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00:00:03,660 --> 00:00:09,990
PROFESSOR: ih bar
d psi dt equal E

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psi where E hat is equal to p
squared over 2m, the operator.

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00:00:19,890 --> 00:00:22,800
That is the
Schrodinger equation.

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The free particle
Schrodinger equation--

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you should realize it's
the same thing as this.

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Because p is h bar over i ddx.

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00:00:35,060 --> 00:00:43,151
And now Schrodinger did the
kind of obvious thing to do.

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He said, well, suppose I have a
particle moving in a potential,

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a potential V of x and t--

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potential.

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Then the total energy is kinetic
energy plus potential energy.

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So how about if we think of
the total energy operator.

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And here is a guess.

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We'll put the just p squared
over 2m, what we had before.

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That's the kinetic
energy of a particle.

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But now add plus V of
x and t, the potential.

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That is reasonable from
your classical intuition.

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The total energy
is the sum of them.

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But it's going to change
the Schrodinger equation

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quite substantially.

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Now, most people,
instead of calling this

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the energy operator,
which is a good name,

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have decided to call
this the Hamiltonian.

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So that's the most popular
name for this thing.

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This is called
the Hamiltonian H.

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And in classical
mechanics, the Hamiltonian

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represents the energy
expressed in terms

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of position and momenta.

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That's what the
Hamiltonian is, and that's

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roughly what we have here.

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The energy is [? in ?]
[? terms ?] [? of ?] momenta

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and position.

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And we're going to soon
be getting to the position

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operator, therefore.

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So this is going to
be the Hamiltonian.

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And we'll put the hat as well.

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So Schrodinger's
inspiration is to say, well,

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this is going to be H hat.

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And I'm going to say that ih bar
d psi dt is equal to H hat psi.

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00:03:05,860 --> 00:03:14,770
Or equivalently, ih bar ddt
of psi of x and t is equal

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to minus h squared over
2m, [? v ?] [? second ?] dx

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squared--

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that's the p squared over 2m--

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plus V of x and t,
all multiplying psi.

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This is it.

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This is the full
Schrodinger equation.

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So it's a very simple departure.

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You see, when you
discover the show

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that the equation for a free
particle, adding the energy

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was not that difficult. Adding
the potential energy was OK.

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We just have to interpret this.

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00:04:06,590 --> 00:04:09,845
And maybe it sounds to
you a little surprising

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that you multiply this by psi.

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But that's the only way it could
be to be a linear equation.

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It cannot be that psi is
acted by this derivative,

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but then you add v.

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It would not be a
linear equation.

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And we've realize that the
structure of the Schrodinger

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equation is d psi dt is equal
to an energy operator times psi.

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The whole game of
quantum mechanics

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00:04:40,390 --> 00:04:45,800
is inventing energy
operators, and then solving

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these equations, then
see what they are.

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So in particular, you
could invent a potential

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and find the equations.

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And, you see, it looks funny.

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You've made a very
simple generalization.

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And now you have an equation.

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And now you can put the
potential for the hydrogen atom

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and calculate, and
see if it works.

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And it does.

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So it's rather unbelievable
how very simple generalizations

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suddenly produce
an equation that

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has the full spectrum
of the hydrogen atom.

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00:05:19,610 --> 00:05:23,540
It has square wells, barrier
penetration, everything.

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00:05:23,540 --> 00:05:29,080
All kinds of dynamics
is in that equation.

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00:05:29,080 --> 00:05:33,410
So we're going to say a few more
things about this equation now.

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And I want you to understand
that the V, at this moment,

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can be thought as an operator.

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This is an operator,
acts on a wave

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function to give you a function.

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00:05:49,270 --> 00:05:52,400
This is a simpler operator.

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00:05:52,400 --> 00:05:54,520
It's a function of x and t.

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00:05:54,520 --> 00:05:57,370
And multiplying by a
function of x and t gives you

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00:05:57,370 --> 00:05:58,870
a function of x and t.

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So it is an operator.

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Multiplying by a given
function is an operator.

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It changes all the functions.

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But it's a very simple one.

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And that's OK, but
V of x and t should

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be thought as an operator.

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00:06:31,150 --> 00:06:35,590
So, in fact, numbers
can be operator.

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Multiplication by a
number is an operator.

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It adds on every function and
multiplies it by a number,

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so it's also an operator.

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But x has showed up.

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So it's a good time to
try to figure out what

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x has to do with these things.

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So that's what we're
going to do now.

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Let's see what's x
have to do with things.

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OK, so functions
of x, V of x and t

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multiplied by wave functions,
and you think of it

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as an operator.

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So let's make this formal.

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Introduce an operator,
X hat, which,

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acting on functions of
x, multiplies them by x.

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00:07:59,380 --> 00:08:05,580
So the idea is that if you
have the operator X hat acting

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00:08:05,580 --> 00:08:09,580
on the function f
of x, it gives you

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another function, which is
the function x times f of x--

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multiplies by x.

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And you say, wow,
well, why do you

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have to be so careful in
writing something so obvious?

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Well, it's a good
idea to do that,

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because otherwise
you may not quite

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realize there's something
very interesting happening

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with momentum and
position at the same time,

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as we will discover now.

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So we have already
found some operators.

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We have operators
P, x, Hamiltonian,

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which is p squared over 2m.

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And now you could
put V of x hat t.

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00:09:03,090 --> 00:09:09,180
You know, if here you
put V of x hat, anyway,

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whatever x hat does
is multiplied by x.

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00:09:12,510 --> 00:09:15,120
So putting V of x hat here--

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you may want to do
it, but it's optional.

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I think we all know
what we mean by this.

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We're just multiplying
by a function of x.

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Now when you have operators,
operators act on wave functions

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and give you things.

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And we mentioned that
operators are associated

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or analogs of matrices.

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And there's one fundamental
property of matrices.

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The order in which you multiply
them makes a difference.

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So we've introduced
two operators, p and x.

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And we could ask whether
the order of multiplication

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matters or not.

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00:10:01,270 --> 00:10:06,740
And this is the way Heisenberg
was lead to quantum mechanics.

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00:10:06,740 --> 00:10:09,030
Schrodinger wrote
the wave equation.

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00:10:09,030 --> 00:10:13,620
Heisenberg looked at operators
and commutation relations

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00:10:13,620 --> 00:10:14,400
between them.

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And it's another way of
thinking of quantum mechanics

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that we'll use.

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So I want to ask the question,
that if you have p and x

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and you have two operators
acting on a wave function,

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does the order matter,
or it doesn't matter?

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00:10:31,420 --> 00:10:33,180
We need to know that.

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This is the basic
relation between p and x.

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So what is the question?

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00:10:40,150 --> 00:10:42,315
The question is, if I have--

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I'll show it like that--

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x and p acting on a
wave function, phi,

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00:10:53,660 --> 00:11:04,260
minus px acting on a wave
function, do I get 0?

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00:11:04,260 --> 00:11:09,350
Do I get the same result or not?

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00:11:12,880 --> 00:11:15,030
This is our question.

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00:11:15,030 --> 00:11:17,510
We need to understand
these two operators

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and see how they are related.

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So this is a very good question.

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So let's do that computation.

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It's, again, one of
those computation that

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is straightforward.

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But you have to be careful,
because at every stage,

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you have to know very
well what you're doing.

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So if you have two
operators like a and b

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acting on a function,
the meaning of this

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is that you have a acting on
what b acting on phi gives you.

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00:12:01,450 --> 00:12:04,600
That's what it means to
have two things acting.

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Your first act with
the thing on the right.

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You then act on the other one.

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00:12:08,630 --> 00:12:11,210
So let's look at this thing--

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xp phi minus px phi.

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00:12:18,820 --> 00:12:20,620
So for the first
one, you would have

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00:12:20,620 --> 00:12:34,215
x times p hat on phi minus p hat
times x on phi, phi of x and t

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00:12:34,215 --> 00:12:35,670
maybe--

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00:12:35,670 --> 00:12:37,061
phi of x and t.

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OK, now what do we have?

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We have x hat acting on this.

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00:12:48,850 --> 00:12:51,100
And this thing, we
already know what it is--

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00:12:51,100 --> 00:12:56,160
h over i ddx of phi of x and t--

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minus p hat and
x, acting on phi,

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00:13:06,970 --> 00:13:12,562
is little x phi of x and t.

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00:13:15,870 --> 00:13:22,050
Now this is already a
function of x and t.

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00:13:22,050 --> 00:13:27,660
So an x on it will
multiply it by x.

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00:13:27,660 --> 00:13:36,440
So this will be h
over i x ddx of phi.

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00:13:36,440 --> 00:13:40,180
It just multiplies it
by x at this moment--

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00:13:40,180 --> 00:13:47,766
minus here we have h
bar over i ddx of x phi.

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00:13:51,580 --> 00:13:58,550
And now you see that when
this derivative acts on phi,

186
00:13:58,550 --> 00:14:00,560
you get a term
that cancels this.

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00:14:03,720 --> 00:14:07,630
But when it acts on x, it
gives you an extra term. ddx

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00:14:07,630 --> 00:14:12,810
of x is minus h over i phi--

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00:14:15,540 --> 00:14:19,590
or ih phi.

190
00:14:23,910 --> 00:14:27,630
So the derivative
acts on x or an phi.

191
00:14:27,630 --> 00:14:29,600
When it acts on phi,
gives you this term.

192
00:14:29,600 --> 00:14:32,910
When it acts on x, gives you
the thing that is left over.

193
00:14:32,910 --> 00:14:37,380
So actually, let me write
this in a more clear way.

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00:14:37,380 --> 00:14:42,630
If you have an operator,
a linear operator A

195
00:14:42,630 --> 00:14:50,910
plus B acting on a function
phi, that's A phi plus B phi.

196
00:14:50,910 --> 00:14:54,640
You have linear
operators like that.

197
00:14:54,640 --> 00:14:56,520
And we have these things here.

198
00:14:56,520 --> 00:15:03,560
So this is actually equal
to x hat p hat minus p hat

199
00:15:03,560 --> 00:15:05,570
x hat on phi.

200
00:15:12,960 --> 00:15:21,540
That's what it means when
you have operators here.

201
00:15:21,540 --> 00:15:25,940
So look what we got, a
very surprising thing.

202
00:15:25,940 --> 00:15:29,060
xp minus px is an operator.

203
00:15:29,060 --> 00:15:30,830
It wants to act on function.

204
00:15:30,830 --> 00:15:33,730
So we put a function
here to evaluate it.

205
00:15:33,730 --> 00:15:35,870
And that was good.

206
00:15:35,870 --> 00:15:41,410
And when we evaluate it, we got
a number times this function.

207
00:15:41,410 --> 00:15:43,810
So I could say--

208
00:15:43,810 --> 00:15:46,295
I could forget about the phi.

209
00:15:49,190 --> 00:16:02,260
I'm simply right that xp
minus px is equal to ih bar.

210
00:16:02,260 --> 00:16:06,100
And although it looks a little
funny, it's perfectly correct.

211
00:16:06,100 --> 00:16:11,780
This is an equality
between operators--

212
00:16:11,780 --> 00:16:17,160
equality between operators.

213
00:16:17,160 --> 00:16:19,960
On the left-hand side, it's
clear that it's an operator.

214
00:16:19,960 --> 00:16:22,800
On the right-hand side,
it's also an operator,

215
00:16:22,800 --> 00:16:27,480
because a number acts as an
operator on any function it

216
00:16:27,480 --> 00:16:29,370
multiplies by it.

217
00:16:29,370 --> 00:16:34,410
So look what you've
discovered, this commutator.

218
00:16:34,410 --> 00:16:36,180
And that's a notation
that we're going

219
00:16:36,180 --> 00:16:43,050
to use throughout this semester,
the notation of the commutator.

220
00:16:43,050 --> 00:16:45,100
Let's introduce it here.

221
00:16:54,910 --> 00:17:00,700
So if you have two
operators, linear operators,

222
00:17:00,700 --> 00:17:05,520
we define the commutator
to be the product

223
00:17:05,520 --> 00:17:10,670
in the first direction minus the
product in the other direction.

224
00:17:10,670 --> 00:17:21,400
This is called the
commutator of A and B.

225
00:17:21,400 --> 00:17:26,230
So it's an operator,
again, but it shows you

226
00:17:26,230 --> 00:17:31,090
how they are non-trivial, one
with respect to the other.

227
00:17:31,090 --> 00:17:34,810
This is the basis, eventually,
of the uncertainty principle.

228
00:17:34,810 --> 00:17:38,140
x and p having a
commutator of this type

229
00:17:38,140 --> 00:17:40,400
leads to the
uncertainty principle.

230
00:17:40,400 --> 00:17:42,760
So what did we learn?

231
00:17:42,760 --> 00:17:49,330
We learned this
rather famous result,

232
00:17:49,330 --> 00:17:56,250
that the commutator of x and p
in quantum mechanics is ih bar.