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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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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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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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00:03:14,770 --> 00:03:20,980
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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00:03:40,960 --> 00:03:47,305
This is the full
Schrodinger equation.
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00:03:51,660 --> 00:03:54,810
So it's a very simple departure.
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You see, when you
discover the show
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00:03:56,430 --> 00:03:59,240
that the equation for a free
particle, adding the energy
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00:03:59,240 --> 00:04:04,440
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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00:04:22,930 --> 00:04:25,400
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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00:05:00,000 --> 00:05:03,710
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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00:05:14,300 --> 00:05:16,610
suddenly produce
an equation that
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00:05:16,610 --> 00:05:19,610
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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00:05:33,410 --> 00:05:42,120
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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00:05:45,490 --> 00:05:47,660
This is an operator,
acts on a wave
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00:05:47,660 --> 00:05:49,270
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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00:05:58,870 --> 00:06:01,780
So it is an operator.
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00:06:01,780 --> 00:06:05,110
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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00:06:10,060 --> 00:06:16,780
And that's OK, but
V of x and t should
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00:06:16,780 --> 00:06:23,740
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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00:06:41,770 --> 00:06:44,890
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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00:06:50,800 --> 00:06:54,620
x has to do with these things.
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00:06:54,620 --> 00:06:56,240
So that's what we're
going to do now.
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00:07:02,360 --> 00:07:09,070
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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00:07:14,640 --> 00:07:17,500
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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00:08:09,580 --> 00:08:14,890
another function, which is
the function x times f of x--
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00:08:14,890 --> 00:08:17,990
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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00:08:33,020 --> 00:08:35,750
realize there's something
very interesting happening
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00:08:35,750 --> 00:08:38,870
with momentum and
position at the same time,
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as we will discover now.
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00:08:40,880 --> 00:08:45,860
So we have already
found some operators.
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00:08:45,860 --> 00:08:54,070
We have operators
P, x, Hamiltonian,
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00:08:54,070 --> 00:08:57,430
which is p squared over 2m.
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00:08:57,430 --> 00:09:03,090
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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00:09:09,180 --> 00:09:12,510
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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00:09:17,790 --> 00:09:21,000
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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00:09:31,500 --> 00:09:33,510
and give you things.
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00:09:33,510 --> 00:09:37,650
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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And this is the way Heisenberg
was lead to quantum mechanics.
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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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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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00:10:33,180 --> 00:10:37,170
This is the basic
relation between p and x.
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So what is the question?
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The question is, if I have--
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I'll show it like that--
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00:10:47,470 --> 00:10:53,660
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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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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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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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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So let's look at this thing--
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xp phi minus px phi.
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So for the first
one, you would have
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x times p hat on phi minus p hat
times x on phi, phi of x and t
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maybe--
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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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00:13:00,470 --> 00:13:06,970
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,
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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.
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00:14:23,910 --> 00:14:27,630
So the derivative
acts on x or an phi.
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00:14:27,630 --> 00:14:29,600
When it acts on phi,
gives you this term.
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00:14:29,600 --> 00:14:32,910
When it acts on x, gives you
the thing that is left over.
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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.
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00:14:50,910 --> 00:14:54,640
You have linear
operators like that.
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00:14:54,640 --> 00:14:56,520
And we have these things here.
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00:14:56,520 --> 00:15:03,560
So this is actually equal
to x hat p hat minus p hat
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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.
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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.
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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.
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00:15:41,410 --> 00:15:43,810
So I could say--
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00:15:43,810 --> 00:15:46,295
I could forget about the phi.
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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--
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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.
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00:17:34,810 --> 00:17:38,140
x and p having a
commutator of this type
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00:17:38,140 --> 00:17:40,400
leads to the
uncertainty principle.
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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,
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00:17:49,330 --> 00:17:56,250
that the commutator of x and p
in quantum mechanics is ih bar.