation has a

436
00:32:17,000 --> 00:32:22,000
constant speed of about 3x10^8
meters per second,

437
00:32:22,000 --> 00:32:26,000
or we call it c.
And what that is,

438
00:32:26,000 --> 00:32:33,000
is the product of the
wavelength times the frequency.

439
00:32:33,000 --> 00:32:38,000
The electromagnetic spectrum,
of course, is infinitely wide.

440
00:32:38,000 --> 00:32:41,000
And here is the electromagnetic
spectrum.

441
00:32:41,000 --> 00:32:46,000
We won't do this in any kind of
detail, but I just want you to

442
00:32:46,000 --> 00:32:50,000
note here that on the long
wavelength end,

443
00:32:50,000 --> 00:32:52,000
we have what we call radio
waves.

444
00:32:52,000 --> 00:32:57,000
And on the short wavelength,
then, we have our gamma rays

445
00:32:57,000 --> 00:33:02,000
and cosmic rays.
And, in the case of the gamma

446
00:33:02,000 --> 00:33:06,000
and the cosmic rays,
because the wavelength is

447
00:33:06,000 --> 00:33:10,000
small, lambda is small,
that means those waves have a

448
00:33:10,000 --> 00:33:14,000
high frequency.
In the case of the radio waves,

449
00:33:14,000 --> 00:33:19,000
because those wavelengths are
long, that means those waves

450
00:33:19,000 --> 00:33:24,000
have a lower frequency because
the frequency times the

451
00:33:24,000 --> 00:33:27,000
wavelength is a constant.
It is this c.

452
00:33:27,000 --> 00:33:32,000
It is 3x10^8 meters per second.
And, of course,

453
00:33:32,000 --> 00:33:35,000
right in here,
a very small region,

454
00:33:35,000 --> 00:33:39,000
narrow region of the
electromagnetic spectrum,

455
00:33:39,000 --> 00:33:44,000
are the light waves that are
sensitive to our eye.

456
00:33:44,000 --> 00:33:49,000
What you do need to know is
that the red wavelengths are

457
00:33:49,000 --> 00:33:53,000
longer and the blue wavelengths
are shorter.

458
00:33:53,000 --> 00:33:58,000
Again, the important thing is
lambda times nu is always,

459
00:33:58,000 --> 00:34:04,000
for every kind of radiation,
equal to a constant.

460
00:34:04,000 --> 00:34:07,000
And that constant is c. Now,

461
00:34:07,000 --> 00:34:11,000
the other thing that you just
need to know is the relative

462
00:34:11,000 --> 00:34:15,000
ordering here in wavelengths.
You do need to know that

463
00:34:15,000 --> 00:34:19,000
microwaves are longer
wavelengths than gamma rays.

464
00:34:19,000 --> 00:34:22,000
All MIT students should know
that.

465
00:34:22,000 --> 00:34:26,000
And one other thing I might
say, because we are going to

466
00:34:26,000 --> 00:34:31,000
talk about this a little later
in the course.

467
00:34:31,000 --> 00:34:34,000
See these microwaves?
Well, molecules will absorb

468
00:34:34,000 --> 00:34:38,000
microwaves, take it in.
That kind of radiation is going

469
00:34:38,000 --> 00:34:43,000
to set the molecule rotating.
Molecules will absorb infrared

470
00:34:43,000 --> 00:34:47,000
radiation, and that kind of
radiation is going to set the

471
00:34:47,000 --> 00:34:51,000
molecules vibrating.
Molecules will absorb visible

472
00:34:51,000 --> 00:34:55,000
and ultraviolet radiation.
What that is going to do is

473
00:34:55,000 --> 00:35:00,000
promote an electron to an
excited state.

474
00:35:00,000 --> 00:35:04,000
Then sometimes those electrons,
in the excited state,

475
00:35:04,000 --> 00:35:08,000
want to relax back down to the
ground state.

476
00:35:08,000 --> 00:35:11,000
When they do so they give off
radiation.

477
00:35:11,000 --> 00:35:15,000
That is the origin of
fluorescence and sometimes

478
00:35:15,000 --> 00:35:19,000
phosphorescence.
Then sometimes molecules will

479
00:35:19,000 --> 00:35:23,000
also fluoresce if they absorb
X-rays, but with X-rays,

480
00:35:23,000 --> 00:35:30,000
if a molecule absorbs them,
it also kicks out an electron.

481
00:35:30,000 --> 00:35:35,000
And we will be looking at that
in a few days to identify the

482
00:35:35,000 --> 00:35:38,000
energy levels in atoms and
molecules.

483
00:35:38,000 --> 00:35:41,000
This, I think,
you are familiar with.

484
00:35:41,000 --> 00:35:47,000
So far I have just told you
what electromagnetic radiation

485
00:35:47,000 --> 00:35:51,000
is, how we characterize it:
speed, frequency,

486
00:35:51,000 --> 00:35:56,000
wavelength, maximum amplitude.
But what I have not shown you,

487
00:35:56,000 --> 00:36:01,000
yet, is any evidence that
indeed light has wave-like

488
00:36:01,000 --> 00:36:06,000
characteristics.
And to do that we are going to

489
00:36:06,000 --> 00:36:10,000
do the experiment that
essentially was done to

490
00:36:10,000 --> 00:36:14,000
demonstrate the wave-like
behavior of light,

491
00:36:14,000 --> 00:36:17,000
and that is Young's two-slit
experiment.

492
00:36:17,000 --> 00:36:22,000
This is the late 1800s.
What was done was to take a

493
00:36:22,000 --> 00:36:25,000
source of monochromatic
radiation.

494
00:36:25,000 --> 00:36:31,000
We are going to use 6
angstroms, 633 nanometers.

495
00:36:31,000 --> 00:36:36,000
It is a helium neon laser.
And it is going to impinge on

496
00:36:36,000 --> 00:36:41,000
just a thin metal plate.
It does not have to be metal.

497
00:36:41,000 --> 00:36:45,000
It can be anything.
But what we did was poke two

498
00:36:45,000 --> 00:36:51,000
holes in it, made two slits.
And naively you might think,

499
00:36:51,000 --> 00:36:56,000
if you looked at a screen out
here, that this screen will

500
00:36:56,000 --> 00:37:03,000
light up in spots that are
directly opposite those slights.

501
00:37:03,000 --> 00:37:07,000
Because, after all,
light travels in straight

502
00:37:07,000 --> 00:37:10,000
lines.
And so if the slits here are

503
00:37:10,000 --> 00:37:15,000
0.005 meters apart,
you might think that the two

504
00:37:15,000 --> 00:37:21,000
bright spots on the slit will be
about 0.02 inches apart.

505
00:37:21,000 --> 00:37:25,000
Well, of course,
that isn't the case.

506
00:37:25,000 --> 00:37:31,000
What you really see is an array
of bright spots.

507
00:37:31,000 --> 00:37:36,000
And Christine has up there in
the projection booth a helium

508
00:37:36,000 --> 00:37:39,000
neon laser that is shinning
behind two slits.

509
00:37:39,000 --> 00:37:43,000
You've got really beautifully
now, Christine.

510
00:37:43,000 --> 00:37:47,000
That is great.
And what you see is that there

511
00:37:47,000 --> 00:37:52,000
is a whole array of spots.
There aren't just two spots.

512
00:37:52,000 --> 00:37:55,000
There is a bunch of spots here,
bright spot,

513
00:37:55,000 --> 00:38:00,000
dark spot, bright spot,
dark spot.

514
00:38:00,000 --> 00:38:05,000
You've also got another pattern
superimposed on that.

515
00:38:05,000 --> 00:38:11,000
It almost looks like you would
see the single slit diffraction,

516
00:38:11,000 --> 00:38:17,000
too, on top of the double slit,
but we won't get into that.

517
00:38:17,000 --> 00:38:22,000
But this is not just two spots.
Let's see if we can try to

518
00:38:22,000 --> 00:38:28,000
understand how this pattern
arises, what this pattern comes

519
00:38:28,000 --> 00:38:32,000
from.
Well, waves have the property

520
00:38:32,000 --> 00:38:37,000
of superposition.
Superposition means that if I

521
00:38:37,000 --> 00:38:42,000
take a wave and have it in
space, but now I take a second

522
00:38:42,000 --> 00:38:48,000
wave and put it in the same
place in space but make it such

523
00:38:48,000 --> 00:38:53,000
that the maxima of both waves
are in the identical place in

524
00:38:53,000 --> 00:38:58,000
space, what I have is a
situation where the two waves

525
00:38:58,000 --> 00:39:04,000
add that property of addition of
waves --

526
00:39:04,000 --> 00:39:08,000
When they are in the same place
in space, that property is

527
00:39:08,000 --> 00:39:11,000
called superposition.
That is the property of waves.

528
00:39:11,000 --> 00:39:15,000
And in this particular case,
we are going to have what we

529
00:39:15,000 --> 00:39:20,000
call constructive interference.
They are going to add up such

530
00:39:20,000 --> 00:39:24,000
that the amplitude here of the
resulting wave is going to be

531
00:39:24,000 --> 00:39:29,000
twice the amplitude of each of
the individual waves.

532
00:39:29,000 --> 00:39:31,000
This is constructive
interference.

533
00:39:31,000 --> 00:39:34,000
On the other hand,
I can have two waves in the

534
00:39:34,000 --> 00:39:37,000
same place in space,
but they can be positioned so

535
00:39:37,000 --> 00:39:41,000
that the maximum of one wave is
at the same point in space as

536
00:39:41,000 --> 00:39:45,000
the minimum of the other.
And because we have these

537
00:39:45,000 --> 00:39:47,000
positive and negative
amplitudes, well,

538
00:39:47,000 --> 00:39:51,000
then these are going to cancel
when they add up and we are

539
00:39:51,000 --> 00:39:55,000
going to have the null result.
We are going to have no

540
00:39:55,000 --> 00:39:57,000
intensity.
That is called destructive

541
00:39:57,000 --> 00:40:03,000
interference.
Well, in order to understand

542
00:40:03,000 --> 00:40:08,000
how this property of
interference gives rise to these

543
00:40:08,000 --> 00:40:13,000
array of bright spots in the two
slit experiment,

544
00:40:13,000 --> 00:40:19,000
let me actually use water waves
as an example to try to

545
00:40:19,000 --> 00:40:26,000
understand why we get this array
of spots, or this row of bright

546
00:40:26,000 --> 00:40:31,000
spots and dark spots.
Here is the beach.

547
00:40:31,000 --> 00:40:35,000
Here is the water.
This is the top view.

548
00:40:35,000 --> 00:40:39,000
Here is the water.
Here is the sand.

549
00:40:39,000 --> 00:40:43,000
Here is where I wanted to be
all weekend.

550
00:40:43,000 --> 00:40:47,000
And the waves are rolling in to
the shore.

551
00:40:47,000 --> 00:40:51,000
There are the wave fronts.
And then suppose I get

552
00:40:51,000 --> 00:40:56,000
ambitious and,
for whatever perverse reason,

553
00:40:56,000 --> 00:41:02,000
I decide to build a barrier to
prevent these waves from coming

554
00:41:02,000 --> 00:41:07,000
onto the beach.
Except I poke two holes in the

555
00:41:07,000 --> 00:41:11,000
barrier, two little holes.
Well, you know what is going to

556
00:41:11,000 --> 00:41:13,000
happen.
When the wave approaches that

557
00:41:13,000 --> 00:41:16,000
barrier, well,
through that little hole a

558
00:41:16,000 --> 00:41:19,000
little bit of the wave is going
to sneak through.

559
00:41:19,000 --> 00:41:23,000
And because that little hole is
really pretty little,

560
00:41:23,000 --> 00:41:27,000
what is going to happen is that
the wave front is going to

561
00:41:27,000 --> 00:41:32,000
spread out isotropically.
And so that wave front is going

562
00:41:32,000 --> 00:41:36,000
to look like a semicircle
centered on that little hole.

563
00:41:36,000 --> 00:41:40,000
And, of course,
this wave front is going to

564
00:41:40,000 --> 00:41:43,000
keep propagating.
And it propagates out.

565
00:41:43,000 --> 00:41:46,000
And then soon enough,
a wavelength later,

566
00:41:46,000 --> 00:41:50,000
another wave sneaks through and
I have two semi-circles.

567
00:41:50,000 --> 00:41:55,000
And the distance between those
two semi-circles is lambda.

568
00:41:55,000 --> 00:42:00,000
That is the wavelength.
That is the wave crest.

569
00:42:00,000 --> 00:42:02,000
That is the maximum of the
wave.

570
00:42:02,000 --> 00:42:04,000
Keep going.
That propagates out.

571
00:42:04,000 --> 00:42:06,000
Keep going.
That propagates out.

572
00:42:06,000 --> 00:42:11,000
Well, at the same time that the
waves are sneaking out through

573
00:42:11,000 --> 00:42:14,000
that little hole,
waves are sneaking out through

574
00:42:14,000 --> 00:42:18,000
this little hole.
And I will color them green.

575
00:42:18,000 --> 00:42:21,000
That wave propagates out and
keeps propagating.

576
00:42:21,000 --> 00:42:25,000
The other one sneaks through
and keeps propagating.

577
00:42:25,000 --> 00:42:30,000
And now let me clean up the
drawing a little bit.

578
00:42:30,000 --> 00:42:32,000
And I am going to call this
slit one.

579
00:42:32,000 --> 00:42:37,000
The green waves are the waves
that have come through slit one.

580
00:42:37,000 --> 00:42:41,000
The blue waves are the ones
that have come through slit two.

581
00:42:41,000 --> 00:42:45,000
And the distance between any
two successive maxima here,

582
00:42:45,000 --> 00:42:49,000
or any two semi-circles is,
of course, lambda.

583
00:42:49,000 --> 00:42:52,000
And lambda is the same for slit
one and slit two.

584
00:42:52,000 --> 00:42:57,000
Now, I want you to look at this
spot that I just circled right

585
00:42:57,000 --> 00:43:00,000
here.
Right here what do you see?

586
00:43:00,000 --> 00:43:02,000
Interference.
Absolutely.

587
00:43:02,000 --> 00:43:05,000
You have two maxima at the same
place in space.

588
00:43:05,000 --> 00:43:09,000
You are going to have
constructive interference right

589
00:43:09,000 --> 00:43:11,000
there.
What about this spot?

590
00:43:11,000 --> 00:43:14,000
Constructive interference.
What about this spot?

591
00:43:14,000 --> 00:43:16,000
Right.
Everywhere along that line you

592
00:43:16,000 --> 00:43:19,000
are going to have constructive
interference.

593
00:43:19,000 --> 00:43:22,000
Now, let me just tell you one
other thing.

594
00:43:22,000 --> 00:43:26,000
We have every constructive
interference all along this

595
00:43:26,000 --> 00:43:30,000
line.
Now look right at this point

596
00:43:30,000 --> 00:43:33,000
here.
What you see is you have the

597
00:43:33,000 --> 00:43:38,000
superposition of the blue wave
that has come from slit two,

598
00:43:38,000 --> 00:43:43,000
and this blue wave has traveled
out from slit two a distance

599
00:43:43,000 --> 00:43:45,000
four lambda.
One, two, three,

600
00:43:45,000 --> 00:43:48,000
four lambda.
That is the radius.

601
00:43:48,000 --> 00:43:52,000
It has traveled out a distance
four lambda.

602
00:43:52,000 --> 00:43:56,000
It is constructively
interfering with a wave coming

603
00:43:56,000 --> 00:44:03,000
from slit one that has traveled
out a distance three lambda.

604
00:44:03,000 --> 00:44:06,000
One, two, three.
The difference in the distance

605
00:44:06,000 --> 00:44:11,000
traveled by those two waves that
are constructively interfering

606
00:44:11,000 --> 00:44:15,000
is one lambda.
Let's keep going in order to

607
00:44:15,000 --> 00:44:18,000
understand this diagram.
Let's look at this spot.

608
00:44:18,000 --> 00:44:23,000
Right here, what do you have?
Constructive interference.

609
00:44:23,000 --> 00:44:27,000
Right here you have
constructive interference.

610
00:44:27,000 --> 00:44:31,000
If you kept going you would
see, everywhere along this line,

611
00:44:31,000 --> 00:44:36,000
constructive interference.
Now let's look at the

612
00:44:36,000 --> 00:44:40,000
difference in the distance
traveled by the waves that are

613
00:44:40,000 --> 00:44:43,000
constructively interfering along
that line.

614
00:44:43,000 --> 00:44:46,000
Well, you see the green wave
here?

615
00:44:46,000 --> 00:44:50,000
The wave that is constructively
interfering is one that has

616
00:44:50,000 --> 00:44:53,000
traveled out a distance two
lambda.

617
00:44:53,000 --> 00:44:56,000
That is, r sub one is equal to
two lambda.

618
00:44:56,000 --> 00:45:00,000
It is interfering
with this wave front

619
00:45:00,000 --> 00:45:05,000
that has traveled out a distance
four lambda.

620
00:45:05,000 --> 00:45:09,000
The difference in the distance
traveled by those two waves is

621
00:45:09,000 --> 00:45:11,000
two lambda.
4 lambda minus 2 lambda equals

622
00:45:11,000 --> 00:45:13,000
2 lambda.
I think on your notes,

623
00:45:13,000 --> 00:45:17,000
it is actually this case that I
have written it down.

624
00:45:17,000 --> 00:45:20,000
Here is another point of
constructive interference.

625
00:45:20,000 --> 00:45:24,000
Here is another point of
constructive interference.

626
00:45:24,000 --> 00:45:27,000
Everywhere along this line,
we have constructive

627
00:45:27,000 --> 00:45:31,000
interference.
And, if you analyze this,

628
00:45:31,000 --> 00:45:35,000
the difference in the distance
traveled would be zero.

629
00:45:35,000 --> 00:45:39,000
What you would expect,
if you were to image this,

630
00:45:39,000 --> 00:45:43,000
you'd expect right here very
bright spot, very bright spot,

631
00:45:43,000 --> 00:45:47,000
very bright spot.
This is going to be symmetric

632
00:45:47,000 --> 00:45:51,000
around the center,
so there will be a bright spot

633
00:45:51,000 --> 00:45:54,000
out here, a bright spot out
there.

634
00:45:54,000 --> 00:46:00,000
Let's look at this actually in
real life in a water tank.

635
00:46:00,000 --> 00:46:03,000
There we go,
up here on the side boards.

636
00:46:03,000 --> 00:46:09,000
Here are the waves coming this
way onto some barrier,

637
00:46:09,000 --> 00:46:13,000
and here are the holes.
Here is one hole.

638
00:46:13,000 --> 00:46:17,000
Here is the other hole.
And then these bright

639
00:46:17,000 --> 00:46:21,000
semi-circles are the wave
fronts.

640
00:46:21,000 --> 00:46:27,000
And what I want you to notice,
and you have to kind of look

641
00:46:27,000 --> 00:46:32,000
out here, right there you see a
whole bunch of very bright

642
00:46:32,000 --> 00:46:37,000
spots.
Well, if this were light and we

643
00:46:37,000 --> 00:46:41,000
had a screen then right here we
would see the screen light up.

644
00:46:41,000 --> 00:46:44,000
And then right here you see
kind of nothing.

645
00:46:44,000 --> 00:46:47,000
That nothing is destructive
interference.

646
00:46:47,000 --> 00:46:52,000
That would be a dark spot if,
in fact, this were light and we

647
00:46:52,000 --> 00:46:55,000
were looking at a screen.
Then here is another very

648
00:46:55,000 --> 00:46:58,000
bright spot.
Here is another very bright

649
00:46:58,000 --> 00:47:02,000
spot.
This is on a website from the

650
00:47:02,000 --> 00:47:07,000
University of Colorado,
which, if you are not familiar

651
00:47:07,000 --> 00:47:11,000
with, is actually kind of a very
neat website.

652
00:47:11,000 --> 00:47:15,000
It has some very elementary
topics in it,

653
00:47:15,000 --> 00:47:19,000
but it also has some topics
that even you would be

654
00:47:19,000 --> 00:47:23,000
interested in.
And that is actually the name

655
00:47:23,000 --> 00:47:27,000
of the website.
And so what is going on here,

656
00:47:27,000 --> 00:47:34,000
in the case of the light,
is just what we have explained.

657
00:47:34,000 --> 00:47:37,000
We've got this line of
constructive interference that

658
00:47:37,000 --> 00:47:41,000
is going to result on the screen
as a very bright spot.

659
00:47:41,000 --> 00:47:45,000
And then another line with
another bright spot and another

660
00:47:45,000 --> 00:47:49,000
line with a very bright spot.
And this is symmetric around

661
00:47:49,000 --> 00:47:52,000
the zero.
Right at this point we have

662
00:47:52,000 --> 00:47:56,000
constructive interference.
In between we have destructive

663
00:47:56,000 --> 00:47:58,000
interference.
Constructive,

664
00:47:58,000 --> 00:48:00,000
destructive,
constructive.

665
00:48:00,000 --> 00:48:05,000
And that is the origin of the
many different bright spots.

666
00:48:05,000 --> 00:48:09,000
And now there is a condition
that has to obtain in order for

667
00:48:09,000 --> 00:48:12,000
there to be maximum constructive
interference,

668
00:48:12,000 --> 00:48:17,000
and that is this condition.
The difference in the distance

669
00:48:17,000 --> 00:48:21,000
traveled of the two waves that
are interfering to give us that

670
00:48:21,000 --> 00:48:25,000
maximum constructive
interference has to be an

671
00:48:25,000 --> 00:48:29,000
integral multiple of the
wavelength.

672
00:48:29,000 --> 00:48:33,000
I will explain this a little
bit more starting on Wednesday.

673
00:48:33,000 --> 00:48:36,000
Okay.
See you then.