WEBVTT

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.

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Prof: I dropped a
bombshell end of last class

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telling you that all the
geometric optics I taught you

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has to be revised,
and before letting you go for

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lunch I said you've got to tell
me why I would do that.

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Student:
Experiments.

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.

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Prof:  Yes,
somebody said it's the

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experiments, and that's what I
want you to remember.

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That's the only way things
change in physics is experiments

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tell you something's wrong with
your theory,

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and in the end if your theory
doesn't agree with the

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experiment it's over.

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And conversely,
if you're a newcomer unknown to

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anybody, make predictions and
they agree with experiment then

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you're a rock star.

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Everything is based on
experiment.

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That's the only way we change
our mind.

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Now you might say,
"Why do you keep doing

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this to us?

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We believe you.

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We write everything down.

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We do the problems and then you
say, 'Oh, these guys are wrong.

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Here's a better theory,' what's
going on?

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Are people really wrong?"

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So I have to be very careful
when I say people are wrong

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because the news leaks to the
press and the media will say,

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"Physicists think they're
always wrong."

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In fact, I've gone on record
here saying, "We're always

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wrong."

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What I mean by that is no
matter how many laws we find one

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day you will find some new
experiments which are not

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explained by these laws.

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That's not really bad news.

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That's what keeps us in
business because we want to find

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something that doesn't fit
anything we know.

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For example,
Newtonian mechanics in some

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sense is wrong because it
doesn't work when velocities

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approach the speed of light,
but it's not wrong in the sense

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that the predictions it made in
its proper domain do not work

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anymore.

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It was supposed to work in a
limited range of experimental

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observations.

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If you push the limit,
if you build accelerators that

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send particles at very high
speeds you may find they don't

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obey Newtonian mechanics.

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Then you get Einstein's special
theory.

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Now if you have a special
theory like Einstein,

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or a new theory that overthrows
the old theory,

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and it explains new phenomena,
there is still one extra

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requirement.

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Can you guess what that might
be with a new theory?

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One thing we demand of every
new theory, yes?

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Student:
That part of it

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coincides with the old theory?

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.

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Prof:  Very good.

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If you go back to the old
experiments, which are explained

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by the old theory,
well your new theory should

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continue to explain that.

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In fact the new theory,
when it's a good one,

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will tell you why for hundreds
of years people fell for the old

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theory and then it'll also do
new stuff.

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If you do new stuff and it
doesn't fit the old stuff that's

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not good.

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So relativity is just like that.

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It works for all velocities up
to the speed of light,

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but if you let v/c go to
zero then you will finally get

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back Newtonian mechanics.

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Similarly, quantum mechanics
works for very,

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very tiny objects,
the atomic scale,

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but when you apply it to big
things you'll find that the

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world begins to look Newtonian.

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Similarly, the end of geometric
optics came from a series of

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experiments,
and I will tell you why we

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didn't realize there was
something wrong with it for a

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while.

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So here's an experiment you
could do.

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Here a screen and there's some
light coming from here,

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and there's a hole,
and behind this screen is

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another screen.

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So I'm not good at drawing,
so you should imagine this is

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coming out of the blackboard,
and the ray of light is coming

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in.

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It'll form a shadow.

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I happen to have in mind a
circular hole here.

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It'll form a disc shape exactly
like that.

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And another thing,
but you will find that as you

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shrink this, make it smaller and
smaller, suddenly that won't be

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true anymore.

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You will find that the light
spreads out like that to bigger

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and bigger areas.

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So it's no longer forming the
geometric image of this

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aperture.

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It's fanning out,
so it's not going to come from

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geometrical optics.

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That's one experiment where
something goes wrong.

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There's a more dramatic
experiment which really puts the

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nail on all kinds of ray
theories, and that's the

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experiment involving
interference.

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So you take a partition with
two holes in it,

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and you send light from here,
and you put a screen behind.

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Are you guys with me?

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This is a cross section of a
plate coming out of the

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blackboard.

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It's got only two holes in it.

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Or here's a better way to think
about it.

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This is the top view of an
experiment.

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I'm looking down at my lab from
the top.

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There's a wall with two holes
in it, and there's the back

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wall, and the light is coming
from here.

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If I close this hole what we
will find typically--let me make

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them slightly bigger to make my
point.

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You'll find some light.

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I didn't draw it properly.

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Most of the brightest areas
will be in front of this hole.

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So let's see.

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It'll tend to be big here and
kind of fall off.

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Now if you block the other one
and you shine light here you get

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something like that.

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It's like two windows in the
room.

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You've got one window,
you get some sunlight.

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You've got another one you get
another sunlight.

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Now the question is if you open
both of them our expectation is

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that there is some energy coming
here,

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some energy coming here,
and if you open both you should

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get the sum of the two energies,
which I'm assuming this is the

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sum of these two graphs plotted
here.

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By the way, do you know how I'm
plotting this?

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I hope you understand that this
graph really means that distance

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is proportional to how bright
this point is.

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See, normally you're used to
drawing graphs like this as a

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function of distance,
but I've turned it around so

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that that is the graph of the
brightness at different points

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on the screen.

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That's what you expect.

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You open two windows you get
the light equal to the sum of

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the two.

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For example,
if you're feeling warm it'll be

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warmer in this region because
you're getting light from two

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sources.

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But this experiment also gives
you different results under

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certain conditions.

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I'll tell you what the
conditions are.

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What you find then if one slid
open it looks like that.

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If the other slid open it looks
like that, but with both open it

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looks like this.

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That's called interference.

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The main point to notice is
there are some places,

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like right here,
where you get less light with

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one more slid open,
okay?

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And more light with only one
open, less light with two open.

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In fact, you can arrange it so
this is really zero.

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I mean, it can come all the way
down here.

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In other words that's the point
on the screen which used to be

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bright with one slit open.

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You open the second one that
point becomes dark.

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That means if you're warm
there, and you tell somebody,

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"Hey, this is good,
open another window,"

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suddenly you get nothing.

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That happens.

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So when did people realize this
is what happens,

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and why did people fall for the
other graph that looks like this

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before?

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The answer is this.

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In order to see this phenomenon
in which it instead of being a

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shadow like this you get a huge
spread,

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or instead of being simply the
sum of these two featureless

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graphs it starts oscillating,
and a certain condition has to

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be satisfied.

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The condition is the following.

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This light, if it has a
wavelength λ,

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the λ, if it is much,
much smaller than any distance

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in the problem then you get
geometrical optics.

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So in this example if this slit
is thousand times the wavelength

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of light then the shadow,
in fact, will be geometrical,

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but as you make the slit
smaller and smaller until it

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becomes comparable to the
wavelength then you'll find it

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begin to fan out.

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Similarly here,
if this distance between the

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two slits,
d, is much,

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much bigger than the wavelength
and each slit is much,

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much bigger than the
wavelength, then you will find

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that you don't see any of this
interference effects.

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Because what happens is
interference effects are always

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there, but they oscillate like
mad on that screen.

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Suppose it oscillates like
this, and this could be 10,000

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oscillations per centimeter,
and your eye is able to look at

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it only one millimeter at a
time,

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then you're averaging over a
region and you only find the

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average of this graph.

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That looks like what you got
without interference.

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So in order to see interference
you need to make your apparatus

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probe the physics at a distance
scale comparable to the

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wavelength.

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So that's the answer.

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We will see all the details,
but it'll turnout that if you

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look at light using any
equipment,

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maybe hole in the wall,
two holes in the wall,

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whatever, whose dimensions are
much bigger than the wavelength

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you will not see the wave theory
of light.

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You'll be able to get away with
the geometric theory.

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But once you probe light at
length comparable to wavelength

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all these funny things will
happen.

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That's why people didn't
realize all these things about

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light.

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For example,
the wavelength of light,

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typical light,
is 5,000 angstroms,

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or 500 nanometers,
that is 500�10^(-9),

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or 5�10^(-7) meters.

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That means if you forget the 5
there are 10 million

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oscillations of the wave within
1 meter,

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so if you have an instrument
that can pick up that kind of

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fine features then you will see
the wave nature,

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but normally you won't.

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So it's only when people probed
light with very,

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very tiny holes placed very
close to each other they saw it.

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The second important thing to
remember is the following.

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Even if you made the holes very
small, and if you shine white

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light on this,
you will not see any of these

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patterns.

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Why is that?

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Can you guess?

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I'm going to tell you
everything, but can you guess

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why?

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Yes?

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You have a guess?

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Student:
Because they're white

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light.

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.

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Prof:  That's
right.

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White light is made of many
colors and many wavelengths,

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so each wavelength will form
its own pattern,

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but the maximum and minimum of
one guy won't coincide with that

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of the other.

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So where this one wants a
maximum that'll want a minimum,

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it'll get washed out.

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So in order to see this effect
you need monochromatic light

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sent by a source which is
sending it at a definite

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frequency and wavelength.

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And these are all things you
have to get right before you

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will see these effects.

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Okay, so now let's ask
ourselves, what is the nature of

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waves or what is the nature of
light that produces these funny

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effects?

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Light, as you know,
obeys a wave equation.

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Let me call ψ
as whatever it is that's doing

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the oscillations,
generic name is ψ.

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For example,
ψ can be the height of this

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string that's vibrating.

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If the string is doing this
from equilibrium ψ

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is the deviation from
equilibrium.

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ψ can be E or
B in the electromagnetic

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case.

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ψ can be the height of
water in a lake above some

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reference level.

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That's the reference level,
and the water does some

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oscillations,
ψ is the deviation from the

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normal.

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In the case of a sound wave,
as you know sound wave is

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deviations in air pressure.

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Again, this graph would stand
for the normal air pressure,

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ambient air pressure,
and when you talk you make some

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oscillations in the density or
in the pressure,

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and there's the deviation in
pressure.

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So I'm going to call ψ
as any one of these things.

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It's something that's
oscillating in the wave.

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And all these ψ's obey the
wave equation.

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The wave equation looks like
d^(2)ψ/dx^(2) minus 1

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over the velocity
d^(2)ψ/dt^(2) = 0.

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The context is different but
the equation's the same.

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The velocity will vary from
problem to problem.

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So I'm going to give a short
name for this.

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I'm going to all it box ψ
= 0.

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Turns out it's not a crazy
notation.

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A lot of people actually use
that.

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Do you know why they draw the
box?

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This is for people who have
done somewhat more advanced

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mathematics.

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There's something called del
squared which stands for the

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three spatial derivatives
squared.

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This has got the fourth one so
you use the box.

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Anyway, it just means I'm tired
of writing this.

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This stands exactly for this.

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That's the main point.

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Now when the wave obeys the
wave equation there's a very,

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very important property.

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The important property is it's
a linear equation.

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It's a linear equation which
means if you multiply ψ

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by a number it also satisfies
the wave equation.

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Because if I multiply by number
3 I can take the 3 inside all

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the derivatives,
and I find 3 times ψ

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is also a solution.

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Another important property is
that if someone gives you a

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solution called
ψ_1,

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another person gives you
another solution to the wave

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equation which is some other
function of x and t called

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ψ_2,
then you can check that if you

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take any number times
ψ_1,

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and any number times
ψ_2,

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and add this,
you will find box of

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aψ_1
bψ_2 is also 0.

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You understand why?

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When the derivatives come it
doesn't care about a.

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It's a constant.

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Box of ψ_1 will
vanish and the box of

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ψ_2 will vanish.

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The whole thing will vanish.

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But the important property is
you can take two solutions,

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ψ_1 and
ψ_2,

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multiply each one a number you
like,

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a constant, and add them,
that's also a solution.

15:47.620 --> 15:50.340
That's the principle of
superposition,

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and its origin is in the fact
that the underlying ψ

15:54.317 --> 15:56.377
obeys a linear equation.

15:56.379 --> 15:57.979
It's very important that it's
linear.

15:57.980 --> 16:01.020
I've told you many times if you
have something like that plus

16:01.017 --> 16:03.947
ψ^(2) = 0 you can verify
that if ψ_1 is a

16:03.952 --> 16:06.942
solution and ψ_2
is a solution if you add them

16:06.940 --> 16:08.610
you will get
ψ_1^(2)

16:08.610 --> 16:11.300
ψ_2^(2),
but that is not the same as

16:11.298 --> 16:13.828
ψ_1 ψ_2
the whole thing squared.

16:13.830 --> 16:17.110
So for nonlinear equations you
cannot add the solutions.

16:17.110 --> 16:22.280
These are linear because they
involve the first part of ψ.

16:22.278 --> 16:28.178
Okay, now what that means is
first of all ψ

16:28.181 --> 16:32.031
can be positive or negative.

16:32.028 --> 16:34.478
What if someone tried to prove
to you that ψ

16:34.476 --> 16:35.566
is always positive?

16:35.570 --> 16:39.800
How will you shoot that person
down?

16:39.799 --> 16:40.739
Do you know?

16:40.740 --> 16:41.430
Yes?

16:41.428 --> 16:42.198
Student:
Well,

16:42.203 --> 16:43.093
if the constant is negative 1.

16:43.090 --> 16:43.200
.

16:43.200 --> 16:44.000
Prof: Very good.

16:44.000 --> 16:46.490
If the constant is negative 1
then minus ψ

16:46.485 --> 16:49.515
is also a solution so no one
can ever convince you ψ

16:49.523 --> 16:51.183
should always be positive.

16:51.178 --> 16:53.688
Well, if ψ
can be positive and negative,

16:53.690 --> 16:56.910
which means generally it can go
up and down in sign,

16:56.908 --> 17:00.638
it cannot stand for things
which are always positive such

17:00.644 --> 17:02.584
as the brightness of light.

17:02.580 --> 17:05.910
The brightness of light is
always positive.

17:05.910 --> 17:07.660
The worst think you can have is
no light.

17:07.660 --> 17:10.720
You cannot have negative
brightness.

17:10.720 --> 17:13.410
Therefore, in the electric
field the brightness is not

17:13.412 --> 17:16.622
measured by the electric field,
the one that oscillates,

17:16.623 --> 17:19.223
but by something quadratic in
the field,

17:19.220 --> 17:21.030
and that's called the intensity.

17:21.028 --> 17:25.248
The intensity is proportional
to square of whatever is

17:25.246 --> 17:26.356
oscillating.

17:26.358 --> 17:30.778
It is the measure of energy
contained in the wave or

17:30.781 --> 17:31.911
brightness.

17:31.910 --> 17:35.180
Intensity being positive
definite has some chance of

17:35.175 --> 17:39.015
being a description of something
that's positive definite.

17:39.019 --> 17:44.419
Not only does it have a chance
that happens to be the case,

17:44.417 --> 17:45.067
okay?

17:45.068 --> 17:48.808
See, in all these problems it's
the square of the function that

17:48.811 --> 17:51.891
stands for something like
brightness, something like

17:51.890 --> 17:53.520
energy per unit second.

17:53.519 --> 17:55.889
They're all given by the square.

17:55.890 --> 17:59.940
So the important thing to
notice is that when you have one

17:59.938 --> 18:03.348
source producing some light,
and a second source producing

18:03.347 --> 18:05.827
some light,
and you turn them both on,

18:05.829 --> 18:09.929
together they'll produce a
field which is the sum of the

18:09.934 --> 18:10.984
two fields.

18:10.980 --> 18:13.920
What you can add is the ψ,
which in this case happens to

18:13.915 --> 18:15.205
be E or B.

18:15.210 --> 18:17.960
You cannot add the intensities.

18:17.960 --> 18:20.910
No one told you the intensities
are additive.

18:20.910 --> 18:24.550
The correct process is if
ψ_1 has an

18:24.546 --> 18:27.836
intensity I_1,
let me call it

18:27.836 --> 18:30.896
ψ_1^(2),
and ψ_2 has an

18:30.903 --> 18:33.283
intensity I_2
which is

18:33.281 --> 18:37.191
ψ_2^(2),
then if you put the two waves

18:37.193 --> 18:41.293
together by producing them due
to the two sources,

18:41.288 --> 18:45.358
I_1 2 is
(ψ_1

18:45.357 --> 18:47.667
ψ_2)^(2).

18:47.670 --> 18:51.440
That, you see,
has got a ψ_1^(2)

18:51.436 --> 18:56.116
and a ψ_2^(2)
2ψ_1ψ_2

18:56.124 --> 19:01.064
which is I_1
I_2  something

19:01.063 --> 19:03.913
which is indefinite in sign.

19:03.910 --> 19:04.780
This can be positive.

19:04.779 --> 19:05.489
This can be negative.

19:05.490 --> 19:07.140
This can be 0.

19:07.140 --> 19:11.060
So I_1 2 is
not equal to I_1

19:11.057 --> 19:12.407
I_2.

19:12.410 --> 19:18.190
It's got the extra stuff,
and that's going to be the

19:18.194 --> 19:22.624
origin of all the things we see
here.

19:22.618 --> 19:25.578
So it turns out for waves
there's two levels as which

19:25.578 --> 19:26.488
things happen.

19:26.490 --> 19:30.170
There's a thing that actually
oscillates and then you have to

19:30.171 --> 19:33.971
square it to get things that you
normally associate with energy

19:33.974 --> 19:35.084
or brightness.

19:35.078 --> 19:39.268
And the additive law applies to
the things that oscillates and

19:39.267 --> 19:40.637
not to its square.

19:40.640 --> 19:44.360
This is the reason why if
you've got two sources each one

19:44.355 --> 19:48.265
of it produces a positive
intensity I_1.

19:48.269 --> 19:50.849
When you turn them both on the
answer is not I_1

19:50.851 --> 19:53.121
I_2,
but this cross term,

19:53.118 --> 19:57.188
and the cross term can come and
neutralize these and make it

19:57.190 --> 19:57.880
vanish.

19:57.880 --> 20:03.300
That's why the experiment when
you open a second hole some

20:03.295 --> 20:07.185
places can become darker than
they are.

20:07.190 --> 20:10.550
Now later on,
I mean, not later on,

20:10.548 --> 20:16.178
later this week when we start
quantum mechanics we'll deal

20:16.182 --> 20:18.062
with the ψ.

20:18.058 --> 20:24.348
That's called the wave function
in quantum mechanics.

20:24.348 --> 20:26.158
Now that's a very bizarre
object.

20:26.160 --> 20:28.970
I don't want to even talk about
it now.

20:28.970 --> 20:32.950
Let me just say that it's
something which is intrinsically

20:32.951 --> 20:33.651
complex.

20:33.650 --> 20:36.720
See, normally for
electromagnetic waves or

20:36.721 --> 20:39.941
harmonic oscillator if you've
got x =

20:39.941 --> 20:44.211
Acosωt we're
used to taking x =

20:44.211 --> 20:49.081
Ae^(iωt) then
taking the real part at the end.

20:49.078 --> 20:50.818
That everyone knows x is
real.

20:50.818 --> 20:53.628
You bring the complex
exponential because it makes it

20:53.632 --> 20:55.582
easier to solve certain
equations.

20:55.578 --> 20:59.418
But in quantum mechanics all of
ψ, real part and imaginary

20:59.420 --> 21:01.120
part together are needed.

21:01.119 --> 21:04.849
It is, by nature, complex.

21:04.848 --> 21:07.108
In fact the equations of
quantum mechanism,

21:07.111 --> 21:09.961
the analog of Newton's Laws,
or the analog of the wave

21:09.963 --> 21:12.283
equation will have an i
in them.

21:12.278 --> 21:15.538
There's an i in the
equation for ψ,

21:15.544 --> 21:17.724
so you cannot get rid of it.

21:17.720 --> 21:22.560
So in those cases what you want
to think about the analog of

21:22.560 --> 21:26.060
intensity cannot be ψ^(2),
because ψ^(2),

21:26.058 --> 21:29.658
if you really mean by ψ^(2)
the real part of ψ

21:29.657 --> 21:32.437
i times the imaginary
part of ψ

21:32.442 --> 21:35.562
whole thing squared,
you see that is

21:35.558 --> 21:40.758
ψ_real^(2) -
ψ_imaginary^(2)

21:40.762 --> 21:44.452
2iψ_real
ψ_

21:44.452 --> 21:46.442
imaginary.

21:46.440 --> 21:48.260
What's wrong with this?

21:48.259 --> 21:51.479
If you want this to stand for
something positive definite this

21:51.483 --> 21:52.703
is definitely not it.

21:52.700 --> 21:54.110
Can you see that?

21:54.108 --> 21:58.648
In the simple case where ψ
is purely imaginary you see

21:58.654 --> 22:01.444
it's negative,
but neither positive,

22:01.443 --> 22:02.883
not even real.

22:02.880 --> 22:06.730
So you must know what I should
do in the case where ψ

22:06.730 --> 22:10.580
is intrinsically complex to
define something positive.

22:10.579 --> 22:15.139
Any guess on what it is?

22:15.140 --> 22:16.180
Yes?

22:16.180 --> 22:17.740
Student:
Take the ________.

22:17.740 --> 22:17.850
.

22:17.851 --> 22:19.521
Prof:  Could take
the absolute value of ψ,

22:19.519 --> 22:25.709
or another way to say that is
you want to take ψ*ψ.

22:25.710 --> 22:29.690
But anyway, I'm doing some of
this today because I want you to

22:29.692 --> 22:32.112
be on top of this before
Wednesday.

22:32.109 --> 22:33.659
I've given you enough warning.

22:33.660 --> 22:37.320
I ruined your spring break by
sending you notes on complex

22:37.318 --> 22:39.308
numbers so you can read them.

22:39.308 --> 22:41.708
If you know it already it's
fine, but you should be able to

22:41.707 --> 22:43.317
understand the meaning of real
part,

22:43.318 --> 22:45.598
imaginary part,
absolute value squared,

22:45.598 --> 22:47.418
things like
E_Iθ,

22:47.420 --> 22:47.960
okay?

22:47.960 --> 22:49.420
That's going to be used a lot.

22:49.420 --> 22:52.200
By anyway, if I take
ψ*ψ that becomes

22:52.204 --> 22:56.004
ψ_real i
times ψ_imaginary

22:56.003 --> 22:58.663
times ψ_real -
i times

22:58.663 --> 23:00.693
ψ_imaginary.

23:00.690 --> 23:02.830
And if you do the whole thing
you will find it's

23:02.828 --> 23:05.468
ψ_real^(2 )
ψ_imaginary^(2).

23:05.470 --> 23:10.480
That's, of course,
never negative.

23:10.480 --> 23:13.410
So when you do complex waves
they will also obey the

23:13.412 --> 23:15.542
superposition principle for
ψ,

23:15.538 --> 23:18.418
but the analog of intensity,
which is the positive definite

23:18.420 --> 23:21.180
number you extract from ψ,
will be that.

23:21.180 --> 23:25.550
You can also write this as
absolute value ψ^(2).

23:25.548 --> 23:27.818
When you do complex don't
forget absolute value.

23:27.818 --> 23:29.878
When you do real either you can
square the ψ

23:29.878 --> 23:31.498
or you can take the absolute
value.

23:31.500 --> 23:33.330
It doesn't make any difference.

23:33.328 --> 23:37.488
For the real function the
simple squared and the absolute

23:37.486 --> 23:40.006
value squared are both the same.

23:40.009 --> 23:45.089
All right, so now I'm going to
start with trying to understand

23:45.086 --> 23:48.996
interference of waves by doing
the following.

23:49.000 --> 23:53.560
Usually you get interference
when two people are sending you

23:53.558 --> 23:54.408
a signal.

23:54.410 --> 23:58.600
One person is sending you a
signal ψ_1 and the

23:58.595 --> 24:01.145
signal may be traveling in
space,

24:01.150 --> 24:04.770
but I want you to sit at once
place and just register what's

24:04.772 --> 24:06.372
coming to where you are.

24:06.368 --> 24:08.348
For example,
you could be in one part of a

24:08.348 --> 24:11.098
lake and someone is rocking the
boat there and sending out

24:11.102 --> 24:13.262
ripples,
the ripples come to where you

24:13.257 --> 24:15.097
are, and let's say it's
periodic,

24:15.098 --> 24:19.758
and at the location where you
are let it be of the form

24:19.758 --> 24:22.258
Acosωt.

24:22.259 --> 24:25.199
So A is the amplitude
and ω is the angular

24:25.196 --> 24:25.846
frequency.

24:25.848 --> 24:28.678
If you like,
it's 2Π times f where

24:28.682 --> 24:30.992
f is the normal
frequency.

24:30.990 --> 24:35.180
Then another wave is coming
towards you.

24:35.180 --> 24:38.660
It is
Acosωt,

24:38.655 --> 24:43.215
but it comes at a different
phase Φ.

24:43.220 --> 24:46.020
Now the phase Φ,
if you've got only a single

24:46.022 --> 24:48.172
wave you can choose Φ
to be 0.

24:48.170 --> 24:50.630
Because Φ
is--what does Φ

24:50.625 --> 24:51.005
do?

24:51.009 --> 24:55.019
cosωt likes to
have a maximum of t = 0,

24:55.019 --> 24:57.349
cos(ωt
Φ) has a

24:57.351 --> 24:58.751
maximum somewhat earlier.

24:58.750 --> 25:02.160
You can always reset your clock
so that whenever it hits a

25:02.161 --> 25:05.871
maximum you press your clock and
set it to 0 you can get rid of

25:05.871 --> 25:06.471
Φ.

25:06.470 --> 25:11.130
But you cannot get rid of Φ
from two wave functions.

25:11.130 --> 25:13.520
You cannot choose them both to
have their maximum at the same

25:13.516 --> 25:14.586
time because they're not.

25:14.588 --> 25:19.288
In fact this fellow goes like
this.

25:19.288 --> 25:24.318
This one has got an extra lead
on it, so it'll sort of start

25:24.321 --> 25:24.921
here.

25:24.920 --> 25:28.340
So it'll go like this.

25:28.338 --> 25:32.258
If you went backward it'll have
its maximum at some negative

25:32.255 --> 25:32.715
time.

25:32.720 --> 25:36.420
So what we want to do is to add
these two signals to get

25:36.423 --> 25:39.053
ψ_1
ψ_2.

25:39.049 --> 25:40.219
Think of this as water waves.

25:40.220 --> 25:42.610
That's the best example.

25:42.608 --> 25:46.058
Okay, someone shaking the
water, ripples are coming out.

25:46.059 --> 25:47.719
They're coming to where you are.

25:47.720 --> 25:50.700
With only the first vibrator
sending the waves you get this.

25:50.700 --> 25:51.930
With the second one you get
this.

25:51.930 --> 25:55.550
But now if both are doing it
then this is what you will get.

25:55.548 --> 26:00.918
And the height of the water
will actually be simply the sum

26:00.923 --> 26:02.873
of the two heights.

26:02.869 --> 26:04.789
It's not obvious but true, okay?

26:04.788 --> 26:08.178
If you agitate the water with
one source and it produces the

26:08.178 --> 26:10.018
height here,
and with the second one that

26:10.019 --> 26:12.179
produces another height,
if you do both it's not

26:12.182 --> 26:14.362
obvious, it's not a logical
necessity,

26:14.358 --> 26:18.528
but it's a fact that the height
of the water at any point will

26:18.531 --> 26:20.791
simply be the sum of these two.

26:20.789 --> 26:22.889
So you've got to add these guys.

26:22.890 --> 26:28.230
So you do that you get
Acosωt

26:28.226 --> 26:34.006
Acos(ωt
Φ). Now

26:34.007 --> 26:40.787
you've got to go back to your
trig identities and remember the

26:40.788 --> 26:45.008
following: cosine α
cosine β

26:45.011 --> 26:50.461
is twice cosine α
β over 2 times cosine

26:50.460 --> 26:54.240
α - β
over 2.

26:54.240 --> 26:56.910
If you've got a formula like
this, of which there are

26:56.912 --> 26:59.172
numerous ones,
and if you have any doubts you

26:59.173 --> 27:00.873
can check some special cases.

27:00.868 --> 27:03.718
For example,
if α is equal to β

27:03.720 --> 27:07.340
I want to get cos α
cos α or twice cosine

27:07.335 --> 27:08.165
α.

27:08.170 --> 27:11.650
On the right hand side I get
twice cosine α

27:11.651 --> 27:14.411
α over 2,
which is cosine α

27:14.405 --> 27:16.795
times cosine of 0 which is 1.

27:16.798 --> 27:19.628
Now it doesn't prove it's the
right formula,

27:19.632 --> 27:23.582
but it will tell you that if it
is wrong in some way then you

27:23.583 --> 27:24.443
can tell.

27:24.440 --> 27:28.500
Anyway, this is the right
formula applied to this problem.

27:28.500 --> 27:34.630
If I apply it to this problem,
I get twice cosine ψ

27:34.625 --> 27:40.345
over 2,
I'm sorry, 2A cosine

27:40.346 --> 27:50.196
ψ over 2 times cosine
ωt Φ/2.

27:50.200 --> 27:51.990
This is just plugging in.

27:51.990 --> 27:57.600
It is an example.

27:57.598 --> 28:02.828
This whole thing looks like a
pre-factor, doesn't depend on

28:02.833 --> 28:05.633
time and this is oscillating.

28:05.630 --> 28:11.670
So this is a problem which
looks like some amplitude cosine

28:11.671 --> 28:13.651
ωt Φ/2.

28:13.650 --> 28:19.390
The amplitude Ã
is 2A cosine Φ/2.

28:19.390 --> 28:24.060
And if I want the amplitude to
be always positive I will take

28:24.058 --> 28:27.948
it to mean the absolute value of
cosine Φ/2.

28:27.950 --> 28:29.610
So here's the answer, guys.

28:29.608 --> 28:32.288
When you add these two
waves--if you draw the pictures

28:32.288 --> 28:35.118
it's kind of confusing,
but this is the simple answer.

28:35.118 --> 28:39.878
It says the total will be a new
wave whose amplitude is

28:45.605 --> 28:48.615
as 0 depending on what the
cosine does.

28:48.618 --> 28:51.168
And then it will also not be in
step with this guy,

28:51.173 --> 28:54.243
not in step with this guy,
it'll be half way between them.

28:54.240 --> 28:58.990
Its phase will be Φ/2.

28:58.990 --> 29:02.620
Now let's check the formula in
a couple of special cases.

29:02.618 --> 29:06.728
If Φ is equal to 0 that
means the two signals are

29:06.730 --> 29:07.680
identical.

29:07.680 --> 29:10.590
We don't need any fancy stuff
to know the answer is

29:10.592 --> 29:12.342
2Acosωt.

29:12.338 --> 29:14.888
If Φ is equal to 0,
that's

29:14.892 --> 29:19.562
Acosωt,
cos 0 is 1 and you get the

29:19.557 --> 29:20.787
2A.

29:20.788 --> 29:24.658
The one other case you can do
very easily in your head is when

29:24.662 --> 29:26.252
Φ is equal to Π.

29:26.250 --> 29:30.860
If Φ is equal to Π
cosine of ωt  Π

29:30.858 --> 29:34.068
is just minus cosine
ωt.

29:34.068 --> 29:38.058
That means this wave starts
like this.

29:38.058 --> 29:40.938
Wherever this is going up this
is going, I'm sorry,

29:40.941 --> 29:42.441
it starts like this one.

29:42.440 --> 29:45.110
So when that one is doing this,
this will do that and it'll

29:45.106 --> 29:46.346
always be opposite to it.

29:46.348 --> 29:49.848
If you add them you'll get
identically 0.

29:49.848 --> 29:54.078
Anyway, they are the two
extreme cases where you can

29:54.077 --> 29:55.817
check your formula.

29:55.818 --> 29:58.758
So this is the only formula you
will need, so you should get

29:58.755 --> 29:59.845
that into your head.

29:59.848 --> 30:02.698
There is some natural frequency
ω and the phase is

30:02.702 --> 30:05.662
increasing at the rate ω
so it's ωt.

30:05.660 --> 30:08.210
The other guy has shifted by
phase Φ.

30:08.210 --> 30:12.950
Their sum has a new amplitude
and a new phase.

30:12.950 --> 30:16.260
I'm doing a special case where
the two waves have the same

30:16.261 --> 30:16.961
amplitude.

30:16.960 --> 30:23.220
If they're different amplitudes
it becomes more messy,

30:23.220 --> 30:26.410
but that's not necessary.

30:26.410 --> 30:30.900
So I'm going to do this once
more, but with complex numbers,

30:30.901 --> 30:35.241
because I want you guys to get
used to complex numbers.

30:35.240 --> 30:38.060
And I think the only way you're
going to get used to anything is

30:38.063 --> 30:39.053
if you use it a lot.

30:39.048 --> 30:41.888
You can sit down and read it at
bedtime.

30:41.890 --> 30:45.010
It may help you go to sleep,
but it doesn't do any good.

30:45.009 --> 30:47.909
So I'm going to keep using the
complex numbers,

30:47.912 --> 30:51.382
because starting Wednesday
we're going to be doing a lot

30:51.382 --> 30:51.952
more.

30:51.950 --> 30:53.830
But you don't need very fancy
complex numbers.

30:53.828 --> 30:56.258
It's at the level that I'm
doing now.

30:56.259 --> 31:03.119
So let us now do exactly this
addition using complex numbers.

31:03.118 --> 31:07.208
So let's take one single wave
first, ψ_1 =

31:07.211 --> 31:09.371
Acosωt.

31:09.368 --> 31:16.448
We agree there's no complex
number here, but let's write it

31:16.452 --> 31:21.952
as the real part
Re(Ae^(iωt)).

31:21.950 --> 31:26.860
Now you remember the complex
number is z = mod

31:26.857 --> 31:30.537
ze^(iθ) looks like
this.

31:30.538 --> 31:34.438
It's got a length equal to mod
z and there's an angle

31:34.442 --> 31:35.702
equal to θ.

31:35.700 --> 31:37.110
That's the complex number
z.

31:37.108 --> 31:40.058
If you want you can think of it
as that.

31:40.058 --> 31:45.508
It starts at the origin and
terminates at the point

31:45.509 --> 31:46.709
z.

31:46.710 --> 31:52.850
Its real part is really this
one, this mod z cosine

31:52.852 --> 31:53.932
θ.

31:53.930 --> 31:55.930
The imaginary part is mod
z sine θ,

31:55.932 --> 31:57.172
but I just want the cosine.

31:57.170 --> 31:59.070
It's the real part of this guy.

31:59.069 --> 32:00.969
So what does this look like?

32:00.970 --> 32:02.380
Well, let me draw you a picture
here.

32:02.380 --> 32:04.640
This looks like this.

32:04.640 --> 32:10.000
It's got length A and
the angle is ωt,

32:10.001 --> 32:14.331
and as time goes by it goes
round and round.

32:14.328 --> 32:21.518
So at any future time it'll be
on this circle.

32:21.519 --> 32:25.599
At this instant its real part
is just this one,

32:25.596 --> 32:28.606
A cosine
ωt.

32:28.608 --> 32:31.778
So if you like,
the complex number has fixed

32:31.778 --> 32:35.168
magnitude, goes round and round
in a circle.

32:35.170 --> 32:39.230
The real part of it has got a
maximum of A here,

32:39.230 --> 32:43.420
a 0 there, and -A,
and that'll be the cosine,

32:43.420 --> 32:46.270
because the horizontal part of
this is A cosine

32:46.270 --> 32:47.240
ωt.

32:47.240 --> 32:50.850
So we can keep track of the
thing we're interested in by

32:50.852 --> 32:54.992
letting this vector rotate in a
circle by looking at a shadow on

32:54.990 --> 32:56.830
the x-axis,
okay?

32:56.828 --> 32:58.588
If you have some light shining
from here you ask,

32:58.593 --> 32:59.993
"What image does it
cast?"

32:59.990 --> 33:04.260
Well, it comes there.

33:04.259 --> 33:07.719
So for a single complex number
it's not very useful,

33:07.718 --> 33:11.108
but now let me bring in a
second complex number.

33:11.108 --> 33:17.168
Let's bring in a second complex
number, so I'll draw a new

33:17.166 --> 33:20.246
figure for the two of them.

33:20.250 --> 33:23.750
So you should understand this
is a dynamic situation.

33:23.750 --> 33:25.760
This is not a fixed vector.

33:25.759 --> 33:29.249
As t increases this is
constantly rotating,

33:29.250 --> 33:29.750
okay?

33:29.750 --> 33:32.830
And as it rotates if you follow
the shadow on the x-axis

33:32.832 --> 33:35.172
you get the real part,
which is our function.

33:35.170 --> 33:39.540
Now if you've got a second
function, so let me take a

33:39.542 --> 33:44.252
circle of radius A,
and here's the first guy who's

33:44.252 --> 33:46.442
angle ωt.

33:46.440 --> 33:51.550
The second one is also on that
circle, but you've got an extra

33:51.547 --> 33:52.717
angle Φ.

33:52.720 --> 33:54.340
You follow that?

33:54.338 --> 33:57.638
So this ψ_1 this
is ψ_2,

33:57.640 --> 33:59.300
actual ψ
is actually the real part of

33:59.295 --> 34:01.895
these two guys,
but I'm going to work with the

34:01.903 --> 34:03.273
complex number first.

34:03.269 --> 34:06.969
We all remember in the end to
take the real part.

34:06.970 --> 34:10.030
And I'm using the fact that the
real part of ψ_1

34:10.027 --> 34:12.927
ψ_2 is the real
part of ψ_1 the

34:12.934 --> 34:14.694
real part of ψ_2.

34:14.690 --> 34:17.430
Therefore if you want to add
two real waves you can add the

34:17.425 --> 34:19.495
complex waves and then take the
real part.

34:19.500 --> 34:21.540
You can take it before or you
can take it after.

34:21.539 --> 34:25.259
Our philosophy is going to be
to add the vectors first then

34:25.262 --> 34:26.612
take the real part.

34:26.610 --> 34:30.660
I want you to think about,
what is the sum of these two

34:30.663 --> 34:32.093
guys going to do?

34:32.090 --> 34:37.490
As usual I'll give you a few
seconds to think about it,

34:37.489 --> 34:38.189
okay?

34:38.190 --> 34:43.730
By vector addition you do this
parallelogram.

34:43.730 --> 34:48.220
You draw one copy of this here
and one copy of that here,

34:48.215 --> 34:50.455
and you join this to that.

34:50.460 --> 34:56.020
That line is ψ_1
ψ_2,

34:56.019 --> 35:02.739
so their sum would rotate on
that circle, but let's look at

35:02.739 --> 35:04.129
the sum.

35:04.130 --> 35:07.470
What's this angle here?

35:07.469 --> 35:11.579
You can tell from the
similarity of this parallelogram

35:11.581 --> 35:15.051
that that angle is Φ/2,
because the angle Φ

35:15.052 --> 35:17.782
between them is split evenly
between the diagonal of the

35:17.784 --> 35:18.634
parallelogram.

35:18.630 --> 35:22.950
So you already tell that the
new wave, which is obtained by

35:22.952 --> 35:26.232
adding these two,
when this guy rotates we'll

35:26.231 --> 35:28.321
have a phase of Φ/2.

35:28.320 --> 35:29.690
And what's its amplitude?

35:29.690 --> 35:32.060
That we're asking,
what's the length of the

35:32.063 --> 35:33.933
diagonal of this parallelogram?

35:38.355 --> 35:42.945
is the square of one guy,
square of the other guy 2 times

35:42.951 --> 35:47.521
length of A length of
A cosine of the angle

35:47.523 --> 35:48.833
between them.

35:48.829 --> 35:54.029
I mean, cosine of this angle,
Φ.

35:54.030 --> 36:01.270
So this becomes 2A
squared times 1 cosine Φ.

36:01.268 --> 36:07.258
Now I remind you of another
trigonometry identity that 1

36:07.257 --> 36:12.807
cosine Φ is 2 times cosine
squared θ/2.

36:21.306 --> 36:25.386
it's 2A times cosine
Φ/2.

36:25.389 --> 36:28.869
So we just reproduced the
answer, but I want you to

36:28.865 --> 36:33.245
understand that the sum of these
two complex numbers is a number

36:33.246 --> 36:35.956
of length 2A cosine
Φ/2.

36:35.960 --> 36:39.510
It goes around at the same
ω, but it is shifted.

36:39.510 --> 36:44.830
Its phase is half way between
the phase of the two waves.

36:44.829 --> 36:49.069
That half comes because in a
parallelogram the diagonal will

36:49.072 --> 36:50.872
bisect the angle Φ.

36:50.869 --> 36:54.559
So this answer is the same as
what we got from trigonometry.

36:54.559 --> 36:55.769
It's not going to give you
anything new,

36:55.768 --> 36:59.198
but it's helpful to think of
our actual ψ_1

36:59.204 --> 37:01.174
ψ_2,
if you like,

37:01.172 --> 37:03.932
is the real part,
which is the projection on the

37:03.925 --> 37:07.535
x-axis,
but it's a lot easier to add

37:07.536 --> 37:11.876
these arrows than to add the
real parts.

37:11.880 --> 37:16.700
Okay, now I'm going to do the
last part of it one more way,

37:16.702 --> 37:19.032
so once more with feeling.

37:19.030 --> 37:21.690
So let me do that,
ψ_1

37:21.690 --> 37:23.170
ψ_2.

37:23.170 --> 37:25.820
This is when I'm not going to
draw any pictures.

37:25.820 --> 37:27.530
Some people like to draw
pictures.

37:27.530 --> 37:29.580
Some people like to do the
algebra.

37:29.579 --> 37:33.079
So I'm one of these algebra
types because when they draw a

37:33.077 --> 37:35.407
picture I don't understand
anything.

37:35.409 --> 37:37.259
So for fellows like me
ψ_1

37:37.259 --> 37:38.969
ψ_2 looks like
this,

37:38.969 --> 37:42.589
ψ_1 is
Ae^(Iωt),

37:42.590 --> 37:45.060
ψ_2 is
Ae^(Iωt

37:45.061 --> 37:53.011
)^(Φ),
which I can write like this.

37:53.010 --> 38:02.640
So let's write it as a common
Ae^(iωt) times 1

38:02.639 --> 38:07.619
e^(i)^(Φ).

38:07.619 --> 38:11.009
This much, I think,
is automatic.

38:11.010 --> 38:14.790
Your spinal cord can do this
calculation, but the next move

38:14.793 --> 38:18.123
requires a little bit of
planning and we do this.

38:18.119 --> 38:23.279
Ae^(iωt) I'm going
to pull half the phase of that

38:23.280 --> 38:27.090
e^(i)^(Φ)
^(/2) times

38:27.088 --> 38:30.978
e^(i)^(Φ)
^(/2) e^(

38:30.980 --> 38:34.450
i)^(Φ)
^(/2).

38:34.449 --> 38:37.469
You can see if I open all the
brackets this goes back to being

38:37.471 --> 38:37.671
1.

38:37.670 --> 38:40.030
This goes back to being
e^(i)^(Φ).

38:40.030 --> 38:44.790
The reason I do this is that I
want everybody to know this is a

38:44.789 --> 38:47.279
famous combination,
e^(i)^

38:47.280 --> 38:49.590
(θ)
e^(-i)^(θ) is

38:49.586 --> 38:54.126
twice cosine θ,
so twice cosine Φ/2.

38:54.130 --> 39:02.390
So the sum of all these complex
numbers became 2A cosine

39:02.387 --> 39:08.377
Φ/2 times e^(iωt
)^(Φ)

39:08.382 --> 39:11.182
^(/2).

39:11.179 --> 39:15.029
You can see that this is the
amplitude.

39:15.030 --> 39:20.290
This is the length of the
vector, and this is angle by

39:20.293 --> 39:22.483
which it's rotating.

39:22.480 --> 39:24.470
Can you see that?

39:24.469 --> 39:28.479
That's like mod ψ_1
ψ_2,

39:28.480 --> 39:31.640
and there's the i times
whatever the phase of

39:31.643 --> 39:34.253
ψ_1
ψ_2 is.

39:34.250 --> 39:37.430
There are many ways to do this,
but having done it three ways

39:37.425 --> 39:39.645
you should at least remember the
answer.

39:39.650 --> 39:43.670
When you add two waves both at
the same frequency ω,

39:43.670 --> 39:45.900
with a relative phase between
them,

39:45.900 --> 39:50.630
their sum oscillates at the
same frequency with the phase

39:50.632 --> 39:55.702
half way between the two with an
amplitude which is 2A

39:55.704 --> 39:57.314
cosine Φ/2.

39:57.309 --> 40:05.929
Now I'm going to start applying
this result to a variety of

40:05.934 --> 40:07.574
problems.

40:07.570 --> 40:11.400
The problem I'm going to do is
the following.

40:11.400 --> 40:14.590
I'm going to take--let me start
fresh here.

40:14.590 --> 40:19.970
Now I'm going to do the
interference problem.

40:19.969 --> 40:29.759
For interference you take these
two slits and some wave come

40:29.759 --> 40:31.749
from here.

40:31.750 --> 40:34.530
It hits the two slits.

40:34.530 --> 40:37.880
Now here's some part of physics
that I'm just going to tell you

40:37.876 --> 40:39.006
without proving it.

40:39.010 --> 40:40.940
I hate doing that,
but once in a while we have to

40:40.938 --> 40:41.338
do that.

40:41.340 --> 40:44.040
I can only appeal to you sense
of logic.

40:44.039 --> 40:48.749
That is the assumption that if
you look at this wall which is

40:48.751 --> 40:52.521
completely covered except for
two tiny holes,

40:52.518 --> 40:56.168
let's say, then any light on
the back,

40:56.170 --> 40:58.820
on the other side,
will be in the form of two

40:58.815 --> 41:01.955
glowing objects,
and with the two holes in the

41:01.963 --> 41:04.593
wall will act like two light
sources.

41:04.590 --> 41:05.740
That's how it works,
for example.

41:05.739 --> 41:08.079
You open the window,
the window looks like a source

41:08.076 --> 41:08.586
of light.

41:08.590 --> 41:11.350
It's coming from somewhere,
but to the people inside the

41:11.349 --> 41:14.309
room that a source of light and
that's a source of light.

41:14.309 --> 41:16.799
And if the things are very
close to each other,

41:16.800 --> 41:21.230
this hole, or very small it'll
look like a point source of

41:21.230 --> 41:23.940
light,
and for the point source of

41:23.943 --> 41:28.643
light the waves will emanate
like this with that as a center,

41:28.639 --> 41:31.719
and another set of waves with
this at the center,

41:31.719 --> 41:36.379
and the two will be in sync
because this wave front hits

41:36.380 --> 41:38.840
them both at the same time.

41:38.840 --> 41:44.130
And I'm going to sit somewhere
here, and I'm going to measure

41:44.132 --> 41:45.722
what's going on.

41:45.719 --> 41:47.159
So you imagine this is water
waves.

41:47.159 --> 41:49.049
This is a top view of the water
wave.

41:49.050 --> 41:51.020
This is a crest and there's a
trough in between,

41:51.018 --> 41:52.578
another crest,
another trough,

41:52.583 --> 41:55.473
another crest,
and I'm sitting here and these

41:55.467 --> 41:58.867
wave are going by,
so two waves will come to me.

41:58.869 --> 42:03.169
One will come from here,
and one will come from there.

42:03.170 --> 42:05.200
Let that distance be r.

42:05.199 --> 42:10.599
Let that distance be r
δ.

42:10.599 --> 42:13.039
So I am sitting here.

42:13.039 --> 42:16.359
So the water here will be
bobbing up and down by an amount

42:16.358 --> 42:19.908
equal to what this guy tells me
to do plus what that guy tells

42:19.911 --> 42:20.671
me to do.

42:20.670 --> 42:24.320
So I'm only asking you,
"What is the amplitude of

42:24.322 --> 42:28.462
vibrations of the water here
with both of them open?"

42:28.460 --> 42:30.640
Well, you've done it many,
many times.

42:30.639 --> 42:36.299
If the first one sends a signal
Acos(kr -

42:36.304 --> 42:40.784
ωt),
by the way, you have seen waves

42:40.779 --> 42:44.969
in one dimension whether it's
cos or sine of kx -

42:44.974 --> 42:46.294
ωt.

42:46.289 --> 42:51.309
That's the wave in which the
fronts look like this.

42:51.309 --> 42:54.309
When you advance the next the
lines of constant phase or

42:54.309 --> 42:56.709
maximum of the cosine are lines
like this.

42:56.710 --> 42:59.960
But this is not a plane wave
it's a circular wave.

42:59.960 --> 43:03.680
It spreads out in all
directions, and the points of

43:03.681 --> 43:06.811
constant phase are circles,
if you like.

43:06.809 --> 43:09.509
See, here if x is a
constant you get a certain

43:09.514 --> 43:12.634
phase, and as you vary x
at a given instant you'll get

43:12.634 --> 43:14.094
the various maxima here.

43:14.090 --> 43:15.860
Here things depend on r.

43:15.860 --> 43:19.180
At a given r the phase
is constant, so if the cosine is

43:19.177 --> 43:22.547
maximized at one value of r
it's maximized at all values

43:22.548 --> 43:23.418
of r.

43:23.420 --> 43:27.900
So all the crests will lie on
that semicircle on this side,

43:27.898 --> 43:31.758
and all the troughs will be
half way in between.

43:31.760 --> 43:37.330
And I remind you that k
is related to the family of

43:37.331 --> 43:41.731
wavelength by this formula,
2Π/λ.

43:41.730 --> 43:43.040
You see that?

43:43.039 --> 43:45.939
If this k was
2Π/λ,

43:45.938 --> 43:49.748
kr looks like
2Πr/λ.

43:49.750 --> 43:53.010
What that tells you is if to
any given r you add a

43:53.010 --> 43:56.390
λ it doesn't make a
difference because you're adding

43:56.385 --> 43:58.885
a 2Π to that trigonometric
function.

43:58.889 --> 44:01.829
That's the relation between
k and λ,

44:01.829 --> 44:02.789
it's inverted.

44:02.789 --> 44:05.729
Okay, that is ψ_1.

44:05.730 --> 44:11.970
ψ_2 has got the
same amplitude cosine kr

44:11.972 --> 44:16.992
kδ - ωt because
this path,

44:16.989 --> 44:19.339
I'm assuming is longer by an
amount δ,

44:19.340 --> 44:23.480
so the r measured from
this hole is going to be

44:23.483 --> 44:24.973
r δ.

44:24.969 --> 44:26.429
So what happens when you add
them?

44:26.429 --> 44:28.609
It's the same trick we have
done many times.

44:28.610 --> 44:32.470
All you have to do is identify
kδ

44:32.474 --> 44:34.454
as this number Φ.

44:34.449 --> 44:37.249
That's the phase difference
between the two.

44:37.250 --> 44:40.930
And that's the phase difference
because even though these

44:40.927 --> 44:44.537
apertures are acting as
synchronized sources of waves it

44:44.541 --> 44:48.421
takes this wave somewhat longer
to get to my point than this

44:48.416 --> 44:53.946
one,
so there is a delay and that's

44:53.952 --> 45:00.052
why it arrives with some shift
Φ.

45:00.050 --> 45:01.620
Yes?

45:01.619 --> 45:03.779
So what's the amplitude here?

45:03.780 --> 45:05.490
Well, we don't have to do this
over and over again.

45:12.010 --> 45:19.950
cosine of Φ/2 becomes cosine
of kδ/2.

45:19.949 --> 45:23.559
So when is the amplitude big
and when is the amplitude zero?

45:23.559 --> 45:25.779
That's what we're asking.

45:25.780 --> 45:30.110
If you go along the
perpendicular bisector here you

45:30.110 --> 45:35.310
can see by symmetry that two
r's are equal and δ

45:35.307 --> 45:39.117
is 0 then amplitude is 2 times
A.

45:39.119 --> 45:42.059
So here the two waves are
completely in step and they give

45:42.056 --> 45:43.806
you a wave of double the height.

45:43.809 --> 45:46.979
And double the height means
four times the intensity because

45:46.976 --> 45:49.926
intensity is proportional to
square of the amplitude.

45:49.929 --> 45:53.579
So this point will have an
amplitude which is 2A.

45:53.579 --> 45:59.379
I'm just going to draw that
part there, but now let's go to

45:59.380 --> 46:04.880
another place here where
kδ/2 is Π/2.

46:04.880 --> 46:07.850
Well, cosine Π/2 is 0.

46:07.849 --> 46:13.739
At that point the amplitude
will vanish, and then you will

46:13.744 --> 46:17.264
get no vibration of water there.

46:17.260 --> 46:19.990
Let's understand precisely why
that is true.

46:19.989 --> 46:25.789
So kδ/2,
k is 2Π/λ

46:25.793 --> 46:32.593
times δ/2 is Π/2
canceling the 2 with the 2 the

46:32.586 --> 46:36.536
Π with the Π
I find δ

46:36.539 --> 46:39.009
is λ/2.

46:39.010 --> 46:42.980
That means this second wave has
to travel a distance half a

46:42.978 --> 46:45.508
wavelength more than the first
one.

46:45.510 --> 46:46.780
I think you can see what that
means.

46:46.780 --> 46:48.850
I mean, here's a wave going
along.

46:48.849 --> 46:51.929
If that guy hits there and
you're half a wavelength behind

46:51.925 --> 46:53.325
this guy will hit there.

46:53.329 --> 46:56.099
When you go a half a wavelength
in a wave you're doing exactly

46:56.103 --> 46:57.973
the opposite of what you're
doing here.

46:57.969 --> 47:00.329
At every instant you can take
any wave you want,

47:00.329 --> 47:04.219
take a flash photograph,
go to some point and go half a

47:04.222 --> 47:07.972
wavelength you'll find you're
doing minus of that.

47:07.969 --> 47:11.039
Because half a wavelength adds
the phase of Π

47:11.039 --> 47:14.309
to the cosine or sine,
and sine when you add a Π

47:14.306 --> 47:15.936
goes to minus itself.

47:15.940 --> 47:19.070
So that's why you get
destructive interference here

47:19.070 --> 47:23.140
because the signal from this one
is telling the water to go up.

47:23.139 --> 47:25.819
The signal from that one is
telling the water to go down so

47:25.815 --> 47:26.965
it doesn't do anything.

47:26.969 --> 47:29.069
A little later the first one
says go down,

47:29.068 --> 47:31.628
second one says go up then also
nothing happens.

47:31.630 --> 47:35.550
So you can get two signals that
can actually cancel each other.

47:35.550 --> 47:39.840
That's the property of waves.

47:39.840 --> 47:42.150
It's not the property of
intensity.

47:42.150 --> 47:44.410
You might naively think,
"Hey, this used to be

47:44.409 --> 47:45.899
bright without the second slit.

47:45.900 --> 47:47.920
It used to be bright without
the first slit.

47:47.920 --> 47:50.110
Maybe with both it'll be twice
as bright,"

47:50.108 --> 47:52.678
and that's wrong because what
adds is not intensity.

47:52.679 --> 47:56.149
What adds is the amplitude,
the wave is the function ψ,

47:56.150 --> 48:00.430
and ψ being positive or
negative has the possibility of

48:00.432 --> 48:03.242
getting canceled by a second
source.

48:03.239 --> 48:06.569
So one source can cancel
another source in ψ

48:06.574 --> 48:09.064
and therefore kill the
intensity.

48:09.059 --> 48:12.049
That's why if you now plot this
graph you'll get a minimum here,

48:12.052 --> 48:14.142
you'll get a maximum there,
and minimum here,

48:14.143 --> 48:15.193
and maximum there.

48:15.190 --> 48:17.310
So, where are the places where
you get a maximum,

48:17.307 --> 48:19.557
and where are the places where
you get a minimum?

48:19.559 --> 48:21.919
So let's write that down.

48:26.181 --> 48:28.741
cosine kδ/2.

48:32.449 --> 48:35.129
cosine, I mean,
let's take the absolute value

48:35.125 --> 48:36.945
of this for the amplitude.

48:36.949 --> 48:43.029
Whenever kδ/2 is
equal to Π,

48:43.030 --> 48:50.510
equal to 3Π plus or minus,
plus or minus 5Π

48:50.512 --> 49:00.362
and so on that means
2Πδ/λ

49:00.356 --> 49:06.636
is equal to Π--
let me see.

49:06.635 --> 49:10.225
I'm doing something wrong here.

49:10.230 --> 49:14.510
I'm trying to ask,
when is it a maximum?

49:14.510 --> 49:19.560
The maximum is when this angle
is either 0 or Π,

49:19.556 --> 49:23.486
that's right,
so why am I getting half a

49:23.494 --> 49:28.344
wavelength is what I'm trying to
understand.

49:28.340 --> 49:33.910
Okay, delta over 2.

49:33.909 --> 49:42.659
So this is 2Π/λ
times δ/2 = 0.

49:42.659 --> 49:43.099
Oh, I'm sorry.

49:43.099 --> 49:44.599
I know what I did wrong.

49:44.599 --> 49:46.819
This whole thing has to be 0.

49:46.820 --> 49:47.960
That's good.

49:47.960 --> 49:49.430
That means δ is 0.

49:49.429 --> 49:52.699
What's the next time it'll be a
maximum?

49:52.699 --> 49:56.829
When this whole thing is equal
to Π, so let me see what I

49:56.829 --> 49:57.879
get for that.

49:57.880 --> 50:03.060
These two cancel and I get
δ is equal to λ.

50:03.059 --> 50:04.749
Can you see that,
2 cancels 2,

50:04.746 --> 50:06.676
Π cancels Π,
δ equals λ,

50:06.679 --> 50:08.419
so you can have δ
equal to 0,

50:08.420 --> 50:11.880
λ plus or minus,
plus or minus.

50:11.880 --> 50:15.260
They're all going to be bright
if you're talking about light.

50:15.260 --> 50:20.450
They're going to be huge waves
if you're talking about water.

50:20.449 --> 50:24.149
On the other hand if this is
equal to Π/2,

50:24.150 --> 50:27.730
3Π/2 etcetera,
plus or minus,

50:27.730 --> 50:32.200
then there's going to be dark,
that means that δ

50:32.197 --> 50:38.147
is equal to λ/2,
3λ/2 etcetera.

50:38.150 --> 50:43.380
So there will be bright and
dark, and bright and dark

50:43.376 --> 50:45.786
fringes on the screen.

50:45.789 --> 50:49.839
If you go to the screen here
the midpoint will be bright,

50:49.842 --> 50:53.752
but it'll be oscillating like
that many, many times.

50:53.750 --> 50:56.870
This was the experiment that
was done by Young.

50:56.869 --> 51:02.969
Young did the experiment with
light and showed that it has an

51:02.971 --> 51:05.311
interference pattern.

51:05.309 --> 51:08.459
He did not know what light was.

51:08.460 --> 51:10.080
He did not produce
electromagnetic waves,

51:10.081 --> 51:11.461
but you don't need to know
that.

51:11.460 --> 51:13.150
You shine the light,
you get the dark and bright,

51:13.150 --> 51:16.080
and dark and bright,
and I will tell you in a minute

51:16.083 --> 51:19.713
how we can find the wavelength
of light from this experiment.

51:19.710 --> 51:22.410
You can probably guess,
but I'll do that in a minute.

51:22.409 --> 51:26.109
He got the wavelength of light,
but he didn't know what it is

51:26.110 --> 51:29.070
that was vibrating,
but he got the wavelength.

51:29.070 --> 51:34.310
So let's ask ourselves exactly
where on this screen I will get

51:34.309 --> 51:37.659
dark and bright,
and dark and bright.

51:37.659 --> 51:41.279
So for that you've got to do a
little more geometry now.

51:41.280 --> 51:46.850
So let's do that geometry here.

51:46.849 --> 51:51.209
So here is the double slit.

51:51.210 --> 51:53.680
I want to go in some direction.

51:53.679 --> 51:57.219
I'm assuming the screen is so
far away that the rays can be

51:57.224 --> 51:58.574
treated as parallel.

51:58.570 --> 52:03.090
And I want to go at an angle
theta so that these lengths are

52:03.086 --> 52:03.696
equal.

52:03.699 --> 52:09.539
This is the extra length delta
here, and this is the distance

52:09.541 --> 52:11.881
between the two slits.

52:11.880 --> 52:17.360
So you can see delta is equal
to dsinθ,

52:17.360 --> 52:20.320
because this angle θ
between the horizontal and the

52:20.324 --> 52:23.134
ray is the same as the
perpendicular of the horizontal

52:23.130 --> 52:25.090
and the perpendicular to the
ray.

52:25.090 --> 52:27.060
It's at that angle.

52:27.059 --> 52:30.339
So dsinθ is
δ,

52:30.340 --> 52:36.230
and that had to be equal to
either 0 or an integer multiple

52:36.233 --> 52:41.803
of λ for a maximum,
or equal to plus or minus

52:41.804 --> 52:47.604
λ/2,
plus or minus 3λ

52:47.603 --> 52:55.123
over 2 half wavelengths for a
minimum.

52:55.119 --> 52:57.419
Can you imagine doing this
calculation?

52:57.420 --> 52:59.300
If someone says,
"Here is light of some

52:59.297 --> 53:02.087
wavelength,
5,000 angstroms or 500

53:02.092 --> 53:07.352
nanometers, the slits are
separated by some microns or

53:07.349 --> 53:11.019
whatever,"
then you can say,

53:11.018 --> 53:15.368
"What's the angle θ
at which I will get a maximum

53:15.373 --> 53:16.483
or a minimum?

53:16.480 --> 53:18.750
Well, you can already do this
calculation.

53:18.750 --> 53:20.670
If you wanted the first maximum
you said

53:20.672 --> 53:23.042
dsinθ equal
to wavelength.

53:23.039 --> 53:25.729
You solve for sine θ
and look at a calculator,

53:25.733 --> 53:27.373
it'll tell you what theta is.

53:27.369 --> 53:29.789
Then you can go and see,
"Maybe I can get two

53:29.788 --> 53:31.758
lambda,"
then that can also give you

53:31.762 --> 53:32.752
another θ.

53:32.750 --> 53:36.010
But remember,
it cannot go on forever because

53:36.012 --> 53:39.752
at some point sine theta,
which will be some multiple of

53:39.751 --> 53:42.081
λ/d where m is an
integer,

53:42.079 --> 53:44.529
if it's bigger than 1 no luck.

53:44.530 --> 53:47.740
Sine theta cannot be bigger
than 1.

53:47.739 --> 53:53.769
So you'll get some number of
this maxima then it'll stop.

53:53.768 --> 53:57.488
That's because if you've got
two signals coming from here to

53:57.494 --> 54:00.844
here only if you go in this
direction will you get the

54:00.840 --> 54:03.480
maximum phase,
the maximum path difference if

54:03.481 --> 54:06.051
you go straight up,
and if that's not enough to be

54:06.047 --> 54:08.587
some multiple of λ
there's no use going any

54:08.590 --> 54:11.290
further because theta you can
get is 90 degrees,

54:11.289 --> 54:14.369
and the biggest path length you
can get is then the space

54:14.373 --> 54:17.703
between the slits d,
and your multiple wavelength

54:17.695 --> 54:19.305
should fit into that.

54:19.309 --> 54:22.899
So I hope all of you understand
this very simple experiment,

54:22.902 --> 54:24.672
the double slit experiment.

54:24.670 --> 54:26.700
So light comes from the left.

54:26.699 --> 54:29.019
These two points act like
sources of light.

54:29.018 --> 54:31.538
They send out spherical or
circular waves.

54:31.539 --> 54:35.749
You go to a screen far away and
you add the contribution from

54:35.748 --> 54:36.448
the two.

54:36.449 --> 54:39.589
If this and that differ in the
path length by an integer number

54:39.594 --> 54:41.374
of wavelengths it doesn't
matter.

54:41.369 --> 54:43.639
They will arrive in step.

54:43.639 --> 54:45.869
I mean, the ninety-sixth
maximum will come when the

54:45.871 --> 54:48.461
ninety-fifth one from this one
comes, but we don't care.

54:48.460 --> 54:50.390
A maximum is a maximum.

54:50.389 --> 54:55.549
If they differ by half a
wavelength or three halves

54:55.552 --> 55:01.342
wavelengths then they will
cancel, and what I said on the

55:01.335 --> 55:04.635
top also goes on the bottom.

55:04.639 --> 55:09.079
Now sometimes there is a little
extra geometry you can do which

55:09.079 --> 55:10.439
is the following.

55:10.440 --> 55:12.910
This may be some homework
problem.

55:12.909 --> 55:19.549
If you put a screen here at the
distance L and the

55:19.548 --> 55:25.148
patterns look like this,
this is the central maximum,

55:25.148 --> 55:27.838
and I ask you,
"Where is the central

55:27.838 --> 55:29.918
minimum measured from
here," well,

55:29.920 --> 55:32.320
you call that distance as
y.

55:32.320 --> 55:36.210
Then you see y/L is
tanθ and

55:36.211 --> 55:41.101
θ should be such
that dsinθ

55:41.099 --> 55:42.589
is λ/2.

55:42.590 --> 55:44.150
Are you with me?

55:44.150 --> 55:46.320
You find the directions of
minima.

55:46.320 --> 55:49.160
Once you find the found the
direction of a maximum or a

55:49.155 --> 55:51.775
minimum it is simple
trigonometry to find that if a

55:51.782 --> 55:54.992
screen is L meters away
how much do you have to move in

55:54.985 --> 55:56.505
the vertical direction.

55:56.510 --> 55:57.820
That's very easy trig.

55:57.820 --> 55:59.290
You can calculate and you can
predict.

55:59.289 --> 56:03.419
That's what Young did because
he knew the distance between the

56:03.418 --> 56:05.178
maximum and the minimum.

56:05.179 --> 56:07.059
He knew how far the screens
were.

56:07.059 --> 56:09.159
He knew the slit separation.

56:09.159 --> 56:14.029
All he did not know was the
wavelength and he got the

56:14.027 --> 56:19.267
wavelength without knowing what
the waves are made of.

56:19.268 --> 56:24.968
Now you can see why you got
fooled in the old days.

56:24.969 --> 56:32.909
If y/L is--so let's make an
estimation of how far apart

56:32.909 --> 56:39.379
these maxima and minima are in
this graph.

56:39.380 --> 56:44.490
Since they repeat let me just
take the first minimum,

56:44.492 --> 56:49.312
and that's a rough measure of
the oscillations.

56:49.309 --> 56:51.579
How far apart are these maxima
and minima?

56:51.579 --> 56:57.379
Well, the first one occurs when
dsinθ is

56:57.378 --> 57:01.188
equal to λ/2,
or sinθ,

57:01.186 --> 57:05.236
which is roughly going to be
θ for small angles is

57:05.239 --> 57:06.859
λ/2d.

57:06.860 --> 57:10.200
Now if you pick any λ
you like, visible light is

57:10.204 --> 57:11.324
5,000 angstroms.

57:11.320 --> 57:16.850
That is 5000�10^(-8
)centimeters.

57:16.849 --> 57:22.099
And suppose your screen is 1
centimeter away?

57:22.099 --> 57:31.379
Then you get 5�10^(-3).

57:31.380 --> 57:34.810
So it's roughly one-thousandth
of a radian is the difference

57:34.809 --> 57:37.019
between the maximum and the
minimum.

57:37.018 --> 57:42.338
The interference pattern,
even though I've shown it this

57:42.338 --> 57:47.268
way here, actually oscillates
very, very rapidly.

57:47.268 --> 57:49.888
Okay, the angle of difference
between the maximum and a

57:49.885 --> 57:52.645
minimum and a maximum and a
minimum is one-thousandth of a

57:52.646 --> 57:53.176
radian.

57:53.179 --> 57:57.549
So even if you go a centimeter
away you will find that your

57:57.545 --> 58:01.605
probes, like your eyeball,
if your eyeball is this big,

58:01.612 --> 58:02.292
okay?

58:02.289 --> 58:04.689
It's going to pick up so many
of these waves at one time.

58:04.690 --> 58:08.820
It cannot tell the maximum and
minimum.

58:08.820 --> 58:12.380
That's why we think there is no
interference because you only

58:12.376 --> 58:13.856
pick the average value.

58:13.860 --> 58:22.210
That's why you have to do very
careful measurements.

58:22.210 --> 58:35.190
Okay, so that is double slit.

58:35.190 --> 58:39.350
Okay, here's another thing
which will be very useful for

58:39.351 --> 58:43.591
people who are doing biology,
or maybe even chemistry.

58:43.590 --> 58:47.940
Let's call it diffraction
grading.

58:47.940 --> 58:51.960
In a diffraction grading you
take a piece of opaque glass

58:51.963 --> 58:56.133
covered with black soot then you
draw fine lines on it,

58:56.130 --> 59:00.310
evenly spaced lines so that the
effect is the following.

59:00.309 --> 59:05.029
This whole thing is dark except
you've made some tiny places

59:05.030 --> 59:09.350
where light can come through
like that, and the light's

59:09.349 --> 59:11.029
coming like this.

59:11.030 --> 59:11.610
Are you with me?

59:11.610 --> 59:16.350
This is my solid screen with a
few places through which light

59:16.349 --> 59:20.379
can come, and every one of them
becomes a source.

59:20.380 --> 59:23.520
It's like a double slit
experiment except you've got

59:23.518 --> 59:25.798
million of lines,
and you want to ask,

59:25.797 --> 59:27.887
"What will they do?"

59:27.889 --> 59:31.299
In the forward direction if you
put an object,

59:31.302 --> 59:35.552
if you put a screen very far
away that's roughly infinity

59:35.550 --> 59:38.660
they all take the same path
distance.

59:38.659 --> 59:40.399
They all go to the same
distance.

59:40.400 --> 59:42.860
Therefore they will all be in
step.

59:42.860 --> 59:46.440
So in the forward direction you
will get a big maximum,

59:46.438 --> 59:50.278
because for every color the
path is the same in the forward

59:50.282 --> 59:51.212
direction.

59:51.210 --> 59:54.390
But think of another direction
which is not the forward

59:54.391 --> 59:55.101
direction.

59:55.099 --> 59:58.629
It's some other angle.

59:58.630 --> 1:00:01.970
Then this guy and this guy are
out of step by an amount

1:00:01.971 --> 1:00:02.591
δ.

1:00:02.590 --> 1:00:04.860
This guy and this guy are out
of step by an amount δ.

1:00:04.860 --> 1:00:07.900
This guy and this guy are out
of step by 2δ,

1:00:07.900 --> 1:00:11.210
so all the sources add up at
face difference of δ,

1:00:11.210 --> 1:00:13.530
2δ, 3δ,
4δ and so on.

1:00:13.530 --> 1:00:16.660
And suppose you want them all
to be still coherent.

1:00:16.659 --> 1:00:19.559
What will you require?

1:00:19.559 --> 1:00:26.139
What will ensure that all those
things are coherent in a certain

1:00:26.144 --> 1:00:27.404
direction?

1:00:27.400 --> 1:00:30.090
Any guess on what δ
should be?

1:00:30.090 --> 1:00:30.760
Yep?

1:00:30.760 --> 1:00:32.440
Student:  It's equal to
the wavelength.

1:00:32.440 --> 1:00:34.890
Prof: If δ
is equal to the wavelength this

1:00:34.891 --> 1:00:36.801
guy and this guy are a
wavelength apart.

1:00:36.800 --> 1:00:39.380
We already know they are in
step from the double slit

1:00:39.376 --> 1:00:40.016
experiment.

1:00:40.018 --> 1:00:42.408
This guy and this guy are a
wavelength apart.

1:00:42.409 --> 1:00:44.439
This guy and this guy are two
wavelengths apart.

1:00:44.440 --> 1:00:46.490
That is just fine.

1:00:46.489 --> 1:00:50.379
Therefore if you pick a
direction in which that delta is

1:00:50.380 --> 1:00:54.410
either a wavelength or an
integer multiple of a wavelength

1:00:54.411 --> 1:00:57.171
all the things will add
coherently.

1:00:57.170 --> 1:01:01.240
Therefore the favorable
direction for a bright maximum

1:01:01.244 --> 1:01:02.684
satisfy,
again,

1:01:02.681 --> 1:01:06.921
dsinθ =
mλ

1:01:06.916 --> 1:01:12.036
where m can be 0 plus or minus
1 plus or minus 2.

1:01:12.039 --> 1:01:16.159
These are the maximum.

1:01:16.159 --> 1:01:20.449
Except here in the forward
direction where θ

1:01:20.447 --> 1:01:23.947
is equal to 0--okay,
here's the point.

1:01:23.949 --> 1:01:26.929
Let us find the first maximum
away from the center.

1:01:26.929 --> 1:01:30.899
The center is not even counted,
center being white.

1:01:30.900 --> 1:01:34.540
All the colors will come to the
forward direction and you will

1:01:34.541 --> 1:01:38.011
get a big bright area here,
but look at what happens to the

1:01:38.005 --> 1:01:39.255
different colors.

1:01:39.260 --> 1:01:41.880
This condition,
let's take m = 1,

1:01:41.880 --> 1:01:44.500
dsinθ =
λ.

1:01:44.500 --> 1:01:46.800
Whose λ
are we talking about,

1:01:46.802 --> 1:01:47.332
right?

1:01:47.329 --> 1:01:49.959
Red light is one λ,
blue light's another λ.

1:01:49.960 --> 1:01:52.860
So each one will pick a
different λ,

1:01:52.855 --> 1:01:56.985
each one will pick a different
direction for this first order

1:01:56.992 --> 1:01:57.822
maximum.

1:01:57.820 --> 1:02:01.110
So that means the white light,
when it comes in this

1:02:01.110 --> 1:02:03.950
direction, will be split into
many colors.

1:02:03.949 --> 1:02:08.279
It is just that they all shrink
to one line here in the center,

1:02:08.284 --> 1:02:11.294
but any other place the lines
will split.

1:02:11.289 --> 1:02:14.549
That's because λ
depends on the color;

1:02:14.550 --> 1:02:17.630
d is, of course,
the spacing between these

1:02:17.628 --> 1:02:20.858
little lines on the grating,
so sine θ

1:02:20.856 --> 1:02:23.106
is a function of λ.

1:02:23.110 --> 1:02:24.240
So it acts like a prism.

1:02:24.239 --> 1:02:26.059
It splits the colors.

1:02:26.059 --> 1:02:30.139
So this is called the first
order maximum.

1:02:30.139 --> 1:02:32.649
Between this and that will be a
minimum.

1:02:32.650 --> 1:02:33.820
Then there'll be another
minimum.

1:02:33.820 --> 1:02:34.770
Then there's a maximum.

1:02:34.768 --> 1:02:38.298
It's harder to calculate the
minimum, very easy to calculate

1:02:38.300 --> 1:02:39.140
the maximum.

1:02:39.139 --> 1:02:40.579
So it's easy to calculate the
maximum.

1:02:40.579 --> 1:02:42.219
That's when everybody has to
add.

1:02:42.219 --> 1:02:46.019
Minimum is more complicated
because everybody has to cancel.

1:02:46.018 --> 1:02:48.168
You've got to add the little
arrows in different directions

1:02:48.168 --> 1:02:49.168
and ask when they cancel.

1:02:49.170 --> 1:02:50.700
You can do it,
but it's a little more

1:02:50.695 --> 1:02:51.285
complicated.

1:02:51.289 --> 1:02:54.549
But we can sort of imagine that
between the maxima would lie the

1:02:54.550 --> 1:02:56.620
minima, but the colors will get
split.

1:02:56.619 --> 1:03:00.369
That's important.

1:03:00.369 --> 1:03:03.889
So this is how people find out.

1:03:03.889 --> 1:03:07.419
So if you send white light it
will split into all the colors.

1:03:07.420 --> 1:03:10.970
If you look at white light
coming from the sun you'll find

1:03:10.972 --> 1:03:14.842
some colors are missing because
what happens is hydrogen in the

1:03:14.836 --> 1:03:18.576
surface of the sun will absorb
some of the colors because its

1:03:18.577 --> 1:03:20.817
atomic--
its atom likes to absorb light

1:03:20.824 --> 1:03:21.744
at a certain color.

1:03:21.739 --> 1:03:25.479
That color is missing,
missing in the spectrum.

1:03:25.480 --> 1:03:27.800
So in this whole area you'll
have a little cavity.

1:03:27.800 --> 1:03:32.260
Somebody is missing and you can
easily tell from the wavelength

1:03:32.257 --> 1:03:35.347
that it's some emission line of
hydrogen.

1:03:35.349 --> 1:03:38.239
That tells you the sun contains
hydrogen.

1:03:38.239 --> 1:03:40.469
That's how people know what
elements are sitting on

1:03:40.467 --> 1:03:41.357
different planets.

1:03:41.360 --> 1:03:44.070
How does anybody know what
they're made of?

1:03:44.070 --> 1:03:47.660
It was not at all clear in the
ancient days that stars and

1:03:47.661 --> 1:03:50.311
planets are made of the same
thing you are,

1:03:50.309 --> 1:03:50.939
right?

1:03:50.940 --> 1:03:52.990
People thought moon was made of
cheese.

1:03:52.989 --> 1:03:55.839
Well, that's pretty much the
same thing that you are,

1:03:55.840 --> 1:03:57.320
but something else,
okay?

1:03:57.320 --> 1:03:57.940
But they didn't know.

1:03:57.940 --> 1:04:00.510
They thought that heavenly
bodies and earthly bodies,

1:04:00.510 --> 1:04:03.970
but now we find it's the same
elements because you can find

1:04:03.971 --> 1:04:07.131
the spectra lines either by
their presence or by their

1:04:07.132 --> 1:04:07.852
absence.

1:04:07.849 --> 1:04:12.069
So this is the kind of
spectroscopy you do.

1:04:12.070 --> 1:04:16.790
Okay, now I'm going to do one
other thing which is extremely

1:04:16.789 --> 1:04:17.749
important.

1:04:17.750 --> 1:04:22.310
That is to explain why when
light comes through a hole of

1:04:22.307 --> 1:04:27.437
some size d instead of forming a
shadow like this it spreads out

1:04:27.436 --> 1:04:30.526
a lot into something much
broader.

1:04:30.530 --> 1:04:31.920
So let's understand that, okay?

1:04:31.920 --> 1:04:35.970
Here is light coming in.

1:04:35.969 --> 1:04:40.119
Now what you can do is this
aperture is going to be bright.

1:04:40.119 --> 1:04:42.519
When you see it from this side
it'll be glowing.

1:04:42.518 --> 1:04:48.828
So let's treat every point of
it as a source of light.

1:04:48.829 --> 1:04:52.189
In the forward direction
everybody's in step,

1:04:52.190 --> 1:04:56.930
but if you go off at some angle
they're not in step because you

1:04:56.934 --> 1:04:59.924
can see this guy is behind that
guy,

1:04:59.920 --> 1:05:01.290
and that guy's behind by that
amount.

1:05:01.289 --> 1:05:05.909
Each one is somewhat behind the
leader.

1:05:05.909 --> 1:05:11.239
Now my claim that if this
number here is λ,

1:05:11.239 --> 1:05:14.719
what do you think will happen?

1:05:14.719 --> 1:05:18.979
In that direction if that
spacing is λ

1:05:18.976 --> 1:05:23.326
is that going to be a maximum
or a minimum?

1:05:23.329 --> 1:05:25.119
Anybody want to guess?

1:05:25.119 --> 1:05:27.089
Pardon me?

1:05:27.090 --> 1:05:27.930
Yes?

1:05:27.929 --> 1:05:28.719
Student:  Minimum.

1:05:28.719 --> 1:05:30.739
Prof: And on what do you
base that?

1:05:30.739 --> 1:05:36.649
Student:  Because the
λ/2 ________ the minimum,

1:05:36.650 --> 1:05:37.440
and...

1:05:37.440 --> 1:05:39.660
Prof: But this thing is
λ.

1:05:39.659 --> 1:05:42.539
Usually λ
between two holes was

1:05:42.538 --> 1:05:45.338
considered good,
but now it's bad.

1:05:45.340 --> 1:05:47.870
The reason is that it's just
not the top guy and the bottom

1:05:47.867 --> 1:05:48.127
guy.

1:05:48.130 --> 1:05:51.110
It's all these little guys in
between.

1:05:51.110 --> 1:05:54.800
So if this and that differ by
λ, by similar triangles,

1:05:54.804 --> 1:05:56.964
you see that's λ
over two?

1:05:56.960 --> 1:06:01.380
So this fellow will cancel that
fellow, and the second guy will

1:06:01.376 --> 1:06:04.436
be canceled by the second guy
from below.

1:06:04.440 --> 1:06:06.330
So when the whole phase shift
is λ,

1:06:06.329 --> 1:06:07.879
or the path difference is
λ,

1:06:07.880 --> 1:06:11.450
you can pair the points into
two at a time each differing by

1:06:11.445 --> 1:06:14.645
λ/2 and we know they will
separately cancel.

1:06:14.650 --> 1:06:17.910
So for a single slit the
fraction

1:06:17.905 --> 1:06:24.105
dsinθ =
λ is the first minimum,

1:06:24.110 --> 1:06:26.180
and after that some funny
things happen which are

1:06:26.177 --> 1:06:30.757
negligible,
but the light spreads out to

1:06:30.762 --> 1:06:32.472
that angle.

1:06:32.469 --> 1:06:35.949
And this is the diffraction of
light by an opening.

1:06:35.949 --> 1:06:36.729
And let's look at it.

1:06:36.730 --> 1:06:41.550
Sine θ
is λ/d.

1:06:41.550 --> 1:06:46.420
Suppose λ
is 10^(-4) and d is 1?

1:06:46.420 --> 1:06:49.680
Sine θ
= theta 10^(-4) that means the

1:06:49.675 --> 1:06:53.855
angular spread is 1 over
ten-thousandth of a radian.

1:06:53.860 --> 1:06:56.660
That's the extent to which
light won't go straight.

1:06:56.659 --> 1:06:58.589
It'll spread out,
but the spread is 1 over

1:06:58.592 --> 1:06:59.962
ten-thousandth of a radian.

1:06:59.960 --> 1:07:02.250
It's very small.

1:07:02.250 --> 1:07:05.520
But take a case where λ
and d become comparable,

1:07:05.519 --> 1:07:12.009
maybe d is equal to 2λ,
then sine θ is one-half.

1:07:12.010 --> 1:07:13.710
That's 30 degrees.

1:07:13.710 --> 1:07:16.510
It's already beginning to
spread out.

1:07:16.510 --> 1:07:20.000
That's why I said earlier that
if you want to see the effect of

1:07:19.996 --> 1:07:23.256
wave optics you need to have the
length in the problem,

1:07:23.260 --> 1:07:26.000
which is the size of the hole,
and the wavelength in the

1:07:25.996 --> 1:07:27.736
problem comparable to each
other.

1:07:27.739 --> 1:07:30.829
If the wavelength is much
smaller than the dimensions of

1:07:30.831 --> 1:07:33.981
your experiment you can get away
with the ray optics,

1:07:33.980 --> 1:07:38.590
and you need wave optics only
when the condition is not met.

1:07:38.590 --> 1:07:41.860
So I remind you one more time
why people thought ray optics

1:07:41.858 --> 1:07:42.478
was good.

1:07:42.480 --> 1:07:45.250
They were not able to produce
slits which are this narrow.

1:07:45.250 --> 1:07:47.720
Most slits you produce by
drilling a hole with your

1:07:47.719 --> 1:07:50.489
electric drill will be many,
many times the wavelength of

1:07:50.485 --> 1:07:50.975
light.

1:07:50.980 --> 1:07:54.200
That means its angular spread
will be slightly non-zero,

1:07:54.201 --> 1:07:57.481
but instead of a shadow that
big you'll get a shadow that

1:07:57.481 --> 1:07:57.951
big.

1:07:57.949 --> 1:07:59.099
You won't know.

1:07:59.099 --> 1:08:01.079
Similarly the interference
pattern,

1:08:01.079 --> 1:08:05.139
the interference pattern for
microscopic separations between

1:08:05.137 --> 1:08:09.327
the slits will be so tight that
your eye cannot detect it even

1:08:09.331 --> 1:08:13.081
if you use light of one color,
and if you use light of many

1:08:13.083 --> 1:08:15.773
colors it'll wash out anyway
because each lambda likes to

1:08:15.769 --> 1:08:19.669
peak at a different angle,
so different colors would wipe

1:08:19.672 --> 1:08:20.332
it out.

1:08:20.328 --> 1:08:23.538
So I wanted to let you know why
it is that people believed in

1:08:23.539 --> 1:08:26.479
geometric optics in the
beginning because they didn't do

1:08:26.483 --> 1:08:27.343
experiments.

1:08:27.340 --> 1:08:29.250
This is why every theory will
fall.

1:08:29.250 --> 1:08:31.600
One day somebody has the
technology to produce slits

1:08:31.600 --> 1:08:33.830
which are very,
very near and to get light

1:08:33.832 --> 1:08:36.802
which is very pure of a definite
color then you will see

1:08:36.801 --> 1:08:37.721
something new.

1:08:37.720 --> 1:08:40.010
Then you've got to figure out
what it is.

1:08:40.010 --> 1:08:41.950
But when Young did the
experiment and saw the

1:08:41.954 --> 1:08:44.524
interference they knew right
away it's waves because people

1:08:44.520 --> 1:08:48.740
had played with water waves,
and they knew with water waves

1:08:48.742 --> 1:08:51.692
that you can have interference.

1:08:51.689 --> 1:08:56.189
Okay, here's another thing you
can do.

1:08:56.189 --> 1:09:00.869
Here is a bunch of atoms at the
surface of some metal.

1:09:00.868 --> 1:09:06.968
Metal's down here,
and I'm shining light.

1:09:06.970 --> 1:09:08.830
You know how reflection works?

1:09:08.829 --> 1:09:10.279
The light hits the atom.

1:09:10.279 --> 1:09:13.609
Atom absorbs the light and it
reemits the light.

1:09:13.609 --> 1:09:14.919
That's what reflection is.

1:09:14.920 --> 1:09:16.820
You don't just bounce off.

1:09:16.819 --> 1:09:19.609
You get absorbed and get
reemitted.

1:09:19.609 --> 1:09:22.999
Now the problem is if you
reemit you reemit in all

1:09:23.000 --> 1:09:23.900
directions.

1:09:23.899 --> 1:09:26.389
You don't just reemit in one
direction.

1:09:26.390 --> 1:09:30.360
The question you can ask is,
why is it that i = r for

1:09:30.355 --> 1:09:31.225
reflection?

1:09:31.229 --> 1:09:34.239
That's what I want to explain
to you.

1:09:34.239 --> 1:09:35.879
So this guy emits something.

1:09:35.880 --> 1:09:37.230
There's a wave emanating from
this.

1:09:37.229 --> 1:09:39.979
This guy emits something,
wave emanating from that.

1:09:39.979 --> 1:09:46.539
So let's take two rays going
off in some direction.

1:09:46.538 --> 1:09:52.108
And let's just ask when these
two guys will be in step in

1:09:52.105 --> 1:09:54.885
terms of the emitted wave.

1:09:54.890 --> 1:09:57.950
So you see the light signal is
coming from here.

1:09:57.949 --> 1:10:00.139
It hits this one first.

1:10:00.140 --> 1:10:01.940
Are you with me?

1:10:01.939 --> 1:10:03.419
So this emits right way.

1:10:03.420 --> 1:10:06.730
Its signal has started taking
off in the final direction.

1:10:06.729 --> 1:10:09.689
This guy is delayed because it
hasn't hit this,

1:10:09.689 --> 1:10:13.939
but once they start moving
toward the eye this is closer

1:10:13.936 --> 1:10:16.736
than this one,
so you see this one and that

1:10:16.742 --> 1:10:19.142
one, if you want to make a real
comparison--

1:10:19.140 --> 1:10:20.410
well, this is bad picture.

1:10:20.408 --> 1:10:23.458
Let me draw you another picture
where there's a good competition

1:10:23.462 --> 1:10:24.192
between them.

1:10:24.189 --> 1:10:25.839
So let's see.

1:10:25.840 --> 1:10:30.340
Suppose I had a really bad idea
and I thought they reflect like

1:10:30.341 --> 1:10:30.851
this.

1:10:30.850 --> 1:10:32.470
I'll show you that won't work.

1:10:32.470 --> 1:10:34.380
That's all I want to do.

1:10:34.380 --> 1:10:39.010
You see what you should do is
draw perpendicular here and

1:10:39.006 --> 1:10:42.636
that's the extra distance this
has to travel,

1:10:42.643 --> 1:10:43.473
right?

1:10:43.470 --> 1:10:45.360
And you will draw a
perpendicular here and let's

1:10:45.363 --> 1:10:47.663
see, what's the extra distance
the other has to travel?

1:10:47.658 --> 1:10:50.598
It has to travel that extra
distance.

1:10:50.600 --> 1:10:54.030
So if you draw those two angles
you will find--let's see which

1:10:54.033 --> 1:10:55.333
angle I want to take.

1:10:55.328 --> 1:10:58.268
If you call this alpha you'll
find d cosine

1:10:58.270 --> 1:11:01.870
α_1 is d
cosine α_2 or

1:11:01.868 --> 1:11:04.628
α_1 =
α_2.

1:11:04.630 --> 1:11:08.010
That means I = r.

1:11:08.010 --> 1:11:08.790
You see that?

1:11:08.788 --> 1:11:11.478
Maybe I should repeat the
picture here so everybody can

1:11:11.479 --> 1:11:11.779
see.

1:11:11.779 --> 1:11:15.459
I just need two of them.

1:11:15.460 --> 1:11:17.950
So one let me draw like this.

1:11:17.948 --> 1:11:23.148
Let me draw the second at what
is definitely not i = r.

1:11:23.149 --> 1:11:27.439
So here I'm saying draw that
perpendicular and draw that

1:11:27.440 --> 1:11:28.690
perpendicular.

1:11:28.689 --> 1:11:32.009
Starting from infinity these
two have the same length,

1:11:32.011 --> 1:11:34.081
those two have the same length.

1:11:34.078 --> 1:11:36.558
This is an extra length
traveled by this guy.

1:11:36.560 --> 1:11:39.800
That's the extra length
traveled by that guy,

1:11:39.795 --> 1:11:40.305
okay?

1:11:40.310 --> 1:11:43.090
So if you call that angle
alpha, you call that angle

1:11:43.085 --> 1:11:45.685
β,
and if this is d,

1:11:45.694 --> 1:11:49.054
you can see that d
cosine α

1:11:49.046 --> 1:11:54.376
must be d cosine β
or α should be β.

1:11:54.380 --> 1:11:55.760
That is a very interesting
result.

1:11:55.760 --> 1:11:59.260
It tells you why i = r
is true.

1:11:59.260 --> 1:12:02.300
See, if light is made up of
particles that bounce off a wall

1:12:02.296 --> 1:12:05.276
and ricochet with i = r
you can understand that from

1:12:05.280 --> 1:12:06.620
conservation momentum.

1:12:06.618 --> 1:12:09.898
But light is a wave and when
the atoms absorb light they emit

1:12:09.900 --> 1:12:12.740
in all directions,
but only in this direction will

1:12:12.743 --> 1:12:15.973
everybody be in step because
once I made sure these two guys

1:12:15.966 --> 1:12:19.296
are in step you can repeat it
for these two and those two.

1:12:19.300 --> 1:12:24.730
You'll find that's the
direction in which every single

1:12:24.728 --> 1:12:26.468
guy is in step.

1:12:26.470 --> 1:12:29.400
So i = r is the
direction in which the

1:12:29.404 --> 1:12:33.344
re-radiated light arrives in
step in only one direction,

1:12:33.340 --> 1:12:36.160
and that's the direction with
the angle of incidence is the

1:12:36.163 --> 1:12:37.433
angle of the reflection.

1:12:37.430 --> 1:12:40.580
You understand why there is a
difference?

1:12:40.578 --> 1:12:44.538
Okay, one last thing I should
mention, because I've given a

1:12:44.542 --> 1:12:47.762
problem for you,
and I just want to tell you one

1:12:47.755 --> 1:12:49.595
trick you may not know.

1:12:49.600 --> 1:12:50.250
And you can read up.

1:12:50.250 --> 1:12:52.590
The book is very good in that.

1:12:52.590 --> 1:12:58.010
If you've got some oil slick on
a rainy day, there's water here.

1:12:58.010 --> 1:13:00.130
This is an oil slick.

1:13:00.130 --> 1:13:02.090
You see all kind of colors in
the oil slick?

1:13:02.090 --> 1:13:04.340
This is to explain why it is.

1:13:04.340 --> 1:13:09.200
So white light comes or any
light comes, bounces off the top

1:13:09.195 --> 1:13:10.015
surface.

1:13:10.020 --> 1:13:13.350
The light also bounces off the
bottom surface.

1:13:13.350 --> 1:13:19.310
And the sum of those two hit
your eye.

1:13:19.310 --> 1:13:23.210
Suppose you want to arrange it
so that blue light cancels out

1:13:23.206 --> 1:13:24.306
in the process?

1:13:24.310 --> 1:13:27.400
What will it take to cancel?

1:13:27.399 --> 1:13:30.589
I claim first of all the
wavelength in this medium is

1:13:30.592 --> 1:13:32.192
n times the wave.

1:13:32.189 --> 1:13:34.799
It's λ/n.

1:13:34.800 --> 1:13:37.540
If this is the wavelength here
the wavelength in a medium of

1:13:37.543 --> 1:13:39.733
refractive and n is
λ/n.

1:13:39.729 --> 1:13:43.189
If this whole thing is equal to
a full wavelength,

1:13:43.194 --> 1:13:46.734
this round trip,
I claim you'll get a dark spot.

1:13:46.729 --> 1:13:49.719
You may think that's wrong
because this guy's gone an extra

1:13:49.720 --> 1:13:50.650
full wavelength.

1:13:50.649 --> 1:13:53.869
That should be additive,
but it turns out that when

1:13:53.868 --> 1:13:57.598
light bounces off in going from
a rare to a dense medium it

1:13:57.601 --> 1:13:59.471
changes by an extra Π.

1:13:59.470 --> 1:14:02.940
At the other end when it goes
from dense to rare it doesn't.

1:14:02.939 --> 1:14:06.609
So you have to add an extra
Π to everything you thought.

1:14:06.609 --> 1:14:09.759
Then you will find out that
when this extra round trip is

1:14:09.762 --> 1:14:12.862
the full wavelength you get
destructive interference.

1:14:12.859 --> 1:14:15.419
So if that occurs for a
wavelength for blue light you

1:14:15.422 --> 1:14:18.182
take blue light out of white
light you get some yellowish

1:14:18.182 --> 1:14:19.122
greenish color.

1:14:19.118 --> 1:14:22.638
That's what you see in the oil
slick.

1:14:22.640 --> 1:14:25.370
And if the oil slick thickness
varies different colors will be

1:14:25.368 --> 1:14:27.158
missing from white at different
point.

1:14:27.158 --> 1:14:30.208
That's why you see all kinds of
different colors in the oil

1:14:30.208 --> 1:14:30.628
slick.

1:14:30.630 --> 1:14:32.720
So I agree I didn't do a good
job of this one,

1:14:32.721 --> 1:14:34.491
but this is the easiest thing
to me.

1:14:34.488 --> 1:14:36.928
The only thing to do,
when you do your homework,

1:14:36.930 --> 1:14:40.390
if you didn't read the section
don't forget there's an extra

1:14:40.389 --> 1:14:43.029
factor of Π
in the phase shift when you hit

1:14:43.027 --> 1:14:44.197
the hard medium.

1:14:44.199 --> 1:14:48.999