WEBVTT

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Prof: So the rules of
quantum mechanics are going to

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be stated for one last time.

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Now we're using the fact that
you've heard them over and over,

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it's going to be more compact
than before.

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So it's really the form of many
postulates, because it's a

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theory which is based on
experiment;

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it's not deduced by mathematics.

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You make up some postulates
that condense all the

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experimental facts.

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There are new mathematical
results you will need,

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but they are separate.

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They are deducible
mathematically.

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They are not part of the
physics postulates.

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The first postulate--by the
way, this is the postulate for

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Physics 201.

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If you know more mathematics
and so on, there's another way

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to write the postulate,
but we're not interested in

00:44.584 --> 00:44.924
that.

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We just want the rules for this
class,

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and I don't even care how much
of this you carry in your head,

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but I want you to know that in
principle,

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every problem I gave you,
every question I asked,

00:56.410 --> 01:00.450
can be answered just by turning
to these postulates,

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so you should know what they
are.

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So the first postulate is that
every particle--

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this is for 1 dimension,
1 particle only--

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is given by a certain wave
function,

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Y(x),
that contains all the possible

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information on that particle.

01:16.819 --> 01:20.289
This is the analog of x
and p in classical

01:20.292 --> 01:21.052
mechanics.

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And we always choose Y
so that Y^(2)dx is

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normalized to 1 in all of space.

01:30.220 --> 01:32.980
That is a convention;
it's not required.

01:32.980 --> 01:35.960
If you take a Y and you
multiply it by a number,

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it stands for the same physical
state.

01:38.170 --> 01:42.200
So you pick the number so that
the squared integral of Y

01:42.196 --> 01:42.956
is 1.

01:42.959 --> 01:52.189
The second thing is,
a state of definite momentum.

01:52.190 --> 01:54.540
In other words,
is there a wave function that

01:54.535 --> 01:57.885
describes a particle guaranteed
to have a definite momentum when

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I measure it?

01:58.800 --> 02:00.740
The answer is yes.

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That function is denoted by
Y _p(x).

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It is not an arbitrary function.

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This Y is whatever you
like.

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This is a particular Y
which assures you that if you

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measure my momentum when I'm in
this state,

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you will get the value
p, and it looks like this -

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e^(ipx/ℏ)
divided by square root of

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the length of that universe in
which you're living.

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That's the second way to define
this function.

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Either I can say this is the
answer, or I can give you a

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little way to find this function
in an equivalent way.

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The equivalent way says states
of definite momentum obey the

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following equation.

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If you solve this equation,
the function you get is in fact

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Y_p.

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So Y_p
can be defined either way,

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either by saying,
solve this equation and then

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the function you get is
Y_p,

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but you can easily verify that
this function in fact is a

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solution to that equation.

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Now there's a new piece of
mathematics that's not connected

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with this.

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This is a postulate.

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Mathematics says,
if you live on a ring,

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you go around in a circle,
and the x goes to x

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L,
you've got to come back to the

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same function.

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That quantizes p, I have
the value as (2ph/L)

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times m,
where m is an integral.

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Let me call it n.

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m is the particle mass.

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That is a mathematical
deduction.

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You don't have to make that a
postulate.

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That's the postulate;
that's a deduction.

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Second postulate says--or third
postulate--a state of definite

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position, x_0.

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Namely, if you want to describe
a particle which is guaranteed

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to be at x_0,
what function describes that.

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I'm going to give the function
the name Y_

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X0.

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You have to be very clear on
what this x_0

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is and what x is,
okay?

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This p is a label that
says something about the wave

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function.

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This is a function of x,
so x varies from - to

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infinity.

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This is a label for the state.

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What's special about this state?

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It's got a momentum p.
What's special about this state?

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It's got a definite location
x_0.

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What is the function?

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I think you should know by now
what the function looks like.

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Anybody know?

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State of definite
x_0?

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Any guess?

04:44.470 --> 04:45.230
Yes?

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Student:
>

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Prof: Yes,
so I'm going to make it a

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little simpler.

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It's the function,
I'm just going to call it big

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spike at x_0.

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Now big spike at
x_0 is a little

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vague and deliberately left so,
because this spike has a little

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bit of non 0 value away from
x_0.

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And you can make it narrower
and narrower and taller and

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taller and that's what has a
technical name,

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but I don't want to go there.

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Let's imagine we can only
measure position to one part in

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10 to the 97 meters.

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Yes?

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Student:  Is it the
Dirac delta function?

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Prof: This one is not,
but eventually it will become

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that.

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But I just want to keep it like
this.

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But instead of saying it's a
spike, I can also say it obeys

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the following equation.

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x Y_x0
(x) =

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x_0 times same
function(ie

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Y_x0
(x)).

05:39.490 --> 05:42.490
So let me take a minute to
discuss both these equations.

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They are very special
mathematical forms,

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and this is just for your
own--you don't have to know this

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part of it.

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This will do for this course.

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If you want to know the whole
story--

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see, normally,
when you take a function

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Y(x),
and you differentiate it,

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you will get a new function,
right?

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Take a sine,
you'll get a cosine.

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If you take x^(2),
you will get 2x.

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So the act of taking
derivatives usually gives you a

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brand new function,
a different function.

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This one says,
except for those constants here

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and here, if you take the
derivative, it should look like

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a multiple of the same function.

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Obviously, that's a very
special function.

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You're saying upon
differentiating,

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it's going to look the same.

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Well, not every function has a
property, but we all know one

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function does,
the exponential function.

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Similarly, if you took a
function, any function Y

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(x) and multiplied it by
x, you will get a new

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function, right?

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Cos x will become x
times cos x.

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Sine x will become x
times sine x.

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e^(x )will become x
times e^(x).

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It becomes a completely new
function.

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But I am demanding that for
this magical function,

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when I multiply by x,
it is essentially a constant

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times the same function.

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You've got to ask yourself,
how can I take a function,

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f(x), multiply it by x
everywhere and it looks the

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same except for multiplicative
factor.

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It looks impossible.

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But the spike is in fact such a
function.

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Let us see why.

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If we take a random function,
if you multiply it by x,

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which looks like this,
it will get taller and taller

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as you go to the right.

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But suppose the function in
question is a spike at

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x_0.

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If you multiply it by x,
you're basically multiplying

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only by x_0,
because it has a life only of

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x_0.

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It doesn't exist anywhere else.

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The only place it's even
non-zero is at the point

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x_0.

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So there's no difference
between multiplying it by

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x, and multiplying it by
x_0.

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Multiplying by x all
over here is a waste,

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because the function is 0
there.

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That's why for that spike
function, this is actually true.

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You see that?

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It's the second way that
characterizes it.

07:57.410 --> 08:00.710
So again it says multiplying
by x is generally an

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operation that changes the
nature of the function,

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but there are special functions
for whom the effect is to simply

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rescale the function by a
number.

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Find me that function.

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That's the state of definite
x_0,

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and the only answer to that is
a spike.

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So we have the option of simply
accepting the left hand side as

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postulate 2 and postulate 3,
but there are equivalent ways

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to say this.

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And I will tell you in a minute
why I'm saying this,

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because you can ask the
following question - what is the

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state of definite energy?

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The state of definite energy,
I'm going to write it this way

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in the right hand side,
is also a solution to some

08:44.532 --> 08:45.842
equation.

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That equation is −(
ℏ^(2)

08:50.899 --> 08:55.959
/2m)(d^(2)
Y_E/dx ^(2))

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V(x)Y_E
(x) =

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EY_E
(x).

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But I'm trying to tell you that
I'm not going to elevate this to

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a postulate,
because I think one of you

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noticed already something very
familiar about this equation.

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Because, you see,
in classical mechanics,

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energy is given by
(p^(2)/2m)

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V(x).

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Do you see that on the left
hand side,

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except for the function
Y, I have what here looks

09:38.758 --> 09:43.118
/dx)^(2)
VY.

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That's the left-hand side.

09:46.620 --> 09:50.430
So that once you know states of
definite p and states of

09:50.428 --> 09:53.008
definite x obey these
equations,

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states of definite anything
else, any function of x

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and p, obeys a
similar equation,

09:58.190 --> 10:00.690
but every p is replaced
by −iℏ

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d/dx,
and every x is just

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simply left as an x.

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So this energy formula does not
require a new axiom,

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but if you didn't follow this
logic,

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you can simply say I have one
more postulate,

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which is that if I have states
of definite energy,

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I must solve that equation.

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But then you will need more and
more postulates,

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because you've got zillions of
possible variables you're

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interested in.

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What if somebody wants to know,
what is x^(2)

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p^(2)?

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That's a variable in classical
mechanics.

10:31.490 --> 10:33.630
But I'm not going to write
another postulate for that,

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because if you wanted
x^(2)  p^(2),

10:35.950 --> 10:38.290
states of definite x^(2)
p^(2),

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it will obey the equation
((-iℏ)^(2)

10:42.743 --> 10:48.133
d^(2)/dx^(2)) x Y
= some constant,

10:48.129 --> 10:50.269
whatever it is, times Y.

10:50.269 --> 10:53.999
Namely, p is going to be
replaced by the square of this

10:53.995 --> 10:54.785
derivative.

10:54.788 --> 10:58.278
Square of a derivative means do
it twice.

10:58.279 --> 11:01.689
So there's a universal rule for
finding states of definite

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anything.

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Let g be a
general variable in

11:04.658 --> 11:05.838
classical mechanics.

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In quantum theory,
states of definite g

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satisfy the equation where you
first write the classical

11:12.008 --> 11:13.548
formula for g.

11:13.548 --> 11:15.848
Maybe g is x^(2)
p^(2).

11:15.850 --> 11:17.910
Replace x simply by
x;

11:17.908 --> 11:21.258
replace p by
-iℏd/dx

11:21.260 --> 11:25.090
and let that multiply as a
function or differentiate some

11:25.091 --> 11:29.591
function,
and that will be a state of

11:29.589 --> 11:32.079
definite g.

11:32.080 --> 11:36.020
So I'm just giving you two ways
to do this postulate.

11:36.019 --> 11:42.579
Either you can say states of
definite energy are solutions to

11:42.577 --> 11:46.487
the following equation,
that's HY= E

11:46.486 --> 11:48.466
Y,
and that's what I've been doing

11:48.466 --> 11:50.506
so far,
because I didn't mind in this

11:50.514 --> 11:53.054
class to introduce it as one
more postulate.

11:53.048 --> 11:55.578
If you're looking further
ahead, you should realize that

11:55.580 --> 11:57.650
it is really not an independent
postulate,

11:57.649 --> 12:00.339
that once you know what happens
to x and once you know

12:00.341 --> 12:02.941
happens to p, you can
predict what the equation will

12:02.942 --> 12:05.552
be for any function of x
and p. So if you see

12:05.546 --> 12:07.516
that analogy and it's helpful to
you,

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you're welcome to the right
hand side of the board.

12:09.778 --> 12:13.358
If you find it complicated,
add on to your list of things

12:13.364 --> 12:15.224
to remember one more thing.

12:15.220 --> 12:18.870
If you want a state of definite
energy, you've got to solve this

12:18.871 --> 12:19.511
equation.

12:19.509 --> 12:24.949
Okay, now we come to another
postulate, postulate 4.

12:24.950 --> 12:26.180
Is that right?

12:26.179 --> 12:30.189
Postulate 4.

12:30.190 --> 12:34.970
If Y(x) is
written as a Σ_g

12:34.966 --> 12:39.376
A_g
Y_g(x)

12:39.384 --> 12:42.814
where g could be
p,

12:42.808 --> 12:45.318
g could be E,
g could be position,

12:45.320 --> 12:49.890
whatever you like,
then the probability that you

12:49.894 --> 12:53.204
will get a value g is
simply

12:53.203 --> 12:59.193
|A_g|^(2),
and A_g is

12:59.186 --> 13:04.246
given by ∫Y_g
'(x)Y

13:04.249 --> 13:06.159
(x)dx.

13:06.158 --> 13:10.008
Perhaps you can see now,
if I put g = p,

13:10.008 --> 13:13.638
this is what I call
|A_p|^(2),

13:13.639 --> 13:17.179
and this will be e^(ipx
)etc.

13:17.178 --> 13:19.248
If I put g = energy,
this will be

13:19.250 --> 13:22.170
A_E,
this will be Y'E.

13:22.168 --> 13:25.058
Whatever variable you want,
you do these integrals and

13:25.059 --> 13:26.969
you get the
probabilities.

13:26.970 --> 13:31.750
Then there's another postulate,
postulate 5.

13:31.750 --> 13:44.650
Postulate 5 says if g is
measured and you get one value,

13:44.649 --> 13:48.949
say g_10--this
is a postulate,

13:48.950 --> 13:50.980
you don't have to write
g_10.

13:50.980 --> 13:53.540
You can say one particular
value--then the

13:53.543 --> 13:55.983
Σ_g
A_g

13:55.981 --> 13:59.861
Y_g
collapses to one term,

13:59.860 --> 14:02.760
just
Y_g0.

14:02.759 --> 14:04.949
g_0 is a
particular value of g,

14:04.951 --> 14:07.061
namely the third value,
the fourth value or the ninth

14:07.062 --> 14:07.472
value.

14:07.470 --> 14:10.900
This says the state collapses.

14:10.899 --> 14:12.679
So originally,
you have a state,

14:12.682 --> 14:15.572
a general one,
which is potentially capable of

14:15.567 --> 14:19.137
giving any possible answer
that's contained in this sum,

14:19.139 --> 14:21.059
with the probability
proportional to the square of

14:21.062 --> 14:21.772
the coefficient.

14:21.769 --> 14:24.019
But once you made a measurement
and you got a particular

14:24.022 --> 14:26.032
answer g, call it
g_0,

14:26.028 --> 14:28.658
all the terms of the sum
disappear except the one term,

14:28.658 --> 14:30.928
corresponding to the one answer
you got.

14:30.928 --> 14:32.948
That's a new property of
quantum mechanics,

14:32.950 --> 14:36.450
that the act of measurement
changes the state from being

14:36.450 --> 14:39.950
able to do many things to being
able to do one thing.

14:39.950 --> 14:42.590
The one thing could be
g, if you measure g.

14:42.587 --> 14:44.037
Namely,
it can be momentum,

14:44.038 --> 14:45.958
it can be energy,
it can be position,

14:45.956 --> 14:48.516
but you've got to be very
careful that if you measure

14:48.523 --> 14:50.993
momentum and got a state of
definite momentum,

14:50.990 --> 14:53.710
then you go and measure
position and got a state of

14:53.707 --> 14:55.897
definite position,
and you go back and measure

14:55.900 --> 14:58.720
momentum,
you won't get the same answer.

14:58.720 --> 15:01.150
States of definite momentum
don't know what the position is,

15:01.145 --> 15:03.855
and states of definite position
don't know what the momentum is.

15:03.860 --> 15:05.720
You will never be able to get
them both.

15:05.720 --> 15:08.330
You've got to pick one or the
other.

15:08.330 --> 15:13.950
By the way, I want to point out
to you one postulate seems to be

15:13.953 --> 15:17.613
somewhat missing,
the oldest thing we ever

15:17.614 --> 15:20.564
learned in this class,
right?

15:20.559 --> 15:21.499
Why didn't I mention that?

15:21.500 --> 15:26.070
That's actually contained here.

15:26.070 --> 15:27.790
It's contained here because if
I say what's the probability

15:27.788 --> 15:29.268
that x has a value
x_0,

15:29.269 --> 15:31.889
I will take integral
Y(x_0

15:31.894 --> 15:34.794
)'Y(x)
dx.

15:34.789 --> 15:36.259
Look at what this is.

15:36.259 --> 15:38.759
Y(x_0)'
is a spike at

15:38.761 --> 15:40.071
x_0.

15:40.070 --> 15:42.110
Y(x) is some
function.

15:42.110 --> 15:45.280
But the only place where the
products survives is near

15:45.283 --> 15:46.663
x_0.

15:46.658 --> 15:49.788
Everywhere else is 0,
so the only place that integral

15:49.792 --> 15:52.502
receives a contribution is
when x is at

15:52.503 --> 15:54.013
x_0.

15:54.009 --> 15:56.839
So up to some constant
proportional to area of this

15:56.837 --> 15:59.327
spike, the answer is
proportional to the wave

15:59.327 --> 16:01.417
function at
x_0.

16:01.418 --> 16:04.978
Therefore--I'm sorry,
I mean to say square this,

16:04.975 --> 16:09.585
and that is what I've been
saying is the rule for position.

16:09.590 --> 16:14.390
So again, for position you can
write a separate postulate,

16:14.386 --> 16:15.646
but it's not.

16:15.649 --> 16:20.619
The postulate that I gave you
here, if you put g =

16:20.615 --> 16:25.845
position, will give you the same
answer, namely this one.

16:25.850 --> 16:28.140
Again, if you don't want to get
into too many details,

16:28.140 --> 16:30.390
I would say the following is
what you should know.

16:30.389 --> 16:32.469
You want a stated definite
momentum,

16:32.470 --> 16:35.330
or the probability for definite
momentum,

16:35.330 --> 16:36.650
take the integral with
Y_p

16:36.650 --> 16:38.980
. For energy,
take Y_E.

16:38.980 --> 16:41.140
For definite position,
forget all integrals.

16:41.139 --> 16:44.299
Just take Y at the
point x and then square

16:44.302 --> 16:44.592
it.

16:44.590 --> 16:47.100
What I'm telling you is taking
Y at the point x

16:47.104 --> 16:49.934
and squaring it is what you
will do if you took the integral

16:49.932 --> 16:52.072
of Y with a spike,
which will filter out,

16:52.067 --> 16:53.497
if you like,
Y at only the one point

16:53.498 --> 16:54.178
x_0.

16:54.178 --> 16:56.618
And if you square it,
you will get Y

16:56.624 --> 16:58.494
(x_0).

16:58.490 --> 17:04.230
And the last postulate,
all-important postulate 6,

17:04.228 --> 17:10.438
says that the state changes
with time according to the

17:13.950 --> 17:16.050
I'm just going to call it
HY,

17:16.048 --> 17:18.978
but from now on,
it will be -d^(2)

17:18.980 --> 17:22.980
Y/dx^(2)
V(x)Y(x).

17:22.979 --> 17:30.319
If I get tired of writing it,
I will call it HY.

17:30.318 --> 17:33.218
That's of course a postulate,
because you can never derive

17:36.022 --> 17:36.942
Newton's laws.

17:36.940 --> 17:39.500
This is, as I said,
one of the most powerful

17:39.496 --> 17:42.926
equations in modern physics,
because it contains all of

17:42.926 --> 17:46.216
non-relativistic quantum
mechanics and all of classical

17:46.224 --> 17:48.524
mechanics,
and all of solid state physics

17:48.521 --> 17:50.351
and super fluids and
superconductors.

17:50.348 --> 17:54.978
Everything comes from this
equation.

17:54.980 --> 17:57.330
A very powerful equation.

17:57.328 --> 18:01.668
Okay, now what did we learn
from this equation?

18:01.670 --> 18:03.930
I will mention one thing that's
very important.

18:03.930 --> 18:05.380
It's a consequence of the
equation,

18:05.380 --> 18:10.270
which is, if you start the
system out at time t = 0,

18:10.269 --> 18:14.409
in a state of definite energy
and you see what happens if I

18:14.413 --> 18:18.603
wait some time,
well, it becomes the following

18:18.604 --> 18:20.544
function at time t.

18:20.538 --> 18:25.438
It's essentially the same
function, multiplied by

18:25.443 --> 18:28.513
[e^(i)]Et/ℏ.

18:28.509 --> 18:30.329
You realize,
generally a wave function,

18:30.332 --> 18:32.922
when you let it evolve,
will flop and wiggle and change

18:32.921 --> 18:34.601
its shape in a complicated way.

18:34.598 --> 18:38.268
But if you release it in this
initial condition,

18:38.266 --> 18:40.836
all that happens to it is that.

18:40.838 --> 18:44.548
These are called stationary
states, because despite the time

18:44.546 --> 18:47.876
dependence I showed you,
the probability for measuring

18:47.875 --> 18:49.945
anything is time independent.

18:49.950 --> 18:52.610
Because when you take the
probability, you find the

18:52.613 --> 18:55.913
absolute value of some integral,
and the absolute value of this

18:55.914 --> 18:57.304
factor just goes away.

18:57.299 --> 18:59.089
So it's time independent.

18:59.088 --> 19:00.688
That's why we're very
interested.

19:00.690 --> 19:04.490
If you want an atom and the
atom has been sitting for a long

19:04.490 --> 19:08.420
time, it will be in one of these
states of definite energy.

19:08.420 --> 19:12.690
So that's the case of starting
in a state of definite energy,

19:12.690 --> 19:14.420
but then I said,
"Maybe I don't want to

19:14.420 --> 19:15.950
start in a state of definite
energy;

19:15.950 --> 19:18.840
I want to start in an arbitrary
state."

19:18.838 --> 19:22.568
At time 0 I'm going to give you
a Y (x ,0) and I'm

19:22.574 --> 19:25.114
going to ask you,
"What is the fate of this

19:25.108 --> 19:26.428
state,
if I wait some time

19:26.430 --> 19:27.240
p?"

19:27.240 --> 19:30.580
The simple answer is,
take Y(x ,0),

19:30.578 --> 19:33.918
calculate
A_E=∫

19:33.923 --> 19:38.893
Y_E'
Y(x,0)dx and then

19:38.886 --> 19:42.986
Y(x,t) =
A_E

19:42.987 --> 19:46.977
Y_E
(x)

19:46.980 --> 19:51.080
e^(−iEt/ℏ).

19:51.078 --> 19:53.458
In other words,
if you take your initial state

19:53.460 --> 19:56.090
and expand it--
this is called expanding it--as

19:56.090 --> 19:59.330
a sum over states of definite
energy with some coefficients

19:59.332 --> 20:02.442
you compute at t = 0,
then at any future time,

20:02.444 --> 20:05.044
the coefficients are given
simply by the initial

20:05.040 --> 20:07.160
coefficients,
times this factor.

20:10.615 --> 20:13.945
equation, because every term in
it is a solution to the

20:13.946 --> 20:14.746
equation.

20:14.750 --> 20:17.550
And at t = 0,
when you drop this guy,

20:17.548 --> 20:21.388
it agrees with the initial
state and that's all you want.

20:21.390 --> 20:23.560
Because in this problem,
because it's first order

20:23.564 --> 20:25.764
equation in time,
if my initial state matches

20:28.680 --> 20:29.710
it is the answer.

20:29.710 --> 20:32.970
There are no two answers.

20:32.970 --> 20:36.660
So I want to give you
one--first let me tell you

20:36.655 --> 20:40.025
something that's going to make
you relax.

20:40.029 --> 20:43.879
Everything that I say after
this moment is not in your exam.

20:43.880 --> 20:46.000
I don't want you to worry about
that, because it's an

20:45.999 --> 20:48.529
interesting thing and I want to
tell you all the things you can

20:48.527 --> 20:49.747
do with quantum mechanics.

20:49.750 --> 20:51.350
I don't want you to worry about
anything.

20:51.348 --> 20:55.518
Just try to follow this and ask
questions, and get a glimpse of

20:55.522 --> 20:58.622
what you could possibly do with
this theory.

20:58.618 --> 21:02.568
First thing you can do is you
can say, "How do I

21:02.569 --> 21:04.469
understand atoms?"

21:04.470 --> 21:08.680
You know the simplest atom in
the world is a hydrogen atom.

21:08.680 --> 21:11.410
It's got a proton and it's got
an electron.

21:11.410 --> 21:13.970
One has charge e,
one has charge -e.

21:13.970 --> 21:17.940
And the proton can be taken to
be so heavy that it is fixed.

21:17.940 --> 21:19.870
Just like when the earth goes
round the sun,

21:19.868 --> 21:21.868
in principle,
the sun is also moving,

21:21.868 --> 21:23.898
but we ignore that,
so we're going to ignore the

21:23.896 --> 21:24.886
motion of the proton.

21:24.890 --> 21:28.950
Then the electron,
in classical mechanics,

21:28.950 --> 21:32.200
it has an energy which is
(p_x^(2)

21:32.195 --> 21:35.225
p_y^(2)
p_z^(2))/2m.

21:35.230 --> 21:39.890
That's the kinetic energy,
and the potential energy

21:39.892 --> 21:45.972
-ze^(2)/r, and r
is √(x^(2) y^(2),

21:45.970 --> 21:48.380
z^(2)). Do you understand
that?

21:48.380 --> 21:52.060
That's the classical formula
for energy.

21:52.058 --> 21:55.608
Then our recipe tells us that
in quantum theory,

21:55.608 --> 22:01.528
states of definite energy will
obey the following equation -

22:01.532 --> 22:06.252
-ℏ^(2)/2m x
(d^(2)

22:06.250 --> 22:10.070
Y/dx^(2)
d^(2)Y

22:10.066 --> 22:13.476
/dy^(2)
d^(2)Y

22:13.480 --> 22:18.700
/dz^(2) ) -
(ze^(2)/√(x^(2) y^(2)

22:18.701 --> 22:23.521
z^(2)))Y=
EY.

22:23.519 --> 22:28.479
You have to solve that equation.

22:28.480 --> 22:31.930
Now don't worry about how you
solve it.

22:35.930 --> 22:40.080
He had to ask a mathematician
friend of his how to solve it.

22:40.078 --> 22:42.878
A mathematician owed him a
favor, because at that time,

22:46.454 --> 22:49.624
wife and you might say,
"Who owed the favor to

22:49.619 --> 22:50.349
whom?"

22:55.311 --> 22:57.121
seduce a pair of underage twins.

22:57.118 --> 23:00.578
These are all very interesting
things you don't know about the

23:00.583 --> 23:04.203
lives of famous people,
but if you read his biography,

23:07.759 --> 23:11.249
was all over the classically
forbidden region.

23:11.250 --> 23:15.140
Anyway, he's a very interesting
person.

23:15.140 --> 23:18.430
But what I want you to know is
that even he couldn't solve the

23:18.432 --> 23:19.082
equations.

23:19.078 --> 23:22.208
I don't care if you solve it or
not, because he had already done

23:22.205 --> 23:23.095
the great thing.

23:23.098 --> 23:25.138
So you go to this equation and
you solve it.

23:25.140 --> 23:26.880
So this math guy helped him
solve it.

23:26.880 --> 23:29.740
Nowadays, undergraduates,
graduates, everybody knows how

23:29.736 --> 23:32.586
to solve it, but the first time
it came, it was quite an

23:32.593 --> 23:33.843
unfamiliar equation.

23:33.838 --> 23:37.998
If you solve this thing,
what you find is that the

23:38.001 --> 23:41.231
energy can take only certain
values.

23:41.230 --> 23:47.200
Those values have some number
in front of them a 13.6 electron

23:47.196 --> 23:47.976
volts.

23:47.980 --> 23:51.620
That 13.6 electron volts is
some combination of the electric

23:51.623 --> 23:55.023
charge of Planck's constant and
mass of the electron.

23:55.019 --> 23:55.739
Forget all that.

23:55.740 --> 23:58.450
Some number,
divided by n^(2) where

23:58.452 --> 24:01.102
n is an integer of 1,2,
3, etc.

24:01.099 --> 24:02.439
That's it.

24:02.440 --> 24:06.460
These are the only allowed
energy levels of hydrogen.

24:06.460 --> 24:09.640
If you solve this equation in 3
dimensions and you demand the

24:09.642 --> 24:12.682
functions vanish at infinity,
rather than blow up at

24:12.680 --> 24:16.340
infinity, you will find you can
get solutions only at certain

24:16.336 --> 24:19.686
special energies and these are
the energies you get.

24:19.690 --> 24:21.490
So let's plot these energies.

24:21.490 --> 24:26.290
Remember this is energy 0 and
this is n = 1.

24:26.289 --> 24:30.659
n = 1 is at -13.6eV.

24:30.660 --> 24:37.360
n = 2 will be -13.6/4eV.

24:37.358 --> 24:40.088
n = 3 will be that thing
divided by 9 and so on.

24:40.089 --> 24:43.719
These are the allowed energies.

24:43.720 --> 24:47.800
See, that's a great result,
because you are able to find

24:47.804 --> 24:52.044
out for the first time what
energies are available to this

24:52.037 --> 24:53.987
atom,
and furthermore,

24:53.987 --> 24:58.587
you also learn that if this
atom were able to absorb light,

24:58.588 --> 25:02.058
it can do that only say by
going from here to here or going

25:02.061 --> 25:05.001
from here to here,
or going from one allowed level

25:05.000 --> 25:06.410
to another allowed level.

25:06.410 --> 25:10.020
And the difference in energy is
E_f -

25:10.017 --> 25:14.097
E_i will be
ℏw of the

25:14.101 --> 25:14.851
photon.

25:14.849 --> 25:17.709
That will allow it to go up.

25:17.710 --> 25:21.080
Or it can also come down,
and if it comes down from some

25:21.077 --> 25:24.167
level,
n = 4 to n = 2,

25:24.171 --> 25:26.211
it may,
let's see, 1,2,

25:26.207 --> 25:30.297
3,4, it can go from 5 to 2,
it will emit some light.

25:30.298 --> 25:32.548
In fact, this is the only way
we know anything about the

25:32.546 --> 25:33.196
hydrogen atom.

25:33.200 --> 25:34.980
No one can see the hydrogen
atom.

25:34.980 --> 25:38.360
You cannot actually see it
because the act of observation

25:38.359 --> 25:39.929
will destroy everything.

25:39.930 --> 25:42.560
But we know it's there;
we know it's doing its thing by

25:42.555 --> 25:45.885
shining light on it and seeing
what it absorbs and what it

25:45.892 --> 25:46.422
emits.

25:46.420 --> 25:49.060
The understanding of the
quantum world is very different

25:49.064 --> 25:50.174
from classical world.

25:50.170 --> 25:51.240
For example,
if Newton says,

25:51.243 --> 25:52.883
"I know the orbits or
ellipses,"

25:52.875 --> 25:53.825
you can go see them.

25:53.828 --> 25:56.308
That's what Kepler did,
looked at them for 40 years,

25:56.308 --> 25:58.398
you can see they go in
elliptical orbits.

25:58.400 --> 26:00.810
For hydrogen,
there are no orbits.

26:00.808 --> 26:04.518
There are energy levels and
there are these corresponding

26:04.521 --> 26:06.511
wave functions,
Y (x,

26:06.510 --> 26:07.440
y ,z).

26:07.440 --> 26:10.730
And you can plot them and you
know if you start out with any

26:10.733 --> 26:13.753
one particular state,
it will stay that way forever.

26:13.750 --> 26:16.780
And the probability density,
I told you, is also fixed in

26:16.780 --> 26:17.160
time.

26:17.160 --> 26:22.570
They are the clouds one finds
in textbook as shape of various

26:22.567 --> 26:24.007
atomic orbits.

26:24.009 --> 26:29.669
Okay, yes, there's only one
other thing I should mention,

26:29.670 --> 26:33.600
which is that even though these
are the only allowed energies,

26:33.598 --> 26:38.308
there's more than one state at
a given energy.

26:38.308 --> 26:41.088
You may remember when I did a
particle on a ring,

26:41.086 --> 26:43.456
at a given energy,
there are two states of

26:43.458 --> 26:44.208
momentum.

26:44.210 --> 26:46.890
One has got momentum = square
root of 2mE,

26:46.890 --> 26:50.830
other has got momentum square
root of -2mE,

26:50.828 --> 26:52.918
because when you take the
square and divide by 2m,

26:52.920 --> 26:54.340
you get the same energy.

26:54.338 --> 26:56.678
So a given energy can have two
states.

26:56.680 --> 26:59.950
Here a given energy can have
many states.

26:59.950 --> 27:02.470
There's only one here and here
you can have 4,

27:02.467 --> 27:03.807
and here you can more.

27:03.808 --> 27:06.538
There are ways to calculate
them.

27:06.538 --> 27:09.678
Then it turns out that even
this 1 is actually 2,

27:09.679 --> 27:13.149
because it's another variable
which does not enter our

27:13.145 --> 27:16.085
physics, called the spin of the
electron.

27:16.088 --> 27:18.918
I don't want to even go there,
but it can do one of two

27:18.917 --> 27:19.387
things.

27:19.390 --> 27:22.230
So everything I do here,
you've got to double.

27:22.230 --> 27:26.210
So really, there are two states
of energy, lowest energy and

27:26.211 --> 27:28.171
there are 8 here and so on.

27:32.017 --> 27:32.757
equation.

27:32.759 --> 27:35.909
All right, so that's an example
of what you can do with quantum

27:35.910 --> 27:36.520
mechanics.

27:36.519 --> 27:39.789
You can solve for the spectrum
of atoms and you can see what

27:39.794 --> 27:42.684
light they will emit,
what light they will absorb.

27:42.680 --> 27:43.820
You can even go beyond that.

27:43.818 --> 27:46.518
You can even ask,
what's the rate at which it

27:46.515 --> 27:47.735
will absorb light?

27:47.740 --> 27:49.340
What's the rate at which it
will emit light?

27:49.338 --> 27:51.408
Everything can be computed,
because you see,

27:51.410 --> 27:54.960
you not only know the energies,
you also know the corresponding

27:54.963 --> 27:58.123
functions and they are very
important in calculating the

27:58.116 --> 28:00.866
rate of absorption and the rate
of emission.

28:00.868 --> 28:03.388
Okay, that's one thing you can
do.

28:03.390 --> 28:09.030
Now I'm going to tell you about
another uncertainty principle

28:09.028 --> 28:12.128
which takes the following form.

28:12.130 --> 28:16.250
It says DE
Dt >

28:16.248 --> 28:18.168
=ℏ.

28:18.170 --> 28:21.590
I've got to tell you what it
means.

28:21.588 --> 28:25.168
It looks a lot like this one,
DpDx

28:25.170 --> 28:28.960
> = ℏ,
but it's very different in

28:28.961 --> 28:32.731
nature, because x is the
position of something.

28:32.730 --> 28:34.290
You can try to measure it.

28:34.288 --> 28:36.408
p is the momentum of
something, you can try to

28:36.410 --> 28:36.930
measure it.

28:36.930 --> 28:38.880
t is not the time of
something.

28:38.880 --> 28:40.310
t is just time.

28:40.308 --> 28:43.168
It's not time of this or that,
and it can be measured

28:43.165 --> 28:44.535
arbitrarily accurately.

28:44.539 --> 28:46.569
That is not the problem.

28:46.568 --> 28:48.628
So I will have to tell you the
meaning of this uncertainty

28:48.627 --> 28:49.167
relationship.

28:49.170 --> 28:50.430
It is different from the others.

28:50.430 --> 28:52.530
There are many such things in
quantum mechanics,

28:52.529 --> 28:56.309
but this is unique in the sense
that unlike x and p

28:56.305 --> 28:58.565
which are physical
variables,

28:58.568 --> 29:00.758
time is not a variable
describing a particle.

29:00.759 --> 29:05.009
It's an independent parameter
along which everything happens.

29:05.009 --> 29:07.009
But I'll tell you what this
means,

29:07.009 --> 29:09.449
but before that,
I want to go back to the

29:09.452 --> 29:12.392
uncertainty principle,
which you notice I never

29:12.394 --> 29:14.814
mention, because it is not a
postulate,

29:14.809 --> 29:16.379
it's a consequence.

29:16.380 --> 29:18.780
And I told you all about trying
to look at something with a

29:18.776 --> 29:20.426
microscope and have you shine
photons.

29:20.430 --> 29:23.080
And a lot of you tried to find
ways to get around that,

29:23.078 --> 29:24.388
and say, "Maybe if you did
this,

29:24.390 --> 29:26.400
maybe if you did that,
you can beat the uncertainty

29:26.396 --> 29:27.116
principle."

29:27.118 --> 29:29.178
So I'm going to put you out of
your misery.

29:29.180 --> 29:32.480
You cannot beat the uncertainty
principle for the following

29:32.478 --> 29:32.988
reason.

29:32.990 --> 29:34.880
Forget about quantum mechanics.

29:34.880 --> 29:39.010
Let's ask the following
question - what is the

29:39.009 --> 29:41.579
wavelength of this signal?

29:41.579 --> 29:46.959
This is 1 meter long.

29:46.960 --> 29:50.720
You might say it's 1 meter,
but that's not a correct

29:50.717 --> 29:51.377
answer.

29:51.380 --> 29:53.890
This signal doesn't have a 1
meter wavelength.

29:53.890 --> 29:57.820
You know who has a 1 meter
wavelength is this guy who goes

29:57.823 --> 29:58.723
on forever.

29:58.720 --> 30:00.380
That is a wave of 1 meter
wavelength.

30:00.380 --> 30:04.710
This was 1 meter for a while,
then completely dead on either

30:04.714 --> 30:05.234
side.

30:05.230 --> 30:07.020
So this does not have a
definite wavelength,

30:07.023 --> 30:08.363
even though you think it does.

30:08.358 --> 30:11.178
A wavelength is a repetitive
phenomenon in space.

30:11.180 --> 30:16.140
It's got to repeat itself
indefinitely to have a unique

30:16.144 --> 30:17.344
wavelength.

30:17.338 --> 30:21.348
So let's take a wave train that
looks like this.

30:21.348 --> 30:24.018
You somehow chop it off nicely
near the end.

30:24.019 --> 30:28.729
It does some number of
oscillations and then it stops.

30:28.730 --> 30:31.970
What is the wavelength
associated with this?

30:31.970 --> 30:35.470
You can associate an
approximate wavelength as

30:35.471 --> 30:36.251
follows.

30:36.250 --> 30:40.370
First let me remind you that we
want to think in terms of

30:40.373 --> 30:42.733
k which is 2p/l.

30:42.730 --> 30:45.980
It's a reciprocal of a
wavelength, and if you remember,

30:45.979 --> 30:49.529
functions look like kx -
wt when we studied waves.

30:49.529 --> 30:52.329
k is that.

30:52.329 --> 30:55.219
What's the meaning of k?

30:55.220 --> 30:58.170
k is the rate of change
of phase of the wave,

30:58.167 --> 31:01.517
because if you go a distance
x, the phase changes by

31:01.519 --> 31:02.329
kx.

31:02.329 --> 31:03.519
You watch the wave change.

31:03.519 --> 31:07.439
Every cycle is worth 2p and it
changes by some amount over some

31:07.442 --> 31:08.142
distance.

31:08.140 --> 31:11.030
k is the ratio of how
much phase change you had

31:11.025 --> 31:13.035
divided by the distance you
travel.

31:13.039 --> 31:13.759
Do you understand that?

31:13.759 --> 31:15.059
That's k.

31:15.058 --> 31:21.978
So let me take this wave and
the k for this will be 2p times

31:21.978 --> 31:28.308
the number of cycles here
divided by the length of that

31:28.310 --> 31:30.070
wave train.

31:30.068 --> 31:32.178
But there is no unique n
you can associate with it.

31:32.180 --> 31:35.230
These are all full cycles,
but once you come near the end

31:35.227 --> 31:38.057
there's a little bit of
confusion as to what to do at

31:38.057 --> 31:38.707
the end.

31:38.710 --> 31:39.840
How do you close it off?

31:39.838 --> 31:43.988
So there's an error or
uncertainty of about 1 cycle

31:43.986 --> 31:46.056
coming from the 2 ends.

31:46.058 --> 31:50.108
So the uncertainty in k
is 2 p/L times the

31:50.107 --> 31:54.817
uncertainty in n which is
1 or 2, but that's all it is.

31:54.819 --> 31:55.369
You understand?

31:55.368 --> 31:57.948
A finite wave train,
when you round out the edges,

31:57.953 --> 32:00.173
it may not have a full number
of cycles.

32:00.170 --> 32:02.820
You taper it off in some
fashion, but here you've got a

32:02.818 --> 32:04.388
well defined number of cycles.

32:04.390 --> 32:07.350
So it's or - 1 coming from the
ends.

32:07.349 --> 32:10.149
So Dk is that.

32:10.150 --> 32:14.080
But this is a particle which is
now confined to a region of

32:14.077 --> 32:18.207
length L because the wave
function is 0 beyond that.

32:18.210 --> 32:21.180
So it's a particle whose
position is known to lie inside

32:21.180 --> 32:23.070
this interval of length
L.

32:23.068 --> 32:26.778
So the uncertainty in position
is L.

32:26.778 --> 32:30.208
Again, uncertainty is a
technical definition,

32:30.212 --> 32:33.492
but this is our qualitative
explanation.

32:33.490 --> 32:35.820
So L is really
Dx.

32:35.819 --> 32:37.759
You can write Dx.

32:37.759 --> 32:41.879
Dk is 2p.

32:41.880 --> 32:45.230
This has nothing to do with
quantum mechanics.

32:45.230 --> 32:47.480
Where did I mention quantum
mechanics?

32:47.480 --> 32:50.370
I'm saying something
intuitively very clear.

32:50.368 --> 32:54.128
If you're defining a
wavelength, you have to let it

32:54.133 --> 32:56.093
run through many cycles.

32:56.088 --> 32:58.358
For example,
if you say this guy goes to New

32:58.356 --> 33:00.726
York every week,
and you observed him for only

33:00.730 --> 33:02.630
one week, that's got no meaning.

33:02.630 --> 33:04.560
God knows what will happen next
week.

33:04.558 --> 33:07.308
But if you studied a person for
50 weeks and found 50 times the

33:07.310 --> 33:09.970
person went to New York,
it's more credible when you say

33:09.970 --> 33:11.950
this person goes to New York
once a week.

33:11.950 --> 33:15.130
So periodic phenomenon,
either in time or in space,

33:15.125 --> 33:18.295
are defined only after many,
many, many periods.

33:18.298 --> 33:21.858
Therefore a wavelength cannot
be defined for an arbitrarily

33:21.856 --> 33:22.896
short interval.

33:22.900 --> 33:25.190
In fact, the longer the
interval, the more wavelengths

33:25.192 --> 33:27.142
you can fit in to say that's my
wavelength.

33:27.140 --> 33:30.140
So wavelength gets more and
more defined as the train gets

33:30.144 --> 33:32.624
longer and therefore the
incompatibility between

33:32.621 --> 33:34.941
wavelength and the length of the
train,

33:34.940 --> 33:37.490
the fact that if it's too short
in one,

33:37.490 --> 33:41.180
it's too big in the other one,
is a classical result.

33:41.180 --> 33:45.260
Quantum mechanics comes in with
a new relation that the momentum

33:45.259 --> 33:48.369
of a particle is connected to
this wave number by

33:48.369 --> 33:49.859
ℏk.

33:49.858 --> 33:52.718
Multiply both sides by
ℏ,

33:52.720 --> 33:54.730
ℏ there and
h here,

33:54.730 --> 33:59.380
then it becomes
DxDp

33:59.383 --> 34:02.633
is roughly about an ℏ.

34:02.630 --> 34:05.800
So once you concede that a
particle has definite momentum

34:05.798 --> 34:09.248
only if it has a definite
wavelength in its wave function,

34:09.250 --> 34:12.980
that wave function has to be
very extended to have a definite

34:12.983 --> 34:15.913
wavelength and therefore a
definite momentum.

34:15.909 --> 34:19.739
So there is no way you can make
it--there's no way you can take

34:19.739 --> 34:23.569
a wave arbitrarily short in its
extent with an arbitrarily well

34:23.568 --> 34:24.988
defined wavelength.

34:24.989 --> 34:28.429
You've got to let it do its
thing many times.

34:28.429 --> 34:32.159
Mathematically what happens is,
there's a mathematical theorem

34:32.161 --> 34:35.951
that says any function you write
down can be built out of waves

34:35.952 --> 34:39.992
of all possible wavelengths with
suitably chosen coefficients.

34:39.989 --> 34:43.359
If your wave has an almost well
defined wavelength,

34:43.360 --> 34:46.480
then the coefficients,
as the function of wavelength,

34:46.480 --> 34:50.690
will have a very sharp peak at
the wavelength corresponding to

34:50.686 --> 34:51.166
this.

34:51.170 --> 34:53.940
And as the wave repeats more
and more and more times,

34:53.936 --> 34:57.286
the coefficients will become
sharper and sharper and sharper.

34:57.289 --> 34:59.229
As the wave becomes very small,
if you say "What's the

34:59.231 --> 35:00.271
wavelength of this guy?"

35:00.269 --> 35:02.789
it will be very broad.

35:02.789 --> 35:04.399
It's nothing to do with quantum
mechanics.

35:04.400 --> 35:07.430
It's just the incompatibility
between two qualities,

35:07.429 --> 35:08.439
you understand?

35:08.440 --> 35:12.460
Wavelength needs some time to
express itself.

35:12.460 --> 35:14.460
You cannot do it in a tiny
region.

35:14.460 --> 35:17.980
So if you want to localize the
particle, how can it tell you in

35:17.983 --> 35:21.053
the tiny region what its
frequency of oscillation is in

35:21.050 --> 35:21.620
space?

35:21.619 --> 35:24.219
So once you concede that,
once you concede that

35:24.219 --> 35:27.439
wavelength is connected to
momentum, you cannot escape the

35:27.440 --> 35:28.910
uncertainty principle.

35:28.909 --> 35:33.339
So now I'm going to come back
to energy and time,

35:33.340 --> 35:36.940
and I come back in the
following way.

35:36.940 --> 35:41.320
Remember, a state of definite
energy E behaves as

35:41.317 --> 35:42.747
follows in time.

35:42.750 --> 35:50.490
It looks like Y(x)
times e^(−iEt/

35:50.487 --> 35:53.017
ℏ).

35:53.018 --> 35:56.618
So state of definite energy,
you know the particle has

35:56.623 --> 36:00.303
definite energy if you observe
the wave function and it

36:00.295 --> 36:01.855
oscillates in time.

36:01.860 --> 36:05.480
And you can write this as
e^(-i)^(w)

36:05.481 --> 36:08.141
^(t),
where w is

36:08.137 --> 36:10.067
E/ℏ.

36:10.070 --> 36:14.030
So your particle that has been
in that state forever has a well

36:14.025 --> 36:15.105
defined energy.

36:15.110 --> 36:18.240
But if a particle was just put
in that state right now,

36:18.239 --> 36:21.099
the wave function really does
this only from t = 0,

36:21.099 --> 36:24.889
before that it was something
else, then this function has not

36:24.885 --> 36:28.665
oscillated enough times for you
to define its time period.

36:28.670 --> 36:31.320
So if you're watching an
oscillation,

36:31.320 --> 36:33.180
let's say it's an oscillation
in time,

36:33.179 --> 36:35.569
and you say,
"What's the time period of

36:35.574 --> 36:36.694
oscillation?"

36:36.690 --> 36:41.760
again, if it extends over a
certain time T,

36:41.755 --> 36:47.645
the frequency w is just the
phase change per unit time.

36:47.650 --> 36:51.260
So it's 2p times the number of
cycles, but there's an

36:51.262 --> 36:55.642
uncertainty in N of order
1 that's 2p/T.

36:55.639 --> 36:58.769
Therefore if the system has
been in existence for a time

36:58.771 --> 37:01.791
T and I call that time as
Dt--

37:01.789 --> 37:03.339
that's the meaning of
Dt--

37:03.340 --> 37:06.960
Dt Dw is 2p,
then quantum mechanics comes in

37:06.960 --> 37:09.630
when you put an
ℏ here,

37:09.630 --> 37:13.720
you put an ℏ
there and you call that the

37:13.722 --> 37:15.512
uncertainty in energy.

37:15.510 --> 37:17.720
In other words,
if someone tells me,

37:17.722 --> 37:20.762
"Go see if that clock is a
regular clock.

37:20.760 --> 37:23.550
Tell me if it's running
rhythmically and

37:23.554 --> 37:25.064
periodically."

37:25.059 --> 37:28.149
Well, if you give me enough
time, I observe it for 100

37:28.153 --> 37:30.143
cycles, I say it's a good clock.

37:30.139 --> 37:32.689
But if you don't give me time
even to finish one cycle,

37:32.690 --> 37:35.100
how can I tell you anything
about its period yet?

37:35.099 --> 37:36.749
It hasn't had time to establish
a period.

37:36.750 --> 37:40.010
Periodic phenomena have to
repeat themselves,

37:40.010 --> 37:41.790
so it takes some time.

37:41.789 --> 37:44.799
Therefore for a state to have a
well-defined period,

37:44.802 --> 37:46.992
that is to say,
a well defined energy,

37:46.989 --> 37:49.589
it has to be in existence for
some time.

37:49.590 --> 37:52.880
Then it has a well-defined
energy.

37:52.880 --> 37:57.090
Okay, therefore you can say
that if you've had time to

37:57.085 --> 38:00.335
measure energy only for a finite
time,

38:00.340 --> 38:05.130
then energy will not be defined
to better than this accuracy,

38:05.130 --> 38:08.650
DE
Dt is of order

38:08.650 --> 38:10.250
ℏ.

38:10.250 --> 38:13.730
Let me give you an example that
is from classical mechanics,

38:13.728 --> 38:16.028
nothing to do with quantum
mechanics.

38:16.030 --> 38:19.820
You take a bunch of reeds,
you attach them to the wall.

38:19.820 --> 38:22.100
Each one has a different length
maybe.

38:22.099 --> 38:26.049
They have many frequencies of
vibration, natural frequencies

38:26.048 --> 38:28.728
of vibration,
depending on the length and

38:28.726 --> 38:29.526
whatnot.

38:29.530 --> 38:33.260
Now if you connect this whole
thing,

38:33.260 --> 38:36.210
apparatus, to some vibrating
gadget,

38:36.210 --> 38:38.890
start shaking the whole thing,
suppose you shake it at a

38:38.887 --> 38:44.277
certain frequency,
omega 0 that matches maybe this

38:44.275 --> 38:44.975
guy.

38:44.980 --> 38:49.160
Namely, it resonates with this
guy, so your expectation is,

38:49.161 --> 38:53.561
this one will oscillate like
crazy and the others will not.

38:53.559 --> 38:55.759
So let me look at this rod from
the edge, okay?

38:55.760 --> 38:57.710
I look at them end on,
they all look like this.

38:57.710 --> 38:59.690
They're coming out of the
blackboard.

38:59.690 --> 39:03.740
So I expect this one to be
resonating wildly up and down

39:03.737 --> 39:08.147
and the others to be ignoring
the signal because they are not

39:08.152 --> 39:10.362
at the resonant frequency.

39:10.360 --> 39:13.900
But what will happen in real
life is that you will take the

39:13.902 --> 39:17.842
system and you start shaking it,
your intention is to shake it

39:17.840 --> 39:21.130
at a very definite frequency
that matches this guy,

39:21.130 --> 39:24.320
but the system does not know
your plans.

39:24.320 --> 39:27.770
After quarter of a cycle,
it knows you've exerted that

39:27.768 --> 39:28.288
force.

39:28.289 --> 39:31.169
It doesn't know you plan to
keep doing this.

39:31.170 --> 39:33.690
So at that time,
what it will do is it will try

39:33.688 --> 39:37.028
to write this as a sum of many
waves of definite frequency.

39:37.030 --> 39:38.950
They will contain many,
many frequencies,

39:38.949 --> 39:41.639
so what you will find is,
this guy also oscillates a

39:41.641 --> 39:43.871
little bit,
that guy also oscillates a

39:43.865 --> 39:46.315
little bit,
and as you wait longer and

39:46.320 --> 39:50.400
longer, so the system caught on
to the fact that you are really

39:50.398 --> 39:53.948
serious and sending a signal of
definite frequency.

39:53.949 --> 39:56.349
And when it has stabilized,
you will find these guys don't

39:56.353 --> 40:00.133
move, these guys don't move;
the guy at the resonant

40:00.126 --> 40:01.976
frequency moves.

40:01.980 --> 40:05.560
So you've got to understand,
it takes some time for the

40:05.556 --> 40:08.666
system to know what frequency
you're sending.

40:08.670 --> 40:12.840
If you have not sent many
cycles, if you have sent only

40:12.835 --> 40:15.485
this much,
it does not consider that a

40:15.487 --> 40:17.277
periodic function,
because to it,

40:17.282 --> 40:19.612
the function would really do
that or it could do something

40:19.612 --> 40:19.942
else.

40:19.940 --> 40:22.420
It goes with the information it
has.

40:22.420 --> 40:24.670
It goes to the mathematical
tables and finds the Fourier

40:24.670 --> 40:26.880
series for this and sees what
frequencies are there.

40:26.880 --> 40:30.480
And anything that's not 0 can
excite all these things.

40:30.480 --> 40:33.650
And after a while,
when the frequency is well

40:33.654 --> 40:35.534
defined, one reed moves.

40:35.530 --> 40:37.240
Same thing happens with atoms.

40:37.239 --> 40:39.489
I don't know,
somewhere I drew a picture of

40:39.494 --> 40:42.614
atoms absorbing energy.So take
this atom here in the ground

40:42.608 --> 40:43.788
state of hydrogen.

40:43.789 --> 40:47.779
Send light of the frequency
just correct to go to the first

40:47.779 --> 40:50.049
excited state,
to n = 2.

40:50.050 --> 40:53.380
You take a laser or something
of that frequency and you hit

40:53.378 --> 40:54.008
the atom.

40:54.010 --> 40:57.240
Take a collection of atoms and
you will think they will all

40:57.239 --> 41:00.359
jump from the ground state to
the first state but nowhere

41:00.358 --> 41:00.858
else.

41:00.860 --> 41:03.600
But you will find the minute
you turn on the laser,

41:03.601 --> 41:06.451
even though the laser dials say
this is my frequency,

41:06.454 --> 41:07.884
atoms don't know that.

41:07.880 --> 41:10.400
So initially,
it will jump like crazy to all

41:10.403 --> 41:13.753
kinds of energy states and only
after sufficient time,

41:13.750 --> 41:16.210
only after many cycles have
happened,

41:16.210 --> 41:17.840
it will realize, "Hey,
this is the frequency this

41:17.840 --> 41:18.710
guy's sending me."

41:18.710 --> 41:22.150
Then it will start going
preferentially to the one state.

41:22.150 --> 41:26.140
Okay, now the final topic I
want to talk about is really the

41:26.144 --> 41:29.254
beginning of a long topic,
but I want to give an

41:29.248 --> 41:32.178
introduction to it because
you're going to need this.

41:32.179 --> 41:35.279
And the question is,
what if I have not just 1

41:35.275 --> 41:37.955
electron but 2,
not 1 particle but 2.

41:37.960 --> 41:40.310
After all, in real life,
there are lots of particles.

41:40.309 --> 41:43.439
What does the quantum mechanics
of more than 1 particle look

41:43.438 --> 41:43.808
like?

41:43.809 --> 41:46.839
So if there are 2 particles,
let's say an electron and a

41:46.838 --> 41:49.828
proton,
you will not be surprised that

41:49.833 --> 41:53.993
you will now have a wave
function of 2 variables,

41:53.989 --> 41:56.509
and if you squared that wave
function,

41:56.510 --> 41:58.800
you will get
Y(x_1)

41:58.797 --> 42:00.757
Y(x_2)
--

42:00.760 --> 42:05.390
Y(x_1),
x_2^(2),

42:05.389 --> 42:08.559
will give you the probability
that the proton is at

42:08.556 --> 42:11.216
x_1 and the
electron is at

42:11.215 --> 42:12.795
x_2.

42:12.800 --> 42:14.770
You've got 2 things,
you've got to give 2

42:14.771 --> 42:16.201
probabilities,
but once again,

42:16.202 --> 42:18.572
probabilities come from
squaring a function.

42:18.570 --> 42:22.330
This has to be a function of 2
variables, because each guy has

42:22.329 --> 42:23.499
his own position.

42:23.500 --> 42:25.940
So you can expect that in
quantum theory of 2 particles,

42:25.940 --> 42:27.870
you'll get a Y that
depends on 2 coordinates,

42:27.869 --> 42:30.009
with 3 particles,
Y that depends on 3.

42:30.010 --> 42:34.220
But let me just stop with 2,
because you can learn some very

42:34.215 --> 42:38.415
profound things in about 5
minutes just by starting here.

42:38.420 --> 42:41.220
Now let's consider,
so let's take a concrete

42:41.219 --> 42:41.869
example.

42:41.869 --> 42:45.839
These guys are both in a box,
let's say.

42:45.840 --> 42:52.590
Let's take a simple case,
Yof x1 x2 = Ya of

42:52.594 --> 42:57.544
x1 Yb of x2,
where a and b are wave

42:57.539 --> 43:03.449
functions of energy in a box
with 1 particle.

43:03.449 --> 43:07.779
For example,
this could be the state a and

43:07.780 --> 43:10.740
this could be the state b.

43:10.739 --> 43:13.129
The probability to find the
proton at some place and the

43:13.126 --> 43:15.686
electron at some place is not
the same as the probability of

43:15.686 --> 43:17.506
finding them with exchanged
positions.

43:17.510 --> 43:20.290
For example,
the chance of finding the

43:20.291 --> 43:24.881
electron here and the proton in
the middle of the box is 0.

43:24.880 --> 43:28.240
The proton's function vanishes
in the middle of the box.

43:28.239 --> 43:31.639
But the probability of finding
the electron in the middle of

43:31.635 --> 43:33.875
the box and the proton here is
not 0.

43:33.880 --> 43:36.260
That's perfectly okay,
because there are two different

43:36.264 --> 43:37.574
outcomes of the experiment.

43:37.570 --> 43:38.880
I look for the particles.

43:38.880 --> 43:40.820
I find the electron here and
the proton there.

43:40.820 --> 43:44.360
I say I found this guy here and
that there, and I do it many,

43:44.358 --> 43:46.598
many times and you give me the
odds.

43:46.599 --> 43:50.049
But something very dramatic
happens if the 2 particles are

43:50.052 --> 43:50.782
identical.

43:50.780 --> 43:54.630
Identical particles in quantum
mechanics have very different

43:54.628 --> 43:57.828
connotations from identical
particles in classical

43:57.826 --> 43:58.736
mechanics.

43:58.739 --> 44:01.109
So in classical mechanics,
suppose you have identical

44:01.110 --> 44:01.750
twins, okay?

44:01.750 --> 44:04.250
They're born and they're
separated, they're moving

44:04.250 --> 44:04.710
around.

44:04.710 --> 44:07.120
Let's say they look identical
in every way.

44:07.119 --> 44:08.709
We can still follow them.

44:08.710 --> 44:11.050
We know this is Joe and this is
Moe.

44:11.050 --> 44:14.370
Follow the two characters no
matter how identical they are,

44:14.371 --> 44:17.181
because we can keep track of
them continuously.

44:17.179 --> 44:21.149
So let's do the following
experiment involving these

44:21.146 --> 44:21.766
twins.

44:21.768 --> 44:26.188
There are 4 doors in this room
and 1 twin comes out like this,

44:26.188 --> 44:28.868
the other twin comes out like
that.

44:28.869 --> 44:31.069
Then they do one of 2 things.

44:31.070 --> 44:37.860
Either they cross over like
this, or this guy goes back to

44:37.858 --> 44:43.218
that door, that guy goes back to
this door.

44:43.219 --> 44:46.099
Now suppose you saw them
running like this in the

44:46.099 --> 44:48.749
beginning,
and you left the room when this

44:48.746 --> 44:51.406
was happening,
and you saw them enter these 2

44:51.407 --> 44:53.867
doors,
you cannot tell which of the 2

44:53.871 --> 44:55.921
happened,
because you've just got 2

44:55.920 --> 44:58.540
identical twins and these 2
identical exit doors.

44:58.539 --> 45:00.059
But somebody knows what happens.

45:00.059 --> 45:02.929
Somebody in that room who was
watching them can clearly tell

45:02.932 --> 45:05.902
you, if this happened or that
happened, because you can follow

45:05.903 --> 45:07.223
the twins at all times.

45:07.219 --> 45:09.219
So even though they're
identical, they're

45:09.219 --> 45:10.119
distinguishable.

45:10.119 --> 45:14.039
They cannot swap roles without
your knowing.

45:14.039 --> 45:18.129
But imagine now that these are
not classical particles but

45:18.128 --> 45:20.708
quantum particles,
like electrons.

45:20.710 --> 45:22.990
For an electron,
you don't have a definite

45:22.994 --> 45:24.504
location or a trajectory.

45:24.500 --> 45:26.340
You only have probabilities.

45:26.340 --> 45:29.520
So you know an electron was
emitted here and emitted here,

45:29.519 --> 45:32.199
and later on was absorbed here
and absorbed here,

45:32.199 --> 45:34.709
and you cannot tell who really
came here.

45:34.710 --> 45:36.820
Was it this guy or was it that
guy?

45:36.820 --> 45:40.160
There's no way to tell.

45:40.159 --> 45:43.139
So when you have identical
particles whose trajectories you

45:43.144 --> 45:45.594
cannot follow,
when you catch a particle here

45:45.586 --> 45:48.636
and a particle there,
you cannot say Joe was here and

45:48.635 --> 45:49.525
Moe was there.

45:49.530 --> 45:51.460
It's not allowed,
because you're not following

45:51.458 --> 45:52.058
their names.

45:52.059 --> 45:54.299
You can only say,
"I found a particle here,

45:54.304 --> 45:55.884
I found a particle there."

45:55.880 --> 45:59.600
Therefore the theory cannot
give different probabilities for

45:59.601 --> 46:02.631
finding particle 1 here and
particle 2 there,

46:02.630 --> 46:04.370
and particle 2 there and
particle 1 here,

46:04.369 --> 46:07.239
because the outcomes are
indistinguishable,

46:07.239 --> 46:10.149
so the probabilities must be
equal.

46:10.150 --> 46:12.600
So for 2 identical
particles, p (x_1

46:12.597 --> 46:15.197
x_2) must =
p( x_2

46:15.197 --> 46:16.367
x_1).

46:16.369 --> 46:20.379
That's not true for these
functions.

46:20.380 --> 46:24.980
Now I'll write a function for
which it's actually true.

46:24.980 --> 46:30.520
Can you make a guess on what it
may be, what kind of function

46:30.516 --> 46:35.496
will respect the fact that if
you swap the coordinates,

46:35.498 --> 46:38.358
probabilities don't change?

46:38.360 --> 46:41.580
We know there's particle in
state a and a particle in state

46:41.577 --> 46:43.017
b and they're identical.

46:43.018 --> 46:48.348
This says particle 1 is in and
2 is in b, but you're not

46:48.347 --> 46:53.287
allowed to say who is who,
so the correct answer is,

46:53.288 --> 46:55.708
you can also do this.

46:55.710 --> 46:59.680
Look at this function now.

46:59.679 --> 47:02.209
Here it's a superposition,
quantum mechanical

47:02.213 --> 47:04.533
superposition of --
1 doing something,

47:04.527 --> 47:06.737
2 doing something else,
the opposite.

47:06.739 --> 47:08.139
You add them together.

47:08.139 --> 47:09.999
Now I invite you to check that
if you exchange

47:10.003 --> 47:11.873
x_1 and
x_2,

47:11.869 --> 47:12.739
see what happens.

47:12.739 --> 47:13.909
This becomes
Y_a(

47:13.914 --> 47:15.594
x_2),
that guy sitting here.

47:15.590 --> 47:16.780
This becomes
Y_b(

47:16.777 --> 47:18.467
x_1),
that guy sitting here.

47:18.469 --> 47:19.389
This becomes
Y_b(

47:19.389 --> 47:20.589
x_2),
that sitting here.

47:20.590 --> 47:24.610
So these 2 terms exchange roles
when you exchange the particles.

47:24.610 --> 47:27.400
Y(x) to
x_1 is in fact

47:27.398 --> 47:29.998
Y(x)--
Y(x_1,

47:30.003 --> 47:33.223
x_2) is the same as
Y(x_2,

47:33.224 --> 47:34.344
x_1).

47:34.340 --> 47:38.350
So that is an allowed state in
quantum mechanics,

47:38.351 --> 47:42.701
where you add them and you
symmetrize the product.

47:42.699 --> 47:43.819
You guys following that?

47:43.820 --> 47:46.690
Now I'm going to try another
combination.

47:46.690 --> 47:52.780
I'm going to put a - sign here.

47:52.780 --> 47:54.960
If you put a - sign,
perhaps you can see if I

47:54.960 --> 47:57.090
exchange the 2 guys and this
goes into that,

47:57.092 --> 47:59.472
that goes into this,
I don't come back to where I

47:59.472 --> 47:59.922
am.

47:59.920 --> 48:04.210
I come back to where the - of
this guy.

48:04.210 --> 48:07.650
If this looked like x -
y, that looks like y -

48:07.646 --> 48:08.086
x.

48:08.090 --> 48:10.690
So you might say,
"Hey, things are not the

48:10.688 --> 48:13.118
same when I exchange the
particles."

48:13.119 --> 48:15.759
But remember,
the physical quantities are not

48:15.764 --> 48:18.594
given by Y but the square
of Y.

48:18.590 --> 48:20.800
The probabilities are not
Y;

48:20.800 --> 48:22.400
the probabilities are
Y^(2).

48:22.400 --> 48:24.710
So even with the - sign,
the probability for

48:24.713 --> 48:26.783
x_1,
x_2 here is also the

48:26.775 --> 48:28.875
same as the probability for
x_2 ,x_1

48:28.876 --> 48:30.016
here,
because when you find the

48:30.016 --> 48:33.626
absolute value,
the - sign goes away.

48:33.630 --> 48:36.730
So in quantum mechanics,
there are 2 options for

48:36.728 --> 48:38.178
identical particles.

48:38.179 --> 48:42.249
Either you can take the product
function and add to it the

48:42.251 --> 48:46.111
product with the reversed
coordinates, or subtract from

48:46.108 --> 48:46.608
it.

48:46.610 --> 48:51.050
And the particles in the world,
all of them decide to go with

48:51.045 --> 48:52.815
one camp or the other.

48:52.820 --> 48:58.540
Particles called bosons always
choose that sign,

48:58.541 --> 49:05.481
and particles called fermions
always choose the - sign.

49:05.480 --> 49:08.030
And every particle is either
boson or fermion.

49:08.030 --> 49:09.550
P mesons are bosons.

49:09.550 --> 49:10.760
Photons are bosons.

49:10.760 --> 49:11.940
Electrons are fermions.

49:11.940 --> 49:13.160
Quarks are fermions.

49:13.159 --> 49:14.579
Gravitons are bosons.

49:14.579 --> 49:16.769
So everybody is one or the
other.

49:16.768 --> 49:19.258
If you put 2 bosons in a box,
their wave function,

49:19.264 --> 49:21.664
when you exchange them,
will remain the same.

49:21.659 --> 49:27.189
If you put 2 fermions in a box,
the wave function will change

49:27.193 --> 49:27.843
sign.

49:27.840 --> 49:30.710
But now here is the beautiful
result.

49:30.710 --> 49:35.880
Take this case for 2 fermions
and ask yourself--this is a

49:35.880 --> 49:39.760
fermionic wave function for 2
particles.

49:39.760 --> 49:43.490
Ask yourself the following
question - can they both be in

49:43.485 --> 49:45.145
the same quantum state?

49:45.150 --> 49:48.120
In other words,
can the state a be the same as

49:48.123 --> 49:49.053
the state b?

49:49.050 --> 49:50.970
Remember, here a was this and b
was that.

49:50.969 --> 49:54.699
I'm asking, can they both be in
the same state?

49:54.699 --> 49:57.289
So I invite you to put a = b
here.

49:57.289 --> 49:59.149
Let's see what happens.

49:59.150 --> 50:00.670
That's a.

50:00.670 --> 50:04.340
That's also a.

50:04.340 --> 50:06.990
What do you get?

50:06.989 --> 50:11.139
Can you see something when both
are a?

50:11.139 --> 50:14.099
You get 0.

50:14.099 --> 50:17.239
So you cannot write a wave
function in which the 2 fermions

50:17.244 --> 50:20.284
are in the same state,
and that's the Pauli principle.

50:20.280 --> 50:23.250
Pauli principle says,
for some of the particles,

50:23.253 --> 50:25.473
they cannot be in the same
state.

50:25.469 --> 50:28.139
Bosons, on the other hand,
can be in the same state and

50:28.137 --> 50:29.667
like to be in the same state.

50:29.670 --> 50:32.750
I don't have time to talk about
that, but this is the Pauli

50:32.751 --> 50:33.391
principle.

50:33.389 --> 50:36.529
And you can show if you've got
3 fermions and 4 fermions and so

50:36.525 --> 50:38.375
on,
you will find out the quantum

50:38.378 --> 50:41.738
mechanical wave functions never
allow any 2 of them to be in the

50:41.742 --> 50:42.492
same state.

50:42.489 --> 50:48.419
And that is the origin of the
entire periodic table of atoms,

50:48.420 --> 50:51.530
because what you do when you've
got many electrons in an atom

50:51.529 --> 50:54.819
is,
you find these energy levels

50:54.822 --> 50:58.902
that you did,
the 1/n^(2) that you

50:58.898 --> 51:00.518
got,
but the nuclear charge of

51:00.519 --> 51:01.419
course may not be 1.

51:01.420 --> 51:05.190
It's some non 0 number,
depending on how many charges

51:05.188 --> 51:06.708
are in the nucleus.

51:06.710 --> 51:09.720
You take these levels,
I don't know how many levels

51:09.722 --> 51:11.952
there are here,
then you start putting

51:11.952 --> 51:13.402
electrons into them.

51:13.400 --> 51:16.790
The first electron will go here.

51:16.789 --> 51:19.469
It turns out they have
something called spin,

51:19.474 --> 51:23.204
so I will just say 1 way goes
with down and 1 way goes with up

51:23.195 --> 51:23.985
and down.

51:23.989 --> 51:26.849
The third electron,
you have to put here.

51:26.849 --> 51:29.549
It has to go to higher energy
state.

51:29.550 --> 51:32.760
If they were bosons,
you can put them all in the

51:32.757 --> 51:34.257
lowest energy state.

51:34.260 --> 51:37.870
That world will look completely
different from the world we live

51:37.873 --> 51:38.163
in.

51:38.159 --> 51:40.889
In the world we live in,
the levels keep filling up as

51:40.887 --> 51:43.097
you put more and more and more
electrons.

51:43.099 --> 51:45.979
They've got to go to higher and
higher energy levels.

51:45.980 --> 51:50.470
You go to higher and higher
energy levels,

51:50.474 --> 51:56.844
what happens once in a while
is, you fill all of this right

51:56.835 --> 52:01.765
now, then that atom becomes very
passive.

52:01.768 --> 52:05.988
It is not interested in either
giving up electrons or taking

52:05.990 --> 52:06.850
electrons.

52:06.849 --> 52:09.639
Whereas if you had one more
electron here,

52:09.635 --> 52:12.825
it's very happy to lose this to
some other atom,

52:12.827 --> 52:16.697
which may have just one
electron in its lowest state.

52:16.699 --> 52:20.719
It's a waste of it to be here;
it will go sit there.

52:20.719 --> 52:23.219
Sometimes they like to give an
electron to another atom,

52:23.219 --> 52:25.189
or if they do,
this will become positively

52:25.193 --> 52:28.083
charged and that will become
negatively charged and they have

52:28.081 --> 52:30.731
an attraction and they stay
together as a molecule.

52:30.730 --> 52:34.280
So you can understand that as
you go on piling more and more

52:34.284 --> 52:38.204
electrons, a time will come when
this level is completely full.

52:38.199 --> 52:43.069
That atom, whatever its place
in the periodic table is,

52:43.072 --> 52:45.602
will be also very passive.

52:45.599 --> 52:48.459
So you will find things which
are electrically very active

52:48.456 --> 52:50.306
with loose electrons in the
upper,

52:50.309 --> 52:52.929
called valence states,
and the inner shells are

52:52.929 --> 52:54.339
filled,
they are not.

52:54.340 --> 52:55.750
They are inert.

52:55.750 --> 52:57.710
So the behavior of active,
inert, active,

52:57.711 --> 53:00.021
inert is periodic,
and that's the periodic table

53:00.018 --> 53:01.048
that is observed.

53:01.050 --> 53:04.050
But you get that from quantum
mechanics in great detail by

53:04.052 --> 53:07.212
solving for the energy levels,
then using the Pauli principle

53:07.211 --> 53:09.631
to put only 1 electron there
every quantum state,

53:09.630 --> 53:14.740
and you can see a lot of this
behavior can be anticipated.

53:14.739 --> 53:16.779
So anyway, these are things you
will learn if you learn

53:16.782 --> 53:18.072
chemistry, if you learn physics.

53:18.070 --> 53:20.930
You will see where everything
comes from.

53:20.929 --> 53:24.659
All right, so this is the end
of the quantum mechanics part.

53:24.659 --> 53:29.369
I'm just going to tell you that
I want to stop here and I'll see

53:29.371 --> 53:31.841
you for the discussion section.

53:31.840 --> 53:34.590
I'm really going to miss my
Mondays and Wednesdays because

53:34.592 --> 53:36.672
for me, that's the best time of
the week.

53:36.670 --> 53:38.970
So really good to be with you
guys.

53:38.969 --> 53:39.779
Thank you.

53:39.780 --> 53:43.000
>

53:43.000 --> 53:48.000