WEBVTT

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Prof: All right,
today's topic is the theory of

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nearly everything,
okay?

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You wanted to know the theory
of everything?

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You're almost there,
because I'm finally ready to

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reveal to you the laws of
quantum dynamics that tells you

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how things change with time.

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So that's the analog of F =
ma.

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and just about anything you see
in this room,

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or on this planet,
anything you can see or use is

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really described by this
equation I'm going to write down

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today.

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It contains Newton's laws as
part of it, because if you can

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do the quantum theory,
you can always find hidden in

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it the classical theory.

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That's like saying if I can do
Einstein's relativistic

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kinematics at low velocities,
I will regain Newtonian

01:00.414 --> 01:01.324
mechanics.

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So everything is contained in
this one.

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There are some things left,
of course, that we won't do,

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but this goes a long way.

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So I'll talk about it probably
next time near the end,

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depending on how much time
there is.

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But without further ado,
I will now tell you what the

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laws of motion are in quantum
mechanics.

01:25.310 --> 01:32.030
So let's go back one more time
to remember what we have done.

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The analogous statement is,
in classical mechanics for a

01:36.882 --> 01:40.062
particle moving in one
dimension,

01:40.060 --> 01:45.070
all I need to know about it
right now is the position and

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the momentum.

01:46.410 --> 01:47.180
That's it.

01:47.180 --> 01:49.460
That's the maximal information.

01:49.459 --> 01:51.669
You can say,
"What about other things?

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What about angular momentum?

01:52.920 --> 01:54.120
What about kinetic energy?

01:54.120 --> 01:55.470
What about potential energy?

01:55.470 --> 01:57.070
What about total energy?"

01:57.069 --> 01:59.969
They're all functions of x
and p.

01:59.970 --> 02:02.130
For example,
in 3 dimensions, x will

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be replaced by r, p
will be replaced by some

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vector
p, and there's a

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variable called angular
momentum,

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but you know that once you
know r

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and p
by taking the cross product.

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That's it.

02:15.490 --> 02:19.000
And you can say,
"What happens when I

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measure any variable for a
classical particle in this

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state, x,p?"

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Well, if you know the location,
it's guaranteed to be x

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100 percent,
momentum is p,

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100 percent.

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Any other function of x
and p,

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like r X
p, is guaranteed to have

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that particular value.

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So everything is completely
known.

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That's the situation at one
time.

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Then you want to say,
"What can you say about

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the future?

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What's the rate of change of
these things?"

02:49.240 --> 02:54.350
And the answer to that is,
d^(2)x/dt^(2) times

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m is the force,
and in most problems you can

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write the force as a derivative
of some potential.

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So if you knew the potential,
1/2kx^(2) or whatever it

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is,
or mgx,

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you can take the derivative on
the right hand side,

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and the left hand side tells
you the rate of change of

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x.

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I want you to note one thing -
we know an equation that tells

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you something about the
acceleration.

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Once the forces are known,
there's a unique acceleration.

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So you are free to give the
particle any position you like,

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and any velocity,
dx/dt.

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That's essentially the momentum.

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You can pick them at random.

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But you cannot pick the
acceleration at random,

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because the acceleration is not
for you to decide.

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The acceleration is determined
by Newton's laws to equal the

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essentially the force divided by
mass.

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That comes from the fact
mathematically that this is a

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second order equation in time,
namely involving the second

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derivative.

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And that, from a mathematical
point of view,

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if the second derivative is
determined by external

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considerations,
initial conditions are given by

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initial x and the first
derivative.

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All higher derivatives are
slaved to the applied force.

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You don't assign them as you
wish.

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You find out what they are from
the equations of motion.

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That's really all of classical
mechanics.

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Now you want to do quantum
mechanics, and we have seen many

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times the story in quantum
mechanics is a little more

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complicated.

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You ask a simple question and
you get a very long answer.

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The simple question is,
how do you describe the

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particle in quantum mechanics in
one dimension?

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And you say,
"I want to assign to it a

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function
Y(x)."

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Y(x) is any
reasonable function which can be

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squared and integrated over the
real line.

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Anything you write down is a
possible state.

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That's like saying,
any x and any p

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are allowed.

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Likewise, Y(x) is
nothing special.

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It can be whatever you like as
long as you can square it and

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integrate it to get a finite
answer over all of space.

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That's the only condition.

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And if your all of space goes
to infinity, then Y

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should vanish a and- infinity.

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That's the only requirement.

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Then you say,
"That tells me everything.

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Why don't you tell me what the
particle is doing?"

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And you can say,
"What do you want to

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know?"

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Well, I want to know where it
is.

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That's when you don't get a
straight answer.

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You are told,
"Well, it can be here,

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it can be there,
it can be anywhere else.

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And the probability density
that it's at point x is

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proportional to the absolute
square of Y."

05:42.319 --> 05:44.449
That means you take the
Y and you square it,

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so it will have nothing
negative in it.

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Everything will be real and
positive, and Y itself

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may be complex.

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But this Y^(2),
I told you over and over,

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is defined to be
Y*Y.

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That's real and positive.

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Then you can say,
"What if I measure

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momentum, what answer will I
get?"

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That's even longer.

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First you are supposed to
expand--I'm not going to do the

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whole thing too many times--
you're supposed to write this

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Y as some coefficient
times these very special

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functions.

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In a world of size L,
you have to write the given

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Y in this fashion,
and the coefficients are

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determined by the integral of
the complex conjugate of this

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function times the function you
gave me, Y(x).

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Now I gave some extra notes,
I think.

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Did people get that?

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Called the "Quantum
Cookbook?"

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That's just the recipe,
you know.

06:46.300 --> 06:51.030
Quantum mechanics is a big fat
recipe and that's all we can do.

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I tell you, you do this,
you get these answers.

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That's my whole goal,
to simply give you the recipe.

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So the recipe says--what's
interesting about quantum

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mechanics,
what makes it hard to teach,

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is that there are some physical
principles which are summarized

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by these rules,
which are like axioms.

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Then there are some purely
mathematical results which are

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not axioms.

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They are consequences of pure
mathematics.

07:19.149 --> 07:22.079
You have to keep in mind,
what is a purely mathematical

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result,
therefore is deduced from the

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laws of mathematics,
and what is a physical result,

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that's deduced from experiment.

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The fact that Y
describes everything is a

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physical result.

07:33.430 --> 07:39.120
Now it tells you to write
Y as the sum of these

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functions,
and then the probability to

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obtain any momentum p is
A_p^(2),

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where A_p is
defined by this.

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The mathematics comes in in the
following way - first question

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is, who told you that I can
write every function Y in

07:56.173 --> 07:57.103
this fashion?

07:57.100 --> 07:58.530
That's called the Fourier's
theorem,

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that guarantees you that in a
circle of size L,

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every periodic function,
meaning that returns to its

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starting value,
may be expanded in terms of

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these functions.

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That's the mathematical result.

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The same mathematical result
also tells you how to find the

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coefficients.

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The postulates of quantum
mechanics tell you two things.

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A_p^(2) is the
probability that you will get

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the value p when you
measure momentum,

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okay?

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That's a postulate,
because you could have written

08:30.639 --> 08:33.689
this function 200 years before
quantum mechanics.

08:33.690 --> 08:37.520
It will still be true,
but this function did not have

08:37.522 --> 08:41.652
a meaning at that time as states
of definite momentum.

08:41.649 --> 08:43.889
How do I know it's a state of
definite momentum?

08:43.889 --> 08:47.129
If every term vanished except
one term,

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that's all you have,
then one coefficient will be

08:50.255 --> 08:53.005
A something = 1,
everything is 0,

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that means the probability for
getting momentum has a non 0

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value only for that momentum.

08:58.980 --> 09:04.720
All other momenta are missing
in that situation.

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Another postulate of quantum
mechanics is that once you

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measure momentum and you get one
of these values,

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the state will go from being a
sum over many such functions,

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and collapse to the one term
and the sum that corresponds to

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the answer you got.

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Then here is another
mathematical result - p is not

09:25.109 --> 09:27.969
every arbitrary real number you
can imagine.

09:27.970 --> 09:30.730
We make the requirement,
if you go around on a circle,

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the function should come back
to the starting value,

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therefore p is
restricted to be

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2pℏ
/L times some

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integer n.

09:39.950 --> 09:42.410
That's a mathematical
requirement, because if you

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think Y ^(2) is the
probability, Y should

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come back to where you start.

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It cannot get two different
values of Y when you go

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around the circle.

09:50.220 --> 09:55.520
That quantizes momentum to
these values.

09:55.519 --> 10:00.169
The last thing I did was to
say, if you measure energy,

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what answer will you get?

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That's even longer.

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There you're supposed to solve
the following equations,

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(((h^(2)/2m)d^(2)
Y_E

10:13.977 --> 10:20.497
/dx^(2)) V(x))
Y_E(x) =

10:20.498 --> 10:24.408
EY_E(x).

10:24.408 --> 10:26.468
In other words,
for energy the answer's more

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complicated, because before,
I can tell you anything.

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I want you to solve this
equation.

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This equation says,
if in classical mechanics the

10:35.557 --> 10:40.097
particle was in some potential
V(x) and the particle had

10:40.101 --> 10:43.131
some mass m,
you have to solve this

10:43.131 --> 10:46.141
equation, then it's a purely
mathematical problem,

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and you try to find all
solutions that behave well at

10:49.268 --> 10:51.358
infinity,
that don't blow up at infinity,

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that vanish at infinity.

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That quantizes E to certain
special values.

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And there are corresponding
functions, Y_E

11:00.653 --> 11:02.283
for each allowed value.

11:02.278 --> 11:05.378
Then you are done,
because then you make a similar

11:05.378 --> 11:08.918
expansion, you write the unknown
Y and namely some

11:08.919 --> 11:11.639
arbitrary Y that's given
to you.

11:11.639 --> 11:13.369
You write it as a
ΣA_E

11:13.369 --> 11:15.809
Y
_E(x),

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where A_E is
found by a similar rule.

11:18.759 --> 11:21.889
Just replace p by
E and replace this

11:21.892 --> 11:23.812
function by these functions.

11:23.808 --> 11:27.138
Then if you square
A_E you will

11:27.139 --> 11:30.399
get the probability you will get
the energy.

11:30.399 --> 11:33.919
So what makes the energy
problem more complicated is that

11:33.918 --> 11:37.688
whereas for momentum we know
once and for all these functions

11:37.690 --> 11:41.400
describe a state of definite
momentum where you can get only

11:41.398 --> 11:44.178
one answer,
states of definite energy

11:44.178 --> 11:47.608
depend on what potential is
acting on the particle.

11:47.610 --> 11:49.900
If it's a free particle,
V is 0.

11:49.899 --> 11:53.859
If it's a particle that in
Newtonian mechanics is a

11:53.861 --> 11:57.031
harmonic oscillator,
V(x) would be

11:57.033 --> 11:59.493
�kx^(2) and so on.

11:59.490 --> 12:02.970
So you should know the
classical potential and you've

12:02.967 --> 12:06.647
got it in for every possible
potential you have to solve

12:06.647 --> 12:07.247
this.

12:07.250 --> 12:10.370
But that's what most people in
physics departments are doing

12:10.373 --> 12:11.383
most of the time.

12:11.379 --> 12:16.489
They're solving this equation
to find states of definite

12:16.493 --> 12:17.333
energy.

12:17.330 --> 12:20.940
So today, I'm going to tell you
why states of definite energy

12:20.937 --> 12:22.077
are so important.

12:22.080 --> 12:23.350
What's the big deal?

12:23.350 --> 12:26.110
Why is state of momentum not so
important?

12:26.110 --> 12:28.550
Why is the state of definite
position not so interesting?

12:28.548 --> 12:31.348
What is privileged about the
states of definite momentum?

12:31.350 --> 12:35.630
And now you will see the role
of energy.

12:35.629 --> 12:41.179
So I'm going to write down for
you the equation that's analog

12:41.181 --> 12:42.941
of F = ma.

12:42.940 --> 12:44.760
So what are we trying to do?

12:44.759 --> 12:51.589
Y(x) is like
x and p.

12:51.590 --> 12:53.800
You don't have a time label
here.

12:53.798 --> 12:57.318
These are like saying at some
time a particle has a position

12:57.316 --> 12:58.326
and a momentum.

12:58.330 --> 13:00.810
In quantum theory or some time,
it has a wave function

13:00.807 --> 13:01.787
Y(x).

13:01.788 --> 13:04.118
But the real question in
classical mechanics is,

13:04.123 --> 13:06.903
how does x vary with
time and how does p vary with

13:06.903 --> 13:07.353
time.

13:07.350 --> 13:10.960
The answer is,
according to F = ma.

13:10.960 --> 13:13.340
And here the question is,
how does Y vary with

13:13.339 --> 13:13.659
time?

13:13.658 --> 13:17.648
First thing you've got to do is
to realize that Y itself

13:17.648 --> 13:19.898
can be a function of time,
right?

13:19.899 --> 13:22.979
That's the only time you've got
to ask, "What does it do

13:22.984 --> 13:23.914
with time?"

13:23.908 --> 13:27.148
So at t = 0 it may look
like this.

13:27.149 --> 13:29.149
A little later,
it may look like that.

13:29.149 --> 13:34.009
So it's flopping and moving,
just like say a string.

13:34.009 --> 13:36.749
It's changing with time and you
want to know how it changes with

13:36.745 --> 13:37.045
time.

13:39.980 --> 13:44.290
It says iℏd
Y(x,t)/dt (it's

13:44.292 --> 13:47.002
partial,
because this depends on

13:46.998 --> 13:52.008
x and t so this is
the t derivative) = the

13:52.008 --> 13:57.858
following,
[-ℏ^(2) /2m

13:57.860 --> 14:06.200
d^(2)Y/dx^(2) V(x)
Y(x,t)].

14:06.200 --> 14:09.960
That's the equation.

14:09.960 --> 14:14.610
x comma t.

14:14.610 --> 14:17.730
So write this down,
because if you know this

14:17.730 --> 14:21.500
equation, you'll be surprised
how many things you can

14:21.504 --> 14:22.524
calculate.

14:22.519 --> 14:25.359
From this follows the spectrum
of the atoms,

14:25.363 --> 14:28.743
from this follows what makes a
material a conductor,

14:28.735 --> 14:31.245
a semiconductor,
a superconductor.

14:36.842 --> 14:38.052
equation.

14:38.048 --> 14:41.038
This is an equation in which
you must notice that we're

14:41.039 --> 14:43.919
dealing for the first time with
functions of time.

14:43.918 --> 14:45.978
Somebody asked me long back,
"Where is time?"

14:45.980 --> 14:49.010
Well, here is how Y
varies with time.

14:49.009 --> 14:53.229
So suppose someone says,
"Here is initial

14:53.227 --> 14:55.757
Y(x) and 0.

14:55.759 --> 14:59.149
Tell me what is Y a
little later,

14:59.153 --> 15:01.593
1 millisecond later."

15:01.590 --> 15:04.480
Well, it's the rate of change
of Y with time multiplied

15:04.479 --> 15:05.379
by 1 millisecond.

15:05.379 --> 15:08.159
The rate of change of Y
at the initial time is obtained

15:08.164 --> 15:10.684
by taking that derivative of
Y and adding to it V

15:10.676 --> 15:12.436
times Y,
you get something.

15:12.440 --> 15:14.900
That's how much Y
changes.

15:14.899 --> 15:17.319
Multiply by Dt,
that's the change in Y.

15:17.320 --> 15:20.360
That'll give you Y at a
later time.

15:20.360 --> 15:22.840
This is the first order
equation in time.

15:22.840 --> 15:26.620
What that means mathematically
is, the initial Y

15:26.620 --> 15:29.070
determines the future
completely.

15:29.070 --> 15:33.020
This is different from position
where you need x and

15:33.024 --> 15:35.484
dx/dt are the initial
time.

15:35.480 --> 15:39.460
The equation will only tell you
what d^(2)x/dt^(2)^( )is.

15:39.460 --> 15:41.830
But in quantum mechanics,
dY /dt

15:41.827 --> 15:43.967
itself is determined,
so you don't get to choose

15:43.970 --> 15:44.380
that.

15:44.379 --> 15:46.679
You just get to choose the
initial Y.

15:46.678 --> 15:50.908
That means an initial wave
function completely determines

15:50.913 --> 15:53.943
the future according to this
equation.

15:53.940 --> 15:55.080
So don't worry about this
equation.

15:55.080 --> 15:59.050
I don't expect you all to see
it and immediately know what to

15:59.054 --> 16:02.704
do, but I want you to know that
there is an equation.

16:02.700 --> 16:03.750
That is known now.

16:03.750 --> 16:06.580
That's the analog of F =
ma. If you solve this

16:06.575 --> 16:09.825
equation, you can predict the
future to the extent allowed by

16:09.833 --> 16:11.033
quantum mechanics.

16:11.028 --> 16:13.458
Given the present,
and the present means

16:13.462 --> 16:15.962
Y(x,0) is
given,

16:15.960 --> 16:17.340
then you go to the math
department and say,

16:17.340 --> 16:18.990
"This is my
Y(x,0).

16:18.994 --> 16:21.174
Please tell me by some trick
what is Y(x) and

16:21.174 --> 16:21.894
t."

16:21.889 --> 16:26.069
It turns out there is a trick
by which you can predict

16:26.072 --> 16:30.652
Y(x) and t.
Note also that this number

16:30.649 --> 16:35.069
i is present in the very
equations of motion.

16:35.070 --> 16:37.960
So this is not like the
i we used in electrical

16:37.956 --> 16:40.786
circuits where we really meant
sines and cosines,

16:40.788 --> 16:42.448
but we took
e^(t)^(q)or

16:42.446 --> 16:44.236
e^(i)^(w)
^(t),

16:44.240 --> 16:47.270
always hoping in the end to
take the real part of the answer

16:47.272 --> 16:50.052
because the functions of
classical mechanics are always

16:50.047 --> 16:50.507
real.

16:50.509 --> 16:53.139
But in quantum theory,
Y is intrinsically

16:53.143 --> 16:56.563
complex and it cannot get more
complex than that by putting an

16:56.562 --> 16:58.862
i in the equations of
motion,

16:58.860 --> 17:00.250
but that's just the way it is.

17:00.250 --> 17:05.520
You need the i to write
the equations.

17:05.519 --> 17:10.419
Therefore our goal then is to
learn different ways in which we

17:10.423 --> 17:11.793
can solve this.

17:11.788 --> 17:13.948
Now remember,
everybody noticed,

17:13.951 --> 17:17.651
this looks kind of familiar
here, this combination.

17:17.650 --> 17:19.010
It's up there somewhere.

17:19.009 --> 17:20.489
It looks a lot like this.

17:20.490 --> 17:22.160
You see that?

17:22.160 --> 17:24.270
But it's not quite that.

17:24.269 --> 17:26.989
That is working on a function
only of x;

17:26.990 --> 17:31.030
this is working on a function
of x and t.

17:31.028 --> 17:33.328
And there are partial
derivatives here and there are

17:33.334 --> 17:34.514
total derivatives there.

17:34.509 --> 17:36.019
They are very privileged
functions.

17:36.019 --> 17:38.199
They describe states of
definite energy.

17:38.200 --> 17:41.570
This is an arbitrary function,
just evolving with time,

17:41.567 --> 17:43.747
so you should not mix the two
up.

17:43.750 --> 17:48.680
This Y is a generic
Y changing with time.

17:48.680 --> 17:52.000
So let's ask,
how can I calculate the future,

17:52.000 --> 17:53.510
given the present?

17:53.509 --> 17:55.979
How do I solve this equation?

17:55.980 --> 17:58.140
So here is what you do.

17:58.140 --> 18:00.330
I'm going to do it at two
levels.

18:00.328 --> 18:03.928
One is to tell you a little bit
about how you get there,

18:03.930 --> 18:05.490
and for those of you who say,
"Look,

18:05.490 --> 18:07.460
spare me the details,
I just want to know the

18:07.458 --> 18:09.288
answer,"
I will draw a box around the

18:09.292 --> 18:11.452
answer,
and you are free to start from

18:11.451 --> 18:11.831
there.

18:11.828 --> 18:15.768
But I want to give everyone a
chance to look under the hood

18:15.772 --> 18:17.612
and see what's happening.

18:17.608 --> 18:21.758
So given an equation like this,
which is pretty old stuff in

18:21.759 --> 18:24.919
mathematical physics from after
Newton's time,

18:24.923 --> 18:28.093
people always ask the following
question.

18:28.088 --> 18:30.868
They say, "Look,
I don't know if I can solve it

18:30.874 --> 18:33.444
for every imaginable initial
condition."

18:33.440 --> 18:35.070
It's like saying,
even the case of the

18:35.065 --> 18:36.765
oscillator,
you may not be able to solve

18:36.773 --> 18:40.313
every initial condition,
you say, "Let me find a

18:40.308 --> 18:44.808
special case where
Y(x, t),

18:44.808 --> 18:47.528
which depends on x and
on t,

18:47.529 --> 18:54.639
has the following simple form -
it's a function of t alone times

18:54.642 --> 18:58.822
a function of x
alone."

18:58.819 --> 18:59.629
Okay?

18:59.630 --> 19:04.030
I want you to know that no one
tells you that every solution to

19:04.030 --> 19:06.090
the equation has this form.

19:06.088 --> 19:08.778
You guys have a question about
this?

19:08.779 --> 19:10.409
Over there.

19:10.410 --> 19:12.020
Okay, good.

19:12.019 --> 19:15.259
All right, so this is an
assumption.

19:15.259 --> 19:19.109
You want to see if maybe there
are answers like this to the

19:19.105 --> 19:19.765
problem.

19:19.769 --> 19:22.949
The only way to do that is to
take their assumed form,

19:22.953 --> 19:26.443
put it into the equation and
see if you can find a solution

19:26.438 --> 19:27.458
of this form.

19:27.460 --> 19:28.940
Not every solution looks like
this.

19:28.940 --> 19:33.580
For example,
you could write

19:33.575 --> 19:37.005
e^((x-t)2).

19:37.009 --> 19:39.289
That's the function of x
and t.

19:39.288 --> 19:42.058
But it's not a function of
x times the function of

19:42.058 --> 19:42.948
t, you see that?

19:42.950 --> 19:45.590
x and t are mixed
up together.

19:45.588 --> 19:48.448
You cannot rip it out at the
two parts, so it's not the most

19:48.454 --> 19:51.264
general thing that can happen;
it's a particular one.

19:51.259 --> 19:54.609
Right now, you are eager to get
any solution.

19:54.608 --> 19:56.258
You want to say,
"Can I do anything?

19:56.259 --> 19:59.299
Can I calculate even in the
simplest case what the future

19:59.299 --> 20:00.929
is, given the present?"

20:00.930 --> 20:04.200
You're asking,
"Can this happen?"

20:04.200 --> 20:09.820
So I'm going to show you now
that the equation does admit

20:09.819 --> 20:12.329
solutions of this type.

20:12.328 --> 20:16.648
So are you guys with me now on
what I'm trying to do?

20:16.650 --> 20:22.390
I'm trying to see if this
equation admits solutions of

20:22.390 --> 20:23.690
this form.

20:23.690 --> 20:27.410
So let's take that and put it
here.

20:27.410 --> 20:31.770
Now here's where you've got to
do the math, okay?

20:31.769 --> 20:33.919
Take this Y and put it
here and start taking

20:33.917 --> 20:34.517
derivatives.

20:34.519 --> 20:37.089
Let's do the left hand side
first.

20:37.088 --> 20:40.438
Left hand side,
I have iℏ.

20:40.440 --> 20:44.920
Then I bring the d by dt to act
on this product.

20:44.920 --> 20:50.100
d by dt partial means only time
has to be differentiated;

20:50.099 --> 20:51.979
x is to be held constant.

20:51.980 --> 20:53.040
That's the partial derivative.

20:53.038 --> 20:54.678
That's the meaning of the
partial derivative.

20:54.680 --> 20:57.500
It's like an ordinary
derivative where the only

20:57.501 --> 21:00.571
variable you'd ever
differentiate is the one in the

21:00.568 --> 21:01.488
derivative.

21:01.490 --> 21:04.280
So the entire
Y(x) doesn't do

21:04.278 --> 21:05.008
anything.

21:05.009 --> 21:06.349
It's like a constant.

21:06.348 --> 21:08.298
So you just put that
Y(x) there.

21:08.298 --> 21:10.638
Then the derivative,
d by dt,

21:10.640 --> 21:14.720
comes here, and I claim it
becomes the ordinary derivative.

21:14.720 --> 21:18.730
That's the left hand side.

21:18.730 --> 21:22.520
You understand that,
why that is true?

21:22.519 --> 21:24.819
Because on a function only of
time there's no difference

21:24.816 --> 21:27.026
between partial derivative and
ordinary derivative.

21:27.029 --> 21:28.849
It's got only one variable.

21:28.848 --> 21:31.608
The other potential variable
this d by dt,

21:31.609 --> 21:34.909
doesn't care about so it's just
standing there.

21:34.910 --> 21:36.990
That's the left hand side.

21:36.990 --> 21:41.630
Now look at the right hand
side, all of this stuff,

21:41.634 --> 21:47.024
and imagine putting for this
function Y this product

21:47.022 --> 21:47.862
form.

21:47.858 --> 21:50.488
Is it clear to you,
in the right hand side the

21:50.491 --> 21:52.541
situation's exactly the
opposite?

21:52.538 --> 21:54.758
You've got all these d by
dx's partials,

21:54.762 --> 21:57.472
they're only interested in this
function because it's got

21:57.470 --> 21:58.630
x dependence.

21:58.630 --> 22:01.990
F(t) doesn't do anything.

22:01.990 --> 22:05.690
All the derivatives,
they go right through F,

22:05.692 --> 22:08.382
so you can write it as
F(t).

22:08.380 --> 22:12.940
Now you have to take the
derivative with respect to

22:12.944 --> 22:13.954
Y.

22:13.950 --> 22:17.930
That looks like
d^(2)Y

22:17.933 --> 22:24.493
/dx^(2)
V(x)Y(x).

22:24.490 --> 22:27.270
If you follow this,
you're almost there,

22:27.273 --> 22:30.133
but take your time to
understand this.

22:30.130 --> 22:32.780
The reason you write it as a
product of two functions is the

22:32.778 --> 22:35.068
left hand side is only
interested in differentiating

22:35.067 --> 22:37.587
the function F,
where it becomes the total

22:37.593 --> 22:38.203
derivative.

22:38.200 --> 22:40.780
The right hand side is only
taking derivatives with respect

22:40.775 --> 22:43.035
to x so it acts on this
part of the function,

22:43.039 --> 22:44.239
it depends on x.

22:44.240 --> 22:48.520
And all partial derivatives
become total derivatives because

22:48.518 --> 22:52.578
if you've got only one variable,
there's no need to write

22:52.578 --> 22:54.318
partial derivatives.

22:54.318 --> 22:59.508
This combination I'm going to
write to save some time as

22:59.507 --> 23:01.297
HY.

23:01.298 --> 23:05.118
Let's just say between you and
me it's a shorthand.

23:05.118 --> 23:11.088
HY is a shorthand
for this entire mess here.

23:11.088 --> 23:12.908
Don't ask me why it looks like
H times Y,

23:12.913 --> 23:13.863
where are the derivatives?

23:13.859 --> 23:16.639
It is a shorthand, okay.

23:16.640 --> 23:20.900
I don't feel like writing the
combination over and over,

23:20.895 --> 23:23.445
I can call it HY.

23:23.450 --> 23:26.090
So what equation do I have now?

23:26.088 --> 23:31.038
I have iℏ
Y(x)dF/dt =

23:31.041 --> 23:36.321
F(t)HY,
where all I want you to notice,

23:36.317 --> 23:41.267
HY depends only
on x.

23:41.269 --> 23:42.619
It has no dependence on time.

23:42.619 --> 23:43.949
Do you see that?

23:43.950 --> 23:52.970
There's nothing here that
depends on time.

23:52.970 --> 23:55.140
Okay, now this is a trick which
if you learned,

23:55.140 --> 23:57.490
you'll be quite pleased,
because you'll find that as you

23:57.489 --> 23:59.969
do more and more stuff,
at least in physics or

23:59.970 --> 24:04.160
economics or statistics,
the trick is a very old trick.

24:04.160 --> 24:06.890
The problem is quantum
mechanics, but the mathematics

24:06.885 --> 24:07.615
is very old.

24:07.619 --> 24:09.399
So what do you do next?

24:09.400 --> 24:13.140
You divide both sides by
YF.

24:13.140 --> 24:25.010
So I say divide by
F(t)Y(x).

24:25.009 --> 24:26.899
What do you think will happen
if I divide by F(t),

24:26.900 --> 24:27.610
Y(x)?

24:27.608 --> 24:29.918
On the left hand side,
I say divide by

24:29.924 --> 24:32.684
YF and the right
hand side,

24:32.680 --> 24:35.140
I say divide by
YF,

24:35.141 --> 24:38.681
can you see that the Y
cancels here,

24:38.680 --> 24:40.340
and you have a 1/F.

24:40.338 --> 24:43.668
F cancels here and you
have a 1/Y.

24:43.670 --> 24:48.450
The equation then says
iℏ(1/

24:48.454 --> 24:53.244
F(t))dF/dt =
1/Y(x)H

24:53.240 --> 24:59.580
Y. I've
written this very slowly because

24:59.584 --> 25:02.854
I don't know,
you'll find this in many

25:02.846 --> 25:05.146
advanced books,
but you may not find it in our

25:05.146 --> 25:05.666
textbook.

25:05.670 --> 25:08.310
So if you don't follow
something, you should tell me.

25:08.308 --> 25:12.648
There's plenty of time to do
this, so I'm in no rush at all.

25:12.650 --> 25:15.680
These are purely mathematical
manipulations.

25:15.680 --> 25:20.230
We have not done anything
involving physics.

25:20.230 --> 25:23.860
You all follow this?

25:23.859 --> 25:25.079
Yes?

25:25.079 --> 25:26.249
Okay.

25:26.250 --> 25:27.650
Now you have to ask yourself
the following.

25:27.650 --> 25:29.240
I love this argument.

25:29.240 --> 25:31.900
Even if you don't follow this,
I'm just going to get it off my

25:31.896 --> 25:33.446
chest,
it is so clever,

25:33.449 --> 25:37.869
and here is the clever part -
this is supposedly a function of

25:37.867 --> 25:39.747
time,
you agree?

25:39.750 --> 25:42.870
All for a function of time.

25:42.869 --> 25:45.699
This is a function of x.

25:45.700 --> 25:47.870
This guy doesn't know what time
it is;

25:47.868 --> 25:49.428
this guy doesn't know what
x is.

25:49.430 --> 25:52.820
And yet they're supposed to be
equal.

25:52.819 --> 25:56.449
What can they be equal to?

25:56.450 --> 25:59.170
They cannot be equal to a
function of time,

25:59.171 --> 26:01.891
because then,
as you vary time--suppose you

26:01.893 --> 26:04.683
think it's a function of time,
suppose.

26:04.680 --> 26:06.060
It's not so.

26:06.058 --> 26:08.678
Then as time varies,
this part is okay.

26:08.680 --> 26:11.940
It can vary with time to match
that, but this cannot vary with

26:11.936 --> 26:14.336
time at all, because there is no
time here.

26:14.339 --> 26:16.919
So this cannot depend on time.

26:16.920 --> 26:19.200
And it cannot depend on
x, because if it was a

26:19.202 --> 26:21.182
function of x that it was
equal to,

26:21.180 --> 26:23.060
as you vary x,
this can change with x

26:23.056 --> 26:23.916
to keep up with that.

26:23.920 --> 26:25.860
This has no x dependence.

26:25.859 --> 26:27.509
It cannot vary with x.

26:27.509 --> 26:31.779
So this thing that they are
both equal to is not a function

26:31.781 --> 26:34.951
of time and it's not a function
of space.

26:34.950 --> 26:36.650
It's a constant.

26:36.650 --> 26:38.220
That's all it can be.

26:38.220 --> 26:43.500
So the constant is going to
very cleverly be given the

26:43.497 --> 26:45.287
symbol E.

26:45.288 --> 26:47.578
We're going to call the
constant E.

26:47.578 --> 26:50.578
It turns out E is
connected to the energy of the

26:50.583 --> 26:51.143
problem.

26:51.140 --> 26:55.070
So now I have two equations,
this = E and that =

26:55.066 --> 26:58.116
E and I'm going to write
it down.

26:58.118 --> 27:06.128
So one of them says i
ℏ (1/F) dF/dt =

27:06.134 --> 27:07.474
E.

27:07.470 --> 27:09.770
So let me bring the F
here.

27:09.769 --> 27:14.079
Other one says H
Y = E

27:14.079 --> 27:20.769
Y. These two
equations, if you solve them,

27:20.773 --> 27:26.563
will give you the solution you
are looking for.

27:26.558 --> 27:28.808
In other words,
going back here,

27:28.810 --> 27:31.250
yes,
this equation does admit

27:31.253 --> 27:34.423
solutions of this form,
of the product form,

27:34.423 --> 27:37.713
provided the function F
you put in the product that

27:37.711 --> 27:40.021
depends on time obeys this
equation,

27:40.019 --> 27:43.339
and the function Y that
depends only on x obeys

27:43.343 --> 27:44.233
this equation.

27:44.230 --> 27:46.650
Remember, HY is
the shorthand for this long

27:46.646 --> 27:47.636
bunch of derivatives.

27:47.640 --> 27:51.680
We'll come to that in a moment,
but let's solve this equation

27:51.681 --> 27:52.221
first.

27:52.220 --> 27:54.100
Now can you guys do this in
your head?

27:54.099 --> 28:03.309
iℏdF/dt = EF.

28:03.308 --> 28:06.538
So it's saying F is a
function of time whose time

28:06.537 --> 28:08.647
derivative is the function
itself.

28:08.650 --> 28:12.060
Everybody knows what such a
function is.

28:12.059 --> 28:14.509
It's an exponential.

28:14.509 --> 28:16.779
And the answer is,
I'm going to write it down and

28:16.784 --> 28:20.434
you can check,
F(t) is F(0)

28:20.428 --> 28:26.748
e^(−iEt/ℏ).
If you want now,

28:26.750 --> 28:29.240
take the derivative and check.

28:29.240 --> 28:31.650
F(0) is some constant.

28:31.650 --> 28:34.840
I call it F(0) because
if t = 0,

28:34.844 --> 28:38.184
this goes away and F(t)
= F(0).

28:38.180 --> 28:39.920
But take the time derivative
and see.

28:39.920 --> 28:41.650
When you take a time derivative
of this,

28:41.650 --> 28:45.110
you get the same thing times
−iE/ℏ,

28:45.108 --> 28:47.528
and when you multiply by
iℏ,

28:47.529 --> 28:50.359
everything cancels except
EF.

28:50.359 --> 28:52.699
So this is a very easy solution.

28:52.700 --> 28:54.950
So let's stop and understand.

28:54.950 --> 28:58.510
It says that if you are looking
for solutions that are products

28:58.505 --> 29:00.565
of F(t) times
f(x),

29:00.568 --> 29:05.398
F(t) necessarily is this
exponential function,

29:05.400 --> 29:08.700
which is the only function you
can have.

29:08.700 --> 29:11.170
But now once you pick that
E, you can pick E

29:11.170 --> 29:13.420
to be whatever you like,
but then you must also solve

29:13.423 --> 29:14.943
this equation at the same time.

29:14.940 --> 29:16.810
But what is this equation.

29:16.808 --> 29:21.008
This says
-ℏ^(2)/2md^(2)

29:21.006 --> 29:25.796
Y/dx^(2)
VY =

29:25.801 --> 29:30.461
EY,
and you guys know who that is,

29:30.462 --> 29:31.022
right?

29:31.019 --> 29:32.889
What is it?

29:32.890 --> 29:39.300
What can you say about the
function that satisfies that

29:39.295 --> 29:40.595
equation?

29:40.599 --> 29:43.029
Have you seen it before?

29:43.029 --> 29:44.149
Yes?

29:44.150 --> 29:45.120
What is it?

29:46.690 --> 29:48.540
Prof: It's the state of
definite energy.

29:48.538 --> 29:51.948
Remember, we said functions of
definite energy obey that

29:51.948 --> 29:52.628
equation.

29:52.630 --> 30:02.420
So that Y is really just
Y_E.

30:02.420 --> 30:04.460
So now I'll put these two
pieces together,

30:04.460 --> 30:09.090
and this is where those of you
who drifted off can come back,

30:12.557 --> 30:15.777
equation in fact admits a
certain solution which is a

30:15.778 --> 30:19.368
product of a function of time
and a function of space.

30:19.368 --> 30:22.428
And what we found by fiddling
around with it is that F(t)

30:22.430 --> 30:24.350
and Y are very
special,

30:24.348 --> 30:28.378
and F(t) must look like
e^(−iEt/

30:28.380 --> 30:32.450
ℏ),
and Y is just our

30:32.450 --> 30:37.030
friend, Y
_E(x), which

30:37.025 --> 30:42.205
are functions associated with a
definite energy.

30:42.210 --> 30:43.110
Yes?

30:43.108 --> 30:46.078
Student:  Is it possible
that there are other solutions,

30:46.077 --> 30:48.897
x/Y(x,
t) that don't satisfy the

30:48.901 --> 30:49.621
conditions?

30:49.618 --> 30:51.948
Prof: Okay,
the question is,

30:51.953 --> 30:55.943
are there other solutions for
which this factorized form is

30:55.936 --> 30:56.826
not true?

30:56.828 --> 31:00.708
Yes, and I will put you out of
your suspense very soon by

31:00.705 --> 31:02.155
talking about that.

31:02.160 --> 31:06.230
But I want everyone to
understand that you can at least

31:09.849 --> 31:10.709
So what does this mean?

31:10.710 --> 31:13.210
I want you guys to think about
it.

31:13.210 --> 31:18.140
This says if you start Y
in some arbitrary configuration,

31:18.140 --> 31:21.010
that's my initial state,
let it evolve with time,

31:25.567 --> 31:26.477
equation.

31:26.480 --> 31:31.280
But if I start it at t =
0, in a state which is a state

31:31.282 --> 31:34.912
of definite energy,
namely a state obeying that

31:34.905 --> 31:35.925
equation.

31:35.930 --> 31:38.550
Then its future is very simple.

31:38.548 --> 31:43.358
All you do is attach this phase
factor, e^(−iEt/

31:43.362 --> 31:44.912
ℏ).

31:44.910 --> 31:47.620
Therefore it's not a generic
solution, because you may not in

31:47.622 --> 31:50.562
general start with a state which
is a state of definite energy.

31:50.558 --> 31:53.128
You'll start with some random
Y(x) and it's made

31:53.130 --> 31:55.110
up of many,
many Y_E's

31:55.114 --> 31:57.494
that come in the expansion of
that Y,

31:57.490 --> 31:59.160
so it's not going to always
work.

31:59.160 --> 32:02.510
But if you picked it so that
there's only one such term in

32:02.510 --> 32:05.270
the sum over E,
namely one such function,

32:05.273 --> 32:07.453
then the future is given by
this.

32:07.450 --> 32:10.120
For example,
if you have a particle in a

32:10.119 --> 32:12.859
box,
you remember the wave function

32:12.861 --> 32:17.541
Y_n looks like
square root of 2/Lsine(n

32:17.535 --> 32:21.605
px/L). An
arbitrary Y doesn't look

32:21.605 --> 32:23.335
like any of these.

32:23.338 --> 32:27.278
These guys, remember,
are nice functions that do many

32:27.281 --> 32:28.421
oscillations.

32:28.420 --> 32:32.860
But if you chose it initially
to be exactly the sine function,

32:32.858 --> 32:35.078
for example,
Y_1,

32:35.077 --> 32:39.807
then I claim as time evolves,
the future state is just this

32:39.810 --> 32:44.530
initial sine function times this
simple exponential.

32:44.529 --> 32:52.929
This behavior is very special
and it called normal modes.

32:52.930 --> 32:56.060
It's a very common idea in
mathematical physics.

32:56.058 --> 32:59.428
It's the following - it's very
familiar even before you did

32:59.434 --> 33:00.604
quantum mechanics.

33:00.598 --> 33:06.198
Take a string tied at both ends
and you pluck the string and you

33:06.202 --> 33:07.362
release it.

33:07.358 --> 33:10.328
Most probably if you pluck it
at one point,

33:10.329 --> 33:14.149
you'll probably pull it in the
middle and let it go.

33:14.150 --> 33:17.350
That's the initial
Y(x,t),

33:17.348 --> 33:19.398
this time for a string.

33:19.400 --> 33:21.240
Pull it in the middle,
let it go.

33:21.240 --> 33:24.560
There's an equation that
determines the evolutions of

33:24.556 --> 33:25.446
that string.

33:25.450 --> 33:27.370
I remind you what that equation
is.

33:27.368 --> 33:31.898
It's d^(2)Y
/dx^(2)=(1/veloc

33:31.901 --> 33:36.331
ity^(2))d^(2)Y
/dt^(2).

33:36.328 --> 33:38.708
That's the wave equation for a
string.

33:38.710 --> 33:41.280
It's somewhat different from
this problem,

33:41.276 --> 33:44.156
because it's a second
derivative in time that's

33:44.155 --> 33:44.965
involved.

33:44.970 --> 33:48.160
Nonetheless,
here is an amazing property of

33:48.163 --> 33:51.513
this equation,
derived by similar methods.

33:51.509 --> 33:53.849
If you pull a string like this
and let it go,

33:53.846 --> 33:55.916
it will go crazy when you
release it.

33:55.920 --> 33:57.070
I don't even know what it will
do.

33:57.068 --> 34:00.438
It will do all kinds of things,
stuff will go back and forth,

34:00.435 --> 34:01.385
back and forth.

34:01.390 --> 34:06.410
But if you can deform the
string at t = 0 to look

34:06.412 --> 34:10.882
exactly like this,
sine(px/L)

34:10.875 --> 34:14.465
times a number A,
that's not easy to do.

34:14.469 --> 34:17.279
Do you understand that to
produce the initial profile,

34:17.282 --> 34:19.992
one hand is not enough,
two hands are not enough?

34:19.989 --> 34:22.889
You've got to get infinite
number of your friends,

34:22.893 --> 34:24.793
who are infinitesimally small.

34:24.789 --> 34:28.359
You line them up along the
string until each one--tell the

34:28.360 --> 34:31.180
person here to pull it exactly
this height.

34:31.179 --> 34:33.199
Person here has to pull it
exactly that height.

34:33.199 --> 34:35.529
You all lift your hands up,
then I follow you.

34:35.530 --> 34:37.240
You follow this perfect sine.

34:37.239 --> 34:40.639
Then you let go.

34:40.639 --> 34:46.589
What do you think will happen
then?

34:46.590 --> 34:50.760
What do you think will be the
subsequent evolution of the

34:50.759 --> 34:51.429
string?

34:51.429 --> 34:52.029
Do you have any guess?

34:52.030 --> 34:52.510
Yes?

34:52.510 --> 34:54.560
Student:
>

34:54.559 --> 35:01.219
Prof: It will go up and
down, and the future of that

35:01.222 --> 35:07.312
string will look like
cosine(pxvt/L).

35:07.309 --> 35:08.149
Look at it.

35:08.150 --> 35:10.360
This is a time dependence,
that t = 0,

35:10.355 --> 35:12.205
this goes away,
this initial state.

35:12.210 --> 35:14.790
But look what happens at a
later time.

35:14.789 --> 35:20.049
Every point x rises and
falls with the same period.

35:20.050 --> 35:22.580
It goes up and down all
together.

35:22.579 --> 35:25.139
That means a little later it
will look like that,

35:25.139 --> 35:26.359
a little later it will look
like that,

35:26.360 --> 35:28.300
then it will look like this,
then it will look like this,

35:28.300 --> 35:30.310
then it will look like that,
then it will go back and forth.

35:30.309 --> 35:32.899
But at every instant,
this guy, this guy,

35:32.900 --> 35:36.270
this guy are all rescaled by
the same amount from the

35:36.268 --> 35:37.498
starting value.

35:37.500 --> 35:39.930
That's called a normal mode.

35:39.929 --> 35:42.259
Now your question was,
are there other solutions to

35:42.262 --> 35:42.872
the string?

35:42.869 --> 35:43.969
Of course.

35:43.969 --> 35:46.279
Typically, if you don't think
about it and pluck the string,

35:46.280 --> 35:49.570
your initial state will be a
sum of these normal modes,

35:49.570 --> 35:51.990
and that will evolve in a
complicated way.

35:51.989 --> 35:54.649
But if you engineered it to
begin exactly this way,

35:54.650 --> 35:58.580
or in any one of those other
functions where you put an extra

35:58.583 --> 36:01.043
n here,
they all have the remarkable

36:01.043 --> 36:03.753
property that they rise and fall
in step.

36:03.750 --> 36:07.650
What we have found here is in
the quantum problem,

36:07.650 --> 36:11.290
if you start the system in that
particular configuration,

36:11.289 --> 36:15.319
then its future has got a
single time dependence common to

36:15.315 --> 36:15.665
it.

36:15.670 --> 36:20.760
That's the meaning of the
factorized solution.

36:20.760 --> 36:22.860
So we know one simple example.

36:22.860 --> 36:24.890
Take a particle in a box.

36:24.889 --> 36:28.529
If it's in the lowest energy
state or ground state the wave

36:28.525 --> 36:30.215
function looks like that.

36:30.219 --> 36:35.049
Then the future of that will be
Y_1/x

36:35.052 --> 36:38.682
and t will be the
Y_1x

36:38.677 --> 36:41.817
times
e^(-iE1t/ℏ),

36:41.820 --> 36:45.460
where E_1 =
ℏ^(2)

36:45.460 --> 36:48.940
p^(2)
1^(2)/2mL^(2).

36:48.940 --> 36:53.160
That's the energy
associated with that function.

36:53.159 --> 36:57.739
That's how it will oscillate.

36:57.739 --> 37:01.479
Now you guys follow what I said
now, with an analogy with the

37:01.481 --> 37:03.541
string and the quantum problem?

37:03.539 --> 37:05.189
They're slightly different
equations.

37:05.190 --> 37:06.860
One is second order,
one is first order.

37:06.860 --> 37:09.940
One has cosines in it,
one has exponentials in it.

37:09.940 --> 37:13.720
But the common property is,
this is also a function of time

37:13.722 --> 37:15.682
times the function of space.

37:15.679 --> 37:26.259
Here, this is a function of
time and a function of space.

37:26.260 --> 37:30.610
Okay, so I'm going to spend
some time analyzing this

37:30.612 --> 37:32.492
particular function.

37:32.489 --> 37:37.949
Y(x) and t =
e^(−iEt/ℏ)^(

37:37.945 --> 37:41.125
)Y
_E(x).

37:41.130 --> 37:45.530
And I'm going to do an abuse in
rotation and give the subscript

37:45.525 --> 37:47.505
E to this guy also.

37:47.510 --> 37:50.090
What I mean to tell you by that
is, this Y,

37:52.731 --> 37:54.421
I invite you to go check it.

37:56.583 --> 37:58.373
equation and you will find it
works.

37:58.369 --> 38:01.599
In the notes I've given you,
I merely tell you that this is

38:04.050 --> 38:06.330
I don't go through this
argument of assuming it's a

38:06.329 --> 38:07.469
product form and so on.

38:07.469 --> 38:08.269
That's optional.

38:08.268 --> 38:10.278
I don't care if you remember
that or not,

38:12.030 --> 38:16.460
and I call it
Y_E because the

38:16.458 --> 38:22.358
function of the right hand side
are identified with states of

38:22.362 --> 38:24.332
definite energy.

38:24.329 --> 38:31.309
Okay, what will happen if you
measure various quantities in

38:31.313 --> 38:32.883
this state?

38:32.880 --> 38:34.680
For example,
what's the position going to

38:34.684 --> 38:34.914
be?

38:34.909 --> 38:37.399
What's the probability for
definite position?

38:37.400 --> 38:39.510
What's the probability for
definite momentum?

38:39.510 --> 38:41.850
What's the probability for
definite anything?

38:41.849 --> 38:43.319
How will they vary with time?

38:43.320 --> 38:47.030
I will show you,
nothing depends on time.

38:47.030 --> 38:49.700
You can say,
"How can nothing depend on

38:49.704 --> 38:50.144
time?

38:50.139 --> 38:52.899
I see time in the function
here."

38:52.900 --> 38:55.100
But it will go away.

38:55.099 --> 38:59.779
Let us ask, what is the
probability that the particle is

38:59.780 --> 39:03.950
at x at time t for a
solution like this?

39:03.949 --> 39:08.029
You know, the answer is
Y *(x,t)

39:08.030 --> 39:13.600
Y(x,t), and
what do you get when you do

39:13.603 --> 39:14.503
that?

39:14.500 --> 39:18.050
You will get Y_E
*(x

39:18.048 --> 39:21.518
)Y
_E(x). Then you

39:21.519 --> 39:25.609
will get the absolute value of
this guy squared,

39:25.610 --> 39:27.850
e^(iEt)^( )times
e^(-iEt),

39:27.849 --> 39:34.709
and that's just 1.

39:34.710 --> 39:38.540
I hope all of you know that
this e^(i)^( )absolute

39:38.541 --> 39:39.981
value squared is 1.

39:39.980 --> 39:42.350
So it does not depend on time.

39:42.349 --> 39:45.869
Even though Y depends on
time, Y *Y has no

39:45.867 --> 39:46.887
time dependence.

39:46.889 --> 39:50.509
That means the probability for
finding that particle will not

39:50.512 --> 39:51.662
change with time.

39:51.659 --> 39:57.049
That means if you start the
particle in the ground state,

39:57.050 --> 40:00.640
Y, and let's say
Y^(2) in fact looks

40:00.641 --> 40:04.931
pretty much the same,
it's a real function,

40:04.925 --> 40:10.215
this probability does not
change with time.

40:10.219 --> 40:13.549
That means you can make a
measurement any time you want

40:13.545 --> 40:16.195
for position,
and the odds don't change with

40:16.195 --> 40:16.745
time.

40:16.750 --> 40:17.950
It's very interesting.

40:17.949 --> 40:20.289
It depends on time and it
doesn't depend on time.

40:20.289 --> 40:22.979
It's a lot like
e^(ipx/ℏ).

40:22.980 --> 40:26.130
It seems to depend on x
but the density does not depend

40:26.130 --> 40:28.610
on x because the
exponential goes away.

40:28.610 --> 40:30.470
Similarly, it does depend on
time.

40:30.469 --> 40:32.989
Without the time dependence,
it won't satisfy

40:36.018 --> 40:38.018
absolute value,
this goes away.

40:38.018 --> 40:41.438
That means for this particle,
I can draw a little graph that

40:41.443 --> 40:43.943
looks like this,
and that is the probability

40:43.938 --> 40:46.258
cloud you find in all the
textbooks.

40:46.260 --> 40:47.510
Have you seen the probability
cloud?

40:47.510 --> 40:51.150
They've got a little atom
that's a little fuzzy stuff all

40:51.150 --> 40:51.930
around it.

40:51.929 --> 40:54.589
They are the states of the
hydrogen atom or some other

40:54.588 --> 40:54.938
atom.

40:54.940 --> 40:56.910
How do you think you get that?

40:56.909 --> 40:59.889
You solve a similar equation,
except it will be in 3

40:59.887 --> 41:02.047
dimensions instead of 1
dimension,

41:02.050 --> 41:08.530
and for V(x), you write
-Ze^(2)/r, if you want

41:08.532 --> 41:12.382
r as x^(2) y^(2)
z^(2).

41:12.380 --> 41:15.600
Ze is the nuclear charge,
and -e is the electron

41:15.599 --> 41:16.099
charge.

41:16.099 --> 41:20.919
You put that in and you solve
the equation and you will find a

41:20.918 --> 41:24.788
whole bunch of solutions that
behave like this.

41:24.789 --> 41:27.059
They are called stationary
states, because in that

41:27.061 --> 41:29.711
stationary state--
see, if a hydrogen atom starts

41:29.710 --> 41:32.280
out in this state,
which is a state of definite

41:32.275 --> 41:33.695
energy,
as time goes by,

41:33.704 --> 41:35.844
nothing happens to it
essentially.

41:35.840 --> 41:38.540
Something trivial happens;
it picks up the phase factor,

41:38.538 --> 41:41.978
but the probability for finding
the electron never changes with

41:41.976 --> 41:42.416
time.

41:42.420 --> 41:45.160
So if you like,
you can draw a little cloud

41:45.161 --> 41:46.991
whose thickness,
if you like,

41:46.987 --> 41:50.967
measures the probability for
finding it at that location.

41:50.969 --> 41:52.069
So that will have all kinds of
shape.

41:52.070 --> 41:54.650
It looks like dumbbells,
pointing to the north pole,

41:54.654 --> 41:57.344
south pole, maybe uniformly
spherical distribution.

41:57.340 --> 42:01.930
They're all the probability of
finding the electron in that

42:01.932 --> 42:05.182
state, and it doesn't change
with time.

42:05.179 --> 42:08.059
So a hydrogen atom,
when you leave it alone,

42:08.056 --> 42:10.796
will be in one of these allowed
states.

42:10.800 --> 42:12.040
You don't need the hydrogen
atom;

42:12.039 --> 42:14.889
this particle in a box is a
good enough quantum system.

42:14.889 --> 42:16.659
If you start it like that,
it will stay like that;

42:16.659 --> 42:19.469
if you start it like that,
it will stay like that,

42:19.474 --> 42:20.974
times that phase factor.

42:20.969 --> 42:25.359
So stationary states are
important, because that's where

42:25.356 --> 42:27.506
things have settled down.

42:27.510 --> 42:31.230
Okay, now you should also
realize that that's not a

42:31.231 --> 42:32.721
typical situation.

42:32.719 --> 42:37.029
Suppose you have in 1
dimension, there's a particle on

42:37.025 --> 42:41.165
a hill, and at t = 0,
it's given by some wave

42:41.168 --> 42:43.928
function that looks like this.

42:43.929 --> 42:47.029
So it's got some average
position, and if you expand it

42:47.032 --> 42:50.432
in terms of exponentials of
p, it's got some range of

42:50.425 --> 42:51.455
momenta in it.

42:51.460 --> 42:54.560
What will happen to this as a
function of time?

42:54.559 --> 42:56.939
Can you make a guess?

42:56.940 --> 43:01.070
Let's say it's right now got an
average momentum to the left.

43:01.070 --> 43:04.730
What do you think will happen
to it?

43:04.730 --> 43:05.370
Pardon me?

43:05.369 --> 43:06.279
Student:  It will move
to the left.

43:06.280 --> 43:07.140
Prof: It will move to
the left.

43:07.139 --> 43:10.589
Except for the fuzziness,
you can apply your classical

43:10.588 --> 43:11.368
intuition.

43:11.369 --> 43:13.369
It's got some position,
maybe not precise.

43:13.369 --> 43:15.369
It's got some momentum,
maybe not precise,

43:15.369 --> 43:17.569
but when you leave something on
top of a hill,

43:17.565 --> 43:19.415
it's going to slide down the
hill.

43:19.420 --> 43:21.900
The average x is going
to go this way,

43:21.898 --> 43:24.208
and the average momentum will
increase.

43:24.210 --> 43:27.280
So that's a situation where the
average of the physical

43:27.282 --> 43:28.992
quantities change with time.

43:28.989 --> 43:31.859
That's because this state is
not a function

43:31.862 --> 43:34.122
Y_E(x).

43:34.119 --> 43:35.379
It's some random function you
picked.

43:35.380 --> 43:38.710
Random functions you picked in
some potential will in fact

43:38.713 --> 43:42.283
evolve with time in such a way
that measurable quantities will

43:42.280 --> 43:43.510
change with time.

43:43.510 --> 43:45.840
The odds for x or the
odds for p,

43:45.836 --> 43:48.206
odds for everything else,
will change with time,

43:48.213 --> 43:48.673
okay?

43:48.670 --> 43:50.880
So stationary states are very
privileged,

43:50.880 --> 43:52.860
because if you start them that
way,

43:52.860 --> 43:55.020
they stay that way,
and that's why when you look at

43:55.021 --> 43:58.051
atoms,
they typically stay that way.

43:58.050 --> 44:01.000
But once in a while,
an atom will jump from one

44:00.998 --> 44:05.038
stationary state to another one,
and you can say that looks like

44:05.036 --> 44:06.316
a contradiction.

44:06.320 --> 44:08.240
If it's stationary,
what's it doing jumping from

44:08.244 --> 44:08.904
here to there?

44:08.900 --> 44:14.340
You know the answer to that?

44:14.340 --> 44:15.760
Why does an atom ever change
then?

44:15.760 --> 44:18.930
If it's in a state of definite
E, it should be that way

44:18.929 --> 44:19.449
forever.

44:19.449 --> 44:22.369
Why do they go up and down?

44:22.369 --> 44:23.339
Want to guess?

44:23.340 --> 44:26.670
Student:
>

44:26.670 --> 44:27.180
Prof: That's correct.

44:27.179 --> 44:28.859
So she said by absorbing
photons.

44:28.860 --> 44:33.020
And what I really mean by that
is, this problem V(x)

44:33.023 --> 44:37.263
involves only the coulomb force
between the electron and the

44:37.260 --> 44:38.050
proton.

44:38.050 --> 44:40.650
If that's all you have,
an electron in the field of a

44:40.654 --> 44:42.664
proton, it will pick one of
these levels,

44:42.657 --> 44:44.157
it can stay there forever.

44:44.159 --> 44:46.489
When you shine light,
you're applying an

44:46.487 --> 44:47.917
electromagnetic field.

44:47.920 --> 44:51.360
The electric field and magnetic
field apply extra forces in the

44:51.358 --> 44:54.518
charge and V(x) should
change to something else.

44:54.518 --> 44:57.108
So that this function is no
longer a state of definite

44:57.106 --> 44:59.696
energy for the new problem,
because you've changed the

44:59.695 --> 45:00.765
rules of the game.

45:00.769 --> 45:03.709
You modified the potential.

45:03.710 --> 45:06.920
Then of course it will move
around and it will change from

45:06.916 --> 45:08.206
one state to another.

45:08.210 --> 45:11.780
But an isolated atom will
remain that way forever.

45:11.780 --> 45:15.330
But it turns out even that's
not exactly correct.

45:15.329 --> 45:19.489
You can take an isolated atom,
in the first excited state of

45:19.492 --> 45:23.592
hydrogen, you come back a short
time later, you'll find the

45:23.585 --> 45:25.345
fellow has come down.

45:25.349 --> 45:27.109
And you say,
"Look, I didn't turn on

45:27.112 --> 45:28.042
any electric field.

45:28.039 --> 45:30.209
E = 0, B = 0.

45:30.210 --> 45:32.810
What made the atom come
down?"

45:32.809 --> 45:37.089
Do you know what the answer to
that is?

45:37.090 --> 45:37.800
Any rumors?

45:37.800 --> 45:38.510
Yes?

45:38.510 --> 45:40.280
Student:  Photon
emission?

45:40.280 --> 45:42.290
Prof: Its photon is
emitted,

45:42.289 --> 45:45.319
but you need an extra thing,
extra electrical magnetic field

45:45.318 --> 45:47.678
to act on it before it will emit
the photon.

45:47.679 --> 45:48.489
But where is the field?

45:48.489 --> 45:49.969
I've turned everything off.

45:49.969 --> 45:53.799
E and B are both
0.

45:53.800 --> 45:59.240
So it turns out that the state
E = B = 0 is like

45:59.237 --> 46:03.867
a state say in a harmonic
oscillator potential x =

46:03.871 --> 46:06.301
p = 0,
sitting at the bottom of the

46:06.304 --> 46:06.544
well.

46:06.539 --> 46:08.769
We know that's not allowed in
quantum mechanics.

46:08.768 --> 46:11.958
You cannot have definite
x and definite p.

46:11.960 --> 46:16.680
It turns out in quantum theory,
E and B are like

46:16.681 --> 46:18.541
x and p.

46:18.539 --> 46:21.039
That means the state of
definite E is not a state

46:21.036 --> 46:22.076
of definite B.

46:22.079 --> 46:24.219
A state of definite B is
not a state of definite

46:24.222 --> 46:24.662
E.

46:24.659 --> 46:26.879
It looks that way in the
macroscopic world,

46:26.882 --> 46:29.902
because the fluctuations in
E and B are very

46:29.900 --> 46:30.430
small.

46:30.429 --> 46:33.469
Therefore, just like in the
lowest energy state,

46:33.469 --> 46:36.869
an oscillator has got some
probability to be jiggling back

46:36.873 --> 46:39.563
and forth in x and also
in p.

46:39.559 --> 46:42.499
The vacuum, which we think has
no E and no B,

46:42.500 --> 46:45.140
has small fluctuations,
because E and B

46:45.135 --> 46:48.275
both vanishing is like x
and p both vanishing.

46:48.280 --> 46:49.200
Not allowed.

46:49.199 --> 46:51.779
So you've got to have a little
spread in both E and both

46:51.780 --> 46:52.240
B.

46:52.239 --> 46:54.859
They're called quantum
fluctuations of the vacuum.

46:54.860 --> 46:57.480
So that's a theory of nothing.

46:57.480 --> 47:00.160
The vacuum you think is the
most uninteresting thing,

47:00.161 --> 47:02.331
and yet it is not completely
uninteresting,

47:02.327 --> 47:04.387
because it's got these
fluctuations.

47:04.389 --> 47:08.049
It's those fluctuations that
tickle the atom and make it come

47:08.047 --> 47:10.607
from an excited state to a
ground state.

47:10.610 --> 47:14.740
Okay, so unless you tamper with
the atom in some fashion,

47:14.742 --> 47:17.622
it will remain in a stationary
state.

47:17.619 --> 47:19.729
Those states are states of
definite energy.

47:22.385 --> 47:23.415
without time in it.

47:23.420 --> 47:25.530
H Y = E
Y is called the time

47:27.099 --> 47:30.529
and that's what most of us do
most of the time.

47:30.530 --> 47:32.830
The problem can be more
complicated.

47:32.829 --> 47:35.089
It can involve two particles,
can involve ten particles.

47:35.090 --> 47:38.190
It may not involve this force,
may involve another force,

47:38.190 --> 47:41.630
but everybody is spending most
of his time or her time solving

47:45.409 --> 47:47.779
because that's where things
will end up.

47:47.780 --> 47:52.410
All right, I've only shown you
that the probability to find

47:52.407 --> 47:56.157
different positions doesn't
change with time.

47:56.159 --> 47:58.949
I will show you the probability
to find different anything

47:58.947 --> 48:00.267
doesn't change with time.

48:00.268 --> 48:03.158
Nothing will change with time,
not just x probability,

48:03.163 --> 48:04.613
so I'll do one more example.

48:04.610 --> 48:09.220
Let's ask, what's the
probability to find a momentum

48:09.224 --> 48:10.224
p?

48:10.219 --> 48:12.189
What are we supposed to do?

48:12.190 --> 48:16.480
We're supposed to take
e^(ipx)^(/

48:16.480 --> 48:22.860
ℏ )times the
function at some time t

48:22.860 --> 48:25.610
and do that integral.

48:25.610 --> 48:28.600
I'm sorry, you should take that
and do that integral,

48:28.603 --> 48:31.313
and then you take the absolute
value of that.

48:31.309 --> 48:33.409
And that's done at every time.

48:33.409 --> 48:37.659
You take the absolute value and
that's the probability to get

48:37.664 --> 48:39.584
momentum p,
right?

48:39.579 --> 48:41.509
The recipe was,
if you want the probability,

48:41.510 --> 48:43.760
take the given function,
multiply to the conjugate of

48:43.755 --> 48:45.955
Y_p and
do the integral,

48:45.960 --> 48:51.590
dL and dx.

48:51.590 --> 48:54.380
Y(x,t) in general
has got complicated time

48:54.382 --> 48:56.162
dependence, but not our
Y.

48:56.159 --> 48:57.479
Remember our Y?

48:57.480 --> 49:03.990
Our Y looks like
Y(x) times e^(

49:03.992 --> 49:07.492
−iEt/ℏ).

49:07.489 --> 49:10.919
But when you take the absolute
value, this has nothing to do

49:10.918 --> 49:11.848
with x.

49:11.849 --> 49:14.629
You can pull it outside the
integral--or let me put it

49:14.626 --> 49:15.356
another way.

49:15.360 --> 49:17.740
Let's do the integral and see
what you get.

49:17.739 --> 49:21.169
You will find
A_p looks like

49:21.172 --> 49:23.752
A_p
(0)e^(−iEt/

49:23.748 --> 49:25.308
ℏ).

49:25.309 --> 49:26.769
Do you see that?

49:26.768 --> 49:30.048
If the only thing that happens
to Y is that you get an

49:30.045 --> 49:33.205
extra factor at later times,
only thing that happens to the

49:33.208 --> 49:36.388
A_p is it gets
the extra factor at later times.

49:36.389 --> 49:42.099
But the probability to find
momentum p is the

49:42.101 --> 49:48.261
absolute value square of that,
and in the absolute value

49:48.260 --> 49:51.620
process, this guy is gone.

49:51.619 --> 49:54.159
You follow that?

49:54.159 --> 49:57.569
Since the wave function changes
by a simple phase factor,

49:57.570 --> 50:01.340
the coefficient to have a
definite momentum also changes

50:01.342 --> 50:05.322
by the same phase factor.This is
called a phase factor,

50:05.320 --> 50:08.100
exponential of modulus 1,
but when you take the absolute

50:08.103 --> 50:12.433
value,
the guy doesn't do anything.

50:12.429 --> 50:14.499
Now you can replace p by
some other variable.

50:14.500 --> 50:15.190
It doesn't matter.

50:15.190 --> 50:17.520
The story is always the same.

50:17.518 --> 50:22.458
So a state of definite energy
seems to evolve in time,

50:22.460 --> 50:24.430
because e^(ipx)^(/
ℏ),

50:24.429 --> 50:26.919
but none of the probabilities
change with time.

50:26.920 --> 50:28.390
It's absolutely stationary.

50:28.389 --> 50:30.539
Just put anything you measure.

50:30.539 --> 50:33.439
That's why those states are
very important.

50:33.440 --> 50:40.920
Okay, now I want to caution you
that not every solution looks

50:40.922 --> 50:42.422
like this.

50:42.420 --> 50:43.280
That's the question you raised.

50:43.280 --> 50:45.490
I'm going to answer that
question now.

50:45.489 --> 50:48.559
Let's imagine that I find two
solutions to the

50:51.360 --> 50:54.340
Solution Y_1
looks like

50:54.335 --> 50:59.765
Y_1(x,t),
looks like e^(ipx)^(/

50:59.773 --> 51:04.263
ℏ
)Y_1,

51:04.264 --> 51:07.494
Y_E1(x).

51:07.489 --> 51:10.629
That's one solution for energy
E_1.

51:10.630 --> 51:16.170
Then there's another solution,
Y_2 (x)

51:16.172 --> 51:21.172
looks like e^(ipx)
^(/ℏ) Y

51:21.170 --> 51:23.080
_E2(x).

51:23.079 --> 51:25.659
This function has all the
properties I mentioned,

51:25.657 --> 51:27.427
namely nothing depends on time.

51:27.429 --> 51:29.839
That has the same property.

51:33.327 --> 51:36.367
linear equation,
it's also true that this

51:36.369 --> 51:38.849
Y, which is
Y_1

51:38.853 --> 51:43.443
Y_2,
add this one to this one,

51:43.440 --> 51:45.800
is also a solution.

51:45.800 --> 51:47.870
I think I have done it many,
many times.

51:47.869 --> 51:50.129
If you take a linear equation,
Y_1 obeys the

51:53.320 --> 51:55.670
Add the left hand side to the
left hand side and right hand

51:55.666 --> 51:57.716
side to the right hand side,
you will find that if

51:57.722 --> 52:00.092
Y_1 obeys it
and Y_2 does,

52:00.090 --> 52:02.280
Y_1
Y_2 also obeys

52:02.284 --> 52:02.504
it.

52:02.500 --> 52:04.690
Not only that,
it can be even more general.

52:04.690 --> 52:07.890
You can multiply this by any
number,

52:07.889 --> 52:10.859
Y_1(x,t),
any constant,

52:10.860 --> 52:12.350
A_2
Y_2

52:12.351 --> 52:13.631
(x,t),
but A_1 and

52:13.628 --> 52:14.668
A_2 don't
depend on time.

52:16.519 --> 52:17.969
Can you see that?

52:17.969 --> 52:21.179
That's superposition of
solutions.

52:21.179 --> 52:22.959
It's a property of linear
equations.

52:26.347 --> 52:29.057
equation, therefore you can add
solutions.

52:29.059 --> 52:32.719
But take a solution of this
form.

52:32.719 --> 52:34.769
Even though
Y_1 is a

52:34.766 --> 52:37.206
product of some F and a
Y,

52:37.210 --> 52:38.690
and Y_2 is a
product of some

52:38.690 --> 52:40.340
F_2 and
Y_2,

52:40.340 --> 52:45.360
the sum is not a product of
some F and some Y.

52:45.360 --> 52:47.800
You cannot write it as a
product, you understand?

52:47.800 --> 52:49.540
That's a product,
and that's a product,

52:49.536 --> 52:52.276
but their sum is not a product,
because you cannot pull out a

52:52.280 --> 52:54.520
common function of time from the
two of them.

52:54.518 --> 52:57.658
They have different time
dependence.

52:57.659 --> 53:01.419
But that is also a solution.

53:01.420 --> 53:06.330
In fact, now you can ask
yourself, what is the most

53:06.327 --> 53:10.937
general solution I can build in
this problem?

53:10.940 --> 53:14.970
Well, I think you can imagine
that I can now write

53:14.967 --> 53:20.057
Y(x) and t
as A_EY

53:20.063 --> 53:23.873
_E(x,t),
sum over all the allowed

53:23.867 --> 53:24.477
Es.

53:28.309 --> 53:29.219
Do you agree?

53:31.829 --> 53:34.009
You add them all up,
multiply by any constant

53:34.010 --> 53:35.990
A_E,
that also satisfies

53:37.480 --> 53:41.160
So now I'm suddenly
manufacturing more complicated

53:41.159 --> 53:42.059
solutions.

53:42.059 --> 53:45.209
The original modest goal was to
find a product form,

53:45.210 --> 53:47.830
but once you got the product
form, you find if you add them

53:47.827 --> 53:49.627
together,
you get a solution that's no

53:49.625 --> 53:52.175
longer a product of x and
a product of t,

53:52.179 --> 53:53.729
function of x and a
function of t,

53:53.730 --> 53:55.680
because this guy has one time
dependence;

53:55.679 --> 53:58.229
another term is a different
time dependence.

53:58.230 --> 54:00.280
You cannot pull them all out.

54:00.280 --> 54:03.140
So we are now manufacturing
solutions that don't look like

54:03.137 --> 54:03.987
their products.

54:03.989 --> 54:07.239
This is the amazing thing about
solving the linear equation.

54:07.239 --> 54:10.479
You seem to have very modest
goals when you start with a

54:10.476 --> 54:12.126
product form,
but in the end,

54:12.125 --> 54:16.065
you find that you can make up a
linear combination of products.

54:16.070 --> 54:21.430
Then the only question is,
will it cover every possible

54:21.434 --> 54:23.824
situation you give me?

54:23.820 --> 54:26.200
In other words,
suppose you come to me with an

54:26.195 --> 54:27.565
arbitrary initial state.

54:27.570 --> 54:29.320
I don't know anything about it,
and you say,

54:29.322 --> 54:30.672
"What is its future?"

54:30.670 --> 54:32.270
Can I handle that problem?

54:32.268 --> 54:35.088
And the answer is,
I can, and I'll tell you why

54:35.090 --> 54:36.010
that is true.

54:36.010 --> 54:41.720
Y(x) and t looks
like A_E.

54:41.719 --> 54:45.949
I'm going to write this more
explicitly as

54:45.945 --> 54:52.125
Y_E(x)e^(-
ipx/ℏ).

54:52.130 --> 54:56.740
Look at this function now at
t = 0.

54:56.739 --> 55:01.809
At t = 0,
I get Y(x) and 0

55:01.807 --> 55:08.417
to be ΣA_E
Y_E(x).

55:08.420 --> 55:12.660
In other words,
I can only handle those

55:12.661 --> 55:17.911
problems whose initial state
looks like this.

55:17.909 --> 55:23.579
But my question is,
should I feel limited in any

55:23.581 --> 55:26.601
way by the restriction?

55:26.599 --> 55:27.759
Do you follow what I'm saying?

55:27.760 --> 55:30.090
Maybe I'll say it one more time.

55:30.090 --> 55:34.020
This is the most general
solution I'm able to manufacture

55:34.023 --> 55:37.343
that looks like this,
A_E

55:37.335 --> 55:42.125
Y_E(x)e^(-
ipx/ℏ).

55:42.130 --> 55:44.860
It's a sum over solution to the
product form with variable,

55:44.858 --> 55:46.738
each one with a different
coefficient.

55:49.820 --> 55:53.080
If I take that solution and
say, "What does it do at

55:53.077 --> 55:54.297
t = 0?"

55:54.300 --> 55:56.610
I find it does the following.

55:56.610 --> 56:04.060
At t = 0,
it looks like this.

56:04.059 --> 56:09.589
So only for initial functions
of this form I have the future.

56:09.590 --> 56:14.330
But the only is not a big only,
because every function you can

56:14.329 --> 56:19.069
give me at t = 0 can
always be written in this form.

56:19.070 --> 56:22.200
It's a mathematical result that
says that just like sines and

56:22.202 --> 56:25.492
cosines and certain exponentials
are a complete set of functions

56:25.490 --> 56:27.110
for expanding any function.

56:27.110 --> 56:30.290
The mathematical theory tells
you that the solutions of H

56:30.289 --> 56:33.489
Y = E Y,
if you assemble all of them,

56:33.487 --> 56:36.977
can be used to build up an
arbitrary initial function.

56:36.980 --> 56:42.500
That means any initial function
you give me, I can write this

56:42.503 --> 56:47.663
way, and the future of that
initial state is this guy.

56:47.659 --> 56:48.329
Yes?

56:48.329 --> 56:52.979
Student:
>

56:52.980 --> 56:53.650
Prof: Yes.

56:53.650 --> 56:56.580
Lots of mathematical
restrictions,

56:56.579 --> 56:57.999
single valued.

56:58.000 --> 57:01.800
Physicists usually don't worry
about those restrictions till of

57:01.800 --> 57:03.580
course they get in trouble.

57:03.579 --> 57:07.229
Then we go crawling back to the
math guys to help us out.

57:07.230 --> 57:10.510
So just about anything you can
write down, by the way physics

57:10.512 --> 57:13.632
works, things tend to be
continuous and differentiable.

57:13.630 --> 57:15.640
That's the way natural things
are.

57:15.639 --> 57:19.279
So for any function we can
think of it is true.

57:19.280 --> 57:21.770
You go the mathematicians,
they will give you a function

57:21.766 --> 57:24.026
that is nowhere continuous,
nowhere differentiable,

57:24.025 --> 57:25.785
nowhere defined,
nowhere something.

57:25.789 --> 57:28.579
That's what makes them really
happy.

57:28.579 --> 57:31.929
But they are all functions the
way they've defined it,

57:31.929 --> 57:33.489
but they don't happen in real
life,

57:33.489 --> 57:36.139
because whatever happens here
influences what happens on

57:36.144 --> 57:38.734
either side of it,
so things don't change in a

57:38.733 --> 57:39.843
discontinuous way.

57:39.840 --> 57:41.300
Unless you apply an infinite
force,

57:41.300 --> 57:43.060
an infinite potential,
infinite something,

57:43.059 --> 57:45.609
everything has got what's
called C infinity,

57:45.610 --> 57:47.990
can differentiate any number of
times.

57:47.989 --> 57:49.449
So we don't worry about the
restriction.

57:49.449 --> 57:54.089
So in the world of physicists'
functions, you can write any

57:54.094 --> 57:57.864
initial function in terms of
these functions.

57:57.860 --> 58:01.240
So let me tell you then the
process for solving the

58:04.559 --> 58:05.529
Are you with me?

58:05.530 --> 58:08.270
You come and give me
Y(x, 0),

58:08.269 --> 58:10.879
and you say,
"As a function of time,

58:10.880 --> 58:13.360
where is it going to end
up?"

58:13.360 --> 58:14.220
That's your question.

58:14.219 --> 58:15.609
That's all you can ask.

58:15.610 --> 58:17.500
Initial state, final state.

58:17.500 --> 58:23.120
This is given, this is needed.

58:23.119 --> 58:26.079
So I'll give you a 3 step
solution.

58:26.079 --> 58:36.169
Step 1, find A_E=
Y_E

58:36.168 --> 58:45.068
*(x)
Y(x,0).

58:45.070 --> 58:54.160
Step 2, Y(x) and
t = this A_E

58:54.164 --> 59:00.534
that you got times
e^(-ipx/

59:00.530 --> 59:08.260
ℏ)Y
_E(x).

59:08.260 --> 59:11.650
So what I'm telling you is,
the fate of a function Y

59:11.648 --> 59:15.268
with the wiggles and jiggles is
very complicated to explain.

59:15.268 --> 59:17.698
Some wiggle goes into some
other wiggle that goes into some

59:17.704 --> 59:19.304
other wiggle as a function of
time,

59:19.300 --> 59:22.760
but there is a basic simplicity
underlying that evolution.

59:22.760 --> 59:24.630
The simplicity is the following.

59:24.630 --> 59:29.270
If at t = 0 you expand
your Y as such a sum,

59:29.268 --> 59:31.798
where the coefficients are
given by the standard rule,

59:31.800 --> 59:35.720
then as time goes away from
t = 0,

59:35.719 --> 59:39.549
all you need to do is to
multiply each coefficient by the

59:39.545 --> 59:43.025
particular term involving that
particular energy.

59:43.030 --> 59:49.200
And that gives you the Y
at later times.

59:49.199 --> 59:54.329
A state of definite energy in
this jargon will be the one in

59:54.327 --> 59:58.497
which every term is absent
except 1, maybe E =

59:58.498 --> 1:00:00.408
E_1.

1:00:00.409 --> 1:00:02.869
That is the kind we study.

1:00:02.869 --> 1:00:06.279
That state has got only 1 term
in the sum and its time

1:00:06.284 --> 1:00:10.024
evolution is simply given by
this and all probabilities are

1:00:10.021 --> 1:00:10.861
constant.

1:00:10.860 --> 1:00:14.080
But if you mix them up with
different coefficients,

1:00:14.083 --> 1:00:16.923
you can then handle any initial
condition.

1:00:16.920 --> 1:00:21.790
So we have now solved really
for the future of any quantum

1:00:21.786 --> 1:00:23.576
mechanical problem.

1:00:23.579 --> 1:00:28.969
So I'm going to give you from
now to the end of class concrete

1:00:28.969 --> 1:00:30.649
examples of this.

1:00:30.650 --> 1:00:35.140
But I don't mind again
answering your questions,

1:00:35.139 --> 1:00:40.869
because it's very hard for me
to put myself in your place.

1:00:40.869 --> 1:00:44.619
So I'm trying to remember when
I did not know quantum

1:00:44.621 --> 1:00:48.231
mechanics, sitting in some
sandbox and some kid was

1:00:48.228 --> 1:00:50.318
throwing sand in my face.

1:00:50.320 --> 1:00:51.300
So I don't know.

1:00:51.300 --> 1:00:55.160
I've lost my innocence and I
don't know how it looks to you.

1:00:55.159 --> 1:00:56.109
Yes.

1:00:56.110 --> 1:00:58.080
Student:  For each of
these problems,

1:00:58.079 --> 1:01:00.869
you have to solve that equation
you gave us before to find the

1:01:00.873 --> 1:01:02.343
form of the
Y_E,

1:01:02.340 --> 1:01:02.890
right?

1:01:02.889 --> 1:01:05.429
Prof: Right.

1:01:05.429 --> 1:01:10.219
So let's do the following
problem.

1:01:10.219 --> 1:01:16.909
Let us take a world in which
everything is inside the box of

1:01:16.909 --> 1:01:18.949
length L.

1:01:18.949 --> 1:01:24.129
And someone has manufactured
for you a certain state.

1:01:24.130 --> 1:01:26.530
Let me come to that case in a
minute.

1:01:26.530 --> 1:01:29.860
Let me take a simple case then
I'll build up the situation you

1:01:29.858 --> 1:01:30.238
want.

1:01:30.239 --> 1:01:35.739
Let's first take a simple case
where at t = 0

1:01:35.735 --> 1:01:40.045
Y(x,0) = the
square root of

1:01:40.047 --> 1:01:45.217
2/Lsine(n
px/L).

1:01:45.219 --> 1:01:51.249
That is just a function with n
oscillations.

1:01:51.250 --> 1:01:54.060
You agree, that's a state of
definite energy.

1:01:54.059 --> 1:01:59.069
The energy of that state,
E_n,

1:01:59.068 --> 1:02:03.638
is ℏ^(2)
p^(2)n^(2)/

1:02:03.641 --> 1:02:05.711
2mL^(2).

1:02:05.710 --> 1:02:08.240
We did that last time.

1:02:08.239 --> 1:02:09.699
And the reason,
why were we so interested in

1:02:09.699 --> 1:02:10.309
these functions?

1:02:10.309 --> 1:02:12.429
Now I can tell you why.

1:02:12.429 --> 1:02:15.459
If this is my initial state,
let me take a particular

1:02:15.460 --> 1:02:17.760
n,
then the state at any future

1:02:17.760 --> 1:02:21.480
time, Y(x,t),
is very simple here,

1:02:21.476 --> 1:02:24.966
the square root of
2/Lsine(n

1:02:24.972 --> 1:02:29.502
px/L), times
e^(-ipx/

1:02:29.496 --> 1:02:33.986
ℏ),
where energy is

1:02:33.994 --> 1:02:39.064
n^(2)p
^(2)

1:02:39.063 --> 1:02:45.773
ℏ^(2)
/2mL^(2).

1:02:45.769 --> 1:02:46.559
That's it.

1:02:46.559 --> 1:02:50.739
That is the function of time.

1:02:50.739 --> 1:02:53.639
All I've done to that initial
state is multiply by

1:02:53.637 --> 1:02:56.417
e^(-iEt),
but E is not some random

1:02:56.418 --> 1:02:57.068
number.

1:02:57.070 --> 1:03:00.540
E is labeled by n
and E_nis whatever you

1:03:00.539 --> 1:03:01.199
have here.

1:03:01.199 --> 1:03:05.009
That's the time dependence of
that state.

1:03:05.010 --> 1:03:09.900
It's very clear that if you
took the absolute value of this

1:03:09.896 --> 1:03:14.696
Y, this guy has absolute
value = 1 at all times.

1:03:14.699 --> 1:03:16.929
It's like saying cos t
depends on time,

1:03:16.929 --> 1:03:20.789
sine t depends on time,
but cos^(2) sine^(2),

1:03:20.789 --> 1:03:23.029
cos^(2 )t sine^(2)
t seems to depend on

1:03:23.025 --> 1:03:25.275
time,
but it doesn't.

1:03:25.280 --> 1:03:28.230
So this seems to depend on time
and it does, but when you take

1:03:28.228 --> 1:03:29.918
the absolute value,
it goes away.

1:03:29.920 --> 1:03:31.490
That's the simplest problem.

1:03:31.489 --> 1:03:34.879
I gave you an initial state,
the future is very simple,

1:03:34.875 --> 1:03:36.125
attach the factor.

1:03:36.130 --> 1:03:39.250
Now let's give you a slightly
more complicated state.

1:03:39.250 --> 1:03:44.810
The more complicated state will
be--I'm going to hide that for

1:03:44.811 --> 1:03:45.361
now.

1:03:45.360 --> 1:03:53.150
Let us take a
Y(x,0) that looks

1:03:53.154 --> 1:04:01.304
like 3 times square root of
2/Lsine(2

1:04:01.302 --> 1:04:08.922
px/L) 4 times
sine...

1:04:08.920 --> 1:04:14.890
This is my initial state.

1:04:14.889 --> 1:04:16.299
What does it look like?

1:04:16.300 --> 1:04:19.420
It's a sum of 2 energy states.

1:04:19.420 --> 1:04:22.690
This guy is what I would call
Y_2 in my

1:04:22.690 --> 1:04:24.890
notation, the second highest
state.

1:04:24.889 --> 1:04:27.669
This guy is
Y_3.

1:04:27.670 --> 1:04:34.410
Everybody is properly
normalized, and these are the

1:04:34.411 --> 1:04:36.031
As.

1:04:36.030 --> 1:04:40.100
So this state,
if you measure its energy,

1:04:40.103 --> 1:04:42.143
what will you get?

1:04:42.139 --> 1:04:48.029
Anybody tell me what answers I
can get if I measure energy now?

1:04:48.030 --> 1:04:52.260
You want to guess what are the
possible energies I could get?

1:04:52.260 --> 1:04:54.200
Yes.

1:04:54.199 --> 1:04:57.399
Any of you, either of you.

1:04:57.400 --> 1:04:59.650
Can you tell?

1:04:59.650 --> 1:05:05.060
No?

1:05:05.059 --> 1:05:05.799
Yes?

1:05:05.800 --> 1:05:10.610
Student:  You can get
h^(2)p^(2) times

1:05:10.614 --> 1:05:13.194
4/2mL^(2),
or times 9.

1:05:13.190 --> 1:05:16.700
Prof: So her answer was,
you can get in my convention

1:05:16.697 --> 1:05:20.257
E_2or
E_3,

1:05:20.260 --> 1:05:22.790
just put n = to 2 0r 3
That's all you have,

1:05:22.789 --> 1:05:26.049
your function written as a sum
over Y_E is

1:05:26.047 --> 1:05:26.887
only 2 terms.

1:05:26.889 --> 1:05:29.449
That means they are the only 2
energies you can get.

1:05:29.449 --> 1:05:31.589
So it's not a state of definite
energy.

1:05:31.590 --> 1:05:34.190
You can get either this answer
or this answer.

1:05:34.190 --> 1:05:36.200
But now you can sort of see,
it's more likely to get this

1:05:36.197 --> 1:05:38.017
guy,
because it has a 4 in front of

1:05:38.019 --> 1:05:40.009
it, and less likely to get this
guy,

1:05:40.010 --> 1:05:44.040
and impossible to get anything
else.

1:05:44.039 --> 1:05:52.309
So the probability for getting
n = 2 is proportional to

1:05:52.309 --> 1:05:56.949
3^(2),
and the probability for getting

1:05:56.952 --> 1:06:01.262
n = 3 is proportional to
4^(2).

1:06:01.260 --> 1:06:04.000
If you want the absolute
probabilities,

1:06:04.000 --> 1:06:07.910
then you can write it as 3^(2)
divided by 3^(2) 4^(2),

1:06:07.909 --> 1:06:15.259
which is 4^(2),
or you can write it as 3^(2)

1:06:15.255 --> 1:06:18.155
4^(2) is 5^(2).

1:06:18.159 --> 1:06:21.349
See, if you square these
probabilities,

1:06:21.349 --> 1:06:23.449
you get 25,3^(2) 4^(2).

1:06:23.449 --> 1:06:26.609
If you want to get 1,
I think you can see without too

1:06:26.614 --> 1:06:29.774
much trouble,
if you rescale the whole thing

1:06:29.766 --> 1:06:33.146
by 1 fifth,
now you'll find the total

1:06:33.146 --> 1:06:35.776
probabilities add up to 1.

1:06:35.780 --> 1:06:38.840
That's the way to normalize the
function, that's the easy way.

1:06:38.840 --> 1:06:41.650
The hard way is to square all
of this and integrate it and

1:06:41.646 --> 1:06:44.056
then set it = to 1 and see what
you have to do.

1:06:44.059 --> 1:06:46.279
In the end, all you will have
to do is divide by 5.

1:06:46.280 --> 1:06:48.330
I'm just giving you a shortcut.

1:06:48.329 --> 1:06:51.419
When you expand the Y in
terms of normalized functions,

1:06:51.422 --> 1:06:53.962
then the coefficient squared
should add up to 1.

1:06:53.960 --> 1:06:56.060
If they don't,
you just divide them by

1:06:56.063 --> 1:06:57.203
whatever it takes.

1:06:57.199 --> 1:07:01.339
So this has got a chance 3 is
to 5 of being this or that

1:07:01.342 --> 1:07:02.022
energy.

1:07:02.018 --> 1:07:05.438
But as a function of time,
you will find here things vary

1:07:05.436 --> 1:07:06.166
with time.

1:07:06.170 --> 1:07:07.610
This is not going to be time
independent.

1:07:07.610 --> 1:07:09.550
I want to show you that.

1:07:09.550 --> 1:07:16.320
So Y(x,t) now is
going to be (3/5√(2/L

1:07:16.322 --> 1:07:21.292
))sine(2
px/L) times

1:07:21.289 --> 1:07:24.449
e^(−i(E2)).

1:07:24.449 --> 1:07:26.379
I don't want to write the full
formula for E_n

1:07:26.382 --> 1:07:27.042
every time.

1:07:27.039 --> 1:07:30.309
I'm just going to call it
E_2,

1:07:30.309 --> 1:07:42.679
(4/5√(2/L))sine
(3px/L) times

1:07:42.677 --> 1:07:50.097
e^(−i(E3)t/ℏ.

1:07:50.099 --> 1:07:55.659
)Now you notice that if I
found the probability to be at

1:07:55.661 --> 1:07:59.341
sum x,
p(x,t), I have to take the

1:07:59.340 --> 1:08:02.640
absolute square of all of this.

1:08:02.639 --> 1:08:06.309
And all I want you to notice is
that the absolute square of all

1:08:06.309 --> 1:08:09.269
of this, you cannot drop these
exponentials now.

1:08:09.268 --> 1:08:11.488
If you've got two of them,
you cannot drop them,

1:08:11.489 --> 1:08:13.569
because when you take
Y_1

1:08:13.570 --> 1:08:15.650
Y_2 absolute
squared,

1:08:15.650 --> 1:08:17.540
Y_1
Y_2,

1:08:17.538 --> 1:08:20.088
you multiply it by
Y_1 conjugate

1:08:20.086 --> 1:08:23.476
Y_2 conjugate,
let's do that.

1:08:23.479 --> 1:08:29.059
So you want to multiply the
whole thing by its conjugate.

1:08:29.060 --> 1:08:33.750
So first you take the absolute
square of this.

1:08:33.750 --> 1:08:39.430
You will get
(9/25)(2/L)sin^(2)(2

1:08:39.426 --> 1:08:42.626
px/L).

1:08:42.630 --> 1:08:45.380
And the absolute value of this
is just 1.

1:08:45.380 --> 1:08:46.130
You see that?

1:08:46.130 --> 1:08:49.290
That is Y_1*
Y_1.

1:08:49.288 --> 1:08:50.828
Then you must take
Y_2*

1:08:50.831 --> 1:08:51.741
Y_2.

1:08:51.739 --> 1:09:00.069
That will be 16/25√(2/L)--
I'm sorry, no square

1:09:00.073 --> 1:09:04.803
root--2/Lsin^(2)(3px/L)
times 1,

1:09:04.800 --> 1:09:08.460
because the absolute value of
this guy with itself is 1.

1:09:08.460 --> 1:09:11.090
But that's not the end.

1:09:11.090 --> 1:09:15.030
You've got 2 more terms which
look like Y_1

1:09:15.033 --> 1:09:18.553
^(*)Y_2
Y_2*

1:09:18.546 --> 1:09:20.336
Y_1.

1:09:20.340 --> 1:09:23.310
I'm not going to work out all
the details, but let me just

1:09:23.307 --> 1:09:25.337
show you that time dependence
exists.

1:09:25.340 --> 1:09:26.750
So if you take
Y_1^(*)

1:09:26.751 --> 1:09:32.651
Y_2,
you will get (3/5√(2/L

1:09:32.645 --> 1:09:40.905
))sine(2
px/L) times e^(

1:09:40.912 --> 1:09:47.562
iE2t/ℏ)^( )times--
sorry.

1:09:47.560 --> 1:09:53.770
3/5 times
(4/5√(2/L))sine

1:09:53.768 --> 1:10:02.658
(3px/L) times
e^((i(E2)−i(E3)t)

1:10:02.662 --> 1:10:08.202
/ℏ) 1 more term.

1:10:08.198 --> 1:10:10.378
I don't care about any of these
things.

1:10:10.380 --> 1:10:14.280
I'm asking you to see,
do things depend on time or

1:10:14.282 --> 1:10:14.762
not?

1:10:14.760 --> 1:10:17.390
This has no time dependence
because the absolute value of

1:10:17.387 --> 1:10:18.137
that vanished.

1:10:18.140 --> 1:10:20.320
This has no time dependence,
the absolute value of this

1:10:20.315 --> 1:10:20.755
vanished.

1:10:20.760 --> 1:10:23.510
But the cross terms,
when you multiply the conjugate

1:10:23.506 --> 1:10:25.816
of this by this,
or the conjugate of this by

1:10:25.822 --> 1:10:27.332
that, they don't cancel.

1:10:27.329 --> 1:10:29.069
That's all I want you to know.

1:10:29.069 --> 1:10:31.729
So I'll get a term like this
this complex conjugate.

1:10:31.729 --> 1:10:33.579
I don't want to write that in
detail.

1:10:33.578 --> 1:10:36.328
If you combine this function
with the conjugate,

1:10:36.329 --> 1:10:39.549
you'll find this the conjugate
will give me a cosine.

1:10:39.550 --> 1:10:54.340
So I should probably write it
on another part of the board.

1:10:54.340 --> 1:10:57.300
So maybe I don't even want to
write it because it's in the

1:10:57.304 --> 1:10:57.724
notes.

1:10:57.720 --> 1:11:00.830
I want you to notice the
following - look at the first

1:11:00.828 --> 1:11:02.998
term, no time dependence,
second term,

1:11:03.000 --> 1:11:04.350
no time dependence.

1:11:04.350 --> 1:11:08.240
The cross term has an e to the
something and I claim the other

1:11:08.242 --> 1:11:11.052
cross term will have e to the -
something.

1:11:11.050 --> 1:11:13.530
e to the i something or
e to the -i something is

1:11:13.532 --> 1:11:14.592
the cosine of something.

1:11:14.590 --> 1:11:16.140
That's all I want you to know.

1:11:16.140 --> 1:11:21.070
So there is somewhere in the
time dependence P(x,t)

1:11:21.070 --> 1:11:25.570
that's got a lot of stuff which
is t independent,

1:11:25.569 --> 1:11:28.239
something that looks like a
whole bunch of numbers,

1:11:28.238 --> 1:11:34.798
times cosine(E_2 -
E_3)t /ℏ.

1:11:34.800 --> 1:11:38.050
That's all I want you to notice.

1:11:38.050 --> 1:11:41.130
That means the probability
density will be oscillating.

1:11:41.130 --> 1:11:45.280
The particle will not be fixed;
it will be bouncing back and

1:11:45.277 --> 1:11:46.977
forth between the walls.

1:11:46.979 --> 1:11:50.629
And the rate at which it
bounces is given by the

1:11:50.628 --> 1:11:55.288
difference in energy between the
two states you formed in the

1:11:55.287 --> 1:11:56.527
combination.

1:11:56.529 --> 1:11:59.229
So this is how a particle in
general--

1:11:59.229 --> 1:12:01.129
if you want,
not the most general one,

1:12:01.130 --> 1:12:04.810
but it's a reasonably general
case where you added some

1:12:04.805 --> 1:12:06.665
mixture of that--
let me see.

1:12:06.670 --> 1:12:08.260
You added 2 to 3.

1:12:08.260 --> 1:12:13.500
You added that state that state.

1:12:13.500 --> 1:12:19.030
You added 4 times fifth of that
times 3 fifths of that as the

1:12:19.027 --> 1:12:20.867
initial condition.

1:12:20.869 --> 1:12:21.299
Okay?

1:12:21.300 --> 1:12:24.240
4 fifths of this one guy with 2
wiggles and 3 fifths with 3

1:12:24.235 --> 1:12:26.105
wiggles and you let it go in
time,

1:12:26.109 --> 1:12:29.419
you will find then if you add
these time dependences,

1:12:29.420 --> 1:12:31.150
there'll be a part that varies
with time.

1:12:31.149 --> 1:12:33.259
So the density will not be
constant now.

1:12:33.260 --> 1:12:37.540
It will be sloshing back and
forth in the box.

1:12:37.539 --> 1:12:39.329
That's a more typical situation.

1:12:39.328 --> 1:12:41.308
So not every state,
initial state,

1:12:41.305 --> 1:12:44.055
is a function of a state of
definite energy.

1:12:44.060 --> 1:12:45.070
It's an admixture.

1:12:45.069 --> 1:12:48.199
I've taken the simplest case
where the admixture has only 2

1:12:48.203 --> 1:12:48.963
parts in it.

1:12:48.960 --> 1:12:51.410
You can imagine taking a
function made of 3 parts and 4

1:12:51.407 --> 1:12:53.937
parts and 10 parts and when you
square them and all that,

1:12:53.944 --> 1:12:55.354
you'll get 100 cross terms.

1:12:55.350 --> 1:12:58.760
They'll all be oscillating at
different rates.

1:12:58.760 --> 1:13:04.400
But the frequencies will always
be given by the difference in

1:13:04.399 --> 1:13:09.569
frequencies, differences in
energies that went into your

1:13:09.570 --> 1:13:10.700
mixture.

1:13:10.698 --> 1:13:12.548
The last problem which I'm not
going to do now,

1:13:12.550 --> 1:13:14.680
I'll do next time,
but I'll tell you what it is I

1:13:14.679 --> 1:13:16.319
want to do,
which is really this

1:13:16.319 --> 1:13:19.699
mathematically more involved,
but the idea is the same.

1:13:19.698 --> 1:13:22.378
Here I gave you the initial
state on a plate.

1:13:22.380 --> 1:13:24.630
I just said,
here it is, 3 fifths times 1

1:13:24.631 --> 1:13:27.951
function of definite energy,
4 fifths times another function

1:13:27.953 --> 1:13:29.253
of definite energy.

1:13:29.250 --> 1:13:32.170
The problem you really want to
be able to solve right now is

1:13:32.166 --> 1:13:34.776
when somebody gives you an
arbitrary initial state and

1:13:34.784 --> 1:13:37.064
says,
"What happens to it?"

1:13:37.060 --> 1:13:42.320
So I'm going to consider next
time a function that looks

1:13:42.322 --> 1:13:47.492
something like this - x
times 1 - x is my

1:13:47.488 --> 1:13:49.688
function at time 0.

1:13:49.689 --> 1:13:52.619
That's a nice function,
because it vanishes at 0,

1:13:52.619 --> 1:13:53.779
it vanishes at 1.

1:13:53.779 --> 1:13:56.599
It's a box of length L =
1.

1:13:56.600 --> 1:14:00.800
It's an allowed initial state.

1:14:00.800 --> 1:14:02.800
Then I can ask,
what does this do.

1:14:02.800 --> 1:14:06.180
So you should think about how
do I predict its future,

1:14:06.180 --> 1:14:07.330
what's involved?

1:14:07.328 --> 1:14:09.668
So you will see that in the
notes.

1:14:09.670 --> 1:14:13.140
So what I'm going to do next
time is to finish this problem

1:14:13.144 --> 1:14:15.784
and give you the final set of
postulates,

1:14:15.779 --> 1:14:18.799
so that you can see what the
rules of quantum mechanics are.

1:14:18.800 --> 1:14:24.000