WEBVTT

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Shall we begin now?

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So as usual,
and as promised,

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I will tell you what we have
done so far, and that's all you

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really need to know,
to follow what's going to

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happen next.

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The first thing we learned is
if you're studying a particle

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living in one dimension,
that's all I'm going to do the

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whole time because it's
mathematically easier and there

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is not many new things you gain
by going to higher dimensions.

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There's a particle somewhere in
this one-dimensional universe.

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Everything you need to know
about that particle is contained

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in a function called the wave
function, and is denoted by the

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symbol ψ.

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By the way, everything I'm
doing now is called kinematics.

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In other words,
kinematics is a study of how to

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describe a system completely at
a given time.

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For example,
in classical mechanics for a

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single particle the complete
description of that particle is

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given by two things,
where is it,

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and what its momentum is.

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Dynamics is the question of how
this changes with time.

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If you knew all you could know
about the particle now can you

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predict the future?

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By predict the future we mean
can you tell me what x

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will be at a later time and
p will be at a later

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time.

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That's Newtonian mechanics.

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So the kinematics is just how
much do you need to know at a

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given time, just x and
p.

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Once you've got x and
p everything follows.

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As I mentioned to you,
the kinetic energy,

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for example,
which you write as

01:45.039 --> 01:47.989
p,
or in higher dimensions the

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angular momentum is some cross
product of position and

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momentum,
so you can get everything of

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interest just by giving the
position and momentum.

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I claim now the equivalent of
this pair of numbers in this

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quantum world is one function,
ψ(x).

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So it's a lot more information
than you had in classical

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physics.

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In classical physics two
numbers tell you the whole

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story.

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Quantum theory says "give
me a whole function",

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and we all know a function is
really infinite amount of

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information because at every
point x the function has a

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height and you've got to give me
all that.

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Only then you have told me
everything.

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That's the definition.

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And the point is ψ
can be real,

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ψ can be complex,
and sometimes ψ

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is complex.

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So we can ask,
you've got this function,

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you say it tells me everything
I can know, well,

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what can I find out from this
function?

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The first thing is that if you
took the absolute square of this

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function, that is the
probability density to find it

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at the point x.

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By that I mean if you multiply
both by some infinitesimal

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Δx that is the
probability that the particle

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will be between x and
x dx.

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That means that you take this
ψ and you square it.

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So you will get something that
will go to 0 here,

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go up, go to zero,
do something like that.

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This is ψ^(2),
and that's your probability

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density and what we mean by
density is,

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if the function p(x) has
an area with the x-axis

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like p(x)dx that's the
actual probability that if you

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look for this guy you'll find
him or her or it in this

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interval,
okay?

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So we will make the requirement
that the total probability to

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find it anywhere add up to 1.

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That is a convention
because--well,

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in some sense.

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It's up to you how you want to
define probability.

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You say, "What are the
odds I will get through this

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course, 50/50?"

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That doesn't add up to 1 that
adds up to 100,

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but it gives you the impression
the relative odds are equal.

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So you can always give odds.

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Jimmy the Greek may tell you
something, 7 is to 4 something's

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going to happen.

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They don't add up to 1 either.

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I mean, 7 divided by 11 is one
thing and 4 divided by 11 is the

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absolute probability.

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So in quantum theory the wave
function you're given need not

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necessarily have the property
that its square integral is 1,

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but you can rescale it by a
suitable number,

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I mean, if it's not 1,
but if it's 100 then you divide

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it by 10 and that function will
have a square integral of 1.

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That's the convention and it's
a convenience,

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and I will generally assume
that we have done that.

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And I also pointed out to you
that the function ψ

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and the function 3 times ψ
stand for the same situation in

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quantum mechanics.

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So this ψ
is not like any other ψ.

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And if ψ
is a water wave 3 inches versus

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30 inches are not the same
situation, they describe

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completely different things.

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But in quantum mechanics ψ
and a multiple of ψ

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have the same physics because
the same relative odds are

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contained in them.

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So given one ψ
you're free to multiply it by

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any number,
in fact, real or complex and

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that doesn't change any
prediction,

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so normally you multiply it by
that number which makes the

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square integral in all of space
equal to 1.

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Such a function is said to be
normalized.

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If it's normalized the
advantage is the square directly

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gives you the absolute
probability density and integral

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of that will give you 1.

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That's one thing we learned.

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You understand now?

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What are the possible functions
I can ascribe with the particle?

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Whatever you like within reason;
it's got to be a single value

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and it cannot have discontinuous
jumps.

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Beyond that anything you write
down is fine.

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That's like saying what are the
allowed positions,

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or allowed momentum for a
particle in classical mechanics?

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Anything, there are no
restrictions except x

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should be real and p should be
real.

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You can do what you want.

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Similarly all possible
functions describe possible

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quantum states.

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It's called a quantum state.

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It's this crazy situation where
you don't know where it is and

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you give the odds by squaring
ψ.

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That's called a quantum state
and it's given by a function

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ψ.

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All right, now I also said
there is one case where I know

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what's going on.

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So let me give you one other
case.

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Maybe I will ask you to give me
one case.

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The particle is known to be
very close to x = 5

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because I just saw it there and
ε later I know it's

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still got to be there because I
just saw it.

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Now what function will describe
that situation?

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You guys know this.

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Want to guess?

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What will ψ
look like so that the particle

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is almost certainly near
x = 5?

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Student:
>

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Prof: Yeah?

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Centered where?

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Student:  Centered at 5.

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Prof: At 5,
everybody agree with that?

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I mean the exact shape we don't
know.

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Maybe that's why you're
hesitating, but here is the

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possible function that describes
a particle that's location isn't

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known to within some accuracy.

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So one look at it,
it tells you,

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"Hey, this guy's close to
5."

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I agree that you can put a few
wiggles on it,

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or you can make it taller or
shorter if you change the shape

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a bit,
but roughly speaking here is

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what functions,
describing particles of

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reasonably well know location,
look like.

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They're centered at the point
which is the well-known

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location.

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On the other hand,
I'm going to call is ψ

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x = 5.

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That is a function,
and the subscript you put on

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the function is a name you give
the function.

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We don't go to a party and say,
"Hi, I am human."

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You say, "I'm so and
so," because that tells you

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a little more than whatever
species you belong to.

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Similarly these are all
normalizable wave functions,

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but x = 5 is one member
of the family,

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which means I'm peaked at
x = 5.

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Another function I mentioned is
the function

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ψ_p(x).

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That's the function that
describes a particle of momentum

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p.

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We sort of inferred that by
doing the double slit

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experiment.

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That function looks like this.

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Some number A times
e^(ipx/ℏ).

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Now you can no longer tell me
you have no feeling for these

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exponentials because it's going
to be all about the exponential.

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I've been warning you the whole
term.

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Get used to those complex
exponentials.

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It's got a real part.

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It's got an imaginary part,
but more natural to think of a

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complex number as having a
modulus and a phase,

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and I'm telling you it's a
constant modulus.

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I don't know what it is.

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But the phase factor should
look like ipx/ℏ.

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So if I wrote a function
e to the i times

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96x/ℏ,
and I said, "What's going

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on?"

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Well, that's a particle whose
momentum is 96.

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So the momentum is hidden in
the function right in the

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exponential.

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It's everything x of the
i, the x and the

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ℏ.

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Whatever is sitting there
that's the momentum.

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I am going to study such states
pretty much all of today.

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So let's say someone says,
"Look, I produced a

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particle in a state of momentum
p.

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Here it is.

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Let's normalize this guy."

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To normalize the guy you've got
to take ψ^(2) and you've got

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to take the dx and you've
got to get it equal to 1.

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If you take the absolute square
of this, A absolute

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square is some fixed number.

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I hope you all know the
absolute value of that is 1

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because that times its
conjugate,

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which is e^(−ipx/ℏ)
will just give you

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e^(0) which is 1.

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I want 1 times dx over
all of space to be equal to 1.

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That's a hopeless task because
you cannot pick an A to

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make that happen,
because all of space,

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the integral of dx over
all of space is the length of

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the universe you're living in,
and if that's infinite no

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finite A will do it.

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So that poses a mathematical
challenge, and people circumvent

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it in many ways.

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One is to say,
"Let's pretend our

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universe is large and
finite."

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It may even be the case because
we don't know.

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And I'm doing quantum mechanics
in which I'm fooling around in a

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tiny region like atoms and
molecules,

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and it really doesn't matter if
the universe even goes beyond

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this room.

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It goes beyond this room.

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It goes beyond the planet.

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It goes beyond the solar system.

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I grant you all that,
but I say allow me to believe

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that if it goes sufficiently far
enough it's a closed universe.

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So a closed universe is like
this.

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A closed one-dimensional
universe is a circle.

11:07.690 --> 11:11.000
In that universe if you throw a
rock it'll come back and hit you

11:10.995 --> 11:11.725
from behind.

11:11.730 --> 11:14.530
In fact you can see the back of
your head in this universe

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because everything goes around
in a circle.

11:16.750 --> 11:21.750
All right, so that's the world
you take.

11:21.750 --> 11:24.320
Now that looks kind of
artificial for the real world,

11:24.320 --> 11:27.000
which we all agree seems to be
miles and miles long,

11:27.000 --> 11:30.010
but I don't care for this
purpose what L is as long

11:30.010 --> 11:30.910
as it's finite.

11:30.908 --> 11:33.748
If L is finite,
any number you like,

11:33.750 --> 11:37.400
then the ψ
of p of x will

11:37.404 --> 11:41.674
look like 1/√L
e^(ipx/ℏ).

11:41.669 --> 11:43.119
Do you agree?

11:43.120 --> 11:46.240
If you take the absolute square
of this you'll get a 1/L.

11:46.240 --> 11:47.420
This'll become 1.

11:47.418 --> 11:51.428
The integral of 1 of L
over the length of space is just

11:51.427 --> 11:51.687
1.

11:51.690 --> 11:55.050
So that's the normalized wave
function.

11:55.048 --> 11:59.248
Now this is also a very
realistic thing if in practice

11:59.253 --> 12:03.223
your particle is restricted to
live in a circle.

12:03.220 --> 12:05.300
Again, there are a lot of
experiments being done,

12:05.298 --> 12:08.488
including at Yale,
where there's a tiny metallic

12:08.486 --> 12:10.406
ring,
a nano scale object,

12:10.407 --> 12:13.987
and the electrons are forced to
lived in that ring.

12:13.990 --> 12:18.240
So the ring has a radius and
this L is just 2ΠR

12:18.238 --> 12:20.588
where R is the radius.

12:20.590 --> 12:23.600
There L is very real.

12:23.600 --> 12:24.710
It's not something you cooked
up.

12:24.710 --> 12:26.950
It's the size of the ring,
but sometimes even if you're

12:26.951 --> 12:29.811
not doing particle in a ring,
if you're doing particle in a

12:29.807 --> 12:32.807
line you just pretend that the
line closes in on itself.

12:32.808 --> 12:36.528
That if you start at the origin
and you go on the two sides it

12:36.528 --> 12:37.258
closes in.

12:37.259 --> 12:41.739
If this is x = 0 you go
right and you go left they all

12:41.741 --> 12:44.881
meet at the back and it's a
closed ring.

12:44.879 --> 12:50.189
So let's imagine what life
looks like for a particle forced

12:50.193 --> 12:54.413
to live in a ring of
circumference L.

12:54.408 --> 12:59.218
The normalized wave function
looks like this of momentum

12:59.217 --> 13:00.177
p.

13:00.178 --> 13:06.268
This is 1/√L
e^(ipx/ℏ).

13:06.269 --> 13:11.109
The probability to find this
guy at some point x which

13:11.105 --> 13:15.615
is the absolute value of
ψ^(2) is just 1/L.

13:15.620 --> 13:18.220
That means the probability is
constant over the entire ring.

13:18.220 --> 13:19.370
We don't know where it is.

13:19.370 --> 13:22.360
You can find it anywhere with
equal probability.

13:22.360 --> 13:26.960
It's always true that if you
know the momentum you don't know

13:26.956 --> 13:28.026
where it is.

13:28.028 --> 13:35.198
Now let's ask one other
question.

13:35.200 --> 13:39.540
Here is the circle in which I'm
living.

13:39.538 --> 13:42.038
If you look at the real part
and imaginary part with their

13:42.042 --> 13:43.932
sines and cosines they kind of
oscillate.

13:43.929 --> 13:44.909
Do you understand that?

13:44.908 --> 13:46.918
When the x varies they
oscillate.

13:46.918 --> 13:50.768
And one requirement you make
that if you go all the way

13:50.774 --> 13:54.634
around and come back it's got to
close in on itself.

13:54.629 --> 13:58.659
Because if you increase
x from anywhere by an

13:58.658 --> 14:01.328
amount L,
which is going all the way

14:01.330 --> 14:03.350
around the circle,
you've got to come back to

14:03.354 --> 14:04.094
where you start.

14:04.090 --> 14:07.570
The function has to come back
where you started meaning it's a

14:07.565 --> 14:08.985
single valued function.

14:08.990 --> 14:11.500
If you say what's ψ
here you've got to get one

14:11.499 --> 14:11.959
number.

14:11.960 --> 14:13.190
That means if you start at some
point,

14:13.190 --> 14:15.450
you go on moving,
you follow the ψ,

14:15.450 --> 14:18.100
you go all the way around and
you come back you shouldn't have

14:18.104 --> 14:18.674
a mismatch.

14:18.669 --> 14:20.019
It should agree.

14:20.019 --> 14:24.669
So the only allowed functions
are those obeying the condition

14:24.673 --> 14:27.623
ψ(x L) =
ψ(x).

14:27.620 --> 14:29.670
You understand that?

14:29.668 --> 14:33.238
Take any point you like,
follow the function for a

14:33.241 --> 14:34.701
distance L.

14:34.700 --> 14:39.250
That means you come back and
the function better come back.

14:39.250 --> 14:43.980
So a function like e to
the −x^(2) is not a

14:43.976 --> 14:47.936
good function to go around a
circle and come back.

14:47.940 --> 14:51.320
If you go in real space,
if you go on all the way it

14:51.323 --> 14:53.783
doesn't come back to where you
are.

14:53.779 --> 14:56.889
So you've got the right
functions which have the

14:56.888 --> 15:00.788
property that when you add L to
them you come back where you

15:00.789 --> 15:01.319
are.

15:01.320 --> 15:09.290
That's a condition of single
valuedness.

15:09.288 --> 15:15.638
So that means in this function
if I take it at some point

15:15.635 --> 15:21.635
e^(ipx) at some point
x then I add to it

15:21.643 --> 15:25.673
L,
namely I'm getting

15:25.668 --> 15:31.768
e^(ipx/ℏ) times
e^(ipL/ℏ).

15:31.769 --> 15:36.059
That has to agree with the
starting value which is

15:36.061 --> 15:37.901
e^(ipx/ℏ).

15:37.899 --> 15:40.469
In other words I'm comparing
the wave function at the point

15:40.471 --> 15:42.071
x and the point x
L.

15:42.070 --> 15:45.390
See, if your world is infinite
this is x,

15:45.386 --> 15:48.136
this is x L,
this is x 2L,

15:48.139 --> 15:50.469
they're all different points.

15:50.470 --> 15:54.070
What the function does here has
nothing to do with what it does

15:54.073 --> 15:55.753
here,
but if you wrapped out the

15:55.754 --> 15:58.474
region back into itself what it
does when you go a distance

15:58.465 --> 16:00.845
L is no longer
independent of what it does at

16:00.847 --> 16:01.967
the starting point.

16:01.970 --> 16:04.460
It has to be the same thing.

16:04.460 --> 16:07.130
Therefore, I'm just applying
the test.

16:07.129 --> 16:10.529
I'm saying take the function at
x L, factorize it this

16:10.528 --> 16:13.758
way, demand that it be equal to
a function of x.

16:13.759 --> 16:22.209
These cancel and you learn then
that e^(ipL/ℏ) should

16:22.210 --> 16:23.180
be 1.

16:23.179 --> 16:24.389
Now how can that be 1?

16:24.389 --> 16:29.769
Well, one way is p = 0,
but I hope you know that it's

16:29.774 --> 16:31.514
not the only way.

16:31.509 --> 16:35.289
Do you know how else it can be
1 should p be 0?

16:35.288 --> 16:38.888
Student:
>

16:38.889 --> 16:39.619
Prof: Pardon me?

16:39.620 --> 16:41.600
Student:
>

16:41.600 --> 16:42.720
Prof: I didn't hear
that, sorry.

16:42.720 --> 16:44.610
Student:  It's infinite.

16:44.610 --> 16:46.240
Prof: I didn't hear that
again.

16:46.240 --> 16:47.910
Student:  It's infinite
>

16:47.908 --> 16:49.818
Prof: You still have to
say it louder.

16:49.820 --> 16:52.330
Student:  If it is
infinite

16:52.332 --> 16:54.392
>

16:54.389 --> 16:55.309
Prof: Any other answers?

16:55.309 --> 16:55.969
Yeah?

16:55.970 --> 16:58.380
Student:  If it's a
multiple of 2Π.

16:58.379 --> 17:00.279
Prof: Is that what you
said multiple of 2Π?

17:00.279 --> 17:01.469
Student:  Sure.

17:01.470 --> 17:02.250
Prof: Good.

17:02.250 --> 17:04.300
I'm glad you didn't say it loud
enough.

17:04.299 --> 17:05.129
Okay good.

17:05.130 --> 17:08.940
It's a multiple of 2Π
because any trigonometry

17:08.938 --> 17:12.158
function if you add 2Π,
or 4Π, or 6Π

17:12.157 --> 17:15.267
whose argument doesn't change
then you should never forget the

17:15.271 --> 17:18.181
fact that this exponential is
the sum of two trigonometric

17:18.181 --> 17:22.311
functions cosine i sine,
and they all come back.

17:22.308 --> 17:25.918
Therefore, it is too strong to
say p should be 0.

17:25.920 --> 17:33.610
p should be such that
pL/ℏ is 2Π

17:33.607 --> 17:40.087
times any integer when the
integer can be 0,

17:40.089 --> 17:46.119
plus minus 1,
plus minus 2 etcetera.

17:46.118 --> 17:55.128
That means p has to be
equal to 2Πℏ/L times

17:55.132 --> 18:02.082
m where m is an
integer, positive,

18:02.076 --> 18:04.436
negative, 0.

18:04.440 --> 18:07.090
Now this is a very big moment
in your life.

18:07.089 --> 18:11.079
Why is it a big moment?

18:11.079 --> 18:12.529
Yes?

18:12.529 --> 18:13.949
You don't know?

18:13.950 --> 18:17.710
Well if you haven't done it
before, this is the first time

18:17.714 --> 18:21.154
you're able to deduce the
quantization of a dynamical

18:21.151 --> 18:22.011
variable.

18:22.009 --> 18:26.109
This is the first time you
realize this is the quantum of

18:26.105 --> 18:27.345
quantum theory.

18:27.348 --> 18:30.158
The allowed momenta for this
particle living in there,

18:30.160 --> 18:32.260
you might think it can zip
around at any speed it likes,

18:32.259 --> 18:39.169
it cannot, especially in a ring
of nano proportions these values

18:39.169 --> 18:42.569
of p are all discrete.

18:42.569 --> 18:43.629
p times..

18:43.630 --> 18:47.450
2Πℏ/L times 0 is
here, times 1 is here,

18:47.445 --> 18:49.725
times 2 is here,
3 is here, -1,

18:49.734 --> 18:50.274
-2.

18:50.269 --> 18:54.859
These are the only allowed
values of p.

18:54.858 --> 18:57.368
So that's the case of
quantization,

18:57.368 --> 19:01.288
and the quantization came from
demanding that the wave function

19:01.294 --> 19:04.844
had a certain behavior that's
mathematically required.

19:04.838 --> 19:08.708
The behavior in question is
single valuedness.

19:08.710 --> 19:10.820
Now I can say this in another
way.

19:10.818 --> 19:13.908
Let's write the same
relationship in another way.

19:13.910 --> 19:16.130
So let me go here.

19:16.130 --> 19:21.400
So I had the allowed values of
p are 2Πℏ/L

19:21.396 --> 19:22.886
times m.

19:22.890 --> 19:27.200
I can write it as p
times L/2Π

19:27.195 --> 19:29.625
is equal to mℏ.

19:29.630 --> 19:35.120
L/2Π
is the radius of the circle.

19:35.118 --> 19:39.328
So I find that p times
R is equal to

19:39.327 --> 19:40.527
mℏ.

19:40.529 --> 19:43.559
You guys know what p
times R is for a particle

19:43.560 --> 19:45.720
moving in a circle at momentum
p?

19:45.720 --> 19:49.550
What is the momentum times the
radial distance to the center?

19:49.548 --> 19:51.198
Student:  Angular
momentum.

19:51.200 --> 19:52.840
Prof: Angular momentum.

19:52.838 --> 19:57.168
That's usually denoted by a
quantity L in quantum

19:57.173 --> 19:58.123
mechanics.

19:58.118 --> 20:03.978
So angular momentum is an
integral multiple of ℏ.

20:03.980 --> 20:06.360
That's something you will find
even in high school people will

20:06.356 --> 20:08.926
tell you angular momentum is an
integral multiple of ℏ.

20:08.930 --> 20:11.770
Where does it come from and how
does it come from quantum

20:11.770 --> 20:12.380
mechanics?

20:12.380 --> 20:17.240
Here is one simple context in
which you can see the angular

20:17.238 --> 20:20.588
momentum is quantized to these
values.

20:20.588 --> 20:25.748
Now what I will do quite often
is to write the state ψ

20:25.750 --> 20:30.530
as ψ of p this way,
e^(2Π), I'm sorry,

20:30.528 --> 20:34.718
e^(ipx/ℏ),
but to remind myself that p

20:34.719 --> 20:39.929
is quantized to be integer
multiples of some basic quantum

20:39.933 --> 20:43.443
where the multiple is just
m,

20:43.440 --> 20:48.850
I may also write the very same
function the ψ's of m

20:48.846 --> 20:52.506
is 1/√L
e^(ix/ℏ),

20:52.509 --> 20:57.489
but for p you put the
allowed value which is

20:57.491 --> 21:00.981
2Πℏ/L times
m.

21:00.980 --> 21:02.510
And what do you get?

21:02.509 --> 21:16.779
You get 1/√L times
e^(2Πimx/L).

21:16.778 --> 21:18.898
They're all oscillating
exponentials,

21:18.901 --> 21:22.321
but you realize that the label
p and the label m are

21:22.317 --> 21:23.317
equally good.

21:23.318 --> 21:25.938
If I tell you what m is
you know what the momentum is

21:25.940 --> 21:27.850
because you just multiply by
this number.

21:27.848 --> 21:32.258
So quite often I'll refer to
this wave function by this label

21:32.259 --> 21:35.419
m which is as good as the label
p.

21:35.420 --> 21:39.400
The nice thing about the label
m is that m ranges

21:39.396 --> 21:40.676
over all integers.

21:40.680 --> 21:42.090
p is a little more
complicated.

21:42.088 --> 21:45.988
p is also quantized,
but the allowed values are not

21:45.992 --> 21:49.352
integers, but integers times
this funny number.

21:49.348 --> 21:50.448
In the limit in which L
is very,

21:50.450 --> 21:52.800
very large, compared to
ℏ,

21:52.798 --> 21:55.278
the spacing between the allowed
values becomes very,

21:55.279 --> 21:59.769
very close, and you may not
even realize that p is

21:59.766 --> 22:02.166
taking only discrete values.

22:02.170 --> 22:04.540
So when you do a microscopic
problem where L is 1

22:04.535 --> 22:06.985
meter the spacing between one
allowed p and another

22:06.986 --> 22:09.476
allowed p will be so
small you won't notice it.

22:09.480 --> 22:12.700
So even if you lived in a real
ring of circumference 1 meter

22:12.695 --> 22:16.075
the momenta that you'll find in
the particle will look like any

22:16.076 --> 22:17.436
momentum is possible.

22:17.440 --> 22:20.370
That's because the allowed
values of p of so densely

22:20.365 --> 22:22.025
close,
packed, that you don't know

22:22.027 --> 22:24.877
whether you've got this one,
or that one or something in

22:24.878 --> 22:27.618
between because you cannot
measure it that well.

22:27.618 --> 22:31.338
So it'll smoothly go into the
classical world of all allowed

22:31.335 --> 22:33.975
momenta if L becomes
macroscopic.

22:33.980 --> 22:36.630
And the notion of macroscopic
is " how big is big".

22:36.630 --> 22:39.680
Well, it should be comparable
to ℏ.

22:39.680 --> 22:47.070
I mean, it should be much
bigger than ℏ then

22:47.066 --> 22:50.756
it'll become continuous.

22:50.759 --> 22:54.979
So now I'm going to ask the
following question.

22:54.980 --> 22:58.780
If I have a particle in this
world,

22:58.779 --> 23:03.239
in this one-dimensional ring
and I plot the wave function,

23:03.240 --> 23:11.400
some function ψ(x),
suppose it's not one of these

23:11.401 --> 23:12.541
functions?

23:12.539 --> 23:14.389
It's not e^(ipx/ℏ).

23:14.390 --> 23:16.690
It's some random thing I wrote
here.

23:16.690 --> 23:22.050
Of course it meets itself when
you go around a circle.

23:22.049 --> 23:23.489
That's a periodic function.

23:23.490 --> 23:24.920
Be very careful.

23:24.920 --> 23:27.280
A periodic function doesn't
mean it oscillates with a

23:27.280 --> 23:27.690
period.

23:27.690 --> 23:30.330
In this case periodic means
when I go around the loop it

23:30.327 --> 23:31.907
comes back to a starting value.

23:31.910 --> 23:35.150
It doesn't do something like
this where the two don't match.

23:35.150 --> 23:36.680
That's what I mean by periodic.

23:36.680 --> 23:39.710
It doesn't mean it's a nice
oscillatory function.

23:39.710 --> 23:43.220
These guys are periodic and
oscillatory with the period.

23:43.220 --> 23:45.990
These are periodic only in the
sense that if you go around the

23:45.994 --> 23:48.684
ring you come back to the
starting value of the function.

23:48.680 --> 23:51.670
So I give you some function
like this and I ask you what's

23:51.674 --> 23:52.994
going on with this guy.

23:52.990 --> 23:56.510
What can you say about the
particle?

23:56.509 --> 24:01.279
So can you tell me anything now
given this function

24:01.281 --> 24:03.001
ψ(x)?

24:03.000 --> 24:08.890
Does it tell you any
information?

24:08.890 --> 24:12.030
You mean if I draw the function
like that, you get no

24:12.030 --> 24:13.360
information from it.

24:13.359 --> 24:15.909
Is that what you're saying?

24:15.910 --> 24:17.160
You must know--yes?

24:17.160 --> 24:19.430
Student:  You square it
you get the probability

24:19.430 --> 24:20.680
Prof: Yes, look.

24:20.680 --> 24:23.430
If you knew that you've got to
say that because if I think that

24:23.430 --> 24:26.270
you didn't realize that I know
we are both in serious trouble.

24:26.269 --> 24:27.529
That's correct.

24:27.528 --> 24:29.828
I want to reinforce the notion
over and over again.

24:29.828 --> 24:31.868
Wave functions should tell you
something.

24:31.868 --> 24:34.098
Square it you get the
probability.

24:34.098 --> 24:36.008
If you don't square it don't
think it's the probability

24:36.006 --> 24:37.026
because it can be negative.

24:37.029 --> 24:38.689
It can even be complex.

24:38.690 --> 24:41.400
So don't forget if you square
it you get the probability.

24:41.400 --> 24:43.560
Define a certain position.

24:43.558 --> 24:45.478
That means if you went around
with this little

24:45.477 --> 24:48.077
Heisenberg microscope
all over the ring and you catch

24:48.077 --> 24:49.577
it and you say,
"Good, I found it

24:49.577 --> 24:50.607
here," and you do that
many,

24:50.609 --> 24:51.979
many times.

24:51.980 --> 24:55.330
By many, many times I mean you
take a million particles in a

24:55.328 --> 24:58.508
million rings each in exactly
this quantum state and make

24:58.508 --> 25:01.318
measurements,
then your histogram will look

25:01.319 --> 25:03.419
like the square of this
function.

25:03.420 --> 25:05.160
But there is more to life than
just saying,

25:05.160 --> 25:07.160
"Where is the
particle?,"

25:07.157 --> 25:09.917
because in classical mechanics
you also ask,

25:09.920 --> 25:12.080
"What is its
momentum."

25:12.078 --> 25:16.428
The question I'm asking you is,
"What is the momentum you

25:16.432 --> 25:20.642
will get when you measure the
momentum of a particle in this

25:20.644 --> 25:21.934
quantum state?

25:21.930 --> 25:24.960
You know the answer only on a
few special cases.

25:24.960 --> 25:27.970
If by luck your function
happened to look like one of

25:27.973 --> 25:31.103
these functions--right now I've
not told you what ψ

25:31.104 --> 25:31.514
is.

25:31.509 --> 25:33.999
It's whatever you like,
but if it looked like one of

25:34.000 --> 25:36.640
these you're in good shape
because then the momentum is

25:36.635 --> 25:38.585
whatever p you find up
here,

25:38.589 --> 25:40.049
but it may not look like that.

25:40.049 --> 25:42.729
This guy doesn't look like that.

25:42.730 --> 25:47.050
So then the question is if you
measure momentum what answer

25:47.051 --> 25:48.171
will you get.

25:48.170 --> 25:54.720
Now that is something--anybody
know what the answer is?

25:54.720 --> 25:58.230
Now I accept your silence
because you're not supposed to

25:58.232 --> 25:59.002
know this.

25:59.000 --> 26:02.430
This is a postulate in quantum
mechanics.

26:02.430 --> 26:03.100
It's not logic.

26:03.099 --> 26:04.219
It's not mathematics.

26:04.220 --> 26:07.720
No one could have told you 300
years ago this is the right

26:07.717 --> 26:08.267
answer.

26:08.269 --> 26:10.649
So this is another postulate
just like saying,

26:10.652 --> 26:13.142
ψ^(2) is probability
defined at x.

26:13.140 --> 26:15.150
There is a new postulate.

26:15.150 --> 26:17.220
It addresses the following
question.

26:17.220 --> 26:19.820
If I shift my attention from
position,

26:19.818 --> 26:23.068
to momentum,
and I ask, "What are the

26:23.074 --> 26:27.524
odds for getting different
answers for momentum?"

26:27.519 --> 26:30.089
That answer's actually
contained in the same function

26:30.093 --> 26:32.323
ψ of x in the
following fashion.

26:32.318 --> 26:38.058
Take this function ψ(x)
and write it as a sum over

26:38.061 --> 26:43.571
p of these functions,
ψ_p of x

26:43.567 --> 26:47.417
with some coefficient
A_p.

26:47.420 --> 26:50.960
I can also write it equally
well as the same function

26:50.957 --> 26:55.037
labeled by integer m and I want
to call the coefficient m.

26:55.038 --> 26:59.168
They're exactly the same thing,
p and m are

26:59.171 --> 27:00.131
synonymous.

27:00.130 --> 27:01.190
The same function.

27:01.190 --> 27:06.430
This function is called
ψ_m because they

27:06.425 --> 27:11.065
contain the same information on
the momentum.

27:11.068 --> 27:14.288
So either you can write it this
way if you want to see the

27:14.291 --> 27:17.561
momentum highlighted,
or you can write the function

27:17.557 --> 27:21.697
this way if you want to see the
quantum number m highlighted,

27:21.700 --> 27:24.050
but they stand for the same
physics.

27:24.048 --> 27:27.928
If m is equal to 4 it means the
cosine and the sine contained in

27:27.932 --> 27:31.882
the complex exponential finish 4
complete oscillations as they go

27:31.875 --> 27:33.165
around the cycle.

27:33.170 --> 27:36.010
All right, so here's what
you're told.

27:36.009 --> 27:39.109
Take the arbitrary periodic
function.

27:39.108 --> 27:43.028
Write it as a sum of these
functions, each of definite

27:43.032 --> 27:45.552
momentum, with some coefficient.

27:45.548 --> 27:49.018
Then the probability that you
will get a momentum p

27:49.019 --> 27:51.479
when you measure it,
which is the same as the

27:51.483 --> 27:53.883
probability you will get the
corresponding m,

27:53.880 --> 28:07.980
is nothing but the absolute
square of this coefficient.

28:07.980 --> 28:12.790
In other words,
anyway, this is a postulate.

28:12.789 --> 28:14.669
Let me repeat the postulate.

28:14.670 --> 28:16.170
Somebody gives you a function.

28:16.170 --> 28:20.010
You write the function as a sum
of all these periodic functions,

28:20.009 --> 28:23.749
each with the index m,
multiply each with a suitable

28:23.751 --> 28:27.431
number so that they add up to
give you the function that's

28:27.429 --> 28:28.719
provided to you.

28:28.720 --> 28:32.880
Once you've done that the
coefficient squared with the

28:32.880 --> 28:37.670
particular value m is the
probability you will get that

28:37.669 --> 28:41.279
value for m or the corresponding
momentum.

28:41.279 --> 28:43.929
So there are two questions one
can ask at this point.

28:43.930 --> 28:47.800
First question is what makes
you think that you can write any

28:47.797 --> 28:51.017
function I give you as a sum of
these functions.

28:51.019 --> 28:52.059
Realize what this means.

28:52.058 --> 28:55.848
I'm saying I can write any
function as

28:55.847 --> 29:01.887
e^(2Πimx/L)√L
times A_m.

29:01.890 --> 29:05.850
I'm writing out this function
explicitly for you.

29:05.848 --> 29:09.268
Now that is a mathematical
result I will not prove here,

29:09.269 --> 29:13.109
it's called Fourier series and
it tells you that every periodic

29:13.106 --> 29:15.296
function,
namely that which comes back to

29:15.299 --> 29:17.779
itself when you go around a
period length L,

29:17.778 --> 29:21.378
can be written as a sum of
these periodic functions with

29:21.380 --> 29:23.410
suitably chosen coefficients.

29:23.410 --> 29:31.420
It can always be done.

29:31.420 --> 29:34.800
That is analogous to the
statement that if you are say

29:34.798 --> 29:38.048
living in three dimensions and
there is a vector

29:38.048 --> 29:41.468
V, and you pick for
yourself three

29:41.467 --> 29:44.967
orthonormal
vectors,

29:44.970 --> 29:48.320
i, j and
k, then any vector

29:48.317 --> 29:52.017
V can be written as
V_x times

29:52.022 --> 29:55.162
i V_y
times j

29:55.156 --> 29:58.146
V_z times
k.

29:58.150 --> 30:00.070
In other words,
I challenge you to write any

30:00.074 --> 30:02.454
arrow in three dimensions
starting from the origin and

30:02.446 --> 30:04.726
pointing in any direction of any
finite length.

30:04.730 --> 30:07.710
I'll build it up for you using
some multiple of i,

30:07.708 --> 30:09.888
some of j,
and some of k.

30:09.890 --> 30:11.270
We know that can be done in
fact.

30:11.269 --> 30:12.899
That's if you want the
technical definition of

30:12.904 --> 30:13.564
three-dimension.

30:13.559 --> 30:14.269
Yes?

30:14.269 --> 30:18.639
Student:  With the
Fourier case would there be one

30:18.643 --> 30:21.223
unique way of writing ________?

30:21.220 --> 30:21.740
Prof: Good point.

30:21.740 --> 30:22.320
That's correct.

30:22.319 --> 30:23.029
There will be a unique way.

30:23.029 --> 30:23.669
Student:  There is?

30:23.670 --> 30:24.550
Prof: There is.

30:24.548 --> 30:28.848
You agree in three dimensions
there is no other mixture except

30:28.845 --> 30:29.615
this one.

30:29.618 --> 30:33.058
If somebody comes with the
second way of writing it you can

30:33.059 --> 30:36.619
show that the second way will
coincide with the first way.

30:36.618 --> 30:40.038
So the expansion of a function,
in what are called these

30:40.038 --> 30:43.578
trigonometric or exponential
functions and it's called the

30:43.582 --> 30:46.932
Fourier series of the function,
has unique coefficients.

30:46.930 --> 30:49.300
And I'll tell you right now
what the formula for the

30:49.295 --> 30:51.645
coefficient is,
but first I'm telling you that

30:51.646 --> 30:54.676
just like it's natural to build
a vector out of some building

30:54.682 --> 30:56.442
blocks
i, j and

30:56.438 --> 31:00.008
k, it's natural to build
up periodic functions with these

31:00.012 --> 31:02.342
building blocks the
ψ_m.

31:02.338 --> 31:05.548
The only difference is there
you need only three guys.

31:05.548 --> 31:08.808
Here you need an infinite
number of them because a range

31:08.814 --> 31:12.024
of m goes from minus to plus
infinity, all integers.

31:12.019 --> 31:14.949
But it's still remarkable that
given all of them you can build

31:14.949 --> 31:17.669
any function you like,
including this thing I just

31:17.671 --> 31:21.141
wrote down arbitrarily,
can be built.

31:21.140 --> 31:23.090
The fact that you can prove it,
that you can do it,

31:23.088 --> 31:25.238
I don't want to prove because
it's kind of tricky,

31:25.240 --> 31:30.060
but I will prove the second
part of it which is given that

31:30.063 --> 31:35.313
such an expansion exists how do
I find these coefficients given

31:35.309 --> 31:36.579
a function?

31:36.578 --> 31:42.138
So let's ask a similar question
in the usual case of vectors.

31:42.140 --> 31:45.320
How do I find the coefficients?

31:45.318 --> 31:48.388
Suppose I write the vector
V as e_1

31:48.390 --> 31:51.190
times V_1
e_2 times

31:51.194 --> 31:53.634
V_2
e_3 times

31:53.630 --> 31:55.060
V_3.

31:55.058 --> 31:55.858
Don't worry
e_1,

31:55.858 --> 31:57.158
e_2 and
e_3 are the

31:57.160 --> 31:58.770
usual guys,
e_1 is

31:58.773 --> 32:01.153
i, e_2
is j and

32:01.148 --> 32:03.048
e_3 is
k.

32:03.048 --> 32:08.728
People like to do that because
in mathematics you may want to

32:08.730 --> 32:12.310
go to 96 dimensions,
but we've only got 26 letters,

32:12.305 --> 32:14.495
so if you stuck to i,
j and k you're

32:14.501 --> 32:17.111
going to have trouble,
but with numbers you never run

32:17.106 --> 32:18.126
out of numbers.

32:18.130 --> 32:21.350
So you label all the dimensions
by some number,

32:21.345 --> 32:24.625
which in this case happens to
go from 1 to 3.

32:24.630 --> 32:27.210
You also know that these
vectors i and j

32:27.205 --> 32:28.995
have some very interesting
properties,

32:29.003 --> 32:30.903
i⋅i is
1.

32:30.900 --> 32:33.010
That's the same as
j⋅j.

32:33.009 --> 32:35.019
That's the same as
k⋅k.

32:35.019 --> 32:37.319
And that
i⋅k

32:37.323 --> 32:39.323
and
i⋅j

32:39.315 --> 32:40.555
are 0 and so on.

32:40.558 --> 32:44.198
Namely the dot product of one
guy with himself is 1,

32:44.198 --> 32:46.908
and any one with anything else
is 0.

32:46.910 --> 32:49.670
That just tells you they all
have unit length and they're

32:49.674 --> 32:50.914
mutually perpendicular.

32:50.910 --> 32:53.370
I want to write this as
e_i

32:53.372 --> 32:55.772
⋅
e_j,

32:55.771 --> 32:57.731
but this could be 1,2 or 3.

32:57.730 --> 32:59.080
That could be 1,2 or 3.

32:59.078 --> 33:02.278
I want to say this is equal to
1 if i is equal to

33:02.278 --> 33:02.918
j.

33:02.920 --> 33:07.050
This is equal to 0 if
i is not equal to

33:07.046 --> 33:07.916
j.

33:07.920 --> 33:09.900
This is a usual vector analysis.

33:09.900 --> 33:13.210
I'm just saying the dot product
of basis vectors has this

33:13.210 --> 33:14.970
property,
1 if they match,

33:14.972 --> 33:18.372
2 if they're different,
so there's a shortcut for this

33:18.374 --> 33:21.594
and that's write the symbol
δ_ij.

33:21.592 --> 33:24.932
δ_ij is
called Kronecker's

33:24.925 --> 33:25.565
delta.

33:25.568 --> 33:30.828
Kronecker's in a lot of things,
but this is one place where his

33:30.828 --> 33:33.288
name has been immortalized.

33:33.288 --> 33:35.768
He just said instead of saying
this all the time,

33:35.769 --> 33:37.379
1 if they're equal,
0 if they're different,

33:37.380 --> 33:40.290
why don't you call it my
symbol, the Kronecker's symbol,

33:40.290 --> 33:41.880
δ_ij.

33:41.880 --> 33:44.800
It's understood that this whole
thing simply says

33:44.798 --> 33:46.498
δ_ij.

33:46.501 --> 33:47.781
This is shorthand.

33:47.779 --> 33:50.499
That means if on the left hand
side there are two guys with

33:50.499 --> 33:53.449
indices i and j if
the indices are equal the right

33:53.452 --> 33:56.162
hand side is 1,
indices are unequal right hand

33:56.160 --> 33:56.810
side is 0.

33:56.808 --> 34:00.428
Do you understand that this
gives you the fact that each

34:00.425 --> 34:04.235
vector is of length 1 and that
each is perpendicular to the

34:04.239 --> 34:05.159
other two.

34:05.160 --> 34:09.400
Now that is what we can use now
to find out.

34:09.400 --> 34:12.780
So let me write the vector
V in this notation as

34:12.782 --> 34:16.352
e_i times
V_ii from 1 to

34:16.353 --> 34:16.733
3.

34:16.730 --> 34:21.050
You're all familiar with this
way to write the sum?

34:21.050 --> 34:23.410
So I come with a certain vector.

34:23.409 --> 34:25.029
The vector is not defined in
any axis.

34:25.030 --> 34:27.650
It's just an arrow pointing in
some direction.

34:27.650 --> 34:29.460
It's got a magnitude and
direction.

34:29.460 --> 34:31.840
And I say, "Can you write
the vector in terms of i,

34:31.844 --> 34:33.004
j and k?"

34:33.000 --> 34:34.940
And the answer is,
"Yes, of course I

34:34.938 --> 34:35.518
can."

34:35.519 --> 34:37.609
So here's your vector V.

34:37.610 --> 34:39.480
Here is, if you like,
e_1,

34:39.476 --> 34:41.606
e_2 and
e_3.

34:41.610 --> 34:42.960
The claim is some mixture of
e_1,

34:42.960 --> 34:44.340
e_2,
e_3 will add

34:44.338 --> 34:44.998
up to this V.

34:45.000 --> 34:47.070
That's granted, but how much?

34:47.070 --> 34:48.110
How much e_1
do I need?

34:48.110 --> 34:49.600
How much e_2
do I need?

34:49.599 --> 34:52.429
There's a very simple trick for
that.

34:52.429 --> 34:54.719
Anybody know what that trick is?

34:54.719 --> 34:59.159
You might know that trick.

34:59.159 --> 35:03.549
You've seen it anywhere?

35:03.550 --> 35:05.180
Here is the trick.

35:05.179 --> 35:08.889
Suppose you want to find
V_2?

35:08.889 --> 35:11.499
You take the dot product of
both these things with

35:11.503 --> 35:12.733
e_2.

35:12.730 --> 35:17.180
Take e_2 dot
this, and take dot product with

35:17.175 --> 35:18.875
e_2.

35:18.880 --> 35:22.910
What happens is you will
get--the dot product can go

35:22.909 --> 35:23.619
inside.

35:23.619 --> 35:25.879
You'll get e_i
⋅

35:25.875 --> 35:28.535
e_2 times
V_i where

35:28.541 --> 35:30.081
i goes from 1 to 3.

35:30.079 --> 35:31.599
What is e_i
⋅

35:31.603 --> 35:32.463
e_2?

35:32.460 --> 35:32.960
e_i
⋅

35:32.956 --> 35:33.766
e_2 is
δ_i2.

35:33.768 --> 35:39.908
That means if this index size
is equal to 2 you get 1.

35:39.909 --> 35:41.719
If not equal to 0 you strike
out.

35:41.719 --> 35:43.389
You get 0.

35:43.389 --> 35:46.509
So of all the three terms in
this only one will survive.

35:46.510 --> 35:49.530
That's the one when i is
equal to 2 in which case you

35:49.525 --> 35:50.185
will get 1.

35:50.190 --> 35:53.930
That's multiplying
V_2 so it'll

35:53.925 --> 35:56.465
give you V_2.

35:56.469 --> 36:00.609
So to find the component number
3 you take the dot product of

36:00.606 --> 36:03.776
the given vector with
e_3,

36:03.780 --> 36:05.590
and that will give you,
you can see,

36:05.590 --> 36:07.870
that will give you
V_3 or

36:07.865 --> 36:10.305
V_2 or whatever
you like.

36:10.309 --> 36:15.429
I'm going to use a similar
trick now in our problem.

36:15.429 --> 36:17.669
The trick I'm going to use is
the following.

36:17.670 --> 36:23.100
So the analogy is just like you
had V = sum over i

36:23.096 --> 36:27.486
e_i times
V_i,

36:27.489 --> 36:30.349
I have ψ
of x is equal to sum

36:30.346 --> 36:34.076
over m of some
A_m times the

36:34.079 --> 36:37.229
function ψ_m of
x.

36:37.230 --> 36:37.860
You understand that?

36:37.860 --> 36:39.940
ψ_m is a
particular function which I

36:39.938 --> 36:44.848
don't want to write every time,
but if you insist it is

36:44.846 --> 36:51.776
2Πimx/L divided by
√L.

36:51.780 --> 36:54.620
So I'm trying to find this guy.

36:54.619 --> 36:55.539
How much is it?

36:55.539 --> 36:57.279
That's the question.

36:57.280 --> 36:59.610
Now here we had a nice rule.

36:59.610 --> 37:01.960
The rule says e_i
⋅

37:01.960 --> 37:04.660
e_j is
δ_ij.

37:04.659 --> 37:07.989
That was helpful in finding the
coefficients.

37:07.989 --> 37:11.219
There's a similar rule on the
right hand side which I will

37:11.224 --> 37:13.784
show you and we can all verify
it together.

37:13.780 --> 37:18.870
The claim is
ψ_m*(x)

37:18.869 --> 37:23.819
times
ψ_n(x) dx

37:23.818 --> 37:32.158
from 0 to L is in
fact δ_mn.

37:32.159 --> 37:36.409
So the basis vectors
ψ_m are like

37:36.405 --> 37:38.525
e_i.

37:38.530 --> 37:40.250
The dot product of two basis
vectors being

37:40.248 --> 37:42.258
δ_ij is
the same here,

37:42.260 --> 37:46.110
like same integral,
but one of them star times the

37:46.114 --> 37:50.914
other one is 1 if it's the same
function and 0 if it's not the

37:50.913 --> 37:52.333
same function.

37:52.329 --> 37:54.599
Let's see if this is true.

37:54.599 --> 37:57.439
If m is equal to
n, can you do this in

37:57.440 --> 37:58.110
your head?

37:58.110 --> 38:02.260
If m is equal to
n what do we have?

38:02.260 --> 38:05.740
This number times this
conjugate is just 1.

38:05.739 --> 38:07.629
You just get 1/L.

38:07.630 --> 38:11.130
An integral of 1/L dx,
so if you want I will write it

38:11.125 --> 38:16.855
here,
it's 2Πi times

38:16.864 --> 38:28.054
n −mx/L dx
from 0 to L.

38:28.050 --> 38:29.390
Do you understand that?

38:29.389 --> 38:31.529
You take the conjugate of the
first function.

38:31.530 --> 38:34.860
That where's it's a -m here,
and take the second function

38:34.864 --> 38:37.134
which is
e^(2Πin)^(/L)

38:37.128 --> 38:38.258
times x.

38:38.260 --> 38:42.030
I'll wait until you have time
to digest that.

38:42.030 --> 38:44.120
The product of
ψ_m* with

38:44.123 --> 38:47.183
ψ_n you combine
the two exponentials,

38:47.179 --> 38:50.459
but the thing that had m
in it has a -m here

38:50.458 --> 38:52.528
because you conjugated
everything.

38:52.530 --> 38:54.330
So I'm saying,
"What is the integral

38:54.329 --> 38:55.229
going to be?"

38:55.230 --> 38:58.290
If n is equal to
m you can see that in

38:58.289 --> 38:59.009
your head.

38:59.010 --> 39:01.410
This is e^(0).

39:01.409 --> 39:02.949
That is just 1.

39:02.949 --> 39:04.299
The integral of dx is
L.

39:04.300 --> 39:05.380
That cancels the L.

39:05.380 --> 39:08.820
You get 1.

39:08.820 --> 39:11.320
That's certainly true if
m is equal to n.

39:11.320 --> 39:15.740
If m is not equal to
n, suppose it is 6?

39:15.739 --> 39:17.599
It doesn't matter.

39:17.599 --> 39:21.439
This will complete 6 full
oscillations in the period,

39:21.440 --> 39:24.810
the sine and the cosine,
but whenever you integrate a

39:24.809 --> 39:28.439
sine or a cosine over some
number of full periods you get

39:28.440 --> 39:28.830
0.

39:28.829 --> 39:32.849
So this exponential,
when integrated over a full

39:32.853 --> 39:37.913
cycle, if it's got a non-zero
exponent integer exponent will

39:37.905 --> 39:39.185
give you 0.

39:39.190 --> 39:43.300
So you see the remarkable
similarity between usual vector

39:43.302 --> 39:45.582
analysis and these functions.

39:45.579 --> 39:47.049
This is an arbitrary vector.

39:47.050 --> 39:49.370
This is an arbitrary function.

39:49.369 --> 39:50.899
e_1,
e_2,

39:50.898 --> 39:52.388
e_3 are basis
vectors;

39:52.389 --> 39:55.359
ψ_m are if you
want basis function.

39:55.360 --> 39:58.220
I can build any vector out of
these unit vectors.

39:58.219 --> 40:01.439
I can build any function out of
these basis functions.

40:01.440 --> 40:04.560
And finally,
if I want to find out a

40:04.557 --> 40:07.407
coefficient, what should I do?

40:07.409 --> 40:08.569
You want to find the
coefficient

40:08.574 --> 40:09.444
A_n.

40:09.440 --> 40:14.330
What did I do here to find
V_j?

40:14.329 --> 40:17.719
You take V⋅
e_j where

40:17.717 --> 40:20.557
j could be whatever
number you picked.

40:20.559 --> 40:23.169
Because if you take the dot
product of the two sides when

40:23.168 --> 40:25.588
you take dot product of
e_j the only

40:25.590 --> 40:27.360
term who survives is i =
j.

40:27.360 --> 40:28.680
That'll give you the
V_j.

40:28.679 --> 40:32.199
So similarly here I claim the
following is true.

40:32.199 --> 40:37.769
If you do the integral of
ψ_n* x

40:37.766 --> 40:43.756
times the given function
dx from 0 to L you

40:43.760 --> 40:47.400
will get A_n.

40:47.400 --> 40:48.530
So once I show this I'm done.

40:48.530 --> 40:51.410
Now if you don't have the
stomach for this proof you don't

40:51.405 --> 40:52.915
have to remember this proof.

40:52.920 --> 40:53.810
That's up to you.

40:53.809 --> 40:56.719
See, I don't know how much you
guys want to know.

40:56.719 --> 40:59.519
I'm trying to keep the stuff I
just tell you without proof to a

40:59.518 --> 40:59.968
minimum.

40:59.969 --> 41:03.059
I felt bad telling you that
every function can be expanded

41:03.059 --> 41:04.849
this way,
but the coefficients being

41:04.851 --> 41:06.741
given by the formula is not too
far away,

41:06.739 --> 41:09.439
so I want to show you how it's
done.

41:09.440 --> 41:12.640
You may not know the details
why this is working,

41:12.639 --> 41:15.469
but you should certainly know
that if you want coefficient

41:15.474 --> 41:18.464
number 13 you've got to take
ψ_13* and multiply

41:18.460 --> 41:20.550
with the given function and
integrate.

41:20.550 --> 41:24.160
That you're supposed to know,
so why does this work?

41:24.159 --> 41:26.189
So let's see what this does.

41:26.190 --> 41:30.410
We are trying to take
ψ_n*(x).

41:30.409 --> 41:35.719
The given function looks like a
sum A_m

41:35.724 --> 41:41.884
ψ_m(x)
dx 0 to L summed over

41:41.875 --> 41:43.225
m.

41:43.230 --> 41:47.740
So this summation you see you
can bring the ψ

41:47.742 --> 41:49.812
in here if you like.

41:49.809 --> 41:52.529
It doesn't matter.

41:52.530 --> 41:56.570
Then do the integral over
x then you will find this

41:56.572 --> 41:59.482
is giving me--maybe I'll write
it here.

41:59.480 --> 42:03.510
That is going to be equal to
sum over m

42:03.514 --> 42:07.374
A_m of
ψ_m of

42:07.371 --> 42:12.221
ψ_n* x
ψ_m(x)

42:12.215 --> 42:14.965
dx,
and that is going to be

42:14.974 --> 42:16.534
δ_mn.

42:16.530 --> 42:19.740
That means I will vanish unless
m equals n,

42:19.737 --> 42:22.827
and when m equals
n I will give you 1.

42:22.829 --> 42:25.699
That means the only term that
survives from all these terms is

42:25.697 --> 42:27.717
the one where m matches
n,

42:27.719 --> 42:39.639
so the thing that comes out is
A_n.

42:39.639 --> 42:45.339
Once again, you will see this
in my notes, but do you have any

42:45.338 --> 42:49.448
idea of what I did or where this
is going?

42:49.449 --> 42:53.189
In quantum theory if you want
to know what'll happen if I

42:53.186 --> 42:57.116
measure momentum for a particle
living in a ring you have to

42:57.123 --> 43:01.063
write the given function in
terms of these special functions

43:01.061 --> 43:04.601
each item defined as a definite
momentum with suitable

43:04.597 --> 43:05.997
coefficients.

43:06.000 --> 43:10.230
The rule for finding
coefficient A_n

43:10.233 --> 43:14.063
is to do this integral of
ψ_n*.

43:14.059 --> 43:15.839
This one, this is the rule.

43:15.840 --> 43:18.700
A_n is the
integral of

43:18.699 --> 43:21.409
ψ_n*ψ
dx.

43:21.409 --> 43:25.299
And once you found the
coefficients for all possible

43:25.297 --> 43:29.467
n then the probability that
you will have some momentum

43:29.474 --> 43:34.014
corresponding to m is
just A_m^(2).

43:34.010 --> 43:35.090
That is a recipe.

43:35.090 --> 43:38.520
What the recipe tells you is if
your function is made up as a

43:38.518 --> 43:42.118
special function with a definite
momentum of course you will get

43:42.117 --> 43:45.087
that momentum as the answer when
you measure it.

43:45.090 --> 43:49.350
If your function is a sum over
many different momentum

43:49.353 --> 43:54.183
functions then you can get any
of the answers in the sum,

43:54.179 --> 43:57.139
but if it had a big coefficient
in the expansion is more likely

43:57.135 --> 43:58.085
to be that answer.

43:58.090 --> 44:00.460
If it had a small coefficient
it's less likely.

44:00.460 --> 44:05.300
If it had no coefficient you
won't get that momentum at all.

44:05.300 --> 44:08.300
That's like saying if you had
ψ(x) it's likely

44:08.298 --> 44:10.658
where it's big,
unlikely where it's small and

44:10.655 --> 44:12.205
impossible where it is 0.

44:12.210 --> 44:13.280
So that's your job.

44:13.280 --> 44:15.830
Anytime someone gives you a
function you have to find these

44:15.827 --> 44:17.667
coefficients
A_n then look

44:17.672 --> 44:18.202
at them.

44:18.199 --> 44:20.339
They'll tell you what the
answer is.

44:20.340 --> 44:21.580
So that's what I'm going to do
now.

44:21.579 --> 44:26.249
I'm going to take some trial
functions and go through this

44:26.248 --> 44:30.838
machinery of finding the
coefficients and reading off the

44:30.835 --> 44:31.815
answers.

44:31.820 --> 44:37.060
So maybe if I do an example
you'll see where this is going.

44:37.059 --> 44:41.539
So let's take an example where
I'm going to pick first of all a

44:41.536 --> 44:45.866
very benign function then maybe
a more difficult function.

44:45.869 --> 44:51.459
The function I want to pick is
this.

44:51.460 --> 44:56.870
Some number n cosine
6Πx/L,

44:56.873 --> 45:01.283
somebody gives you that
function.

45:01.280 --> 45:04.390
That is not a state of definite
momentum because it is not

45:04.394 --> 45:06.914
e to the i
something x.

45:06.909 --> 45:09.469
So we already know when you
measure momentum you won't get a

45:09.465 --> 45:10.155
unique answer.

45:10.159 --> 45:12.929
You'll get many answers,
but what are the many answers?

45:12.929 --> 45:18.009
What are the many odds is what
we're asking.

45:18.010 --> 45:21.740
I forgot to mention one thing
in my postulate.

45:21.739 --> 45:25.949
What I forgot to mention is
that for all this to be true it

45:25.954 --> 45:29.974
is important that the functions,
the momentum functions,

45:29.974 --> 45:33.054
are all normalized and the
given function is also

45:33.052 --> 45:33.952
normalized.

45:33.949 --> 45:37.049
You should first normalize your
function then expand it in terms

45:37.045 --> 45:39.645
of these normalized functions of
definite momentum.

45:39.650 --> 45:42.710
Only then the squares of the
coefficient are the absolute

45:42.706 --> 45:43.576
probabilities.

45:43.579 --> 45:46.799
By that I mean if you do this
calculation,

45:46.800 --> 45:50.810
and you then went and added all
these A_m_

45:50.809 --> 45:55.369
squares,
you will find amazingly it adds

45:55.365 --> 45:56.365
up to 1.

45:56.369 --> 46:00.289
Because you can show
mathematically that if

46:00.289 --> 46:05.889
ψ*ψ dx is 1 then
the coefficients of expansion

46:05.889 --> 46:08.409
squared will also be 1.

46:08.409 --> 46:11.519
So if you took a normalized
ψ then the probabilities you

46:11.523 --> 46:14.853
get are absolute probabilities
because they will add up to 1.

46:14.849 --> 46:21.809
There's also a mathematical
result which I am not showing.

46:21.809 --> 46:23.859
All right, so let's go to this
problem.

46:23.860 --> 46:28.140
So the first job is,
normalize your ψ.

46:28.139 --> 46:30.959
So how to normalize this
function I'm going to demand

46:30.960 --> 46:33.620
that if you square ψ
and you integrate it,

46:33.619 --> 46:40.969
so I say N^(2) times
integral of cosine square as

46:40.969 --> 46:47.919
6Πx/L dx from 0 to
L should be 1.

46:47.920 --> 46:51.380
Now you don't have to look up
the table of integrals because

46:51.376 --> 46:54.656
when you take a cosine squared
or a sine squared over any

46:54.655 --> 46:57.815
number of full cycles the
average value is a half.

46:57.820 --> 47:02.820
That means over the length
L this integral will be

47:02.818 --> 47:03.978
L/2.

47:03.980 --> 47:08.700
So you want N^(2)L/2 to
be 1 or you want n to be square

47:08.702 --> 47:10.332
root of 2/L.

47:10.329 --> 47:18.469
So the normalized function
looks like square root of

47:18.472 --> 47:24.222
2/L cosine
6Πx/L.

47:24.219 --> 47:27.079
So that's the first job.

47:27.079 --> 47:30.699
Second is you can ask now what
are the coefficients

47:30.702 --> 47:32.372
A_n.

47:32.369 --> 47:38.109
So A_n is
going to be integral

47:38.108 --> 47:43.848
1/√L
e^(-2Πinx/L)

47:43.847 --> 47:51.707
times this function square root
of 2/L cosine 6Πx/L

47:51.708 --> 47:53.328
dx.

47:53.329 --> 47:53.959
Are you with me?

47:53.960 --> 47:55.060
That's the rule.

47:55.059 --> 47:56.819
Take the function,
multiply it by

47:56.820 --> 48:00.260
ψ_n*,
which is this guy here,

48:00.255 --> 48:04.545
this is the ψ_n*,
this is the ψ

48:04.547 --> 48:07.227
that's given to you,
and you're integrating the

48:07.233 --> 48:08.453
product in the integral.

48:08.449 --> 48:10.309
That'll give you
A_n.

48:10.309 --> 48:13.709
But I claim,
yeah, if you want you can do

48:13.710 --> 48:17.450
this integrals,
but I think there's a quicker

48:17.454 --> 48:19.074
way to do that.

48:19.070 --> 48:23.200
Have any idea of what the
quicker way just by looking at

48:23.204 --> 48:23.584
it?

48:23.579 --> 48:25.929
In other words,
if you can guess the expansion

48:25.925 --> 48:27.275
I'll say I will take it.

48:27.280 --> 48:35.700
Can you guess what the answer
looks like without doing the

48:35.695 --> 48:36.725
work?

48:36.730 --> 48:41.500
In other words I want you to
write this function as a sum

48:41.501 --> 48:45.081
over exponentials just by
looking at it.

48:45.079 --> 48:49.039
Can you tell me what it is?

48:49.039 --> 48:54.019
I want you to write it in the
form sum over m

48:54.018 --> 48:57.728
A_m
e^(2Πimx/L).

48:57.730 --> 49:01.990
You can find the
A_m by doing

49:01.987 --> 49:06.717
all this nasty work,
but I'm saying in this problem

49:06.717 --> 49:09.647
there's a much quicker way.

49:09.650 --> 49:14.200
You see it anywhere?

49:14.199 --> 49:17.959
What's the relation between
trigonometric functions and

49:17.958 --> 49:19.628
exponential functions?

49:19.630 --> 49:20.260
Yep?

49:20.260 --> 49:21.980
Student:  Euler's
formula?

49:21.980 --> 49:22.880
Prof: And what does it
say?

49:22.880 --> 49:27.680
Student:  e^(ix)
= cosine x i sine

49:27.675 --> 49:28.535
x.

49:28.539 --> 49:31.349
Prof: Yes,
but now I want to go the other

49:31.351 --> 49:31.711
way.

49:31.710 --> 49:34.140
It is certainly true that
cosine θ,

49:34.139 --> 49:37.989
I'm sorry, e^(iθ)
is cosine θ

49:37.985 --> 49:39.945
I sine θ.

49:39.949 --> 49:44.049
I told you, you forget the
formula at your own peril.

49:44.050 --> 49:48.900
The conjugate of that is
e^(-iθ) is cosine

49:48.896 --> 49:51.856
θ -i sine θ.

49:51.860 --> 49:56.140
If you add these two you'll
find cosine θ

49:56.143 --> 50:00.243
is (e^(iθ)
e^(-iθ))/2,

50:00.239 --> 50:04.969
and if you subtract them and
divide by 2i you'll find

50:04.965 --> 50:08.805
sine θ is e to the minus
over 2i.

50:08.809 --> 50:12.919
I wanted you to be familiar
with complex numbers enough so

50:12.920 --> 50:17.100
that when you see a cosine the
two exponentials jump out at

50:17.103 --> 50:17.683
you.

50:17.679 --> 50:20.399
Otherwise you will be doing all
these hard integrals that you

50:20.398 --> 50:21.258
don't have to do.

50:21.260 --> 50:25.810
That's like saying sine squared
cosine squared is 1,

50:25.809 --> 50:28.219
but you don't look it up.

50:28.219 --> 50:29.329
That's something you know.

50:29.329 --> 50:31.999
What you look up is your Social
Security number or mother's

50:32.003 --> 50:32.653
maiden name.

50:32.650 --> 50:34.460
You're allowed to forget those
things.

50:34.460 --> 50:35.970
You look in a book.

50:35.969 --> 50:37.229
That's the name.

50:37.230 --> 50:40.350
But this you have to know
because you cannot go anywhere

50:40.347 --> 50:43.577
without this because if you
don't know it all the time you

50:43.579 --> 50:46.299
must have done these
trigonometry calculations in

50:46.302 --> 50:47.042
school.

50:47.039 --> 50:49.539
You've got to plug in the right
stuff at the right time for

50:49.536 --> 50:50.696
things to all cancel out.

50:50.699 --> 50:51.989
You cannot say,
"I will look it up,"

50:51.992 --> 50:53.172
because you don't know what to
look up.

50:53.170 --> 50:58.480
So everything should be in your
head, and the minimum you should

50:58.480 --> 51:00.840
know is this,
the minimum.

51:00.840 --> 51:09.800
Therefore, I can come to this
function here and I can write it

51:09.804 --> 51:13.344
in terms of--
what I'm telling you is

51:13.338 --> 51:16.598
ψ(x),
which was given to us,

51:16.599 --> 51:21.949
is equal to the square root of
2/L times cosine

51:21.951 --> 51:29.531
6Πx/L,
but I'm going to write it as

51:29.532 --> 51:41.452
square root of 2/L times
½e^(6Πix/L)

51:41.452 --> 51:46.422
e^(-6Πix/L).

51:46.420 --> 51:53.130
So that I will write it very
explicitly as 1/√2 times

51:53.130 --> 51:59.960
e^(6Πix/L)/√L
(1/√2)e^(-6Πix/L)

51:59.958 --> 52:04.008
divided by √L.

52:04.010 --> 52:08.080
In other words,
what I've done is rather than

52:08.081 --> 52:13.731
doing integrals I just massaged
the given function and managed

52:13.726 --> 52:18.536
to write it as a sum over
normalized functions,

52:18.539 --> 52:22.579
as I said, with definite
momentum with some coefficients.

52:22.579 --> 52:25.119
In other words,
here it is in the form that we

52:25.123 --> 52:25.523
want.

52:25.518 --> 52:34.738
A_m
e^(2Πimx/L).

52:34.739 --> 52:39.279
You agree that I have got it to
the form I want?

52:39.280 --> 52:40.670
You see that?

52:40.670 --> 52:43.700
You take the cosine,
write it as sum of

52:43.695 --> 52:45.915
exponentials,
then put factors of

52:45.920 --> 52:48.670
√L so that that's a
normalized function,

52:48.670 --> 52:52.720
and that's a normalized
function, and everybody else is

52:52.724 --> 52:53.554
A.

52:53.550 --> 52:56.150
So what do you find here?

52:56.150 --> 53:00.040
What are the A's for
this problem?

53:00.039 --> 53:03.419
By comparing the two what do
you find are the A's in

53:03.423 --> 53:04.303
this problem?

53:04.300 --> 53:07.530
What is A_14?

53:07.530 --> 53:09.910
Let me ask you this.

53:09.909 --> 53:16.889
If you compare this to this one
what m are you getting?

53:16.889 --> 53:19.299
Do that in your head.

53:19.300 --> 53:27.490
If you compare this to this one
what is m?

53:27.489 --> 53:29.459
Student:  3.

53:29.460 --> 53:29.830
Prof: Pardon me?

53:29.829 --> 53:30.299
Student:  3.

53:30.300 --> 53:33.280
Prof: m is 3 for this
guy,

53:33.280 --> 53:37.130
so this is really
ψ_3 and this is

53:37.130 --> 53:42.320
ψ_-3 because you
can see there are two momenta in

53:42.320 --> 53:43.660
the problem.

53:43.659 --> 53:46.539
And you can stare at them,
you can see right away.

53:46.539 --> 53:51.819
Therefore, A_3
is 1/√2,

53:51.817 --> 53:57.327
and A_-3 is
also 1/√2.

54:10.409 --> 54:19.909
All other A's are 0
because they don't have a role.

54:19.909 --> 54:22.549
So you might think every term
must appear.

54:22.550 --> 54:23.420
It need not.

54:23.420 --> 54:27.440
So this is a good example that
tells you only two of the m's

54:27.440 --> 54:29.690
make it to the final summation.

54:29.690 --> 54:33.100
All the other m's are 0 and
they happen to come with equal

54:33.103 --> 54:34.903
coefficients,
1 over root 2.

54:34.900 --> 54:37.700
The square of that is 1 over 2,
and you can see these

54:37.697 --> 54:39.577
probabilities nicely add up to
1.

54:39.579 --> 54:43.089
I told you if you normalize the
initial function the probability

54:43.092 --> 54:44.992
for everything will add up to 1.

54:44.989 --> 54:47.209
So this is a particle.

54:47.210 --> 54:50.590
If a particle is in a wave
function cosine 6Πx/L,

54:50.590 --> 54:54.490
which is a real wave function,
and you can plot that guy going

54:54.487 --> 54:57.137
around the circle,
it describes a particle whose

54:57.135 --> 54:58.715
momentum,
if you measure,

54:58.724 --> 55:01.354
will give you only 1 of 2
answers,

55:01.349 --> 55:09.609
m = 3 is p =
2Πℏ/L times 3 or - 3

55:09.614 --> 55:14.464
plus or minus 6Πℏ/L.

55:14.460 --> 55:15.350
You can see that too.

55:15.349 --> 55:17.069
I mean, just look at this
function.

55:17.070 --> 55:22.650
If you call it e^(ipx) p
is 6Πℏ/L,

55:22.646 --> 55:28.436
so this particle has only two
possible answers when you

55:28.436 --> 55:30.686
measure momentum.

55:30.690 --> 55:34.040
So it is not as good as the
single exponential which has

55:34.043 --> 55:35.633
only one momentum in it.

55:35.630 --> 55:39.070
This is made up of two possible
momenta, but you won't get

55:39.072 --> 55:40.042
anything else.

55:40.039 --> 55:44.229
You will not get any other
momentum if you measure this.

55:44.230 --> 55:48.100
So if you got m = 14 as
a momentum there's something

55:48.096 --> 55:48.626
wrong.

55:48.630 --> 55:54.590
It's still probabilities,
but it tells you that there is

55:54.585 --> 55:58.805
non-0 probability only for these
two.

55:58.809 --> 56:02.299
Now in a minute I will take on
a more difficult problem where

56:02.300 --> 56:05.860
you cannot look at the answer,
you cannot look at the function

56:05.860 --> 56:08.930
ψ and just by fiddling with
it bring it to this form.

56:08.929 --> 56:09.609
You understand?

56:09.610 --> 56:12.930
It is very fortunate for you
that I gave you cosine which is

56:12.932 --> 56:16.542
just the sum of two exponentials
so the two what are called plane

56:16.536 --> 56:20.026
wave exponentials are staring at
you and you pick them up.

56:20.030 --> 56:23.140
I can write other functions,
crazy functions for which you

56:23.139 --> 56:26.139
will have to do the integral to
find the coefficient.

56:26.139 --> 56:30.629
But I'm going to tell you one
other postulate of quantum

56:30.630 --> 56:31.610
mechanics.

56:31.610 --> 56:34.060
In the end I will assemble all
the postulates for you.

56:34.059 --> 56:35.709
I'm going to tell you one more
postulate.

56:35.710 --> 56:39.110
It's called the measurement
postulate.

56:39.110 --> 56:44.250
The measurement postulate says
that if you make a measurement,

56:44.250 --> 56:48.010
and you found the particle to
be at a certain location

56:48.007 --> 56:49.997
x,
then right after the

56:49.996 --> 56:53.566
measurement the wave function of
the particle will be a spike at

56:53.574 --> 56:56.704
the point x because you
know that you found it at

56:56.699 --> 57:00.159
x it means if that has
any meaning at all if you repeat

57:00.164 --> 57:03.524
the measurement immediately
afterwards you've got to get the

57:03.516 --> 57:04.706
same answer.

57:04.710 --> 57:08.530
Therefore, your initial
function could have looked like

57:08.525 --> 57:12.765
that, but after the measurement
it collapses to a function at

57:12.766 --> 57:15.096
the point where you found it.

57:15.099 --> 57:17.539
This is called the collapse of
the wave function.

57:17.539 --> 57:22.409
It goes from being able to be
found anywhere to being able to

57:22.409 --> 57:25.979
be found only where you found it
just now.

57:25.980 --> 57:28.030
It won't stay that way for
long, but right after

57:28.030 --> 57:29.560
measurement that's what it'll
be.

57:29.559 --> 57:32.999
But the state of the system
changes following the

57:32.996 --> 57:33.996
measurement.

57:34.000 --> 57:36.700
And if you measured x
it'll turn into a wave

57:36.697 --> 57:39.557
function with well defined
x which we know is a

57:39.559 --> 57:42.419
spike at x equal to
wherever you found it.

57:42.420 --> 57:47.860
Similarly, if you measured
momentum and you found it at--

57:47.860 --> 57:49.960
if you measure momentum,
first of all you'll get only

57:49.960 --> 57:52.530
one of two answers,
m = 3 or m = -3.

57:52.530 --> 57:56.260
If you got m = 3 the
state after the measurement will

57:56.264 --> 57:57.344
reduce to this.

57:57.340 --> 57:59.500
This guy will be gone.

57:59.500 --> 58:04.680
This will be the state.

58:04.679 --> 58:07.939
So from being able to have two
momenta which are equal and

58:07.938 --> 58:11.428
opposite the act of measurement
will force it to be one or the

58:11.427 --> 58:11.997
other.

58:12.000 --> 58:14.280
It can give you either one,
but once you've got it that's

58:14.280 --> 58:14.810
the answer.

58:14.809 --> 58:16.199
It's like the double slit.

58:16.199 --> 58:18.409
It can be here and it can be
there.

58:18.409 --> 58:20.769
It is not anywhere in
particular, but if you shine

58:20.766 --> 58:23.316
light and you catch it,
right after the measurement it

58:23.317 --> 58:25.287
is in front of one slit or the
other.

58:25.289 --> 58:26.929
Think of this as double slit.

58:26.929 --> 58:30.179
There's some probability for 3
and some for -3,

58:30.177 --> 58:33.987
but if you catch it at 3 it'll
collapse to this one.

58:33.989 --> 58:37.599
So what happens is in the sum
over many terms the answer will

58:37.597 --> 58:40.967
correspond to one of them,
and whichever one you got only

58:40.965 --> 58:42.765
that one term will remain.

58:42.768 --> 58:46.508
Everything will be deleted from
the wave function.

58:46.510 --> 58:49.960
So the act of measurement
filters out from the sum,

58:49.960 --> 58:53.630
the one term corresponding to
the one answer that you got,

58:53.630 --> 58:57.980
this is called the collapse of
the wave function.

58:57.980 --> 59:01.480
So in classical mechanics when
you measure the position of a

59:01.481 --> 59:03.381
particle nothing much happens.

59:03.380 --> 59:05.700
It doesn't even know you
measured it because there are

59:05.704 --> 59:08.034
ways to measure it without
affecting it in any way.

59:08.030 --> 59:10.930
So if had a momentum p
at a position x before

59:10.925 --> 59:13.815
the measurement it's the answer
right after the measurement

59:13.822 --> 59:16.222
because you can do noninvasive
measurements.

59:16.219 --> 59:20.059
In quantum theory there are no
such measurements in general.

59:20.059 --> 59:23.449
In general the measurement will
change the wave function from

59:23.451 --> 59:26.731
being in one of many options to
the one option you got,

59:26.730 --> 59:29.070
but the answer depends on what
you measure.

59:29.070 --> 59:32.040
With the same wave function on
a ring, let's say,

59:32.043 --> 59:35.703
if you measure position and you
found it here that'll be the

59:35.697 --> 59:36.377
answer.

59:36.380 --> 59:40.310
If you measured momentum and
you got 5 it'll be something

59:40.306 --> 59:42.266
with 5 oscillations in it.

59:42.268 --> 59:44.878
So it'll collapse to that
particular function and what it

59:44.882 --> 59:46.892
collapses it depends on what you
measure.

59:46.889 --> 59:49.479
And for a long time I'm going
to focus only on x and

59:49.478 --> 59:49.968
p.

59:49.969 --> 59:51.949
Of course there are other
things you can measure and I

59:51.952 --> 59:53.302
don't want to go there right
now,

59:53.300 --> 59:56.920
but if you measure x it
collapses to a spike at that

59:56.922 --> 59:57.612
location.

59:57.610 --> 1:00:01.720
If you measure p it
collapses to the one term

1:00:01.722 --> 1:00:05.592
wherever that was,
that one e^(ipx) in the

1:00:05.594 --> 1:00:06.244
sum.

1:00:06.239 --> 1:00:09.819
So another interesting thing is
the only measurement,

1:00:09.820 --> 1:00:13.890
only answers you will get in
the measurement are the allowed

1:00:13.885 --> 1:00:15.465
values of momentum.

1:00:15.469 --> 1:00:18.509
You'll never get a momentum
that's not allowed.

1:00:18.510 --> 1:00:22.130
And once you get one of the
allowed answers there are two

1:00:22.125 --> 1:00:23.735
things I'm telling you.

1:00:23.739 --> 1:00:27.729
One is the probability you will
get that answer is proportional

1:00:27.731 --> 1:00:31.081
to the square of that
coefficient in the expansion of

1:00:31.079 --> 1:00:32.559
the given function.

1:00:32.559 --> 1:00:35.109
In our problem there are only
two non-zero coefficients.

1:00:35.110 --> 1:00:37.470
Both happen to be 1/√2.

1:00:37.469 --> 1:00:38.639
It happened to be an equal
mixture,

1:00:38.639 --> 1:00:41.309
but you can easily imagine some
other problem where this is

1:00:41.306 --> 1:00:42.896
1/√6,
and something that is

1:00:42.900 --> 1:00:44.300
1/√6,
and something that is

1:00:44.298 --> 1:00:46.338
1/√3,
they should all add up to 1

1:00:46.338 --> 1:00:48.048
when you square and add them.

1:00:48.050 --> 1:00:51.820
Then you can get all those
answers with those

1:00:51.818 --> 1:00:53.188
probabilities.

1:00:53.190 --> 1:00:56.550
So the last thing I'm going to
do is just one more example of

1:00:56.547 --> 1:00:59.907
this where you actually have to
do an integral and you cannot

1:00:59.905 --> 1:01:02.475
just read off the answer by
looking at it.

1:01:02.480 --> 1:01:10.580
Then we're done with this whole
momentum thing.

1:01:10.579 --> 1:01:15.199
I will write the postulates
later, so I don't want to write

1:01:15.202 --> 1:01:18.632
it in my handwriting,
but one final time.

1:01:18.630 --> 1:01:20.850
Measurement of x
collapses the function

1:01:20.853 --> 1:01:22.883
ψ(x) to a spike at
x.

1:01:22.880 --> 1:01:25.130
Measurement of p
collapses ψ

1:01:25.130 --> 1:01:28.740
to that particular plane wave
with that particular p or

1:01:28.742 --> 1:01:30.462
m on the exponent.

1:01:30.460 --> 1:01:32.640
And after that that's what's
taken to be.

1:01:32.639 --> 1:01:35.259
People always ask,
"How do we know what state

1:01:35.257 --> 1:01:37.017
the quantum system is in?"

1:01:37.018 --> 1:01:38.938
Who tells you what
ψ(x) is?

1:01:38.940 --> 1:01:40.820
It's the act of measurement.

1:01:40.820 --> 1:01:44.590
If you measure the guy and you
found him at x = 5 the

1:01:44.588 --> 1:01:47.078
answer is ψ
is a big spike there.

1:01:47.079 --> 1:01:49.869
If you measure momentum and
you've got m = 3 the

1:01:49.865 --> 1:01:53.215
answer is that particular wave
function with 3 on the exponent.

1:01:53.219 --> 1:01:58.919
So measurements are a way to
prepare states.

1:01:58.920 --> 1:02:03.000
By the way, it's very important
that if you had a system like

1:02:03.000 --> 1:02:06.740
this the probability for getting
a certain x,

1:02:06.739 --> 1:02:10.039
say at this x,
is proportional to the square

1:02:10.038 --> 1:02:11.158
of that number.

1:02:11.159 --> 1:02:14.869
Once you took the state and
didn't measure x,

1:02:14.869 --> 1:02:18.139
but measured momentum it'll
become one of these oscillatory

1:02:18.139 --> 1:02:20.339
functions with a definite
wavelength.

1:02:20.340 --> 1:02:23.280
It's a complex exponential,
but I'm showing you the real

1:02:23.284 --> 1:02:23.664
part.

1:02:23.659 --> 1:02:27.209
Now if you measure position,
in fact the square of this will

1:02:27.208 --> 1:02:27.808
be flat.

1:02:27.809 --> 1:02:30.879
Remember e^(ipx) is flat.

1:02:30.880 --> 1:02:33.970
So in classical mechanics if I
measure position,

1:02:33.967 --> 1:02:37.317
and I measure momentum,
and I measure position again

1:02:37.317 --> 1:02:39.877
I'll keep getting the same
answers.

1:02:39.880 --> 1:02:41.380
I see where it is.

1:02:41.380 --> 1:02:43.040
I see how fast it's moving.

1:02:43.039 --> 1:02:44.359
Again, I see where it is.

1:02:44.360 --> 1:02:47.980
If I do all of this in rapid
succession you will get the same

1:02:47.981 --> 1:02:49.491
answer xp xp xp.

1:02:49.489 --> 1:02:52.629
In quantum theory once you
measure x it'll become a

1:02:52.628 --> 1:02:54.058
big spike at that point.

1:02:54.059 --> 1:02:56.639
You can get all kinds of
answers p.

1:02:56.639 --> 1:02:59.199
If you measure p you've
got an answer.

1:02:59.199 --> 1:03:01.419
That answer is a new wave
function which is completely

1:03:01.416 --> 1:03:01.706
flat.

1:03:01.710 --> 1:03:03.730
That means if you measure
x you'll no longer get

1:03:03.728 --> 1:03:04.438
the old x.

1:03:04.440 --> 1:03:07.030
In fact you can get any
x.

1:03:07.030 --> 1:03:09.970
That's why you can never filter
out a state with well defined

1:03:09.969 --> 1:03:12.709
x and p because
states of well defined x

1:03:12.713 --> 1:03:15.303
are very spiky,
and states of well-defined

1:03:15.300 --> 1:03:18.630
p are very broad,
so you cannot have it both ways.

1:03:18.630 --> 1:03:22.520
And that's what I want to show
you in this final example.

1:03:22.518 --> 1:03:26.348
I'm going to take the following
wave function on this ring.

1:03:26.349 --> 1:03:31.599
The function I'm going to take
is, ψ is some

1:03:31.601 --> 1:03:35.951
ne^(−α|x|),
mod x [

1:03:35.952 --> 1:03:37.852
|x| ]means this is
x = 0.

1:03:37.849 --> 1:03:42.129
To the right it falls
exponentially and to the left it

1:03:42.132 --> 1:03:43.912
falls exponentially.

1:03:43.909 --> 1:03:45.379
Is it clear what I'm saying
here?

1:03:45.380 --> 1:03:48.170
This is my ring and I'm trying
to plot the function.

1:03:48.170 --> 1:03:52.280
It is highest at the origin and
falls exponentially equally for

1:03:52.280 --> 1:03:54.470
positive and negative x.

1:03:54.469 --> 1:03:58.839
So that's the meaning of mod
x.

1:03:58.840 --> 1:04:02.800
So how far can you go before
ψ becomes negligible?

1:04:02.800 --> 1:04:06.660
Well, that's when e^(-x)
is a big number.

1:04:06.659 --> 1:04:10.189
So I'm going to assume that
when you go all the way around

1:04:10.186 --> 1:04:12.656
half the circle,
I'm going to assume that

1:04:12.661 --> 1:04:15.261
αL is much bigger
than 1.

1:04:15.260 --> 1:04:18.550
That means its function dies
very quickly spreading

1:04:18.545 --> 1:04:20.775
negligible beyond some distance.

1:04:20.780 --> 1:04:24.180
How far does a function live?

1:04:24.179 --> 1:04:28.549
Roughly speaking you can go a
distance Δx so that

1:04:28.552 --> 1:04:31.782
α times Δx is
roughly 1.

1:04:31.780 --> 1:04:35.830
Because if you plot this
exponential you ask,

1:04:35.829 --> 1:04:38.609
when does it come to say half
its value or one fourth of its

1:04:38.614 --> 1:04:41.414
value,
you'll find it's some number of

1:04:41.409 --> 1:04:42.709
order 1/α.

1:04:42.710 --> 1:04:50.160
So this is a particle whose
position has an uncertainty of

1:04:50.157 --> 1:04:52.507
order 1/α.

1:04:52.510 --> 1:04:55.410
So you can make it very narrow
in space so you know pretty much

1:04:55.414 --> 1:04:57.294
where it is or you can make it
broad,

1:04:57.289 --> 1:04:58.909
but I want to consider only
those problems,

1:04:58.909 --> 1:05:03.019
where even if it's broad it's
dead by the time you go to the

1:05:03.018 --> 1:05:04.758
other side to the back.

1:05:04.760 --> 1:05:08.160
That's for mathematical
convenience.

1:05:08.159 --> 1:05:12.229
So this is the state and I want
to ask myself any question we

1:05:12.231 --> 1:05:12.911
can ask.

1:05:12.909 --> 1:05:16.059
First question I can ask is if
I look for its position what

1:05:16.059 --> 1:05:16.819
will I find?

1:05:16.820 --> 1:05:18.280
I think we have done it many,
many times.

1:05:18.280 --> 1:05:26.620
You square the guy you get
n^(2)e^(-2α|x|).

1:05:26.619 --> 1:05:27.529
So that's the shape.

1:05:27.530 --> 1:05:28.790
That shape looks the same.

1:05:28.789 --> 1:05:31.739
If you square an exponential
you get another exponential.

1:05:31.739 --> 1:05:38.949
But now let me demand that this
integral dx be equal to

1:05:38.947 --> 1:05:39.417
1.

1:05:39.420 --> 1:05:41.230
So what is that integral?

1:05:41.230 --> 1:05:44.370
I hope you can see that this
function is an even function of

1:05:44.369 --> 1:05:46.339
x because it's mod
x.

1:05:46.340 --> 1:05:52.450
So it's double the answer I get
for positive x.

1:05:52.449 --> 1:05:55.069
Oh yeah, sorry,
not minus infinity.

1:05:55.070 --> 1:05:59.520
It's −L/2 to
L/2.

1:05:59.518 --> 1:06:03.058
You can go this way L/2
and we can go that way

1:06:03.063 --> 1:06:03.953
L/2.

1:06:03.949 --> 1:06:09.799
So it's n^(2) is 2 times
e to the -α

1:06:09.800 --> 1:06:13.700
x 2α
x dx.

1:06:13.699 --> 1:06:15.169
Student:  Should it be L
over 2?

1:06:15.170 --> 1:06:16.050
Prof: Pardon me?

1:06:16.050 --> 1:06:17.830
Student:  Should it be L
over 2.

1:06:17.829 --> 1:06:20.619
Prof: Yes, thank you.

1:06:20.619 --> 1:06:24.919
So I'm going to now assume that
in the upper limit of the

1:06:24.918 --> 1:06:29.908
integration instead of going up
to L/2 I go to infinity.

1:06:29.909 --> 1:06:33.809
It doesn't matter because this
guy is dead long before that.

1:06:33.809 --> 1:06:35.099
That's why I made that choice.

1:06:35.099 --> 1:06:40.459
So that I'm going to write this
2 times N^(2) times 0 to

1:06:40.463 --> 1:06:43.583
infinity e^(-2αx)
dx.

1:06:43.579 --> 1:06:45.289
That's a pretty trivial
integral.

1:06:45.289 --> 1:06:47.889
It's just 1/2α.

1:06:47.889 --> 1:06:49.559
If you want I will write it out.

1:06:49.559 --> 1:06:54.919
It's
e^(-2αx)/-2α

1:06:54.918 --> 1:06:59.238
from 0 to infinity.

1:06:59.239 --> 1:07:01.659
At the upper limit when you put
infinity you get 0.

1:07:01.659 --> 1:07:04.499
At the lower limit when you put
x = 0 you get 1.

1:07:04.500 --> 1:07:09.870
There's a minus sign from this
and you get

1:07:09.867 --> 1:07:12.877
N^(2)/α.

1:07:12.880 --> 1:07:18.640
It should be 1,
or N is equal to the

1:07:18.637 --> 1:07:20.967
√α.

1:07:20.969 --> 1:07:23.839
So my normalized wave function,
this is the first order of

1:07:23.835 --> 1:07:30.125
business,
is square root of α

1:07:30.125 --> 1:07:36.415
e^(−α|x )^(|).

1:07:36.420 --> 1:07:38.050
I just squared it and I
integrated it.

1:07:38.050 --> 1:07:41.130
The only funny business I did
was instead of cutting off the

1:07:41.130 --> 1:07:44.260
upper integral at L/2 I
cut if off at infinity because

1:07:44.264 --> 1:07:46.884
e^(−L/2) is
e to the minus nine

1:07:46.876 --> 1:07:48.306
million,
let's say.

1:07:48.309 --> 1:07:51.469
So I don't care if it's nine
million or infinity.

1:07:51.469 --> 1:07:55.489
To make the life simpler I just
did it that way.

1:07:55.489 --> 1:08:02.729
Now I want to ask you,
what is A(p)?

1:08:02.730 --> 1:08:07.980
A(p),
the coefficient of the

1:08:07.981 --> 1:08:15.861
expansion remember is
e^(−ipx/ℏ)/√L

1:08:15.858 --> 1:08:24.038
times ψ(x) dx
from −L/2 to

1:08:24.043 --> 1:08:26.673
L/2.

1:08:26.670 --> 1:08:30.120
Now I'm going to work with p
rather than m.

1:08:30.119 --> 1:08:31.209
I will go back and forth.

1:08:31.210 --> 1:08:34.230
You should use to the notion
that the momentum can be either

1:08:34.233 --> 1:08:37.263
labeled by the actual momentum
or the quantum number m which

1:08:37.256 --> 1:08:39.866
tells you how much momentum you
have in multiples of

1:08:39.869 --> 1:08:41.099
2Πℏ/m.

1:08:41.100 --> 1:08:42.710
So you've got to do this
integral.

1:08:42.710 --> 1:08:44.650
So let's write this integral.

1:08:44.649 --> 1:08:51.869
This looks like
−L/2 to L/2.

1:08:51.868 --> 1:08:59.868
In fact I'm going to change
this integral to minus infinity

1:08:59.867 --> 1:09:07.037
to plus infinity because this
function e^(ipx/ℏ)

1:09:07.037 --> 1:09:13.377
times
e^(−α|x|) dx.

1:09:13.380 --> 1:09:16.600
Do you understand that these
limits can be made to be plus

1:09:16.604 --> 1:09:20.114
and minus infinity because area
under a graph that's falling so

1:09:20.113 --> 1:09:22.233
rapidly,
whether it's between minus and

1:09:22.225 --> 1:09:24.285
plus L/2 or minus and
plus infinity,

1:09:24.289 --> 1:09:25.509
is going to be the same.

1:09:25.510 --> 1:09:29.890
It's just that this integral is
so much easier to do.

1:09:29.890 --> 1:09:34.700
Now you cannot jump out and do
this integral because it's a mod

1:09:34.698 --> 1:09:35.938
x here.

1:09:35.939 --> 1:09:38.609
So mod x is not
x, it is x when

1:09:38.610 --> 1:09:41.440
it's positive and it's -x
when it's negative.

1:09:41.439 --> 1:09:44.199
So you've got to break this
integral into two parts.

1:09:44.198 --> 1:09:49.148
One part where x is
positive from 0 to infinity,

1:09:49.150 --> 1:09:52.730
I'm sorry, I also forgot a
root, α

1:09:52.726 --> 1:09:55.016
and A√L.

1:09:55.020 --> 1:09:56.560
Do you see that?

1:09:56.560 --> 1:10:04.060
So this is really square root
of α/L times

1:10:04.060 --> 1:10:12.120
e^(-αx)
e^(−ipx/ℏ) dx

1:10:12.118 --> 1:10:19.898
plus another integral from
minus infinity to 0 e^(

1:10:19.899 --> 1:10:27.539
αx) times
e^(−ipx/ℏ) dx.

1:10:27.538 --> 1:10:30.048
I split the integral into two
parts.

1:10:30.050 --> 1:10:31.760
So I didn't make a mistake here.

1:10:31.760 --> 1:10:35.810
This really is e^( αx)
because x is

1:10:35.805 --> 1:10:36.625
negative.

1:10:36.630 --> 1:10:38.710
So what one does in such
situations,

1:10:38.710 --> 1:10:40.740
I'm going to do it quickly and
you can go home and check it,

1:10:40.738 --> 1:10:43.228
just calculus,
change the variable from

1:10:43.228 --> 1:10:45.518
x to -x
everywhere.

1:10:45.520 --> 1:10:49.100
In the terms of new variable
this will make a -x.

1:10:49.100 --> 1:10:51.310
That'll become x.

1:10:51.310 --> 1:10:54.180
dx will become minus of
the new variable.

1:10:54.180 --> 1:10:56.740
The limits will be plus
infinity to 0,

1:10:56.739 --> 1:11:00.339
and you can flip that for
another change of sign 0 to

1:11:00.340 --> 1:11:01.240
infinity.

1:11:01.238 --> 1:11:03.368
So that was a very rapid
slight-of-hand,

1:11:03.368 --> 1:11:05.168
but I don't want to delay that.

1:11:05.170 --> 1:11:08.900
This is just--you go home and
if you want check that if

1:11:08.904 --> 1:11:11.814
x goes to -x you
get that.

1:11:11.810 --> 1:11:17.200
So you notice this is the
complex conjugate of this one.

1:11:17.198 --> 1:11:19.978
Whatever function I'm
integrating here is the

1:11:19.979 --> 1:11:23.329
conjugate of it because this is
real, αx,

1:11:23.327 --> 1:11:25.977
and -ipx has become
ipx.

1:11:25.979 --> 1:11:29.359
So if I find the first part of
the integral I just take that

1:11:29.363 --> 1:11:31.373
times its conjugate and I'm
done.

1:11:31.369 --> 1:11:33.219
So what do I get for that?

1:11:33.220 --> 1:11:37.880
I get square root of
α/L times--

1:11:37.880 --> 1:11:43.240
remember integral
e^(-αx)

1:11:43.244 --> 1:11:48.044
dx from 0 to infinity is
1/α,

1:11:48.038 --> 1:11:55.768
but what I have here is dx
e^(-α) ip/ℏ

1:11:55.769 --> 1:12:05.269
x_0 to infinity
plus the complex conjugate.

1:12:05.270 --> 1:12:08.580
Now you may be very nervous
about doing this integral with a

1:12:08.582 --> 1:12:11.952
complex number in it because
real we all know the integral is

1:12:11.949 --> 1:12:14.139
just 1/α
e^(-α).

1:12:14.140 --> 1:12:17.220
It turns out that it's true
even if it's got an imaginary

1:12:17.220 --> 1:12:19.860
part as long as you have a
positive real part.

1:12:19.859 --> 1:12:22.159
In other words,
the answer here doesn't depend

1:12:22.161 --> 1:12:23.441
on this guy being real.

1:12:23.439 --> 1:12:29.139
So it's really α
over L times 1 over

1:12:29.141 --> 1:12:35.561
α ip/ℏ plus a
complex conjugate which is

1:12:35.557 --> 1:12:40.427
α −
ip/ℏ.

1:12:40.430 --> 1:12:45.600
Now you should be able to
combine these two denominators

1:12:45.595 --> 1:12:51.045
and you get square root of
α/L divided by α^(2)

1:12:51.045 --> 1:12:56.395
plus p^(2) over
ℏ^(2) times 2α.

1:12:56.399 --> 1:12:58.649
Again, this is something you
can go and check,

1:12:58.653 --> 1:13:01.463
but I don't want to wait until
everyone can do this thing

1:13:01.456 --> 1:13:03.756
because I want to tell you the
punch line.

1:13:03.760 --> 1:13:05.730
So this is what it looks like.

1:13:05.729 --> 1:13:09.089
A(p) looks like a whole
bunch of numbers I'm not going

1:13:09.086 --> 1:13:11.656
to worry about,
but look at the denominator.

1:13:11.658 --> 1:13:15.438
It's α^(2) p^(2)
over ℏ^(2),

1:13:15.443 --> 1:13:18.843
or if you want multiply by
ℏ^(2).

1:13:18.840 --> 1:13:23.600
There are some other numbers
I'm not interested in.

1:13:23.600 --> 1:13:24.980
The numbers are not important.

1:13:24.979 --> 1:13:29.739
How does it vary with p
is all I'm asking you to think

1:13:29.743 --> 1:13:30.383
about.

1:13:30.380 --> 1:13:34.840
I'm sorry, A(p), this is
just A(p),

1:13:34.836 --> 1:13:39.756
but I want the A(p)
squared that looks like the

1:13:39.756 --> 1:13:41.516
square of this.

1:13:41.520 --> 1:13:46.820
All I want you to notice is
that this function is peaked at

1:13:46.818 --> 1:13:52.298
p = 0 and falls very
rapidly as p increases.

1:13:52.300 --> 1:13:54.790
When p is 0 you've got
the biggest height.

1:13:54.788 --> 1:13:57.478
When does it become half as big
or one fourth as big?

1:13:57.479 --> 1:14:01.169
Roughly when p^(2) is
equal to h^(2)α^(2)

1:14:01.172 --> 1:14:04.562
because that's when these
numbers become comparable,

1:14:04.560 --> 1:14:09.320
therefore when p is
ℏα this function

1:14:09.322 --> 1:14:14.592
will have a denominator which is
twice what it had here or maybe

1:14:14.586 --> 1:14:15.836
one fourth.

1:14:15.840 --> 1:14:18.120
I'm not worried about factors
like 1 and 2.

1:14:18.118 --> 1:14:22.628
The point is in momentum space,
in momentum you can get all

1:14:22.628 --> 1:14:26.838
kinds of values of p,
but the odds decrease very

1:14:26.840 --> 1:14:30.440
rapidly for p bigger than
ℏα.

1:14:30.439 --> 1:14:33.299
So the most likely value is 0,
but that's the spread.

1:14:33.300 --> 1:14:36.620
And Δp is
ℏα,

1:14:36.618 --> 1:14:39.908
or α times
Δp is ℏ,

1:14:39.908 --> 1:14:45.738
and that is the uncertainty
principle because α

1:14:45.735 --> 1:14:48.645
is just Δx.

1:14:48.649 --> 1:14:50.189
By the way, I will publish
these notes too,

1:14:50.192 --> 1:14:52.252
so you don't have to worry if
you didn't write everything

1:14:52.248 --> 1:14:52.578
down.

1:14:52.579 --> 1:14:53.759
I suggest you--yes?

1:14:53.760 --> 1:14:56.850
Student:  Can you go
over again what cc is?

1:14:56.850 --> 1:14:58.320
Prof: Complex conjugate.

1:14:58.319 --> 1:14:59.839
That's what I did.

1:14:59.840 --> 1:15:03.160
Whatever number this one is the
other guy is obtained by

1:15:03.158 --> 1:15:05.148
changing i to -i.

1:15:05.149 --> 1:15:07.379
Look, all I want you to notice
is this.

1:15:07.380 --> 1:15:12.850
I took a function whose width
is roughly 1/α.

1:15:12.850 --> 1:15:15.560
Then I looked at what kind of
momenta I can get.

1:15:15.560 --> 1:15:19.780
Then I find that the narrower
the function bigger the spread

1:15:19.780 --> 1:15:22.500
in the possible momenta you can
get.

1:15:22.500 --> 1:15:25.640
So squeezing it in x
broadens it out in p.

1:15:25.640 --> 1:15:27.900
And that's the origin of the
uncertainty principle.

1:15:27.899 --> 1:15:29.349
It's simply a mathematical
result.

1:15:29.350 --> 1:15:32.390
The functions which are narrow
in x have a Fourier

1:15:32.391 --> 1:15:34.621
series which is very broad in
p.

1:15:34.618 --> 1:15:37.638
The quantum mechanics relates
p to momentum and

1:15:37.636 --> 1:15:39.796
therefore the uncertainty
principle.

1:15:39.800 --> 1:15:42.040
So anyway, I'll give you some
homework on this,

1:15:42.041 --> 1:15:44.671
and you can also fill in the
blanks of this derivation,

1:15:44.671 --> 1:15:46.281
which I think is very useful.

1:15:46.279 --> 1:15:50.999