WEBVTT

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Prof: So I got an
interesting email this weekend

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from two of your buddies,
Jerry Wang and Emma Alexander.

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The title of the email is this.

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It says Casaplanka,
so I already know these guys

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are up to no good.

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And the question is the
following.

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They said, "You've written
a wave function ψ(x) and

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A(p).

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There seems to be no reference
of time, so where is time in

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this," and they couldn't
stop there.

00:44.100 --> 00:47.070
That would be a good place to
stop but they went on to say,

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"Are you saying the ψ
is just a ψ

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as time goes by?"

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So that's the kind of stuff
that appeals to me,

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so I don't care if you don't
learn any quantum mechanics,

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but if you can do this kind of
stuff I'm not worried about you.

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But now I have to give a
serious answer to that serious

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question,
which I sort of mentioned

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before, which is that everything
I've done so far is for one

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instant in time.

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I hope I made it very clear.

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So here's the analogy.

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If someone comes to you and
says, "Tell me all about

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Newtonian mechanics.

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How does it work?

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What's the scheme like?"

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You say, "At any instant
someone has to give you the x

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and p of that particle at that
time," and that's all you

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need to know.

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That's all you can ask and
that's all you need to know.

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Everything about that one
particle is completely given by

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this x and p.

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That is called a state.

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That's the complete description
of any system is called at

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state,
and a state in classical

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mechanics for one particle in
one dimension is just a pair of

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numbers.

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It's a state because given that
I can predict the future for

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you.

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I can predict the future
because if I knew the initial

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momentum, namely initial
velocity.

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I know how fast the guy is
moving, and I know where it is

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at the initial x.

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A tiny amount of time later
it'll be at x vt,

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but t is very small.

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Then since I know the
acceleration from Newton's Laws,

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which is rate of change of
velocity,

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if I knew the initial velocity
or initial momentum I can get

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the momentum a little later.

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Then in that manner I can inch
forward in time.

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So you need Newton's Laws,
and Newton's Laws tell you,

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if you like,
rate of change of momentum is

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the force.

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So if you like you can write
this mdx/dt squared is the

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force.

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So this part is called dynamics
and this part is called

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kinematics.

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So dynamics is how do things
change with time?

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Kinematics is what do you need
to know about the particle?

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In the quantum version of this
if you say,

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"What constitutes complete
knowledge of the

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particle?,"
the answer is this function

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ψ(x).

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If you knew this function you
know all that there is to know

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about the particle.

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We're not saying it's real.

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It can be complex.

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Then you can ask,
"If you knew the ψ

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at one time what's it going to
be later on?"

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That's the analog of this one.

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So I have not come to that yet.

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So this is a pretty long story.

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It takes so long just to tell
you what to do at one time.

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Then I'm going to tell you how
to go from now to later.

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In Newtonian mechanics it's
just a one-word answer,

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x and p now is the whole story.

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So anyway, since we've got four
lectures including today I will,

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of course, tell you before it's
all over what is the analog of

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this.

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Namely I'll tell you a formula
for dψ/dt.

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How does ψ change with time?

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So let's summarize what we know.

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I want to do this every class
so that it gets into your

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system.

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If it is true that ψ(x)
tells me everything I need to

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know let's ask questions of this
ψ.

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We can say, "Where is the
particle?"

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So there is some ψ.

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You're told then the
probability that you will find

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it somewhere is the absolute
value.

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Probability density is
proportional to ψ^(2).

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So it doesn't tell you where
the particle is.

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It gives you the odds,
and the odds are this.

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Then you can say,
"Okay, x is just one of

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the variables of interest to me.

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How about momentum?

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If I measure momentum what
answers will I get,

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and what are the odds for the
different answers?"

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That question has a longer
answer and it goes as follows.

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Take the ψ
that's given to you,

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and write is as a sum with some
coefficients of functions that I

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call ψ_p(x),
but I will tell you what they

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are.

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They are e^(ipx/ℏ) by square
root of L.

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Then I said,
"If you manage to write

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the ψ that's given to you in
this form,

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summing over all the allowed
values of p,

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then the probability that you
will get one of those allowed

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values is equal to the absolute
square of that

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coefficient."

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And then final question can be,
"How do I know what

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A(p) is given some
function ψ?,"

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and the answer is A(p)
is the integral of the complex

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conjugate of this ψ(x) dx.

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In our case it was a ring of
length L.

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Then it's 0 to L,
otherwise it's whatever your

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allowed region is.

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Yep?

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Student:  Can you
explain ________ one more time

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what...

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Prof: What this guy
means?

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Student:  Yeah.

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Prof: So let me give an
example.

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Suppose ψ
of p was cosine

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4Πx over L?

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Then that looks like
e^(4Πix/L) e^(-4Πix/L)

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divided by 2.

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Let me put the 2 here,
put the 2 here.

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Then my ψ
is not quite in the standard

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form because I don't have the
square root of Ls.

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So let me put a square root of
L here, a square root of L here

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and also put one outside.

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So my ψ looks like a square
root of L.

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Let me see.

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Oh, I know what the problem is.

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This ψ is not normalized
because the rule is if you

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don't--
first let's say it's not

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normalized,
so don't worry about any of

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these numbers.

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Can you see that it's made up
of two such functions with equal

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weight?

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That means if you measure the
momentum you'll only get the

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answer corresponding to what's
here and what's here.

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If you compare it to
e^(ipx/ℏ) you can see the p

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that goes here is really
4Πℏ/L or -4Πℏ/L.

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So a particle in this state can
have only two values of momentum

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if you measure it,
and the probability for the two

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you can see by symmetry must be
50/50 for each because they come

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with equal weight.

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But if you don't want to do
that you can,

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if you like, like this ψ,
normalize it,

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and then we square it,
integrate it and put a number

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in front so it comes out to have
length 1.

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Then the coefficients here
directly when squared will give

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you probabilities.

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Yes?

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Oh, a lot of questions.

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Yeah?

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Student:  So
A_p has to be a real

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number, right?

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Prof: No.

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A_p does not have to
be a real number just like

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ψ(x) does not have to be a
real number A_p--

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because I'm taking the absolute
value of A^(2).

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So even if it's a complex
number the absolute value of A

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will always come out to be real
and positive.

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Is that your question?

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See A_p is not a
probability.

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The absolute value squared of
A_p is a probability

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which, of course,
must be positive.

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But A does not have to be
positive, does not have to be

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real.

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Yeah?

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Student:  You said
A_p like the integral

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if you're multiplying ψ*
times ψ would that be in

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real numbers?

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Prof: You should be very
careful.

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Student:  Really?

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Prof: A particular ψ
times that same ψ* is a

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real positive number.

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This is not the same function.

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I think I remember now from
previous years that students

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will get confused on this.

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This ψ(x) is the function
that was given to me of which it

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was asked,
"If I measure momentum

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what answer will I get?"

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This ψ_p is not
the function that was given to

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me.

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This is another function that
describes a particle whose

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momentum is p.

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So you've got to multiply that
function with the given function

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to get these coefficients.

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So you can change your mind and
say, "Oh,

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I'm not interested in that
ψ.

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I want to know the answer to
that ψ."

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Well, you take that ψ
and put it here.

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These guys will not change.

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Yeah?

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Student:  I'm sort of
confused.

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Right next to what you were
explaining you have written

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ψ of p.

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Prof: Oh, I'm sorry.

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No, no, no.

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You have every right to be
confused.

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I did not mean that.

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This is ψ of x.

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Student:  And is that
just ψ of x or that

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a ψ_p also?

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Prof: Here?

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Student:  That's just
the ψ of x...

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Prof: Yes,
this is ψ

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of x.

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Student:  Okay.

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Prof: This
ψ(x)--well,

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if I'm more disciplined,
I can do that for you.

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Maybe I should do it one more
time.

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It doesn't hurt.

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So let's so let's say ψ(x)
is cosine 4Πx/L.

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Now first job is to rescale the
ψ so that if you square it

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and integrate it you get 1.

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I know the answer to that is
square root of 2/L.

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You can check that.

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If you take this guy and you
square it and integrate it it'll

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work because the square of this
will be 2/L,

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and the average value of cosign
squared is half.

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And we integrate it from 0 to L
it will cancel.

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You'll get 1.

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If you do this particular
function then all you have to do

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is write this in terms of--
first write this one,

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4Πix/L divided by 2
e^(-4Πix/L) divided by 2

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because exponential is cosine
θ is e iθ

11:23.408 --> 11:27.508
e to the -iθ
over 2.

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Now I'm going to write this as
1/√2 times e^(4Πix/L)

11:35.990 --> 11:42.350
divided by square root of L
e^(-4ΠIx/L).

11:42.350 --> 11:45.880
Divided by square root of L.

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I'm just manipulating the
function that was given to me.

11:49.159 --> 11:51.019
Why do I like this form?

11:51.019 --> 11:56.229
Because this is of the form I
wanted, A of p times ψ(p)

11:56.229 --> 12:00.629
where ψ(p) I this and that's
also ψ(p).

12:00.629 --> 12:05.119
It's a normalized function
associated with momentum p.

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By comparing this you can see
that A is equal to 1 over square

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root of 2 for p = 4Πℏ/L,
and A's also equal to

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1/√2 for p = -4Πℏ/L.

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How do I know what p is?

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States of definite momentum
look like e^(ipx/ℏ).

12:27.740 --> 12:34.570
So whatever multiplies ix/ℏ
is the momentum that you can see

12:34.567 --> 12:36.357
it's 4Πℏ.

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You see what I'm saying?

12:37.029 --> 12:41.199
If you want to put an ℏ down
here and put it on the top and

12:41.200 --> 12:42.730
make the comparison.

12:42.730 --> 12:46.840
So these are the only two
values for momentum.

12:46.840 --> 12:51.910
So this is a state which is
simultaneously in the state of

12:51.909 --> 12:55.379
momentum 4Πℏ/L,
and -4Πℏ/L.

12:55.379 --> 12:57.189
It has no definite momentum yet.

12:57.190 --> 13:00.950
If you measure it then you can
get this answer with probability

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1 over root 2 squared,
which is 1 half,

13:03.798 --> 13:08.618
or you can get this answer 1
over root 2 squared which is 1

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half and no probability for
anything else.

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You cannot get any other value.

13:14.220 --> 13:16.280
See, normally when you take a
ψ(x),

13:16.279 --> 13:21.249
a generic ψ(x) that I draw,
it typically has some non-zero

13:21.248 --> 13:24.068
value in all of space,
so you can find it anywhere.

13:24.070 --> 13:27.440
But this guy has only two
values of p in the sum and only

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those will be possible.

13:28.950 --> 13:30.240
Yeah?

13:30.240 --> 13:33.130
Student:  ________
you're drawing ℏ over there

13:33.130 --> 13:33.720
________.

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Prof: This guy here?

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Student:  Yeah.

13:35.269 --> 13:37.129
Prof: Yes,
I borrowed an ℏ,

13:37.129 --> 13:40.369
put it on top and bottom so you
can compare the expression.

13:40.370 --> 13:44.330
I'm just saying compare,
if you want without these ℏs,

13:44.330 --> 13:48.650
compare this expression to that
and you can see p is equal to

13:48.650 --> 13:49.300
this.

13:49.298 --> 13:52.698
Now this problem was simple in
the sense that you didn't have

13:52.703 --> 13:55.763
to do that integral,
but last time before end of

13:55.761 --> 13:58.231
class I took the following
function,

13:58.230 --> 14:05.540
ψ(x) =
e^(−α|x|).

14:05.538 --> 14:11.368
On a ring the function looked
like this, falling very rapidly

14:11.369 --> 14:16.809
and dying within a width roughly
Δx = 1/α.

14:16.809 --> 14:21.189
Beyond that it is gone.

14:21.190 --> 14:25.440
Then if you square that and
normalize it in the circle you

14:25.441 --> 14:28.501
find this is the correct
normalization.

14:28.500 --> 14:30.830
In other words,
if you took this ψ

14:30.833 --> 14:33.803
and you square and integrate it
you'll get 1.

14:33.798 --> 14:37.558
If you're very,
very careful it is not strictly

14:37.563 --> 14:42.153
1 because this function the x
goes from −L/2 to L/2

14:42.147 --> 14:46.807
whereas in the integral I took
it to go from minus to plus

14:46.812 --> 14:48.042
infinity.

14:48.038 --> 14:50.628
That's because this function is
falling so rapidly.

14:50.629 --> 14:53.649
The function is falling so
rapidly you don't care if you

14:53.649 --> 14:55.789
cut if off at L/2 or go to
infinity,

14:55.788 --> 14:57.458
so I did that to simplify the
math,

14:57.460 --> 14:59.650
but the idea is the same.

14:59.649 --> 15:05.649
Then if you want to know what's
the probability to get some

15:05.645 --> 15:11.635
number p you've got to take A of
p a square root of α

15:11.642 --> 15:17.952
times e^(ipx/ℏ) square root
of L times e^(−α|x|)

15:17.948 --> 15:18.878
dx.

15:18.879 --> 15:20.299
You understand that?

15:20.298 --> 15:24.678
This is the ψ
with the root α

15:24.677 --> 15:25.597
in it.

15:25.600 --> 15:27.170
That's my ψ.

15:27.169 --> 15:35.999
That's my ψ_p*(x).

15:36.000 --> 15:38.740
If you come to me with a new
function tomorrow,

15:38.743 --> 15:41.133
I don't know,
maybe a function that looks

15:41.129 --> 15:44.349
like this, a constant in some
region and 0 beyond.

15:44.350 --> 15:47.050
We'll put that function here
and do these integrals.

15:47.048 --> 15:49.568
You'll get a different set of A
of p's.

15:49.570 --> 15:53.790
So for every function ψ(x)
that's a set of A_p's

15:53.789 --> 15:56.419
that you find by doing the
integral.

15:56.419 --> 15:57.639
Yeah?

15:57.639 --> 16:03.419
Student:  So if you
measure the momentum and you get

16:03.418 --> 16:08.298
one of those answers so it
becomes that answer?

16:08.298 --> 16:08.938
Prof: Right.

16:08.937 --> 16:09.907
So I'm coming to that point.

16:09.908 --> 16:13.178
So the second thing is another
postulate.

16:13.178 --> 16:15.678
In the end I'll give you the
list of postulates.

16:15.678 --> 16:20.198
And the postulate says that if
you measure the momentum you'll

16:20.198 --> 16:24.788
get only those values for which
the corresponding A_p

16:24.792 --> 16:27.632
is not 0,
and the ones for which it is

16:27.625 --> 16:30.825
not 0 the probability for
getting the number is the

16:30.831 --> 16:33.911
absolute value squared of that
A_p.

16:33.908 --> 16:38.868
And right after the measurement
the wave function will change

16:38.865 --> 16:43.325
from this sum over all kinds of
momenta to the one term

16:43.325 --> 16:46.955
corresponding to the one answer
you got.

16:46.960 --> 16:49.450
You understand that?

16:49.450 --> 16:52.470
So in this simple example there
are only two values for

16:52.471 --> 16:53.871
p you can get.

16:53.870 --> 16:55.840
No other values are possible.

16:55.840 --> 16:57.110
They're all theoretically
allowed.

16:57.110 --> 16:59.380
For example,
6Πℏ/L is a perfectly

16:59.380 --> 17:02.820
allowed momentum in that ring
because it corresponds to a

17:02.817 --> 17:05.717
periodic function,
but that's not contained in

17:05.723 --> 17:07.443
this particular wave function.

17:07.440 --> 17:12.140
This wave function is built out
of only two of them,

17:12.141 --> 17:15.461
namely 4Πℏ/L and
-4Πℏ/L.

17:15.460 --> 17:20.370
Once you measure it suppose you
got -4ℏ/L.

17:20.368 --> 17:23.508
The whole wave function,
which I had here in the simple

17:23.510 --> 17:25.490
example, has got only two terms.

17:25.490 --> 17:29.940
This part will simply
disappear, and the function

17:29.944 --> 17:33.384
after the measurement will be
this.

17:33.380 --> 17:36.070
The rough logic is that if I
measure momentum,

17:36.068 --> 17:38.978
and I got an answer,
that's got to mean something in

17:38.978 --> 17:42.398
any real sense that that is the
momentum of the particle,

17:42.400 --> 17:46.110
therefore it has to be true if
I immediately remeasure it.

17:46.108 --> 17:49.328
Immediately remeasure momentum,
and I want to get exactly the

17:49.326 --> 17:51.726
same answer,
that means the function after

17:51.731 --> 17:55.051
the first measurement should
contain only one momentum in its

17:55.046 --> 17:59.656
expansion,
so it'll reduce to that one

17:59.656 --> 18:00.506
term.

18:00.509 --> 18:01.719
Yep?

18:01.720 --> 18:04.010
Student:  Is there a
difference between ψ(x) and

18:04.006 --> 18:06.446
negative ψ(x) because it
seems like everything's just...

18:06.450 --> 18:07.570
Prof: That is correct.

18:07.568 --> 18:13.488
I think I also explained the
other day that in quantum theory

18:13.490 --> 18:19.210
ψ(x) and say 92i
times ψ(x) are treated as

18:19.213 --> 18:25.043
physically equivalent because
they contain the same relative

18:25.036 --> 18:27.006
probabilities.

18:27.009 --> 18:30.659
Of course if ψ(x) had been
properly normalized to 1 this

18:30.655 --> 18:32.815
guy would not be normalized to
1.

18:32.818 --> 18:35.608
It's as if you live in a world
where all you care about is the

18:35.606 --> 18:37.846
direction of the vector,
but not the length of the

18:37.846 --> 18:38.346
vector.

18:38.349 --> 18:39.929
We don't live in such a world.

18:39.930 --> 18:41.470
If you say, "Where is Stop
& Shop,"

18:41.474 --> 18:42.864
you say,
"Well, go in that

18:42.863 --> 18:45.073
direction,"
and you're not told how far,

18:45.068 --> 18:46.978
well, you're missing some
information.

18:46.980 --> 18:51.320
Imagine a world where all you
need to know is which way to go.

18:51.318 --> 18:53.298
In fact, maybe if you go to
some town and ask people where

18:53.304 --> 18:55.144
is something they'll say,
"Go in that

18:55.141 --> 18:57.511
direction,"
that's useful information.

18:57.509 --> 18:59.979
It's as if the direction is the
only thing that matters,

18:59.983 --> 19:01.383
not the length of the vector.

19:01.380 --> 19:05.080
The analogous thing here is if
you multiply the function by any

19:05.076 --> 19:07.576
number you don't change the
information.

19:07.578 --> 19:10.438
They're all considered as the
same state.

19:10.440 --> 19:13.290
And I told you from this huge
family of functions all

19:13.291 --> 19:16.311
describing the same state I will
pick that function,

19:16.308 --> 19:23.638
that member whose square
integral happens to be 1.

19:23.640 --> 19:24.660
Yep?

19:24.660 --> 19:27.050
Student:  So after the
wave function breaks down how

19:27.050 --> 19:29.650
long do you have to wait for the
wave function to rematerialize,

19:29.648 --> 19:30.348
or does it...

19:30.348 --> 19:31.298
Prof: Oh,
it's not break down.

19:31.298 --> 19:42.448
It's another wave function
because e^(ipx/ℏ),

19:42.450 --> 19:46.040
which is ψ(p),
this guy is as much a wave

19:46.040 --> 19:49.960
function as any other ψ(x)
you write down.

19:49.960 --> 19:51.920
So these are not different
creatures.

19:51.920 --> 19:53.250
It's like saying the following.

19:53.250 --> 19:54.980
I think the analogy may be
helpful to you.

19:54.980 --> 19:56.320
It may not be.

19:56.318 --> 19:59.338
ψ is like some vector in
three dimensions,

19:59.337 --> 20:02.757
and you know there are these
three unit vectors,

20:02.759 --> 20:06.799
i, j and
k, and you can write the

20:06.795 --> 20:09.835
vector as some
V_x times

20:09.839 --> 20:13.109
i V_y
times j

20:13.114 --> 20:16.164
V_z times
k where

20:16.160 --> 20:20.040
V_y =
V⋅j

20:20.042 --> 20:21.492
etcetera.

20:21.490 --> 20:24.890
Think of ψ
as a vector and each of these

20:24.894 --> 20:29.174
directions is corresponding to a
possible momentum,

20:29.170 --> 20:31.080
and you want to expand that
vector in terms of these

20:31.076 --> 20:31.446
vectors.

20:31.450 --> 20:34.650
This is ψ_p for
p = p_1.

20:34.650 --> 20:38.300
This is ψ_p for
p = p_2 and so

20:38.296 --> 20:38.596
on.

20:38.598 --> 20:41.258
So it's how much of the vector
is in each direction that

20:41.255 --> 20:43.565
determines the likelihood you'll
get that answer,

20:43.573 --> 20:45.073
that answer or that answer.

20:45.068 --> 20:47.878
But right after the
measurement, if you've caught it

20:47.882 --> 20:50.152
in this direction,
the entire vector,

20:50.148 --> 20:53.478
only the component of the
vector in the direction where

20:53.478 --> 20:55.388
you got your answer remains.

20:55.390 --> 20:57.260
The rest of it gets chopped out.

20:57.259 --> 20:59.989
It's really like Polaroid
glasses.

20:59.990 --> 21:03.440
If you've got Polaroid glasses
light can come in polarized this

21:03.443 --> 21:06.683
way or polarized that way,
but once it goes through the

21:06.681 --> 21:10.271
glass it's polarized only in the
one direction corresponding to

21:10.271 --> 21:12.011
the way the polaroid works.

21:12.009 --> 21:15.259
If it filters light with the e
field going this way the light

21:15.255 --> 21:17.685
on the other side will have only
up and up.

21:17.690 --> 21:20.620
So measurement is like a
filtering process.

21:20.618 --> 21:26.068
It filters out of the sum over
many terms the one term which

21:26.067 --> 21:29.757
corresponds to the one answer
you got.

21:29.759 --> 21:33.379
So I don't mind telling this to
you any number of times,

21:33.381 --> 21:36.741
but that is the way that
quantum mechanics works.

21:36.740 --> 21:41.170
Similarly, if you're chosen to
measure position starting with

21:41.166 --> 21:44.196
this ψ,
and you found the guy here,

21:44.199 --> 21:46.819
once you found it--
see, before you found it the

21:46.820 --> 21:47.840
odds may vary like this.

21:47.838 --> 21:50.618
Once you found it it's there,
and if you remeasure it

21:50.622 --> 21:53.032
infinitesimally later it'd
better be there.

21:53.029 --> 21:57.199
So the only function which has
the property is the big spike at

21:57.203 --> 21:58.823
wherever you found it.

21:58.818 --> 22:02.438
That's called a collapsible of
the wave function.

22:02.440 --> 22:05.580
The real difference is the
following.

22:05.578 --> 22:08.078
I gave you an example of
somebody tracking me to see

22:08.082 --> 22:08.722
where I am.

22:08.720 --> 22:13.480
I drew a probability graph,
and the probability graph may

22:13.476 --> 22:14.916
look like this.

22:14.920 --> 22:17.030
And if you caught me here,
if someone says,

22:17.025 --> 22:19.075
"Where will I find him
next?"

22:19.078 --> 22:22.948
The answer is right there
because the probability has

22:22.948 --> 22:27.188
collapsed from being anywhere to
where we just saw him.

22:27.190 --> 22:31.920
The only difference between me
and the quantum particle is if

22:31.921 --> 22:34.761
you got me here I really was
here.

22:34.759 --> 22:37.769
You couldn't have gotten me
anywhere else at that right now.

22:37.769 --> 22:40.889
Can anybody find me anywhere
else except right now here?

22:40.890 --> 22:42.340
You cannot.

22:42.338 --> 22:44.968
That's because my location's
being constantly measured.

22:44.970 --> 22:48.240
Photons are bouncing off to see
where I am, so my measurement is

22:48.238 --> 22:49.378
constantly measured.

22:49.380 --> 22:52.600
A quantum particle with a
function like this,

22:52.603 --> 22:56.123
which is then found here,
was not really here.

22:56.118 --> 22:58.778
It was in this state of limbo
which has no analog in which it

22:58.775 --> 22:59.965
could have been anywhere.

22:59.970 --> 23:02.530
It's the act of measuring it
that nailed it to there,

23:02.528 --> 23:05.378
but then right after that if
you measure it we expect it to

23:05.384 --> 23:06.324
be still there.

23:06.318 --> 23:07.948
Now the time over which you can
do that is

23:07.946 --> 23:08.936
infinitesimal.

23:08.940 --> 23:11.970
If you wait long enough the
wave function will change;

23:11.970 --> 23:14.700
I will tell you the laws for
the dynamics that will tell you

23:14.701 --> 23:17.611
how it will change,
but we all believe that if a

23:17.605 --> 23:21.445
measurement is immediately
repeated for that same variable

23:21.445 --> 23:23.665
we should get the same answer.

23:23.670 --> 23:27.700
Need not be,
it could have been even worse,

23:27.695 --> 23:30.855
but at least that much is true.

23:30.858 --> 23:34.228
So it's not going to get any
more familiar.

23:34.230 --> 23:37.680
It's a strange thing,
but hopefully you will know

23:37.680 --> 23:39.190
what the rules are.

23:39.190 --> 23:42.610
When I said no one understands
quantum mechanics what I meant

23:42.612 --> 23:45.182
was, of course,
by now you know the recipe.

23:45.180 --> 23:47.520
It doesn't mean you like it,
or it doesn't mean it looks

23:47.515 --> 23:48.785
like anything in daily life.

23:48.788 --> 23:51.698
Things in daily life they have
a location before you measure

23:51.703 --> 23:54.573
it, while you measure it and
right after you measure it.

23:54.568 --> 23:57.388
They're always in one place
doing one thing.

23:57.390 --> 24:01.720
Only in the quantum world they
can be in many places at the

24:01.722 --> 24:02.622
same time.

24:02.618 --> 24:05.068
But you should be very careful
if this is a wave function for

24:05.067 --> 24:05.637
an electron.

24:05.640 --> 24:07.530
The charge of the electron is
not spread out.

24:07.529 --> 24:09.019
I told you that.

24:09.019 --> 24:12.649
An electron is one little guy
you'll only catch in one place.

24:12.650 --> 24:15.730
It's the wave function that's
spread out.

24:15.730 --> 24:19.040
Anyway, I think what'll happen
is you will have to do a lot of

24:19.038 --> 24:20.848
problems,
and you will have to talk to a

24:20.849 --> 24:23.189
lot of people,
and you have to read a lot of

24:23.192 --> 24:23.612
stuff.

24:23.608 --> 24:26.458
You can teach quantum mechanics
for a whole semester,

24:26.455 --> 24:28.695
sometimes I taught it for a
whole year.

24:28.700 --> 24:30.650
There's more and more stuff.

24:30.650 --> 24:34.230
I want you at least to know how
it works.

24:34.230 --> 24:37.340
In real life when you go
forward I don't think you need

24:37.335 --> 24:38.135
all of this.

24:38.140 --> 24:39.960
If you go into physics,
or something,

24:39.958 --> 24:41.878
or chemistry,
they'll teach you quantum

24:41.877 --> 24:43.947
mechanics again,
but I wanted people doing

24:43.949 --> 24:46.629
something else;
they are the people I want to

24:46.634 --> 24:47.704
send a message to.

24:47.700 --> 24:49.220
Is here's a part of the world.

24:49.220 --> 24:51.960
If you ever hear the word
quantum, where does the word

24:51.964 --> 24:53.264
quantization come from?

24:53.259 --> 24:55.259
What's the funny business in
the microscopic world?

24:55.259 --> 24:58.009
I want you to have a felling
for how that works.

24:58.009 --> 25:00.219
And I claim that the
mathematics you need is not very

25:00.221 --> 25:00.521
much.

25:00.519 --> 25:05.289
You should know how to do
integrals, and you should know

25:05.288 --> 25:07.888
what e^(iθ) is.

25:07.890 --> 25:11.810
So the postulates right now are
the state has given me a

25:11.805 --> 25:13.865
function ψ
of x.

25:13.868 --> 25:17.458
State of momentum,
definite momentum looks like

25:17.455 --> 25:20.565
this,
and if you want the odds for

25:20.565 --> 25:25.975
any particular momentum expand
it in this fashion which is the

25:25.978 --> 25:31.568
same as A of p
ψ_p of x,

25:31.568 --> 25:33.508
but A of p,
again, is that integral.

25:33.509 --> 25:35.669
I don't want to write it over.

25:35.670 --> 25:37.980
Then if you measure p
you'll get one of the answers

25:37.980 --> 25:40.410
with the probability given by
the square of the corresponding

25:40.413 --> 25:43.253
number,
and right after the measurement

25:43.250 --> 25:45.070
the function collapses.

25:45.068 --> 25:48.178
So it always collapses to
whatever variable you measure.

25:48.180 --> 25:49.960
Yes, another question?

25:49.960 --> 25:51.960
Student:  It looks like
the opposite.

25:51.960 --> 25:55.650
If you collapse it and you
measure x then what

25:55.648 --> 25:57.208
happens to p?

25:57.210 --> 25:57.730
Prof: Very good.

25:57.730 --> 26:00.590
His question was,
"If you collapse it to

26:00.588 --> 26:03.318
x what happens to
p?"

26:03.319 --> 26:03.939
I'm glad you asked.

26:03.940 --> 26:06.040
I mean, that is the real
problem.

26:06.039 --> 26:07.109
You had the same question?

26:07.108 --> 26:07.598
Student:  No.

26:07.596 --> 26:09.076
I was going to say well don't
you not know about p?

26:09.079 --> 26:09.749
Prof: That is correct.

26:09.750 --> 26:13.430
So I will repeat what she said,
and I'll repeat what you said.

26:13.430 --> 26:16.760
I want to have this discussion
with you guys because it's very

26:16.758 --> 26:19.758
important because everyone's
thinking the same thing.

26:19.759 --> 26:23.019
If I first take a generic
ψ, so I want everyone to

26:23.021 --> 26:26.901
know what the answer's going to
be to any of these questions.

26:26.900 --> 26:29.410
So I take this ψ
somebody prepared for me.

26:29.410 --> 26:32.140
Let's not worry about how that
person knew this is ψ,

26:32.138 --> 26:32.868
you're given.

26:32.868 --> 26:35.888
The electron is in this stage
ψ.

26:35.890 --> 26:39.130
If you measure x,
and you got x equal to

26:39.132 --> 26:42.682
this point the function,
of course, becomes a big spike.

26:42.680 --> 26:45.660
This spike in principle should
be infinitesimally thin,

26:45.661 --> 26:46.711
but I don't care.

26:46.710 --> 26:49.970
Let this be the width of a
proton.

26:49.970 --> 26:52.690
If after that you say,
"What momentum will I

26:52.693 --> 26:55.423
get," well,
you know what you have to do.

26:55.420 --> 26:59.270
You've got to write spike
function equal to sum over these

26:59.267 --> 27:00.547
functions, right?

27:00.548 --> 27:02.928
You've got to take the integral
of the spike with this

27:02.925 --> 27:04.535
exponential and do that
integral,

27:04.538 --> 27:06.788
and you'll get a bunch of
numbers A of p for

27:06.785 --> 27:07.765
all values of p.

27:07.769 --> 27:11.969
And then if you measure
momentum, and you got the one

27:11.965 --> 27:15.915
corresponding to p =
6Πℏ/L,

27:15.920 --> 27:19.710
the state which contains many,
many things will reduce to the

27:19.709 --> 27:23.309
one term corresponding to
p = 6Πℏ/L.

27:23.308 --> 27:26.818
If you plot that function it
will be some oscillatory

27:26.817 --> 27:27.557
function.

27:27.558 --> 27:31.268
The real part and imaginary
part will both oscillate with

27:31.266 --> 27:33.646
some wavelength given by
p.

27:33.650 --> 27:34.830
Be very careful.

27:34.828 --> 27:36.958
This is not the absolute value
of ψ.

27:36.960 --> 27:38.860
It's the real or imaginary part.

27:38.859 --> 27:40.449
They're both sines and cosines.

27:40.450 --> 27:44.680
Absolute value will be flat.

27:44.680 --> 27:48.060
So it'll go from a particle of
known location to a particle

27:48.058 --> 27:51.088
whose probability's completely
flat on the circle.

27:51.089 --> 27:52.219
You understand?

27:52.220 --> 27:57.210
The wave function can look like
6Πi/L square root of

27:57.209 --> 28:00.399
L right after the
measurement.

28:00.400 --> 28:02.650
Let's call is ψ_6.

28:02.650 --> 28:05.770
The absolute value of ψ
is a constant,

28:05.773 --> 28:08.823
but the real and imaginary
parts of ψ

28:08.821 --> 28:11.491
oscillate with some wavelength.

28:11.490 --> 28:15.080
So right after the measurement
of momentum you don't know where

28:15.080 --> 28:17.740
the guy is, and you say,
"Let me find this

28:17.744 --> 28:18.734
fellow."

28:18.730 --> 28:21.870
You catch it somewhere then
it's a spike at that point,

28:21.865 --> 28:24.705
but then you have no guarantee
on the momentum.

28:24.710 --> 28:27.750
So you can never produce for me
something of perfectly well

28:27.748 --> 28:30.838
defined position and momentum
because once you squeeze it in

28:30.838 --> 28:33.038
x it gets broad it in
p.

28:33.038 --> 28:36.028
Once you squeeze it in p
it gets broad in x.

28:36.029 --> 28:39.179
This is really mathematical
property of Fourier analysis,

28:39.180 --> 28:41.590
that functions which are very
narrow in x when you do

28:41.589 --> 28:46.719
the Fourier expansion have many,
many wavelengths in them.

28:46.720 --> 28:50.490
And likewise,
a function with a very well

28:50.493 --> 28:54.363
defined wavelength,
because it's a complex

28:54.364 --> 28:58.804
exponential, has a magnitude
which is flat.

28:58.798 --> 29:03.408
So now I'm going to ask the
following question.

29:03.410 --> 29:07.320
So when I did ψ(x)
the probability for x was

29:07.320 --> 29:09.310
very easy, squared the ψ.

29:09.308 --> 29:11.048
When I said,
"Okay, I want to look at

29:11.048 --> 29:12.658
momentum,"
the answer was long and

29:12.659 --> 29:14.709
complicated,
namely, take these exponential

29:14.708 --> 29:15.958
functions,
write the ψ

29:15.963 --> 29:19.213
in terms of those,
find the coefficient, etcetera.

29:19.210 --> 29:23.360
Now I can say I want some other
variable I'm interested in.

29:23.358 --> 29:26.928
I want to know what happens if
I measure energy.

29:26.930 --> 29:30.220
Energy is a very,
very important variable.

29:30.220 --> 29:33.910
It's very, very important
because it turns out that if a

29:33.906 --> 29:37.456
particle starts out in a state
of definite energy,

29:37.460 --> 29:42.510
I will show that to you later,
it remains in that state.

29:42.509 --> 29:45.409
That's the only state that will
remain the way it is.

29:45.410 --> 29:48.440
If you start in a state of
definite momentum two seconds

29:48.438 --> 29:51.738
later it can have a different
momentum or it can be a mixture

29:51.742 --> 29:53.122
of different momenta.

29:53.118 --> 29:55.758
But if it starts in the state
of definite energy it will

29:55.756 --> 29:56.616
remain that way.

29:56.619 --> 29:57.419
That's not obvious.

29:57.420 --> 29:58.930
I'm going to prove that to you
later.

29:58.930 --> 29:59.990
That's why it's very important.

29:59.990 --> 30:03.470
So most atoms are in a state of
definite energy and they can

30:03.465 --> 30:06.235
stay that way forever,
but once in a while when

30:06.241 --> 30:09.501
they're tickled by something
they will either absorb light or

30:09.497 --> 30:10.797
they will emit light.

30:10.799 --> 30:12.379
So we draw a picture like this.

30:12.380 --> 30:15.540
We will see that the allowed
energies of the systems are some

30:15.535 --> 30:16.425
special values.

30:16.430 --> 30:18.580
Not every value's allowed.

30:18.578 --> 30:22.758
And this can be called
n=1, n=2,

30:22.761 --> 30:24.901
n=3, etcetera.

30:24.900 --> 30:26.890
And an atom,
for example,

30:26.885 --> 30:31.185
can sometimes jump from doing
that to doing that,

30:31.190 --> 30:35.200
and in that process it will
emit an energy which is

30:35.203 --> 30:39.383
E(n=3) −
E(n=1).

30:39.380 --> 30:43.740
That difference of energy will
come in the form of a photon and

30:43.736 --> 30:47.176
the energy of the photon is
ℏω,

30:47.180 --> 30:50.660
or if you like,
2Πℏf where f

30:50.659 --> 30:53.889
is what you and I call
frequency.

30:53.890 --> 30:56.330
And from the frequency you can
find the wavelength.

30:56.328 --> 31:00.628
The wavelength is just the
velocity of light divided by

31:00.625 --> 31:01.575
frequency.

31:01.578 --> 31:04.868
So an atom will have only
certain allowed energies and

31:04.866 --> 31:08.706
when it jumps form one allowed
energy to another allowed energy

31:08.711 --> 31:11.381
it will emit a photon whose
frequency--

31:11.380 --> 31:14.430
in fact, you should probably
call this frequency

31:14.431 --> 31:18.331
f_31 meaning
what I get when I jump from the

31:18.326 --> 31:19.816
level 3 to level 1.

31:19.818 --> 31:23.528
Similarly, if you shine light
on this atom it won't take any

31:23.534 --> 31:24.294
frequency.

31:24.288 --> 31:27.588
It'll only take those
frequencies that connect it from

31:27.590 --> 31:30.520
one allowed energy to another
allowed energy.

31:30.519 --> 31:32.979
That's the fingerprint of the
atom.

31:32.980 --> 31:36.950
Both emission and absorption
betray the atom.

31:36.950 --> 31:39.520
That's how we know what atoms
there are in this star,

31:39.521 --> 31:41.651
or that star,
or what the composition is.

31:41.650 --> 31:45.610
No one's gone to any of these
stars, but we know because of

31:45.614 --> 31:48.764
the light they emit,
and it's all controlled by

31:48.758 --> 31:49.508
energy.

31:49.509 --> 31:53.219
So the question I'm going to
ask is here's the function

31:53.221 --> 31:54.461
ψ(x).

31:54.460 --> 32:00.340
Someone gave it to me in some
context, and I say if I measure

32:00.343 --> 32:05.053
the energy of this particle what
are the answers,

32:05.049 --> 32:07.599
and what are the odds?

32:07.598 --> 32:10.158
So how do you think that will
play out?

32:10.160 --> 32:11.980
You have to make a guess.

32:11.980 --> 32:14.770
Suppose you're inventing
quantum mechanics and someone

32:14.767 --> 32:17.767
says, "What do you think is
going to be the deal with

32:17.765 --> 32:18.655
energy?"

32:18.660 --> 32:23.760
You know what the scenario
might look like?

32:23.759 --> 32:24.719
You can take a guess.

32:24.720 --> 32:27.730
I mean, as I told you many
times I don't expect you to

32:27.732 --> 32:31.202
invent quantum mechanics on the
fly, but you should be able to

32:31.201 --> 32:31.771
guess.

32:31.769 --> 32:36.469
What form do you think the
answer will take?

32:36.470 --> 32:38.000
You want to guess?

32:38.000 --> 32:40.350
Student:  ________
they'll be ________?

32:40.349 --> 32:42.919
Prof: Okay,
but if I want to know what

32:42.923 --> 32:47.863
energies I can get and with what
odds in analogy with momentum

32:47.858 --> 32:50.528
what do you think will happen?

32:50.529 --> 32:51.399
Yep?

32:51.400 --> 32:54.450
Student:  Would you use
the formula p^(2)/2m?

32:54.450 --> 32:56.650
Prof: He said use
p^(2)/2m.

32:56.650 --> 32:57.820
That's a good answer.

32:57.818 --> 33:03.108
His answer was we know that the
energy is equal to 1 half m v

33:03.108 --> 33:06.988
squared which I can write as
p^(2)/2m,

33:06.988 --> 33:07.868
right?

33:07.868 --> 33:10.978
So you're saying if I measure
the momentum and I got a certain

33:10.984 --> 33:12.524
answer, well,
the energy's that

33:12.518 --> 33:13.538
p^(2)/2m.

33:13.538 --> 33:18.388
That's actually correct except
the energy of a particle is not

33:18.394 --> 33:21.024
always just the kinetic energy.

33:21.019 --> 33:23.429
For a free particle this is the
kinetic energy.

33:23.430 --> 33:28.020
For a particle moving in a
potential you know that you have

33:28.018 --> 33:29.678
to add V(x).

33:29.680 --> 33:34.380
That is the total energy.

33:34.380 --> 33:37.840
Now that's when we have a
problem.

33:37.838 --> 33:40.418
In classical mechanics once I
measure the x and

33:40.424 --> 33:43.554
p of the particle I don't
have to make another measurement

33:43.547 --> 33:44.227
of energy.

33:44.230 --> 33:45.370
Do you understand that?

33:45.368 --> 33:49.208
I just plug the values I got
into this formula.

33:49.210 --> 33:51.330
For example,
particle connected to a spring

33:53.246 --> 33:55.766
k is the force constant
of the spring,

33:55.769 --> 33:58.899
and the kinetic energy's always
p^(2)/2m.

33:58.900 --> 34:00.980
Say if I measured p and
I got some number,

34:00.980 --> 34:03.450
and measured x I got
some number, I can put that in

34:03.451 --> 34:04.971
the formula and get the energy.

34:04.970 --> 34:07.950
You don't have to do another
energy measurement.

34:07.950 --> 34:09.940
And you don't have to do an
angular momentum measurement

34:09.943 --> 34:10.273
either.

34:10.268 --> 34:12.578
In higher dimensions angular
momentum is

34:12.579 --> 34:15.709
r x p,
and if you already measured the

34:15.708 --> 34:19.198
position and you measured the
momentum just take the cross

34:19.204 --> 34:19.944
product.

34:19.940 --> 34:23.230
So in classical mechanics you
only need to measure x

34:23.228 --> 34:24.078
and p.

34:24.079 --> 34:27.349
In quantum mechanics he made a
pretty good guess that if you

34:27.349 --> 34:29.789
measured p,
the p^(2)/2m is the

34:29.789 --> 34:30.399
energy.

34:30.400 --> 34:34.470
That is true if the particle is
not in a potential.

34:34.469 --> 34:38.639
But if the particle is in a
potential can you tell me how to

34:38.641 --> 34:41.401
compute p^(2)/2m
V(x)?

34:41.400 --> 34:44.710
You realize you cannot really
compute it because if you knew

34:44.713 --> 34:47.693
the p exactly you have no
idea where it is,

34:47.690 --> 34:50.770
and if you knew the x exactly
you don't know what the momentum

34:50.766 --> 34:51.016
is.

34:51.018 --> 34:53.888
Maybe you know a little bit of
both, but still,

34:53.887 --> 34:56.317
how are you going to find the
energy?

34:56.320 --> 35:01.670
So the answer is you have to do
a separate energy measurement.

35:01.670 --> 35:03.550
You cannot infer that from x
and p,

35:03.547 --> 35:05.967
because first of all you cannot
even get a pair of x and

35:05.974 --> 35:07.114
p at a given time.

35:07.110 --> 35:08.830
I hope I convinced you.

35:08.829 --> 35:10.749
You measure this guy you screw
up that guy.

35:10.750 --> 35:12.970
Measure that one you mess up
this one.

35:12.969 --> 35:15.369
You can never get a state of
well defined x and

35:15.365 --> 35:16.175
p anyway.

35:16.179 --> 35:20.799
So the way to find energy is to
do a whole other calculation.

35:20.800 --> 35:24.230
So I will tell you what the
answer is, and hopefully you

35:24.226 --> 35:28.206
will realize it's not completely
different from the recipe we had

35:28.213 --> 35:28.903
before.

35:28.900 --> 35:31.910
I'm going to give you a rule
for the functions which

35:31.911 --> 35:35.281
correspond to a state in which
the particle has a definite

35:35.277 --> 35:36.457
energy E.

35:36.460 --> 35:41.520
Let's not worry about how you
get it, some function.

35:41.518 --> 35:45.958
You find all those functions,
or you're given all those

35:45.961 --> 35:50.571
functions, then can you imagine
what will happen next?

35:50.570 --> 35:55.750
If I give you all those
functions what do you think the

35:55.748 --> 35:57.858
rule is going to be?

35:57.860 --> 35:58.660
Yep?

35:58.659 --> 36:01.779
Student:  You separate
it into a sum of all the

36:01.775 --> 36:03.475
energies with coefficients.

36:03.480 --> 36:04.370
Prof: Very good.

36:04.369 --> 36:05.459
Let me repeat what she said.

36:05.460 --> 36:08.420
I hope at least some of you
were thinking about the same

36:08.423 --> 36:08.913
answer.

36:08.909 --> 36:12.179
Her answer is take the function
ψ,

36:12.179 --> 36:14.539
write it in terms of these
functions ψ

36:14.538 --> 36:18.098
E of x with some
coefficient A_E

36:18.103 --> 36:21.213
summing this over the allowed
values of E,

36:21.210 --> 36:24.600
whatever they may be.

36:24.599 --> 36:27.639
And now that you said that what
do you think

36:27.644 --> 36:31.474
A_E is going to
be in a given case?

36:31.469 --> 36:32.549
Would you like to continue?

36:32.550 --> 36:37.230
Student:  It's going to
be the integral or the wave

36:37.230 --> 36:39.530
function ________________.

36:39.530 --> 36:50.910
Prof: That's right.

36:50.909 --> 36:52.429
That is correct.

36:52.429 --> 36:54.809
The recipe's almost complete
except you don't know what these

36:54.813 --> 36:57.163
functions are,
but if you knew these functions

36:57.164 --> 36:59.274
you have to write the given
function,

36:59.268 --> 37:01.618
given wave function,
as a sum of these functions

37:01.619 --> 37:03.319
with some suitable coefficients.

37:03.320 --> 37:05.760
Coefficients are found by the
same rule,

37:05.760 --> 37:09.710
and then the probability that
you'll find an energy E

37:09.708 --> 37:11.138
is,
again,

37:11.141 --> 37:20.941
A_E^(2). And
everything else will be also

37:20.943 --> 37:22.323
true.

37:22.320 --> 37:25.140
Once you measure energy you've
got energy corresponding to

37:25.143 --> 37:27.473
E_1,
or E_2 or

37:27.471 --> 37:28.711
E_3.

37:28.710 --> 37:30.330
Let's say you've got
E_3,

37:30.331 --> 37:31.401
the third possible value.

37:31.400 --> 37:35.510
The whole wave function will
collapse from being a sum over

37:35.510 --> 37:39.550
many things to just this one
guy, E_3.

37:39.550 --> 37:40.530
The collapse is the same.

37:40.530 --> 37:42.530
The probability rule is the
same.

37:42.530 --> 37:46.170
The only thing you don't know
is who are these functions ψ

37:46.172 --> 37:47.012
of E?

37:47.010 --> 37:48.260
You understand?

37:48.260 --> 37:51.280
So, again, the analogy's the
following.

37:51.280 --> 37:55.470
There is a vector that we call
ψ.

37:55.469 --> 37:57.169
Sometimes you want to write it
in terms of

37:57.170 --> 37:59.810
i, j,
and k.

37:59.809 --> 38:00.799
They are like the A(p).

38:00.800 --> 38:04.140
Sometimes you may pick three
other mutually perpendicular

38:04.137 --> 38:06.877
vectors, i',
j' and k'.

38:06.880 --> 38:09.270
And if you know those
coefficients you'll get the

38:09.268 --> 38:11.158
probability for some other
variable.

38:11.159 --> 38:14.679
So you're expanding the same
function over and over in many

38:14.677 --> 38:18.617
possible ways depending on what
variable is of interest to you.

38:18.619 --> 38:21.809
If it's momentum you expand it
in terms of exponential ipx of

38:21.809 --> 38:22.129
ℏ.

38:22.130 --> 38:28.150
If it's energy you expand it in
terms of these functions.

38:28.150 --> 38:31.980
So the question is what is the
recipe going to be for these

38:31.981 --> 38:34.891
functions ψ_E of
x.

38:34.889 --> 38:38.219
By what means do I find them?

38:38.219 --> 38:40.679
Now you're getting more and
more and more recipes every day,

38:40.684 --> 38:42.234
but it's going to stop pretty
soon.

38:42.230 --> 38:44.500
This is about the last of the
recipes.

38:44.500 --> 38:48.800
Even this recipe I'll tell you
how to get from a master recipe,

38:48.795 --> 38:52.815
so it's not that many recipes,
but I have to reveal that to

38:52.815 --> 38:54.335
you one at a time.

38:54.340 --> 38:57.260
Now you can say,
"Okay, what function do

38:57.255 --> 39:01.295
you want me to use for every
energy, what is this function?

39:01.300 --> 39:07.720
After all, when it was momentum
you came right out and gave this

39:07.722 --> 39:08.642
answer.

39:08.639 --> 39:12.469
Why don't you do the same thing
here," and that's a problem

39:12.465 --> 39:12.885
here.

39:12.889 --> 39:15.629
The problem is the energy of a
particle depends on what

39:15.628 --> 39:18.668
potential it is in because it's
got a kinetic and a potential

39:18.672 --> 39:21.022
part,
so I cannot give you a

39:21.016 --> 39:25.186
universal answer for
ψ_E(x).

39:25.190 --> 39:28.900
I will have to first ask you;
"Tell me the potential the

39:28.903 --> 39:30.323
particle is in."

39:30.320 --> 39:33.620
Once I know the potential I
will give you the recipe.

39:33.619 --> 39:37.149
Imagine the potential has been
given to me, for example,

39:40.356 --> 39:41.956
what's called a well.

39:41.960 --> 39:45.260
You make a hole in the ground
or you build a little barrier.

39:45.260 --> 39:48.310
That's a possible potential.

39:48.309 --> 39:50.759
You can have a potential that
looks like harmonic isolator.

39:50.760 --> 39:53.060
That's a possible potential.

39:53.059 --> 39:57.239
You can have an electron and a
hydrogen atom -1/r.

39:57.239 --> 39:58.649
That's a possible potential.

39:58.650 --> 40:01.200
There are many-potentials,
and the answer's going to vary

40:01.195 --> 40:01.965
on the problem.

40:01.969 --> 40:05.949
There's no universal answer for
the energy functions.

40:05.949 --> 40:08.789
You tell me what the electron's
doing, what field it is in,

40:08.791 --> 40:10.311
what field of force it is in.

40:10.309 --> 40:12.779
Then for each field of force,
or for each potential,

40:12.782 --> 40:14.192
there's a different answer.

40:14.190 --> 40:16.860
And here is the master formula.

40:16.860 --> 40:20.120
This is the great
Schr�dinger equation.

40:20.119 --> 40:22.229
So the answer looks like this.

40:22.230 --> 40:26.360
The function ψ_E
obeys the following equation,

40:26.360 --> 40:35.250
-ℏ^(2)/2m times
the second derivative of ψ

40:35.251 --> 40:41.671
V(x) times
ψ(x) is equal to

40:41.666 --> 40:46.766
E times ψ(x).

40:46.769 --> 40:47.849
Do not worry.

40:47.849 --> 40:49.929
I will see you through this
equation.

40:49.929 --> 40:52.169
Everything you need to know I
will tell you,

40:52.166 --> 40:55.126
but you should not be afraid of
what the equation says.

40:55.130 --> 40:58.200
It says those functions that
are allowed corresponding to

40:58.199 --> 41:01.379
definite energy will have the
property that if you took the

41:01.378 --> 41:04.198
second derivative,
multiplied it by this number,

41:04.199 --> 41:07.209
add it to that V(x)
times ψ_E you'll

41:07.210 --> 41:08.340
get some function.

41:08.340 --> 41:14.350
That function should be some
number times the very same

41:14.346 --> 41:15.566
function.

41:15.570 --> 41:18.550
If you can find me those
functions then you will find out

41:18.552 --> 41:21.162
when I will show you
mathematically that there are

41:21.161 --> 41:24.231
many solutions to the equation,
but they don't occur for every

41:24.231 --> 41:24.541
energy.

41:24.539 --> 41:27.759
Only some energies are allowed,
and the energies are usually

41:27.757 --> 41:29.717
labeled by some integer
n,

41:29.719 --> 41:31.619
and for every n,
1,2, 3,4,

41:31.619 --> 41:33.909
you'll get a bunch of energies,
ψ_E1,

41:33.905 --> 41:35.685
ψ _E2,
ψ_E3,

41:35.688 --> 41:37.608
3, and you write those
functions down.

41:37.610 --> 41:39.530
Then you can do everything
thing I said.

41:39.530 --> 41:42.590
But the only thing is here you
have to do some hard work.

41:42.590 --> 41:46.240
Whereas for momentum I gave you
that and for a state of definite

41:46.244 --> 41:49.324
positions x = x_0
I told you spike at

41:49.318 --> 41:50.768
x_0.

41:50.769 --> 41:53.799
You didn't have to do much work.

41:53.800 --> 42:02.690
For this you have to solve an
equation before you an even

42:02.693 --> 42:08.733
start, but we'll see how to do
that.

42:08.730 --> 42:15.600
So the first problem I want to
solve is the problem where there

42:15.596 --> 42:17.586
is no potential.

42:17.590 --> 42:21.900
That is called a free particle.

42:21.900 --> 42:24.860
A free particle is one for
which V is 0.

42:24.860 --> 42:31.750
And let me imagine it's living
on this line,

42:31.753 --> 42:39.453
the circle of length
2ΠR = L.

42:39.449 --> 42:41.989
Oh, by the way,
I should mention something

42:41.985 --> 42:42.415
else.

42:42.420 --> 42:46.050
In terms of all the postulates
you notice I never mentioned the

42:46.052 --> 42:49.482
uncertainty principle today,
ΔxΔp should

42:49.480 --> 42:50.930
be bigger than ℏ.

42:50.929 --> 42:54.129
I didn't mention it as a
postulate because once you tell

42:54.130 --> 42:57.210
me ψ is given by wave
function and that states are

42:57.213 --> 43:00.533
definite momentum have definite
wavelength it follows from

43:00.530 --> 43:03.910
mathematics that you cannot have
a function of well defined

43:03.905 --> 43:07.335
periodicity and wavelength also
localized in space.

43:07.340 --> 43:09.240
It's a mathematical consequence.

43:09.239 --> 43:12.979
Similarly, who told me that I
can expand every given ψ

43:12.980 --> 43:15.080
as a sum over these functions?

43:15.079 --> 43:18.499
There's a very general
mathematical theorem that tells

43:18.501 --> 43:22.051
you in what situations you can
actually expand any given

43:22.054 --> 43:24.964
function in terms of a set of
functions,

43:24.960 --> 43:27.300
namely are they like unit
vector i,

43:27.295 --> 43:28.715
j and k.?

43:28.719 --> 43:31.229
Suppose I had only i and
j and I don't have

43:31.226 --> 43:31.716
k.

43:31.719 --> 43:35.009
I cannot expand ever vector in
3D using i and j.

43:35.010 --> 43:37.720
So you've got to make sure
you've got enough basis

43:37.724 --> 43:40.944
functions and the theory tells
you that if you find all the

43:40.936 --> 43:43.926
solutions to that equation
together they can expand any

43:43.929 --> 43:44.759
function.

43:44.760 --> 43:47.810
Similarly the rule for
expansion is also not arbitrary.

43:47.809 --> 43:48.679
It all comes from that.

43:48.679 --> 43:50.899
It's a very,
very beautiful theory of the

43:50.896 --> 43:53.056
mathematics behind quantum
mechanics.

43:53.059 --> 43:56.529
If you learn linear algebra one
day, or if you've already

43:56.527 --> 43:58.877
learned it, it's all linear
algebra.

43:58.880 --> 44:03.010
So every great discovery in
physics is accompanied by some

44:03.007 --> 44:05.177
mathematical stuff you need.

44:05.179 --> 44:07.729
Like all of Newtonian mechanics
requires calculus.

44:07.730 --> 44:10.390
Without calculus you cannot do
Newtonian mechanics.

44:10.389 --> 44:14.759
Maxwell's theory for
electromagnetism requires vector

44:14.764 --> 44:15.694
calculus.

44:15.690 --> 44:18.710
Einstein's non-relativistic
theory doesn't require anything;

44:18.710 --> 44:21.470
it's algebra,
but the general theory requires

44:21.467 --> 44:25.227
what's called tensor calculus,
and quantum mechanics requires

44:25.228 --> 44:26.418
linear algebra.

44:26.420 --> 44:28.440
And string theory we don't know
what it requires.

44:28.440 --> 44:30.980
People are still discovering
new mathematics.

44:30.980 --> 44:34.440
But it's very true that very
often new mathematics is needed

44:34.440 --> 44:36.610
to express the new laws of
physics.

44:36.610 --> 44:39.560
And if you don't know the laws
you may find out you're not able

44:39.556 --> 44:40.456
to write it down.

44:40.460 --> 44:41.740
If you didn't know what a
gradient was,

44:41.739 --> 44:43.809
or if you did not know what a
curl is,

44:43.809 --> 44:47.879
and so on, then you cannot
write down the laws of

44:47.876 --> 44:50.246
electricity and magnetism.

44:50.250 --> 44:53.200
So we have to solve this
equation,

44:53.199 --> 44:55.539
and I'm going to solve it for
the easiest problem in the

44:55.536 --> 44:57.886
world,
a particle moving on a ring of

44:57.889 --> 45:00.649
length L with no
potential energy.

45:00.650 --> 45:02.680
So what does that equation look
like?

45:02.679 --> 45:12.769
It says -ℏ;/2m
d^(2)ψ/dx^(2) no

45:12.773 --> 45:24.283
potential = E times
ψ_E(x).

45:24.280 --> 45:26.850
So let me rearrange the
equation so it looks like this,

45:26.849 --> 45:35.049
d^(2)ψ/dx^(2)
k^(2)ψ

45:35.052 --> 45:44.112
= 0 where k^(2) is
defined to be 2mE/ℏ^(2)

45:44.108 --> 45:48.038
= k^(2).

45:48.039 --> 45:51.709
All I've done is just taken
everything to one side and

45:51.711 --> 45:55.321
multiplied everything by
2m/ℏ^(2) and called

45:55.315 --> 45:57.875
that combination as
k^(2).

45:57.880 --> 45:59.460
So who is this number k?

45:59.460 --> 46:02.280
Let's see.

46:02.280 --> 46:04.780
The energy is
ℏk^(2)/2m,

46:04.784 --> 46:08.064
but we also know energy's
p^(2)/2m.

46:08.059 --> 46:14.419
So this number k will
turn out to be just momentum

46:14.422 --> 46:17.152
divided by ℏ.

46:17.150 --> 46:19.330
Well, momentum has not entered
the picture, but we will see.

46:19.329 --> 46:22.349
Let's solve this equation now.

46:22.349 --> 46:29.029
I say the solution to this
equation is ψ(x) = to

46:29.034 --> 46:34.804
any number A times
e^(ikx) any number

46:34.795 --> 46:38.825
B times e^(-ikx).

46:38.829 --> 46:40.899
Let's see if that is true.

46:40.900 --> 46:42.320
Take two derivatives of ψ.

46:42.320 --> 46:43.850
What do you get?

46:43.849 --> 46:47.539
You understand every time I
take a derivative you pull down

46:47.543 --> 46:48.503
an ik?

46:48.500 --> 46:51.590
If you pull it twice you'll
pull an ik squared which

46:51.594 --> 46:54.254
is -k^(2)ψ,
and the same thing will happen

46:54.251 --> 46:56.631
to this term that it'll pull
down a minus ik,

46:56.630 --> 46:59.240
but if you do it twice you'll
again get −k^(2).

46:59.239 --> 47:02.859
So both of them will have the
property that the second

47:02.864 --> 47:05.264
derivative of ψ
will be equal to

47:05.257 --> 47:08.817
-k^(2)ψ,
and that is the equation you

47:08.820 --> 47:09.850
want to solve.

47:09.849 --> 47:12.859
And A and B is
whatever you like.

47:12.860 --> 47:17.660
A and B are not
fixed by the equation because

47:17.657 --> 47:22.537
for any choice of A and
any choice of B this'll

47:22.538 --> 47:23.268
work.

47:23.269 --> 47:26.349
So let me write it as follows.

47:26.349 --> 47:28.629
Ae to the i.

47:28.630 --> 47:35.510
Let me write k,
k was a shorthand for

47:35.510 --> 47:43.770
square root of 2mE/ℏ^(2)
x B to the -i

47:43.766 --> 47:49.266
square root of 2mE/ℏ^(2)
x.

47:49.268 --> 47:52.178
I'm trying to show you that
these are really functions of

47:52.175 --> 47:54.455
definite energy
ψ_E and here is

47:54.458 --> 47:55.858
how the energy appears.

47:55.860 --> 47:58.480
So what does it look like to
you?

47:58.480 --> 48:03.280
Have you seen these functions
before?

48:03.280 --> 48:03.810
Yes?

48:03.809 --> 48:04.899
What does it look like?

48:04.900 --> 48:08.200
Student:  One of the
spring wave theories?

48:08.199 --> 48:08.869
Prof: Pardon me?

48:08.869 --> 48:10.239
Student:  One of the
spring wave theories?

48:10.239 --> 48:12.919
Prof: The equation is
like the spring equation.

48:12.920 --> 48:21.440
That is absolutely correct,
but what does this function

48:21.436 --> 48:23.326
look like?

48:23.329 --> 48:26.699
That function look like
something to you?

48:26.699 --> 48:27.509
Yep?

48:27.510 --> 48:29.600
Student:  Isn't that a
cosine function?

48:29.599 --> 48:32.759
Prof: It's a cosine only
if A is equal to B.

48:32.760 --> 48:33.950
Forget the sines and cosines.

48:33.949 --> 48:35.319
They are your old flames.

48:35.320 --> 48:37.260
What's their new quantum flame?

48:37.260 --> 48:42.900
What's the function that means
a lot more in quantum mechanics

48:42.898 --> 48:45.208
than sines and cosines?

48:45.210 --> 48:50.610
No?

48:50.610 --> 48:55.450
I think I told you long back
that e to the i

48:55.454 --> 49:00.054
times dog x over ℏ
is a state where the

49:00.045 --> 49:02.505
momentum is equal to dog.

49:02.510 --> 49:05.090
In other words,
you can put anything you want

49:05.092 --> 49:06.152
in the exponent.

49:06.150 --> 49:10.110
If a function looks like that,
that fellow there is the

49:10.112 --> 49:10.922
momentum.

49:10.920 --> 49:14.530
So this is a state of momentum,
of definite momentum.

49:14.530 --> 49:21.000
In fact, look at this,
e to the i square

49:20.996 --> 49:27.706
root of 2mE/ℏ x
B times e to the

49:27.708 --> 49:29.538
minus that.

49:29.539 --> 49:31.499
Do you understand?

49:31.500 --> 49:34.580
This must be the momentum.

49:34.579 --> 49:37.379
It is the momentum because
e^(ipx/ℏ) is a state of

49:37.376 --> 49:40.146
momentum p,
but the momentum here can

49:40.150 --> 49:45.130
either have one value,
square root of 2mE,

49:45.134 --> 49:52.994
or it can have another value
minus square root of 2mE.

49:52.989 --> 49:58.849
So what do you think the
particle is doing in these

49:58.851 --> 50:00.261
solutions?

50:00.260 --> 50:01.230
Yep?

50:01.230 --> 50:03.470
Student:  It's jumping
back and forth _______________

50:03.472 --> 50:04.312
one of those states.

50:04.309 --> 50:08.559
Prof: But in any one of
them what is it doing here?

50:08.559 --> 50:10.569
What is the sign of the
momentum here?

50:10.570 --> 50:12.010
Student:  Oh, positive.

50:12.010 --> 50:14.700
Prof: Positive,
and here it's got negative

50:14.702 --> 50:15.322
momentum.

50:15.320 --> 50:16.990
And how much momentum does it
have?

50:16.989 --> 50:19.829
The momentum it has,
if you look at any of these

50:19.826 --> 50:23.596
things,
is that p^(2)/2m

50:23.601 --> 50:29.941
= E is what is satisfied
by the p that you have

50:29.943 --> 50:30.793
here.

50:30.789 --> 50:33.019
In other words,
what I'm telling you is in

50:33.019 --> 50:35.639
quantum theory,
in classical theory if I said,

50:35.635 --> 50:38.195
"I've got a particle of
energy E,

50:38.199 --> 50:40.499
what is its momentum?"

50:40.500 --> 50:42.040
You would say, "Well,
E is

50:42.041 --> 50:45.911
p^(2)/2m,
therefore p is equal to

50:45.905 --> 50:48.995
plus or minus square root of
2mE,"

50:49.003 --> 50:52.863
because in one dimension when I
give you the energy,

50:52.860 --> 50:55.220
the Kinetic energy if you like,
the particle has to have a

50:55.215 --> 50:57.895
definite speed,
but it can be to the left or it

50:57.896 --> 50:59.206
can be to the right.

50:59.210 --> 51:00.670
And the momentum is not
arbitrary.

51:00.670 --> 51:03.890
If the energy is E the
momentum has to satisfy the

51:03.891 --> 51:06.541
condition p^(2)/2m
= E.

51:06.539 --> 51:09.889
That's also exactly what's
happening in the quantum theory.

51:09.889 --> 51:14.149
The state of definite energy is
a sum over two possible things,

51:14.150 --> 51:17.930
one where the momentum is the
positive value for root of

51:17.929 --> 51:20.809
2mE;
other is a negative value,

51:20.807 --> 51:24.577
and these are the two allowed
values even in classical

51:24.577 --> 51:27.917
mechanics for a particle of
definite energy.

51:27.920 --> 51:30.610
But what's novel in quantum
mechanics,

51:30.610 --> 51:34.310
whereas in classical mechanics
if it's got energy E it

51:34.309 --> 51:37.329
can only be going clockwise or
anti-clockwise,

51:37.329 --> 51:40.929
but this fellow can be doing
both because it's not in a state

51:40.929 --> 51:42.909
of clockwise or anti-clockwise.

51:42.909 --> 51:46.379
In fact, the probability for
clockwise is proportional to

51:46.380 --> 51:47.310
A^(2).

51:47.309 --> 51:49.929
Probability for anti-clockwise
is proportional to B^(2).

51:49.929 --> 51:53.019
I've not normalized it,
but the relative odds are

51:53.021 --> 51:57.011
simply proportional to A^(2)
and B^(2). So that's

51:57.014 --> 52:00.884
what is bizarre about quantum
mechanics that the particle has

52:00.880 --> 52:03.070
indefinite sign of momentum.

52:03.070 --> 52:03.860
Yes?

52:03.860 --> 52:07.050
Student:  Do we have
problem with m because

52:07.047 --> 52:09.607
m is too small to
________ properly?

52:09.610 --> 52:10.410
Prof: Which one?

52:10.409 --> 52:13.529
Student:  With the mass
is there going to be some kind

52:13.534 --> 52:14.944
of issue if that's small?

52:14.940 --> 52:15.760
Prof: What about the
m?

52:15.760 --> 52:16.770
I'm sorry.

52:16.768 --> 52:18.718
Student:  m
stands for the mass,

52:18.722 --> 52:19.072
right?

52:19.070 --> 52:20.570
Prof: Yeah,
m is the mass of the

52:20.567 --> 52:21.397
particle, that's right.

52:21.400 --> 52:23.950
Student:  But is it too
small in some way for our

52:23.952 --> 52:26.462
equations to work because we had
that issue in previous

52:26.460 --> 52:27.110
equations?

52:27.110 --> 52:28.690
Prof: When did we have
the issue?

52:28.690 --> 52:31.780
m is whatever the mass
of the particle is.

52:31.780 --> 52:32.540
It can be small.

52:32.539 --> 52:35.549
It can be large.

52:35.550 --> 52:36.340
I didn't understand.

52:36.340 --> 52:37.400
Student:
>

52:37.400 --> 52:38.330
Prof: No,
no, go ahead.

52:38.333 --> 52:38.883
I want to know.

52:38.880 --> 52:45.980
Student:
>

52:45.980 --> 52:48.760
Prof: There are no
restrictions on the correctness

52:48.762 --> 52:49.262
of this.

52:49.260 --> 52:53.500
If the particle weighs a
kilogram then you will find

52:53.500 --> 52:56.410
that--well, we're coming to
this.

52:56.409 --> 52:59.689
So you find that these are the
allowed values of p.

52:59.690 --> 53:02.160
There are two values,
but p itself is not

53:02.155 --> 53:04.615
arbitrary, p itself is
not continuous.

53:04.619 --> 53:06.459
Maybe that's what you meant.

53:06.460 --> 53:09.580
There's a restriction on the
allowed values of p,

53:09.583 --> 53:13.223
and therefore a restriction on
the allowed values of E.

53:13.219 --> 53:16.829
All I'm telling you now is that
if you want to solve that

53:16.829 --> 53:20.629
equation it is obviously made up
of sines and cosines as you

53:20.632 --> 53:24.182
recognize from the oscillator,
or in the quantum world it's

53:24.184 --> 53:26.684
more natural to write them in
terms of exponentials,

53:26.679 --> 53:30.059
e^(ikx) and
e^(-ikx) where k is not

53:30.059 --> 53:33.179
independent of e,
k satisfies this

53:33.181 --> 53:36.001
condition, and if you call
ℏk as p,

53:36.000 --> 53:37.560
p satisfies this
condition.

53:37.559 --> 53:41.189
This is just a classical
relation between energy and

53:41.186 --> 53:41.966
momentum.

53:41.969 --> 53:46.399
The other difference is that
not every value of momentum is

53:46.396 --> 53:47.156
allowed.

53:47.159 --> 53:50.819
Not every value of momentum is
allowed for the same reason as

53:50.822 --> 53:53.512
when I did particles of definite
momentum.

53:53.510 --> 53:58.070
In other words,
if the particle is living on a

53:58.070 --> 54:03.640
circle and the state of energy
E it's given by

54:03.643 --> 54:08.513
Ae^(ipx/ℏ)
Be^(−ipx/ℏ) where

54:08.507 --> 54:15.397
p is related to E
by E = p^(2)/2m.

54:15.400 --> 54:18.880
We have the requirement that
when you go around a circle

54:18.880 --> 54:21.730
you've got to come back to where
you start.

54:21.730 --> 54:23.530
And that condition,
if you remember,

54:23.530 --> 54:28.340
says that p times
L over ℏ has to

54:28.342 --> 54:32.762
be a multiple of some integer,
or that the allowed values of

54:32.760 --> 54:35.190
p are labeled by some
index n,

54:35.190 --> 54:38.350
which is 2Πℏ/L
times n.

54:38.349 --> 54:41.489
I don't want to use m
because m stands for

54:41.485 --> 54:44.045
particle mass,
n is the integer now.

54:44.050 --> 54:46.730
In other words,
when we studied the state of

54:46.728 --> 54:49.468
definite momentum,
namely the first function,

54:49.467 --> 54:52.767
we realized even then that
p is quantized.

54:52.768 --> 54:56.638
Because of the single valued
condition p is quantized.

54:56.639 --> 55:00.849
And if energy functions are
made up of such functions they

55:00.849 --> 55:03.139
also have to be singlevalued.

55:03.139 --> 55:08.949
That means the p here or
the -p here both have to

55:08.954 --> 55:12.524
satisfy the condition given by
this.

55:12.518 --> 55:18.528
Therefore, the allowed energies
are also labeled by an integer

55:18.534 --> 55:24.254
n and they are really
p n squared over

55:24.253 --> 55:29.483
2m where p
n is 2Πℏ n

55:29.478 --> 55:31.548
over L .

55:31.550 --> 55:32.620
You see that?

55:32.619 --> 55:37.299
This is the quantization of
energy.

55:37.300 --> 55:41.780
So a particle in a ring has
only these allowed values of

55:41.777 --> 55:42.507
energy.

55:42.510 --> 55:48.160
So in a way this problem is
somewhat easy because it's a

55:48.164 --> 55:49.814
free particle.

55:49.809 --> 55:53.859
Once you've understood the
particles in terms of allowed

55:53.860 --> 55:58.570
momenta it turns out the allowed
momentum states are also allowed

55:58.574 --> 55:59.904
energy states.

55:59.900 --> 56:03.150
The allowed momentum states,
you remember,

56:03.152 --> 56:06.882
they either look this or they
look like this.

56:06.880 --> 56:10.390
And I can pick A and
B to be arbitrary,

56:10.389 --> 56:13.379
so one choice is to pick
A equal to

56:13.382 --> 56:17.252
1/√L and just take
e^(ipx/ℏ).

56:17.250 --> 56:22.260
Other is to pick B equal
to 1/√L and pick

56:22.262 --> 56:27.752
e^(−ipx/ℏ),
but you can also mix them up.

56:27.750 --> 56:29.620
You don't have to mix them.

56:29.619 --> 56:33.929
If you don't mix them up you
have a particle here which has a

56:33.934 --> 56:37.534
well-defined energy and a
well-defined momentum.

56:37.530 --> 56:39.760
This guy also has a
well-defined energy and a

56:39.762 --> 56:40.982
well-defined momentum.

56:40.980 --> 56:43.750
This guy only has a
well-defined energy,

56:43.748 --> 56:47.788
but not a well-defined momentum
because there's a two-fold

56:47.793 --> 56:49.643
ambiguity in momentum.

56:49.639 --> 56:50.329
You understand?

56:50.329 --> 56:52.319
Even in classical mechanics
it's true.

56:52.320 --> 56:54.830
If I give you the momentum you
can find the energy.

56:54.829 --> 56:57.789
If I give you the energy you
cannot find the momentum because

56:57.786 --> 57:00.836
there are two square roots you
can take because p^(2)/2m

57:00.842 --> 57:02.802
is E,
p is plus or minus

57:02.795 --> 57:03.995
square root of 2mE.

57:04.000 --> 57:08.350
It's the same uncertainty even
in classical mechanics.

57:08.349 --> 57:11.209
So what happens in quantum
theory is if you pick any

57:11.211 --> 57:14.521
particle of definite momentum on
the ring it'll already have

57:14.523 --> 57:17.723
definite energy which is simply
that momentum squared over

57:17.722 --> 57:18.622
2m.

57:18.619 --> 57:20.799
You don't need to find a new
function.

57:20.800 --> 57:24.340
What is novel is that since the
energy depends only on

57:24.342 --> 57:27.882
p^(2) you can take a
function with one value of

57:27.884 --> 57:30.174
p,
and you can take a function

57:30.172 --> 57:32.732
with the opposite value of
p and add them,

57:32.730 --> 57:36.790
there will still be a state of
definite energy because whether

57:36.789 --> 57:40.579
it's doing this or whether it's
doing that the energy will

57:40.583 --> 57:44.783
always be p^(2)/2m
and the minus signs drop out of

57:44.777 --> 57:45.507
that.

57:45.510 --> 57:50.280
So what is novel here is what's
called degeneracy.

57:50.280 --> 57:53.720
Degeneracy is the name,
but there's more than one

57:53.724 --> 57:57.964
solution for a given value of
the variable you're interested

57:57.956 --> 57:58.456
in.

57:58.460 --> 58:01.110
You saw the energy looks like
this.

58:01.110 --> 58:06.550
Therefore, you'll find there's
a state E_0

58:06.552 --> 58:09.932
which is 0 squared over
2m.

58:09.929 --> 58:15.249
Then there are two states
E_1.

58:15.250 --> 58:19.470
One has got momentum p
going clockwise and one has

58:19.474 --> 58:21.894
momentum going anti-clockwise.

58:21.889 --> 58:27.779
So they look like
(1/√L)

58:27.784 --> 58:31.554
e^(2Πix/L).

58:31.550 --> 58:33.370
That's this guy.

58:33.369 --> 58:37.659
Then you can have
(1/√L)

58:37.655 --> 58:43.245
e^(-2Πix/L) which is
the other guy.

58:43.250 --> 58:47.510
So at every-energy,
allowed energy except 0,

58:47.507 --> 58:50.277
there'll be two solutions.

58:50.280 --> 58:54.050
There are two quantum states
with the same energy.

58:54.050 --> 58:56.830
When you studied hydrogen atom
in high school maybe you

58:56.826 --> 58:59.136
remember there are these shells
with 2, and 4,

58:59.139 --> 59:00.579
and 8, and 10 and so on.

59:00.579 --> 59:03.559
They are called degeneracies
where the energy is not enough

59:03.559 --> 59:05.099
to tell you what it's doing.

59:05.099 --> 59:07.439
There at every-energy it can
have a different angular

59:07.443 --> 59:07.943
momentum.

59:07.940 --> 59:11.950
Here at every-energy it can
have two different momenta,

59:11.954 --> 59:14.264
clockwise and anti-clockwise.

59:14.260 --> 59:15.010
Yep?

59:15.010 --> 59:19.060
Student:  Going back can
you explain how you got the

59:19.063 --> 59:22.073
________ of p L over ℏ
________?

59:22.070 --> 59:22.670
Prof: Here?

59:22.670 --> 59:24.450
Student:  To the left.

59:24.449 --> 59:25.939
Prof: Here?

59:25.940 --> 59:26.990
You mean this one?

59:26.989 --> 59:28.329
Student:  I don't where
that came from.

59:28.329 --> 59:30.969
Prof: It came just like
in the momentum problem.

59:30.969 --> 59:34.529
If you've got a function like
this you have a right to demand

59:34.525 --> 59:37.775
that if you go a distance
L around the circle you

59:37.784 --> 59:39.744
come back to where you start.

59:39.739 --> 59:44.489
So if you take any x here and
add to it an L it should not

59:44.485 --> 59:46.145
make a difference.

59:46.150 --> 59:49.900
And what you're adding is
pL/ℏ and that better be

59:49.900 --> 59:51.350
a multiple of 2Π.

59:51.349 --> 59:53.929
It's the same single valuedness
condition.

59:53.929 --> 59:56.989
So the momentum problem pretty
much does this problem for you,

59:56.985 --> 59:59.485
all the singlevalued stuff we
dealt with before.

59:59.489 --> 1:00:03.819
What is novel here is that
demanding energy have one value

1:00:03.817 --> 1:00:07.307
fixes the momentum to be one of
two values,

1:00:07.309 --> 1:00:10.639
and that double valuedness is
the same as in classical theory

1:00:10.644 --> 1:00:13.704
that a particle of energy
E can have two possible

1:00:13.702 --> 1:00:16.652
momenta plus or minus square
root of 2mE,

1:00:16.650 --> 1:00:21.580
and the quantum theory,
then, is the state of E

1:00:21.581 --> 1:00:26.421
is a sum of one value of
p and the other value

1:00:26.418 --> 1:00:30.698
of p with any coefficient
you like.

1:00:30.699 --> 1:00:35.079
So if this atom makes a jump
from,

1:00:35.079 --> 1:00:37.399
or this system makes a jump
from somewhere there to

1:00:37.400 --> 1:00:40.370
somewhere there you can find the
frequency of the photons it will

1:00:40.373 --> 1:00:42.513
emit because here are my allowed
energies.

1:00:42.510 --> 1:00:45.620
E_n is this.

1:00:45.619 --> 1:00:51.419
Suppose it jumps from
n=4 to n=3.

1:00:51.420 --> 1:00:56.310
The energy that's liberated
that goes to the photon is

1:00:56.306 --> 1:01:01.286
ℏω will be
4Π^(2) ℏ^(2) over

1:01:01.286 --> 1:01:05.986
2mL squared times 4
squared - 3 squared.

1:01:05.989 --> 1:01:08.519
I'm just using this formula
with n=4,

1:01:08.518 --> 1:01:10.988
and n=3,
and take the difference.

1:01:10.989 --> 1:01:13.029
It's 16 - 9 which 7.

1:01:13.030 --> 1:01:16.590
You plug all that in you can
solve for the ω,

1:01:16.590 --> 1:01:20.600
or if you like frequency you
can write it as 2Πℏf

1:01:20.599 --> 1:01:22.939
and you can find the
frequencies.

1:01:22.940 --> 1:01:24.250
This is actually true.

1:01:24.250 --> 1:01:27.490
If you've got a charged
particle moving in a ring and

1:01:27.492 --> 1:01:31.362
you want to excite it from one
state to a higher state you will

1:01:31.358 --> 1:01:34.848
have to give it only one of
these frequencies so that the

1:01:34.851 --> 1:01:37.591
frequency you give it,
ℏω,

1:01:37.585 --> 1:01:40.155
must match the energy
difference of the particle.

1:01:40.159 --> 1:01:42.669
You understand?

1:01:42.670 --> 1:01:44.770
So I've drawn these levels here.

1:01:44.768 --> 1:01:48.308
If you want the electrons,
say, in a metallic ring going

1:01:48.309 --> 1:01:50.819
in the lowest possible energy
state,

1:01:50.820 --> 1:01:53.180
if you want to jump,
if you want to crank it up to

1:01:53.175 --> 1:01:56.055
the next level you've got to
have photons at that energy,

1:01:56.059 --> 1:01:57.569
or that energy.

1:01:57.570 --> 1:01:59.680
Well, that happens to be the
same as this,

1:01:59.681 --> 1:02:02.311
but these are the only
frequencies it'll absorb from

1:02:02.306 --> 1:02:02.716
you.

1:02:02.719 --> 1:02:08.079
And when it cools down it'll
emit back those frequencies,

1:02:08.081 --> 1:02:11.531
and that's something you can
test.

1:02:11.530 --> 1:02:18.160
By the way, do you know why
there's only one state at

1:02:18.164 --> 1:02:22.634
E_0 and not
two?

1:02:22.630 --> 1:02:23.470
Yep?

1:02:23.469 --> 1:02:26.329
Student:  Because 0
equals -0.

1:02:26.329 --> 1:02:27.599
Prof: That's right.

1:02:27.599 --> 1:02:30.929
The solution plus or minus
2m square root of

1:02:30.932 --> 1:02:34.802
2mE has only one-answer
when E is 0 because 0

1:02:34.800 --> 1:02:37.600
momentum and -0 momentum are the
same.

1:02:37.599 --> 1:02:41.609
Otherwise, any finite positive
momentum has a partner which is

1:02:41.605 --> 1:02:43.045
minus that momentum.

1:02:43.050 --> 1:02:46.320
There's a very interesting
piece of work being done

1:02:46.318 --> 1:02:50.178
experimentally at Yale which is
the claim that if you took a

1:02:50.177 --> 1:02:53.767
metallic ring in a magnetic
field it will have a current

1:02:53.773 --> 1:02:57.453
going one way or the other way,
unbalanced current,

1:02:57.445 --> 1:02:59.565
and it's not driven by a
battery.

1:02:59.570 --> 1:03:02.580
It's not driven by anything.

1:03:02.579 --> 1:03:05.789
Normally if you took an
ordinary ring the lowest energy

1:03:05.793 --> 1:03:09.373
state will be a field of zero
current while you go one way or

1:03:09.365 --> 1:03:11.445
the other,
but if you put it in the

1:03:11.454 --> 1:03:14.544
magnetic field one can show it
likes to go one way or the other

1:03:14.541 --> 1:03:14.891
way.

1:03:14.889 --> 1:03:17.879
And now measurements are being
done at Yale where you can

1:03:17.882 --> 1:03:20.932
actually measure the tiny
current due to one electron,

1:03:20.929 --> 1:03:24.929
or one net electron going one
way or the other.

1:03:24.929 --> 1:03:28.309
So this L is either a
mathematical convenience if

1:03:28.311 --> 1:03:31.821
you're talking about free space
and you want to be able to

1:03:31.817 --> 1:03:35.077
normalize your wave functions,
or it really is the

1:03:35.079 --> 1:03:36.949
circumference of a real system.

1:03:36.949 --> 1:03:41.129
Now that we can probe nano
systems very well we can vary

1:03:41.132 --> 1:03:45.472
the L and we can find out
all the energy levels.

1:03:45.469 --> 1:03:51.719
All right, so now I'm going to
do the one problem which is

1:03:51.715 --> 1:03:56.425
really a very standard
pedagogical exercise,

1:03:56.427 --> 1:04:01.027
and that's called a particle in
a box.

1:04:01.030 --> 1:04:05.410
I remember this example the
first time I remember seeing

1:04:05.405 --> 1:04:08.825
quantization,
which is more interesting than

1:04:08.826 --> 1:04:09.936
on a ring.

1:04:09.940 --> 1:04:11.670
A box is the following.

1:04:11.670 --> 1:04:16.860
If you dig a hole in the ground
and you are standing somewhere

1:04:16.855 --> 1:04:21.865
here you realize you're kind of
trapped unless you can scale

1:04:21.869 --> 1:04:23.059
this wall.

1:04:23.059 --> 1:04:25.879
Now, you can call that as a
ground level and think of it as

1:04:25.882 --> 1:04:29.102
a hole in the ground,
or you can think of this as the

1:04:29.096 --> 1:04:32.526
ground level and that's the
height of your barrier.

1:04:32.530 --> 1:04:38.420
So imagine a particle living in
a barrier that looks like this.

1:04:38.420 --> 1:04:40.530
This is the potential energy.

1:04:40.530 --> 1:04:41.750
This is like the height above
the ground,

1:04:41.750 --> 1:04:43.400
if you like,
and it has a height

1:04:43.400 --> 1:04:47.110
V_0 t here,
and is 0 here,

1:04:47.106 --> 1:04:55.186
and it goes from some
−L/2 to L/2.

1:04:55.190 --> 1:05:00.030
If the barrier
V_0 goes to

1:05:00.027 --> 1:05:03.367
infinity it's called a box.

1:05:03.369 --> 1:05:07.009
So a barrier that goes to
infinity, I'll just show you the

1:05:07.007 --> 1:05:08.857
part that you can see here.

1:05:08.860 --> 1:05:12.530
Here V is infinity,
and here V is infinity

1:05:12.527 --> 1:05:14.327
and inside V is 0.

1:05:14.329 --> 1:05:16.719
That's a particle in a box.

1:05:16.719 --> 1:05:20.429
So if the particle goes hits
against the wall it cannot,

1:05:20.429 --> 1:05:24.409
no matter how fast it's moving,
go over the top because it's

1:05:24.409 --> 1:05:25.759
infinitely high.

1:05:25.760 --> 1:05:27.670
And that's not realistic.

1:05:27.670 --> 1:05:32.360
Every barrier is finite,
but just to teach you the

1:05:32.360 --> 1:05:36.860
principles we always pick the
simple example.

1:05:36.860 --> 1:05:39.450
So now I want to solve this
problem.

1:05:39.449 --> 1:05:42.969
What are the allowed wave
functions, ψ,

1:05:42.972 --> 1:05:45.912
for a particle in this
potential?

1:05:45.909 --> 1:05:51.149
So let's go back to the same
equation which says

1:05:51.152 --> 1:05:57.402
-ℏ^(2)/2m
d^(2)ψ/dx^(2)

1:05:57.400 --> 1:06:00.970
= E - V ψ.

1:06:00.969 --> 1:06:05.779
What I've done is I've taken
V to the other side.

1:06:05.780 --> 1:06:09.510
Now the potential is a constant
in the three regions.

1:06:09.510 --> 1:06:12.040
This is region I where the
potential is 0.

1:06:12.039 --> 1:06:14.639
This is region II where the
potential is infinity.

1:06:14.639 --> 1:06:19.229
This is region III where the
potential is infinity.

1:06:19.230 --> 1:06:25.590
I'm going to first look at the
region here, region III.

1:06:25.590 --> 1:06:30.880
What's the solution going to
look like in region III is what

1:06:30.882 --> 1:06:32.052
I'm asking.

1:06:32.050 --> 1:06:34.700
So in this region the energy's
something I don't know what the

1:06:34.697 --> 1:06:35.607
allowed values are.

1:06:35.610 --> 1:06:39.100
Whatever the value is V is an
extremely large number.

1:06:39.099 --> 1:06:40.629
Can you see that?

1:06:40.630 --> 1:06:43.470
In this region don't let the
barrier be infinity.

1:06:43.469 --> 1:06:47.179
Imagine it's one zillion,
a very high barrier.

1:06:47.179 --> 1:06:48.749
Then what's the solution?

1:06:48.750 --> 1:06:59.750
The solution will look like
d^(2)ψ/dx^(2)

1:06:59.753 --> 1:07:09.243
is = 2m/ℏ^(2) times
V - E ψ.

1:07:09.239 --> 1:07:11.029
Then the solution to that is
very easy,

1:07:11.030 --> 1:07:20.200
ψ = Ae^(κx)
Be^(-κx) where

1:07:20.195 --> 1:07:28.315
κ is equal to all of this,
square root of

1:07:28.318 --> 1:07:34.528
2m/ℏ^(2) times V -
E.

1:07:34.530 --> 1:07:39.920
I'm saying this is what I'm
calling κ^(2).

1:07:39.920 --> 1:07:43.460
Look, I'm asking you give me a
function whose second derivative

1:07:43.461 --> 1:07:45.521
is some number times the
function.

1:07:45.518 --> 1:07:49.228
Well, that's obviously
exponential and it's the real

1:07:49.231 --> 1:07:52.941
exponential because it's a
positive number times the

1:07:52.943 --> 1:07:53.893
function.

1:07:53.889 --> 1:07:56.559
Now A and B are
free parameters,

1:07:56.559 --> 1:07:58.289
whatever they are it'll solve
the equation,

1:07:58.289 --> 1:08:01.749
but we don't want a function
that's growing exponentially

1:08:01.751 --> 1:08:05.521
when you go to infinity because
that means that particle would

1:08:05.521 --> 1:08:09.231
rather be at infinity than in
your box or near your box.

1:08:09.230 --> 1:08:13.630
So for that mathematical,
for that physical reason we

1:08:13.625 --> 1:08:15.395
junk this function.

1:08:15.400 --> 1:08:19.070
You pick it on physical grounds
as not having a part growing

1:08:19.074 --> 1:08:21.634
exponentially when you go to
infinity,

1:08:21.630 --> 1:08:24.760
but you do admit a part that's
falling exponentially when you

1:08:24.761 --> 1:08:25.651
go to infinity.

1:08:25.649 --> 1:08:26.949
That's okay.

1:08:26.949 --> 1:08:29.119
But how fast is it falling?

1:08:29.118 --> 1:08:32.368
It's going like B e to
the minus some blah,

1:08:32.365 --> 1:08:35.475
blah, blah times square root of
V - E.

1:08:35.479 --> 1:08:37.839
That's all I want you to look
at.

1:08:37.840 --> 1:08:39.800
Forget all the ℏ's and
m's.

1:08:39.800 --> 1:08:42.400
Make V larger,
and larger, and larger,

1:08:42.395 --> 1:08:45.045
and just tell me what you think
it will do.

1:08:45.050 --> 1:08:47.480
If V is larger and
larger it's like this

1:08:47.484 --> 1:08:50.984
e^(-αx) I wrote for
you where α's very large.

1:08:50.979 --> 1:08:54.169
So this function will fall
faster, and faster,

1:08:54.171 --> 1:08:58.571
and faster, and in the limit in
which V goes to infinity

1:08:58.570 --> 1:08:59.920
it will vanish.

1:08:59.920 --> 1:09:07.260
That basically means the
particle cannot be found outside

1:09:07.259 --> 1:09:08.569
the box.

1:09:08.569 --> 1:09:13.409
So your function ψ
is 0 here and 0 here.

1:09:13.408 --> 1:09:17.358
Because if you made the barrier
height finite you will find it's

1:09:17.360 --> 1:09:19.870
falling exponentially on either
side,

1:09:19.868 --> 1:09:22.098
but the exponential becomes
narrower and narrower as the

1:09:22.099 --> 1:09:23.599
barrier becomes higher and
higher,

1:09:23.600 --> 1:09:27.980
and in the limit in which the
wall is infinitely tall there is

1:09:27.978 --> 1:09:29.268
nothing outside.

1:09:29.270 --> 1:09:35.910
The wave function is non-0 only
inside.

1:09:35.909 --> 1:09:37.779
So what's the solution inside?

1:09:37.779 --> 1:09:39.109
Let me call this 0.

1:09:39.109 --> 1:09:41.029
Let me call this L.

1:09:41.029 --> 1:09:43.949
Inside the box there is no
potential,

1:09:43.948 --> 1:09:50.838
so this equation is
d^(2)ψ/dx^(2)

1:09:50.840 --> 1:09:55.250
(2mE/ℏ^(2))ψ
= 0.

1:09:55.250 --> 1:09:59.580
It's like a free particle in
the box, but it cannot leave the

1:09:59.577 --> 1:10:00.007
box.

1:10:00.010 --> 1:10:04.210
This is what I called k,
remember?

1:10:04.210 --> 1:10:07.690
So now I'm going to purposely
write the solution in terms of

1:10:07.694 --> 1:10:09.234
trigonometric functions.

1:10:09.229 --> 1:10:11.439
You'll see in a minute why.

1:10:11.439 --> 1:10:16.409
So I'm going to write this
Ae^(ikx)

1:10:16.413 --> 1:10:18.723
Be^(-ikx).

1:10:18.720 --> 1:10:21.520
You can see if I take any of
these solutions it's going to

1:10:21.520 --> 1:10:25.270
satisfy the equation,
but k better be related to E in

1:10:25.265 --> 1:10:29.265
this form,
2mE/ℏ^(2) is equal to

1:10:29.273 --> 1:10:30.683
k^(2).

1:10:30.680 --> 1:10:35.680
Sorry that's k^(2).

1:10:35.680 --> 1:10:39.020
Because two derivatives of this
will give me

1:10:39.015 --> 1:10:43.665
−k^(2), and if you
put that particular value for

1:10:43.671 --> 1:10:47.631
−k^(2) here these
two will cancel.

1:10:47.630 --> 1:10:53.140
But I'm somehow going for--I'm
sorry, I'm going to not write it

1:10:53.143 --> 1:10:57.593
this way, but write it as
C cosine kx

1:10:57.590 --> 1:11:00.080
D sine kx.

1:11:00.078 --> 1:11:02.578
Do you realize I can always go
back and forth between

1:11:02.582 --> 1:11:05.282
exponentials and trigonometric
functions because one is a

1:11:05.277 --> 1:11:07.007
linear combination of the other?

1:11:07.010 --> 1:11:09.490
If you want,
write this as cos i sine

1:11:09.494 --> 1:11:12.794
and cos - i sine and
rearrange the coefficients.

1:11:12.788 --> 1:11:14.428
It'll look like something,
something cosine.

1:11:14.430 --> 1:11:16.530
I want to call that a C,
and something,

1:11:16.529 --> 1:11:18.819
something sine which I want to
call this D,

1:11:18.815 --> 1:11:20.725
D and C may be
complex.

1:11:20.729 --> 1:11:23.859
I'm not saying anything,
but you can write a solution

1:11:23.860 --> 1:11:27.650
either in terms of the sines and
cosines or E to the plus

1:11:27.654 --> 1:11:29.044
or minus something.

1:11:29.039 --> 1:11:31.409
So here's the function ψ.

1:11:31.408 --> 1:11:35.788
It looks like I have an answer
for every energy I want,

1:11:35.788 --> 1:11:39.878
because pick any energy you
like, find the corresponding

1:11:39.877 --> 1:11:44.547
k,
you put it here and you're done.

1:11:44.550 --> 1:11:49.720
But that's not allowed because
we have an extra condition which

1:11:49.715 --> 1:11:54.625
is the ψ was identically 0
here, ψ was identically 0

1:11:54.630 --> 1:11:55.380
here.

1:11:55.380 --> 1:11:57.410
We're going to demand that at
the two ends,

1:11:57.408 --> 1:11:59.218
it can do whatever it wants in
the middle,

1:11:59.220 --> 1:12:02.480
it must vanish at the two ends
for the continuity of ψ

1:12:02.476 --> 1:12:05.046
because if ψ
had two values you're getting

1:12:05.047 --> 1:12:08.017
two different probabilities for
the same point,

1:12:08.020 --> 1:12:09.600
so that's not allowed.

1:12:09.600 --> 1:12:14.000
So ψ must have--must match
at the two ends.

1:12:14.000 --> 1:12:16.480
But look at this function.

1:12:16.479 --> 1:12:20.919
It's got to vanish at the left
end at x = 0 and it's got to

1:12:20.918 --> 1:12:22.908
vanish at the right end.

1:12:22.908 --> 1:12:28.578
At x = 0 you can see
ψ of 0 is simply C

1:12:28.578 --> 1:12:31.008
because sine vanishes.

1:12:31.010 --> 1:12:35.650
Cosine is 1 and C has to
be 0 then because this guy has

1:12:35.654 --> 1:12:39.084
no business being non-zero on
the left end.

1:12:39.079 --> 1:12:42.519
That's good.

1:12:42.520 --> 1:12:45.540
This guy vanishes at the left
end so I allow it,

1:12:45.537 --> 1:12:47.527
but I have another condition.

1:12:47.529 --> 1:12:51.749
It should also vanish at the
right end.

1:12:51.750 --> 1:12:56.700
If it should vanish at the
right end I demand that sine

1:12:56.704 --> 1:13:01.844
kL should be 0 because
sine kL vanishing is

1:13:01.844 --> 1:13:02.674
fine.

1:13:02.670 --> 1:13:05.460
Then it'll vanish at both ends.

1:13:05.460 --> 1:13:14.430
But sine kL = 0 means kL
is a multiple of Π.

1:13:14.430 --> 1:13:20.120
That means k is
nΠ/L.

1:13:20.118 --> 1:13:25.368
This means the allowed values
of k in the problem are very

1:13:25.367 --> 1:13:30.607
special so here is n = 1,
ψ looks like

1:13:30.614 --> 1:13:37.194
sin(Πx/L),
and equal to 2 looks like

1:13:37.190 --> 1:13:42.850
sin(2Πx/L),
and so on with some numbers in

1:13:42.849 --> 1:13:45.379
front which I have not chosen
yet.

1:13:45.380 --> 1:13:47.270
If you plot them they look like
this.

1:13:47.270 --> 1:13:48.480
That's one guy.

1:13:48.479 --> 1:13:50.939
At a higher energy I've got
that guy.

1:13:50.939 --> 1:13:55.529
Then I've got that and so on
because these are exactly like

1:13:55.529 --> 1:13:59.329
waves on a string,
a violin string clamped at two

1:13:59.327 --> 1:14:00.037
ends.

1:14:00.038 --> 1:14:03.348
In fact, this wave equation's
identical to the wave equation

1:14:03.353 --> 1:14:04.143
on a string.

1:14:04.140 --> 1:14:06.870
The only requirement is that
the string is clamped at two

1:14:06.868 --> 1:14:07.208
ends.

1:14:07.210 --> 1:14:09.210
Here the ψ
is clamped at the two ends

1:14:09.212 --> 1:14:12.222
because it's got to vanish on
either side outside the box.

1:14:12.220 --> 1:14:15.180
So the allowed wavelengths are
the same except here the

1:14:15.180 --> 1:14:18.360
wavelength is connected to the
momentum that's connected to

1:14:18.359 --> 1:14:21.269
energy and E,
you remember,

1:14:21.273 --> 1:14:24.953
is
ℏ^(2)k^(2)/2m

1:14:24.953 --> 1:14:29.943
then becomes
ℏ^(2)/2m times

1:14:29.940 --> 1:14:33.740
k will be
n^(2)Π

1:14:33.740 --> 1:14:36.710
^(2)/L^(2).

1:14:36.710 --> 1:14:40.280
So this particle in a box can
have only these particular

1:14:40.275 --> 1:14:40.985
energies.

1:14:40.988 --> 1:14:42.238
So let me write it for you
nicely.

1:14:42.238 --> 1:14:48.788
The allowed energies are
ℏ^(2) Π^(2)/2

1:14:48.786 --> 1:14:55.196
mL squared times an
integer n squared,

1:14:55.198 --> 1:14:58.538
and the corresponding wave
functions look like this.

1:14:58.538 --> 1:15:01.728
They're waves in which you've
got half an oscillation,

1:15:01.726 --> 1:15:04.486
or 2 half oscillations and 3
half oscillations,

1:15:04.493 --> 1:15:07.143
but you've got to start and
finish at 0.

1:15:07.140 --> 1:15:08.760
So it's the quantization.

1:15:08.760 --> 1:15:11.300
This is why when Schr�dinger
came up with this equation

1:15:11.302 --> 1:15:14.162
everybody embraced it right away
because you suddenly understood

1:15:14.162 --> 1:15:15.392
why energy's quantized.

1:15:15.390 --> 1:15:18.510
You're trying to fit some
number of waves into an interval

1:15:18.506 --> 1:15:21.676
and only some multiple of half
wavelengths are allowed,

1:15:21.680 --> 1:15:23.640
but wavelength translates into
momentum.

1:15:23.640 --> 1:15:25.170
That translates into energy.

1:15:25.170 --> 1:15:29.050
Suddenly you understand the
quantization of energy.

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So it's got 1 state
n = 1,

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and only one state
n = 2,

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one state n =
3.

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I'm going to come back to this
next time, but you should think

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about this.

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