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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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1911. --> 00:12:13


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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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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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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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That is where we are going.
Let's start off by talking

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about radiation or light.
We are going to talk about its

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wave-like properties,
then Wednesday we are going to

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talk about the particle-like
properties of light,

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and Friday we are going to talk
about the wave-like properties

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of matter.
That is where we are going.

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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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A water wave,
for example,

223
00:16:45 --> 00:16:49
is a periodic variation of the
level of water.

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At some points in space,
the water level is high.

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

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

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periodic variation of the
density of air.

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At some points in space,
the air is very dense.

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

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Well, light or radiation is a
period variation of an electric

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

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Electric field versus position.
There is a periodic variation

233
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of the electric field.
Now, exactly what is an

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electric field?
Some of you know this,

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some of you don't,
but an electric field is

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literally the space through
which the Coulomb force

237
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operates.
For example,

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if we have a negatively charged
plate and a positively charged

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

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

241
00:18:06 --> 00:18:11
operating because we have two
plates here that are oppositely

242
00:18:11 --> 00:18:14
charged.
The electric field is the space

243
00:18:14 --> 00:18:17
through which the Coulomb force
operates.

244
00:18:17 --> 00:18:22
If we put a positive charge in
that space, you know what is

245
00:18:22 --> 00:18:25
going to happen.
In this coordinated system,

246
00:18:25 --> 00:18:30
the positive charge is going to
float up.

247
00:18:30 --> 00:18:34
Because the negatively charged
plate is up above.

248
00:18:34 --> 00:18:40
If we reversed the potential
difference and put a positively

249
00:18:40 --> 00:18:44
charged particle in this
electric field,

250
00:18:44 --> 00:18:48
in this space,
it is going to move down

251
00:18:48 --> 00:18:53
because now the negatively
charged plate is lower.

252
00:18:53 --> 00:18:59
This electric field here has
not only magnitude --

253
00:18:59 --> 00:19:02
You can imagine here the
magnitude is given by the

254
00:19:02 --> 00:19:05
difference in the potentials of
these plates.

255
00:19:05 --> 00:19:09
The larger the difference,
the larger the magnitude.

256
00:19:09 --> 00:19:12
But it also has direction.
In one case,

257
00:19:12 --> 00:19:15
it is pointed this way.
In the other case,

258
00:19:15 --> 00:19:19
it is pointed that way.
And that is reflected here on

259
00:19:19 --> 00:19:22
this plot of the electric field
here.

260
00:19:22 --> 00:19:26
What you see is that right
here, the magnitude of the

261
00:19:26 --> 00:19:31
electric field is small.
As you move along in x,

262
00:19:31 --> 00:19:34
that magnitude increases,
goes to a maximum,

263
00:19:34 --> 00:19:39
then turns around and at some
point literally is zero.

264
00:19:39 --> 00:19:44
And then the electric field
changes direction and its

265
00:19:44 --> 00:19:48
magnitude increases in the
opposite direction.

266
00:19:48 --> 00:19:51
Increases, increases,
gets to a point,

267
00:19:51 --> 00:19:55
then turns around and becomes
zero again.

268
00:19:55 --> 00:20:00
If you have a charge in a
radiation field and you put it

269
00:20:00 --> 00:20:05
right here --
Well, it would be pulled in one

270
00:20:05 --> 00:20:08
direction.
If you put it over here,

271
00:20:08 --> 00:20:12
it would be pulled in the other
direction.

272
00:20:12 --> 00:20:16
We have a magnitude and we have
a direction.

273
00:20:16 --> 00:20:21
Now, not only is light a
periodic variation of the

274
00:20:21 --> 00:20:26
electric field in space,
it is also a periodic variation

275
00:20:26 --> 00:20:32
of the electric field in time.
That is, this is a picture of

276
00:20:32 --> 00:20:35
that field, that one instant in
time.

277
00:20:35 --> 00:20:39
We will call it t equals 0.

278
00:20:39 --> 00:20:42
However, that electric field
moves.

279
00:20:42 --> 00:20:45
It propagates.
And the distance,

280
00:20:45 --> 00:20:50
or the time it takes for the
electric field here to move over

281
00:20:50 --> 00:20:54
one wavelength,
I have shown this as a star.

282
00:20:54 --> 00:20:59
The time it takes for this
maximum to go from here to here,

283
00:20:59 --> 00:21:05
one wavelength,
is defined as one period.

284
00:21:05 --> 00:21:09
And a period is given by one
over nu,

285
00:21:09 --> 00:21:12
where nu is the frequency of
the radiation.

286
00:21:12 --> 00:21:16
It is the number of cycles per
second.

287
00:21:16 --> 00:21:20
In other words,
if you were sitting here at x

288
00:21:20 --> 00:21:24
equals 0,
you were tied at x equals 0,

289
00:21:24 --> 00:21:30
and you were just watching this
electric field come by.

290
00:21:30 --> 00:21:34
You would see a maximum in that
electric field,

291
00:21:34 --> 00:21:40
one maximum every second if the
frequency is one Hertz.

292
00:21:40 --> 00:21:44
In other words,
the frequency is the number of

293
00:21:44 --> 00:21:49
maxima you would see pass by you
per second.

294
00:21:49 --> 00:21:54
Well, we have a unit to
characterize frequency.

295
00:21:54 --> 00:22:00
I call it cycles per second.
It is cycles per second,

296
00:22:00 --> 00:22:06
but the formal unit is Hertz.
Hertz is inverse seconds.

297
00:22:06 --> 00:22:10
We leave out the number of
cycles.

298
00:22:10 --> 00:22:16
The number of cycles is implied
in the unit of Hertz.

299
00:22:16 --> 00:22:22
To give another example,
here, suppose we had some

300
00:22:22 --> 00:22:30
radiation and the frequency of
that radiation was one Hertz.

301
00:22:30 --> 00:22:34
Suppose we had an electron,
an electron is charged,

302
00:22:34 --> 00:22:40
and we put here at x equals 0
and we tie it at x equals 0.

303
00:22:40 --> 00:22:44
What is going to happen to this
electron?

304
00:22:44 --> 00:22:50
Well, what is going to happen
is that this electron is going

305
00:22:50 --> 00:22:56
to be pulled down and then it is
going to be pushed back up once

306
00:22:56 --> 00:23:00
every second because the
frequency, here,

307
00:23:00 --> 00:23:04
is one Hertz.
It is charged,

308
00:23:04 --> 00.
and we are tying it at x equals

309
0. --> 00:23:07


310
00:23:07 --> 00:23:13
It is going to go like this.
It is going to oscillate once

311
00:23:13 --> 00:23:17
every second if this frequency
is one hertz.

312
00:23:17 --> 00:23:22
Here it goes.
An electron is pulled down and

313
00:23:22 --> 00:23:26
then pushed back up once every
second.

314
00:23:26 --> 00:23:30
Now, we, of course,
can write an equation to

315
00:23:30 --> 00:23:36
describe this oscillation of the
electric field in both space and

316
00:23:36 --> 00:23:41
time.
x is the position variable,

317
00:23:41 --> 00:23:45
t, the time variable,
and I have written it down

318
00:23:45 --> 00:23:49
here.
I will explain this more in

319
00:23:49 --> 00:23:53
just a moment,
but what I also want to point

320
00:23:53 --> 00:23:57
out is that an oscillating
electric field always,

321
00:23:57 --> 00:24:01
always, always has
perpendicular to it an

322
00:24:01 --> 00:24:06
oscillating magnetic field.
That is well described by

323
00:24:06 --> 00:24:10
Maxwell's equations.
Again, you are going to see

324
00:24:10 --> 00:24:13
that in 8.02.
And the magnetic field here has

325
00:24:13 --> 00:24:18
the same essentially function
form and characteristics as this

326
00:24:18 --> 00:24:20
electric field.
And, because it does,

327
00:24:20 --> 00:24:24
I am just going to talk about
the electric field.

328
00:24:24 --> 00:24:27
Here is the expression for the
magnetic field.

329
00:24:27 --> 00:24:32
I just call it H.
But, again, it is a function of

330
00:24:32 --> 00:24:36
position and time.
Here is an illustration,

331
00:24:36 --> 00:24:40
just the variation of the
electric field.

332
00:24:40 --> 00:24:45
Light, radiation is actually a
variation in space and time of

333
00:24:45 --> 00:24:49
both the electric and a magnetic
field.

334
00:24:49 --> 00:24:53
That is why it is
electromagnetic radiation.

335
00:24:53 --> 00:24:56
Now, let me show you on the
8.02 website.

336
00:24:56 --> 00:25:01
Let me get that rolling.
There it is.

337
00:25:01 --> 00:25:04
Now we have to start it.
All right.

338
00:25:04 --> 00:25:07
One of these is the electric
field.

339
00:25:07 --> 00:25:10
The other one is the magnetic
field.

340
00:25:10 --> 00:25:14
This is a simulation that 8.02
has made for you.

341
00:25:14 --> 00:25:18
You can go and look at it on
the 8.02 website,

342
00:25:18 --> 00:25:22
but you can see it propagating
here in time,

343
00:25:22 --> 00:25:28
and you can see its variation
in space of this electromagnetic

344
00:25:28 --> 00:25:32
field.
Let's look at this functional

345
00:25:32 --> 00:25:35
form just a little more
carefully, just to make sure

346
00:25:35 --> 00:25:39
everybody is on the same page.
I think many of you have seen

347
00:25:39 --> 00:25:42
this before.
What we are going to do,

348
00:25:42 --> 00:25:46
because we have two variables,
is we are going to hold one

349
00:25:46 --> 00:25:50
variable constant and plot it as
a function of the other

350
00:25:50 --> 00:25:53
variable, just to explain the
parameters that go into this

351
00:25:53 --> 00:25:57
functional form.
At time t equals 0,

352
00:25:57 --> 00:26:01
if in this equation here I
stick in t equals 0,

353
00:26:01 --> 00:26:03
I have a form that looks like
this.

354
00:26:03 --> 00:26:07
It is just the cosine function
in x.

355
00:26:07 --> 00:26:11
And you can see that the
amplitude goes from positive A

356
00:26:11 --> 00:26:14
to minus A.
And so what you see is that

357
00:26:14 --> 00:26:19
this A in front of the cosine,
the physical meaning of it is

358
00:26:19 --> 00:26:23
just the maximum amplitude.
If you were given a functional

359
00:26:23 --> 00:26:27
form with a number in front of a
cosine, well,

360
00:26:27 --> 00:26:32
you could read off the
amplitude immediately.

361
00:26:32 --> 00:26:36
The other parameter that
characterizes this wave is the

362
00:26:36 --> 00:26:40
wavelength.
It is the distance between two

363
00:26:40 --> 00:26:44
successive maxima or two
successive minima.

364
00:26:44 --> 00:26:50
And you can also see here that
the field is going to be at its

365
00:26:50 --> 00:26:55
maximum amplitude whenever this
x is an integral multiple of the

366
00:26:55 --> 00:26:59
wavelength, lambda,
2 lambda, 3 lambda,

367
00:26:59 --> 00:27:04
or minus lambda,
minus 2 lambda or zero.

368
00:27:04 --> 00:27:08
If you were given a waveform
and there was a number in front

369
00:27:08 --> 00:27:12
of the x, you can almost,
by inspection,

370
00:27:12 --> 00:27:17
tell what the wavelength is.
That number would be equal to 2

371
00:27:17 --> 00:27:20
pi over lambda.
Now what we are going to do is

372
00:27:20 --> 00:27:25
hold x constant and set it equal
to zero, and then plot this

373
00:27:25 --> 00:27:30
functional form as a function of
time.

374
00:27:30 --> 00:27:36
Again, we have the cosine
function, oscillates from plus A

375
00:27:36 --> 00:27:39
to minus A.
Now the time between two

376
00:27:39 --> 00:27:46
successive maxima or minima is
what we spoke earlier of as the

377
00:27:46 --> 00:27:50
period.
It is the time for one cycle.

378
00:27:50 --> 00:27:56
In other words,
is it one over the frequency.

379
00:27:56 --> 00:28:01
And you get the maxima then
whenever the time is an integral

380
00:28:01 --> 00:28:06
multiple of the period,
whenever time is 1 over nu,

381
00:28:06 --> 00:28:09
2 over nu, 3 over nu,
or minus 1 over nu,

382
00:28:09 --> 00:28:11
minus 2 over nu,
or zero.

383
00:28:11 --> 00:28:14


384
00:28:14 --> 00:28:18
These are the
characteristics of the

385
00:28:18 --> 00:28:22
functional form,
amplitudes, wavelengths,

386
00:28:22 --> 00:28:26
frequencies.
Now, I told you that the period

387
00:28:26 --> 00:28:31
was given by 1 over nu.

388
00:28:31 --> 00:28:36
Let's just do a quick proof
that the period is actually 1

389
00:28:36 --> 00:28:42
over nu, one over frequency.
How are we going to do that?

390
00:28:42 --> 00:28:48
Well, what I said was the
definition for a period was the

391
00:28:48 --> 00:28:53
time it takes the wave to move
one wavelength.

392
00:28:53 --> 00:28:59
If this is the wave at t equals
0, this then coming up here

393
00:28:59 --> 00:29:05
should be the wave at one period
later.

394
00:29:05 --> 00:29:09
And so, if we moved over
exactly one cycle,

395
00:29:09 --> 00:29:16
what this means is that at one
period later the functional form

396
00:29:16 --> 00:29:21
ought to look exactly like it
did at t equals 0.

397
00:29:21 --> 00:29:27
If I take my general expression
for the waveform and plug in t

398
00:29:27 --> 00:29:33
equals 1 over nu,
I get this.

399
00:29:33 --> 00:29:36
What you can see at first
glance is that it doesn't really

400
00:29:36 --> 00:29:39
look like this,
or at least not just yet,

401
00:29:39 --> 00:29:43
but we are going to make it
look like this and we are going

402
00:29:43 --> 00:29:46
to do so legally.
What are we going do?

403
00:29:46 --> 00:29:48
This just repeats that
equation.

404
00:29:48 --> 00:29:51
You can already see we have
some cancellation here.

405
00:29:51 --> 00:29:55
These two nu's go away,
so I just have cosine ((2 pi x)

406
00:29:55 --> 00:30:00
over lambda minus 2 pi).

407
00:30:00 --> 00:30:03
In order to simplify this,
I am going to need a

408
00:30:03 --> 00:30:06
trigonometric identity,
which you may or may not

409
00:30:06 --> 00:30:11
remember, cosine (alpha minus
beta) is the cosine alpha times

410
00:30:11 --> 00:30:15
cosine beta plus the sine of the
alpha sine beta.

411
00:30:15 --> 00:30:19


412
00:30:19 --> 00:30:23
I am going to let 2(pi)x be
alpha, and beta will be 2pi.

413
00:30:23 --> 00:30:27
I am going to plug that in.
Here, we can see some nice

414
00:30:27 --> 00:30:31
simplification.
This cosine 2pi,

415
00:30:31 --> 00:30:35
of course, is one.
The sign of

416
00:30:35 --> 00:30:40
2pi is zero, this term goes
away, and what I have left is A

417
00:30:40 --> 00:30:45
cosine 2 pi x over lambda at t
equals 1 over nu.

418
00:30:45 --> 00:30:48
And, indeed,

419
00:30:48 --> 00:30:52
that is the same functional
field as the field at t equals

420
00:30:52 --> 00.


421
0. --> 00:30:55
That is our proof that the

422
00:30:55 --> 00:31:00
period is equal to 1 over nu.

423
00:31:00 --> 00:31:03
Now, this wave also propagates
in space.

424
00:31:03 --> 00:31:06
It moves.
It goes from here to here.

425
00:31:06 --> 00:31:10
And another important
characteristic of

426
00:31:10 --> 00:31:15
electromagnetic reaction is the
speed with which it propagates.

427
00:31:15 --> 00:31:20
Let's just quickly calculate
what that speed is.

428
00:31:20 --> 00:31:23
We have enough information to
do that.

429
00:31:23 --> 00:31:27
Speed is always distance
traveled divided by time

430
00:31:27 --> 00:31:32
elapsed.
And we said that at t equals 0,

431
00:31:32 --> 00:31:36
this is what our waveform
looked like.

432
00:31:36 --> 00:31:41
We also said that one period
later, this is what our waveform

433
00:31:41 --> 00:31:45
looked like.
We know at one period that the

434
00:31:45 --> 00:31:48
waveform moved over one
wavelength.

435
00:31:48 --> 00:31:53
The speed is the distance
traveled, which is a wavelength,

436
00:31:53 --> 00:31:57
divided by the time elapsed,
which is 1 over nu,

437
00:31:57 --> 00:32:03
the period.
Therefore, the speed is lambda

438
00:32:03 --> 00:32:07
times nu.
That is the speed with which

439
00:32:07 --> 00:32:11
this wave propagates.
And, of course,

440
00:32:11 --> 00:32:17
you already know that all
electromagnetic radiation has a

441
00:32:17 --> 00:32:22
constant speed of about 3x10^8
meters per second,

442
00:32:22 --> 00:32:26
or we call it c.
And what that is,

443
00:32:26 --> 00:32:33
is the product of the
wavelength times the frequency.

444
00:32:33 --> 00:32:38
The electromagnetic spectrum,
of course, is infinitely wide.

445
00:32:38 --> 00:32:41
And here is the electromagnetic
spectrum.

446
00:32:41 --> 00:32:46
We won't do this in any kind of
detail, but I just want you to

447
00:32:46 --> 00:32:50
note here that on the long
wavelength end,

448
00:32:50 --> 00:32:52
we have what we call radio
waves.

449
00:32:52 --> 00:32:57
And on the short wavelength,
then, we have our gamma rays

450
00:32:57 --> 00:33:02
and cosmic rays.
And, in the case of the gamma

451
00:33:02 --> 00:33:06
and the cosmic rays,
because the wavelength is

452
00:33:06 --> 00:33:10
small, lambda is small,
that means those waves have a

453
00:33:10 --> 00:33:14
high frequency.
In the case of the radio waves,

454
00:33:14 --> 00:33:19
because those wavelengths are
long, that means those waves

455
00:33:19 --> 00:33:24
have a lower frequency because
the frequency times the

456
00:33:24 --> 00:33:27
wavelength is a constant.
It is this c.

457
00:33:27 --> 00:33:32
It is 3x10^8 meters per second.
And, of course,

458
00:33:32 --> 00:33:35
right in here,
a very small region,

459
00:33:35 --> 00:33:39
narrow region of the
electromagnetic spectrum,

460
00:33:39 --> 00:33:44
are the light waves that are
sensitive to our eye.

461
00:33:44 --> 00:33:49
What you do need to know is
that the red wavelengths are

462
00:33:49 --> 00:33:53
longer and the blue wavelengths
are shorter.

463
00:33:53 --> 00:33:58
Again, the important thing is
lambda times nu is always,

464
00:33:58 --> 00:34:04
for every kind of radiation,
equal to a constant.

465
00:34:04 --> 00:34:07
And that constant is c. Now,

466
00:34:07 --> 00:34:11
the other thing that you just
need to know is the relative

467
00:34:11 --> 00:34:15
ordering here in wavelengths.
You do need to know that

468
00:34:15 --> 00:34:19
microwaves are longer
wavelengths than gamma rays.

469
00:34:19 --> 00:34:22
All MIT students should know
that.

470
00:34:22 --> 00:34:26
And one other thing I might
say, because we are going to

471
00:34:26 --> 00:34:31
talk about this a little later
in the course.

472
00:34:31 --> 00:34:34
See these microwaves?
Well, molecules will absorb

473
00:34:34 --> 00:34:38
microwaves, take it in.
That kind of radiation is going

474
00:34:38 --> 00:34:43
to set the molecule rotating.
Molecules will absorb infrared

475
00:34:43 --> 00:34:47
radiation, and that kind of
radiation is going to set the

476
00:34:47 --> 00:34:51
molecules vibrating.
Molecules will absorb visible

477
00:34:51 --> 00:34:55
and ultraviolet radiation.
What that is going to do is

478
00:34:55 --> 00:35:00
promote an electron to an
excited state.

479
00:35:00 --> 00:35:04
Then sometimes those electrons,
in the excited state,

480
00:35:04 --> 00:35:08
want to relax back down to the
ground state.

481
00:35:08 --> 00:35:11
When they do so they give off
radiation.

482
00:35:11 --> 00:35:15
That is the origin of
fluorescence and sometimes

483
00:35:15 --> 00:35:19
phosphorescence.
Then sometimes molecules will

484
00:35:19 --> 00:35:23
also fluoresce if they absorb
X-rays, but with X-rays,

485
00:35:23 --> 00:35:30
if a molecule absorbs them,
it also kicks out an electron.

486
00:35:30 --> 00:35:35
And we will be looking at that
in a few days to identify the

487
00:35:35 --> 00:35:38
energy levels in atoms and
molecules.

488
00:35:38 --> 00:35:41
This, I think,
you are familiar with.

489
00:35:41 --> 00:35:47
So far I have just told you
what electromagnetic radiation

490
00:35:47 --> 00:35:51
is, how we characterize it:
speed, frequency,

491
00:35:51 --> 00:35:56
wavelength, maximum amplitude.
But what I have not shown you,

492
00:35:56 --> 00:36:01
yet, is any evidence that
indeed light has wave-like

493
00:36:01 --> 00:36:06
characteristics.
And to do that we are going to

494
00:36:06 --> 00:36:10
do the experiment that
essentially was done to

495
00:36:10 --> 00:36:14
demonstrate the wave-like
behavior of light,

496
00:36:14 --> 00:36:17
and that is Young's two-slit
experiment.

497
00:36:17 --> 00:36:22
This is the late 1800s.
What was done was to take a

498
00:36:22 --> 00:36:25
source of monochromatic
radiation.

499
00:36:25 --> 00:36:31
We are going to use 6
angstroms, 633 nanometers.

500
00:36:31 --> 00:36:36
It is a helium neon laser.
And it is going to impinge on

501
00:36:36 --> 00:36:41
just a thin metal plate.
It does not have to be metal.

502
00:36:41 --> 00:36:45
It can be anything.
But what we did was poke two

503
00:36:45 --> 00:36:51
holes in it, made two slits.
And naively you might think,

504
00:36:51 --> 00:36:56
if you looked at a screen out
here, that this screen will

505
00:36:56 --> 00:37:03
light up in spots that are
directly opposite those slights.

506
00:37:03 --> 00:37:07
Because, after all,
light travels in straight

507
00:37:07 --> 00:37:10
lines.
And so if the slits here are

508
00:37:10 --> 00:37:15
0.005 meters apart,
you might think that the two

509
00:37:15 --> 00:37:21
bright spots on the slit will be
about 0.02 inches apart.

510
00:37:21 --> 00:37:25
Well, of course,
that isn't the case.

511
00:37:25 --> 00:37:31
What you really see is an array
of bright spots.

512
00:37:31 --> 00:37:36
And Christine has up there in
the projection booth a helium

513
00:37:36 --> 00:37:39
neon laser that is shinning
behind two slits.

514
00:37:39 --> 00:37:43
You've got really beautifully
now, Christine.

515
00:37:43 --> 00:37:47
That is great.
And what you see is that there

516
00:37:47 --> 00:37:52
is a whole array of spots.
There aren't just two spots.

517
00:37:52 --> 00:37:55
There is a bunch of spots here,
bright spot,

518
00:37:55 --> 00:38:00
dark spot, bright spot,
dark spot.

519
00:38:00 --> 00:38:05
You've also got another pattern
superimposed on that.

520
00:38:05 --> 00:38:11
It almost looks like you would
see the single slit diffraction,

521
00:38:11 --> 00:38:17
too, on top of the double slit,
but we won't get into that.

522
00:38:17 --> 00:38:22
But this is not just two spots.
Let's see if we can try to

523
00:38:22 --> 00:38:28
understand how this pattern
arises, what this pattern comes

524
00:38:28 --> 00:38:32
from.
Well, waves have the property

525
00:38:32 --> 00:38:37
of superposition.
Superposition means that if I

526
00:38:37 --> 00:38:42
take a wave and have it in
space, but now I take a second

527
00:38:42 --> 00:38:48
wave and put it in the same
place in space but make it such

528
00:38:48 --> 00:38:53
that the maxima of both waves
are in the identical place in

529
00:38:53 --> 00:38:58
space, what I have is a
situation where the two waves

530
00:38:58 --> 00:39:04
add that property of addition of
waves --

531
00:39:04 --> 00:39:08
When they are in the same place
in space, that property is

532
00:39:08 --> 00:39:11
called superposition.
That is the property of waves.

533
00:39:11 --> 00:39:15
And in this particular case,
we are going to have what we

534
00:39:15 --> 00:39:20
call constructive interference.
They are going to add up such

535
00:39:20 --> 00:39:24
that the amplitude here of the
resulting wave is going to be

536
00:39:24 --> 00:39:29
twice the amplitude of each of
the individual waves.

537
00:39:29 --> 00:39:31
This is constructive
interference.

538
00:39:31 --> 00:39:34
On the other hand,
I can have two waves in the

539
00:39:34 --> 00:39:37
same place in space,
but they can be positioned so

540
00:39:37 --> 00:39:41
that the maximum of one wave is
at the same point in space as

541
00:39:41 --> 00:39:45
the minimum of the other.
And because we have these

542
00:39:45 --> 00:39:47
positive and negative
amplitudes, well,

543
00:39:47 --> 00:39:51
then these are going to cancel
when they add up and we are

544
00:39:51 --> 00:39:55
going to have the null result.
We are going to have no

545
00:39:55 --> 00:39:57
intensity.
That is called destructive

546
00:39:57 --> 00:40:03
interference.
Well, in order to understand

547
00:40:03 --> 00:40:08
how this property of
interference gives rise to these

548
00:40:08 --> 00:40:13
array of bright spots in the two
slit experiment,

549
00:40:13 --> 00:40:19
let me actually use water waves
as an example to try to

550
00:40:19 --> 00:40:26
understand why we get this array
of spots, or this row of bright

551
00:40:26 --> 00:40:31
spots and dark spots.
Here is the beach.

552
00:40:31 --> 00:40:35
Here is the water.
This is the top view.

553
00:40:35 --> 00:40:39
Here is the water.
Here is the sand.

554
00:40:39 --> 00:40:43
Here is where I wanted to be
all weekend.

555
00:40:43 --> 00:40:47
And the waves are rolling in to
the shore.

556
00:40:47 --> 00:40:51
There are the wave fronts.
And then suppose I get

557
00:40:51 --> 00:40:56
ambitious and,
for whatever perverse reason,

558
00:40:56 --> 00:41:02
I decide to build a barrier to
prevent these waves from coming

559
00:41:02 --> 00:41:07
onto the beach.
Except I poke two holes in the

560
00:41:07 --> 00:41:11
barrier, two little holes.
Well, you know what is going to

561
00:41:11 --> 00:41:13
happen.
When the wave approaches that

562
00:41:13 --> 00:41:16
barrier, well,
through that little hole a

563
00:41:16 --> 00:41:19
little bit of the wave is going
to sneak through.

564
00:41:19 --> 00:41:23
And because that little hole is
really pretty little,

565
00:41:23 --> 00:41:27
what is going to happen is that
the wave front is going to

566
00:41:27 --> 00:41:32
spread out isotropically.
And so that wave front is going

567
00:41:32 --> 00:41:36
to look like a semicircle
centered on that little hole.

568
00:41:36 --> 00:41:40
And, of course,
this wave front is going to

569
00:41:40 --> 00:41:43
keep propagating.
And it propagates out.

570
00:41:43 --> 00:41:46
And then soon enough,
a wavelength later,

571
00:41:46 --> 00:41:50
another wave sneaks through and
I have two semi-circles.

572
00:41:50 --> 00:41:55
And the distance between those
two semi-circles is lambda.

573
00:41:55 --> 00:42:00
That is the wavelength.
That is the wave crest.

574
00:42:00 --> 00:42:02
That is the maximum of the
wave.

575
00:42:02 --> 00:42:04
Keep going.
That propagates out.

576
00:42:04 --> 00:42:06
Keep going.
That propagates out.

577
00:42:06 --> 00:42:11
Well, at the same time that the
waves are sneaking out through

578
00:42:11 --> 00:42:14
that little hole,
waves are sneaking out through

579
00:42:14 --> 00:42:18
this little hole.
And I will color them green.

580
00:42:18 --> 00:42:21
That wave propagates out and
keeps propagating.

581
00:42:21 --> 00:42:25
The other one sneaks through
and keeps propagating.

582
00:42:25 --> 00:42:30
And now let me clean up the
drawing a little bit.

583
00:42:30 --> 00:42:32
And I am going to call this
slit one.

584
00:42:32 --> 00:42:37
The green waves are the waves
that have come through slit one.

585
00:42:37 --> 00:42:41
The blue waves are the ones
that have come through slit two.

586
00:42:41 --> 00:42:45
And the distance between any
two successive maxima here,

587
00:42:45 --> 00:42:49
or any two semi-circles is,
of course, lambda.

588
00:42:49 --> 00:42:52
And lambda is the same for slit
one and slit two.

589
00:42:52 --> 00:42:57
Now, I want you to look at this
spot that I just circled right

590
00:42:57 --> 00:43:00
here.
Right here what do you see?

591
00:43:00 --> 00:43:02
Interference.
Absolutely.

592
00:43:02 --> 00:43:05
You have two maxima at the same
place in space.

593
00:43:05 --> 00:43:09
You are going to have
constructive interference right

594
00:43:09 --> 00:43:11
there.
What about this spot?

595
00:43:11 --> 00:43:14
Constructive interference.
What about this spot?

596
00:43:14 --> 00:43:16
Right.
Everywhere along that line you

597
00:43:16 --> 00:43:19
are going to have constructive
interference.

598
00:43:19 --> 00:43:22
Now, let me just tell you one
other thing.

599
00:43:22 --> 00:43:26
We have every constructive
interference all along this

600
00:43:26 --> 00:43:30
line.
Now look right at this point

601
00:43:30 --> 00:43:33
here.
What you see is you have the

602
00:43:33 --> 00:43:38
superposition of the blue wave
that has come from slit two,

603
00:43:38 --> 00:43:43
and this blue wave has traveled
out from slit two a distance

604
00:43:43 --> 00:43:45
four lambda.
One, two, three,

605
00:43:45 --> 00:43:48
four lambda.
That is the radius.

606
00:43:48 --> 00:43:52
It has traveled out a distance
four lambda.

607
00:43:52 --> 00:43:56
It is constructively
interfering with a wave coming

608
00:43:56 --> 00:44:03
from slit one that has traveled
out a distance three lambda.

609
00:44:03 --> 00:44:06
One, two, three.
The difference in the distance

610
00:44:06 --> 00:44:11
traveled by those two waves that
are constructively interfering

611
00:44:11 --> 00:44:15
is one lambda.
Let's keep going in order to

612
00:44:15 --> 00:44:18
understand this diagram.
Let's look at this spot.

613
00:44:18 --> 00:44:23
Right here, what do you have?
Constructive interference.

614
00:44:23 --> 00:44:27
Right here you have
constructive interference.

615
00:44:27 --> 00:44:31
If you kept going you would
see, everywhere along this line,

616
00:44:31 --> 00:44:36
constructive interference.
Now let's look at the

617
00:44:36 --> 00:44:40
difference in the distance
traveled by the waves that are

618
00:44:40 --> 00:44:43
constructively interfering along
that line.

619
00:44:43 --> 00:44:46
Well, you see the green wave
here?

620
00:44:46 --> 00:44:50
The wave that is constructively
interfering is one that has

621
00:44:50 --> 00:44:53
traveled out a distance two
lambda.

622
00:44:53 --> 00:44:56
That is, r sub one is equal to
two lambda.

623
00:44:56 --> 00:45:00
It is interfering
with this wave front

624
00:45:00 --> 00:45:05
that has traveled out a distance
four lambda.

625
00:45:05 --> 00:45:09
The difference in the distance
traveled by those two waves is

626
00:45:09 --> 00:45:11
two lambda.
4 lambda minus 2 lambda equals

627
00:45:11 --> 00:45:13
2 lambda.
I think on your notes,

628
00:45:13 --> 00:45:17
it is actually this case that I
have written it down.

629
00:45:17 --> 00:45:20
Here is another point of
constructive interference.

630
00:45:20 --> 00:45:24
Here is another point of
constructive interference.

631
00:45:24 --> 00:45:27
Everywhere along this line,
we have constructive

632
00:45:27 --> 00:45:31
interference.
And, if you analyze this,

633
00:45:31 --> 00:45:35
the difference in the distance
traveled would be zero.

634
00:45:35 --> 00:45:39
What you would expect,
if you were to image this,

635
00:45:39 --> 00:45:43
you'd expect right here very
bright spot, very bright spot,

636
00:45:43 --> 00:45:47
very bright spot.
This is going to be symmetric

637
00:45:47 --> 00:45:51
around the center,
so there will be a bright spot

638
00:45:51 --> 00:45:54
out here, a bright spot out
there.

639
00:45:54 --> 00:46:00
Let's look at this actually in
real life in a water tank.

640
00:46:00 --> 00:46:03
There we go,
up here on the side boards.

641
00:46:03 --> 00:46:09
Here are the waves coming this
way onto some barrier,

642
00:46:09 --> 00:46:13
and here are the holes.
Here is one hole.

643
00:46:13 --> 00:46:17
Here is the other hole.
And then these bright

644
00:46:17 --> 00:46:21
semi-circles are the wave
fronts.

645
00:46:21 --> 00:46:27
And what I want you to notice,
and you have to kind of look

646
00:46:27 --> 00:46:32
out here, right there you see a
whole bunch of very bright

647
00:46:32 --> 00:46:37
spots.
Well, if this were light and we

648
00:46:37 --> 00:46:41
had a screen then right here we
would see the screen light up.

649
00:46:41 --> 00:46:44
And then right here you see
kind of nothing.

650
00:46:44 --> 00:46:47
That nothing is destructive
interference.

651
00:46:47 --> 00:46:52
That would be a dark spot if,
in fact, this were light and we

652
00:46:52 --> 00:46:55
were looking at a screen.
Then here is another very

653
00:46:55 --> 00:46:58
bright spot.
Here is another very bright

654
00:46:58 --> 00:47:02
spot.
This is on a website from the

655
00:47:02 --> 00:47:07
University of Colorado,
which, if you are not familiar

656
00:47:07 --> 00:47:11
with, is actually kind of a very
neat website.

657
00:47:11 --> 00:47:15
It has some very elementary
topics in it,

658
00:47:15 --> 00:47:19
but it also has some topics
that even you would be

659
00:47:19 --> 00:47:23
interested in.
And that is actually the name

660
00:47:23 --> 00:47:27
of the website.
And so what is going on here,

661
00:47:27 --> 00:47:34
in the case of the light,
is just what we have explained.

662
00:47:34 --> 00:47:37
We've got this line of
constructive interference that

663
00:47:37 --> 00:47:41
is going to result on the screen
as a very bright spot.

664
00:47:41 --> 00:47:45
And then another line with
another bright spot and another

665
00:47:45 --> 00:47:49
line with a very bright spot.
And this is symmetric around

666
00:47:49 --> 00:47:52
the zero.
Right at this point we have

667
00:47:52 --> 00:47:56
constructive interference.
In between we have destructive

668
00:47:56 --> 00:47:58
interference.
Constructive,

669
00:47:58 --> 00:48:00
destructive,
constructive.

670
00:48:00 --> 00:48:05
And that is the origin of the
many different bright spots.

671
00:48:05 --> 00:48:09
And now there is a condition
that has to obtain in order for

672
00:48:09 --> 00:48:12
there to be maximum constructive
interference,

673
00:48:12 --> 00:48:17
and that is this condition.
The difference in the distance

674
00:48:17 --> 00:48:21
traveled of the two waves that
are interfering to give us that

675
00:48:21 --> 00:48:25
maximum constructive
interference has to be an

676
00:48:25 --> 00:48:29
integral multiple of the
wavelength.

677
00:48:29 --> 00:48:33
I will explain this a little
bit more starting on Wednesday.

678
00:48:33 --> 48:36
Okay.
See you then.