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PROFESSOR: Today we begin
with the harmonic oscillator.

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And before we get into
the harmonic oscillator,

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I want to touch on
a few concepts that

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have been mentioned in class
and just elaborate on them.

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It is the issue of nodes,
and how solutions look at,

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and why solutions have
more and more nodes, why

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the ground state has no nodes.

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00:00:52,360 --> 00:00:55,220
This kind of stuff.

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00:00:55,220 --> 00:01:02,600
So these are just a collection
of remarks and an argument

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00:01:02,600 --> 00:01:07,830
for you to understand a
little more intuitively why

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00:01:07,830 --> 00:01:11,620
these properties hold.

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00:01:11,620 --> 00:01:15,430
So one first thing
I want to mention

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is, if you have a Schrodinger
equation for an energy

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eigenstate.

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00:01:21,020 --> 00:01:28,040
Schrodinger equation for
an energy eigenstate.

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00:01:34,180 --> 00:01:43,060
You have an equation of the
from minus h squared over 2m.

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00:01:43,060 --> 00:01:53,230
d second dx squared psi of
x plus v of x psi of x equal

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00:01:53,230 --> 00:01:57,360
e times psi of x.

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00:01:57,360 --> 00:02:00,030
Now the issue of
this equation is

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00:02:00,030 --> 00:02:05,820
that you're trying to solve for
two things at the same time.

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00:02:05,820 --> 00:02:09,550
If you're looking at what
we call bound states, now

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00:02:09,550 --> 00:02:11,570
what is a bound state?

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00:02:11,570 --> 00:02:18,040
A bound state is something that
is not extended all that much.

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00:02:18,040 --> 00:02:22,270
So a bound state will
be a wave function

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00:02:22,270 --> 00:02:29,210
that goes to 0 as the absolute
value of x goes to infinity.

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00:02:29,210 --> 00:02:34,930
So it's a probability function
that certainly doesn't

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00:02:34,930 --> 00:02:36,980
extend all the way to infinity.

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00:02:36,980 --> 00:02:39,090
It just collapses.

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00:02:39,090 --> 00:02:41,000
It's normalizable.

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00:02:41,000 --> 00:02:42,790
So these are bound
states, and you're

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00:02:42,790 --> 00:02:45,560
looking for bound
states of this equation.

39
00:02:45,560 --> 00:02:49,680
And your difficulty is
that you don't know psi,

40
00:02:49,680 --> 00:02:52,160
and you don't know E either.

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00:02:52,160 --> 00:02:57,700
So you have to solve a problem
in which, if you were thinking

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00:02:57,700 --> 00:03:00,570
oh this is just a plain
differential equation,

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00:03:00,570 --> 00:03:03,530
give me the value of E.
We know the potential,

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00:03:03,530 --> 00:03:04,650
just calculate it.

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00:03:04,650 --> 00:03:07,880
That's not the way it
works in quantum mechanics,

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00:03:07,880 --> 00:03:11,450
because you need to have
normalizable solutions.

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00:03:11,450 --> 00:03:15,380
So at the end of the day,
as will be very clear today,

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00:03:15,380 --> 00:03:19,210
this E gets fixed.

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00:03:19,210 --> 00:03:22,740
You cannot get
arbitrary values of E's.

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00:03:22,740 --> 00:03:27,060
So I want to make a couple of
remarks about this equation.

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00:03:27,060 --> 00:03:29,635
Is that there's this
thing that can't happen.

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00:03:34,230 --> 00:03:48,300
Certainly, if v of x is a smooth
potential, then if you observer

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00:03:48,300 --> 00:03:52,602
that the wave function
vanishes at some point,

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00:03:52,602 --> 00:03:55,880
and the derivative
of the wave function

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00:03:55,880 --> 00:03:59,600
vanishes at that same
point, these two things

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00:03:59,600 --> 00:04:05,070
imply that psi of
x is identically 0.

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00:04:05,070 --> 00:04:08,000
And therefore it
means that you really

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00:04:08,000 --> 00:04:09,530
are not interested in that.

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00:04:09,530 --> 00:04:12,430
That's not a solution of
the Schrodinger equation.

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00:04:12,430 --> 00:04:16,530
Psi equals 0 is obviously solves
this, but it's not interesting.

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00:04:16,530 --> 00:04:19,250
It doesn't represent
the particle.

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00:04:19,250 --> 00:04:23,740
So what I claim here
is that, if it happens

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00:04:23,740 --> 00:04:25,540
to be that you're
solving the Schrodinger

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00:04:25,540 --> 00:04:28,035
problem with some
potential that is smooth,

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00:04:28,035 --> 00:04:30,960
you can take derivatives of it.

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00:04:30,960 --> 00:04:34,670
And then you encounter that
the wave function vanishes

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at some point, and its slope
vanishes at that same point.

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Then the wave function
vanishes completely.

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00:04:41,970 --> 00:04:45,120
So you cannot have
a wave function,

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00:04:45,120 --> 00:04:48,210
a psi of x that
does the following.

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00:04:48,210 --> 00:04:51,510
Comes down here.

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00:04:51,510 --> 00:04:54,850
It becomes an inflection
point and goes down.

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00:04:54,850 --> 00:04:56,105
This is not allowed.

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00:05:03,160 --> 00:05:07,610
If the wave function
vanishes at some point,

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then the wave function
is going to do this.

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00:05:11,940 --> 00:05:16,450
It's going to hit at an angle,
because you cannot have that

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00:05:16,450 --> 00:05:20,640
the wave function is 0 and needs
the derivative 0 at the same

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00:05:20,640 --> 00:05:21,460
point.

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00:05:21,460 --> 00:05:23,330
And the reason is simple.

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00:05:23,330 --> 00:05:25,390
I'm not going to
prove it now here.

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00:05:25,390 --> 00:05:29,930
It is that you have a second
order differential equation,

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00:05:29,930 --> 00:05:32,200
and a second order
differential equation

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00:05:32,200 --> 00:05:35,680
is completely determined
by knowing the function

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00:05:35,680 --> 00:05:38,800
at the point and the the
derivative at the point.

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00:05:38,800 --> 00:05:42,570
And if both are 0s, like
the most trivial kind

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00:05:42,570 --> 00:05:46,610
of initial condition, the only
solution consistent with this

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00:05:46,610 --> 00:05:49,610
is psi equals 0 everywhere.

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00:05:49,610 --> 00:05:51,600
So this can't happen.

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00:05:51,600 --> 00:05:54,460
And it's something good
for you to remember.

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00:05:54,460 --> 00:05:57,040
If you have to do a
plot of a wave function,

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00:05:57,040 --> 00:05:59,240
you should never have this.

92
00:05:59,240 --> 00:06:04,855
So this is what we call the
node in a wave function.

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00:06:10,280 --> 00:06:13,780
It's a place where the
wave function vanishes,

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00:06:13,780 --> 00:06:15,800
and the derivative
of the wave function

95
00:06:15,800 --> 00:06:17,360
better not vanish at that point.

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00:06:21,720 --> 00:06:28,590
So this is one claim that
it's not hard to prove,

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00:06:28,590 --> 00:06:30,830
but we just don't try to do it.

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00:06:30,830 --> 00:06:35,270
And there's another claim that
I want you to be aware of.

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00:06:35,270 --> 00:06:43,470
That for bound states
in one dimension,

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00:06:43,470 --> 00:06:50,650
in the kind of thing that we're
doing now in one dimension,

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00:06:50,650 --> 00:06:55,435
no degeneracy is possible.

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00:07:01,410 --> 00:07:03,160
What do I mean by that?

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00:07:03,160 --> 00:07:07,100
You will never find
two bound states

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00:07:07,100 --> 00:07:09,640
of a potential that
are different that

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00:07:09,640 --> 00:07:10,730
have the same energy.

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00:07:10,730 --> 00:07:14,620
It's just absolutely impossible.

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00:07:14,620 --> 00:07:19,076
It's a very wonderful
result, but also I'm

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00:07:19,076 --> 00:07:20,200
not going to prove it here.

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00:07:20,200 --> 00:07:24,070
Maybe it will be given as an
exercise later in the course.

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00:07:24,070 --> 00:07:27,830
And it's discussed
in 805 as well.

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00:07:27,830 --> 00:07:31,540
But that's another statement
that is very important.

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00:07:31,540 --> 00:07:33,630
There's no degeneracy.

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00:07:33,630 --> 00:07:38,610
Now you've looked at
this simple potential,

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00:07:38,610 --> 00:07:45,910
the square well infinite one.

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00:07:45,910 --> 00:07:47,430
And how does it look?

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00:07:47,430 --> 00:07:53,670
You have x going from 0 to a.

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00:07:53,670 --> 00:07:56,220
And the potential is 0.

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00:07:56,220 --> 00:07:58,660
From 0 to a is
infinite otherwise.

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00:07:58,660 --> 00:08:02,540
The particle is bound
to stay inside the two

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00:08:02,540 --> 00:08:04,920
walls of this potential.

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00:08:04,920 --> 00:08:08,830
So in here we've plotted the
potential as a function of x.

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00:08:08,830 --> 00:08:11,270
Here is x.

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00:08:11,270 --> 00:08:13,890
And the wave functions are
things that you know already.

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00:08:13,890 --> 00:08:14,649
Yes?

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00:08:14,649 --> 00:08:17,190
AUDIENCE: Is it true if you have
two wells next to each other

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00:08:17,190 --> 00:08:20,299
that there's still no degeneracy
if it's an infinite barrier?

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00:08:20,299 --> 00:08:22,090
PROFESSOR: If there's
two wells and there's

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00:08:22,090 --> 00:08:24,370
an infinite barrier
between them,

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00:08:24,370 --> 00:08:26,600
it's like having two universes.

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00:08:26,600 --> 00:08:31,250
So it's not really a
one dimensional problem.

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00:08:31,250 --> 00:08:33,870
If you have an infinite
barrier like two worlds

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00:08:33,870 --> 00:08:35,250
that can talk to each other.

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00:08:35,250 --> 00:08:37,750
So yes, then you
would have degeneracy.

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00:08:37,750 --> 00:08:42,570
It's like saying you can have
here one atom of hydrogen

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00:08:42,570 --> 00:08:46,200
or something in one energy
level, here another one.

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00:08:46,200 --> 00:08:47,550
They don't talk to each other.

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00:08:47,550 --> 00:08:49,520
They're degenerate states.

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00:08:49,520 --> 00:08:53,375
But in general, we're talking
about normal potentials

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00:08:53,375 --> 00:08:55,770
that are preferably smooth.

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00:08:55,770 --> 00:08:59,170
Most of these things are true
even if they're not smooth.

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00:08:59,170 --> 00:09:01,140
But it's a little more delicate.

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00:09:01,140 --> 00:09:02,840
But certainly two
potentials that

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00:09:02,840 --> 00:09:05,030
are separated by
an infinite barrier

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00:09:05,030 --> 00:09:10,440
is not part of what we
really want to consider.

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00:09:10,440 --> 00:09:16,870
OK so these wave functions
start with m equals 0, 1, 2,

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00:09:16,870 --> 00:09:25,040
3, and psi ends of x are
square root of 2 over a sin n

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00:09:25,040 --> 00:09:28,340
plus 1 pi x over a.

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00:09:28,340 --> 00:09:30,700
Things that you've seen already.

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00:09:30,700 --> 00:09:36,030
And En, the energies
are ever growing

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00:09:36,030 --> 00:09:41,680
as a function of the integer
n that characterizes them.

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00:09:41,680 --> 00:09:44,650
2ma squared.

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00:09:44,650 --> 00:09:49,750
And the thing that you
notice is that psi 0

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00:09:49,750 --> 00:09:52,790
has technically no nodes.

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00:09:52,790 --> 00:09:55,120
That is to say these
wave functions have

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00:09:55,120 --> 00:09:58,420
to varnish at the end, because
the potential becomes infinite,

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00:09:58,420 --> 00:10:01,340
which means a particle
really can go through.

157
00:10:01,340 --> 00:10:03,370
The wave function
has to be continuous.

158
00:10:03,370 --> 00:10:06,570
There cannot be any wave
function to the left.

159
00:10:06,570 --> 00:10:08,730
So it has to vanish here.

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00:10:08,730 --> 00:10:11,350
These things don't
count as nodes.

161
00:10:11,350 --> 00:10:16,020
It is like a bound state
has to vanish at infinity.

162
00:10:16,020 --> 00:10:18,470
And that's not what
we count the node.

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00:10:18,470 --> 00:10:20,910
A node is somewhere
in the middle

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00:10:20,910 --> 00:10:24,030
of the range of x where
the wave function vanishes.

165
00:10:24,030 --> 00:10:25,640
So this is the ground state.

166
00:10:25,640 --> 00:10:30,750
This is psi zero has no nodes.

167
00:10:30,750 --> 00:10:35,500
Psi one would be something
like that, has one node.

168
00:10:39,170 --> 00:10:42,900
And try the next ones.

169
00:10:42,900 --> 00:10:44,330
They have more and more nodes.

170
00:10:44,330 --> 00:10:47,015
So psi n has n nodes.

171
00:10:51,600 --> 00:10:58,480
And the interesting thing
is that this result actually

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00:10:58,480 --> 00:11:02,550
is true for extremely
general potentials.

173
00:11:02,550 --> 00:11:05,200
You don't have to just
do the square well

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00:11:05,200 --> 00:11:08,240
to see that the ground
state has no nodes.

175
00:11:08,240 --> 00:11:12,580
That first excited state was one
node, and so on and so forth.

176
00:11:12,580 --> 00:11:16,580
It's true in general.

177
00:11:16,580 --> 00:11:18,710
This is actually a
very nice result,

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00:11:18,710 --> 00:11:21,590
but its difficult to prove.

179
00:11:21,590 --> 00:11:26,370
In fact, it's pretty
hard to prove.

180
00:11:26,370 --> 00:11:28,590
So there is a nice argument.

181
00:11:28,590 --> 00:11:32,540
Not 100% rigorous,
but thoroughly nice

182
00:11:32,540 --> 00:11:34,720
and really physical
that I'm going

183
00:11:34,720 --> 00:11:37,890
to present to you why
this result is true.

184
00:11:37,890 --> 00:11:40,030
So let's try to do that.

185
00:11:40,030 --> 00:11:41,430
So here is the general case.

186
00:11:41,430 --> 00:11:46,470
So I'm going to take
a smooth v of x.

187
00:11:46,470 --> 00:11:48,160
That will be of this kind.

188
00:11:48,160 --> 00:11:55,310
The potential, here is
x, and this potential

189
00:11:55,310 --> 00:12:01,210
is going to be like that.

190
00:12:01,210 --> 00:12:02,325
Smooth all over.

191
00:12:05,460 --> 00:12:09,440
And I will actually
have it that it actually

192
00:12:09,440 --> 00:12:12,630
goes to infinity as
x goes to infinity.

193
00:12:12,630 --> 00:12:15,290
Many of these things are
really not necessary,

194
00:12:15,290 --> 00:12:19,610
but it simplifies our life.

195
00:12:24,290 --> 00:12:33,272
OK, so here is a result
that it's known to be true.

196
00:12:33,272 --> 00:12:36,390
If this thing grows to
infinity, the potential never

197
00:12:36,390 --> 00:12:41,210
stops growing, you get
infinite number of bound states

198
00:12:41,210 --> 00:12:42,680
at fixed energies.

199
00:12:42,680 --> 00:12:45,380
One energy, two
energy, three energy,

200
00:12:45,380 --> 00:12:47,330
infinite number of bound states.

201
00:12:50,710 --> 00:12:53,125
Number of bound states.

202
00:12:57,020 --> 00:12:58,550
That's a fact.

203
00:12:58,550 --> 00:13:01,810
I will not try to prove it.

204
00:13:01,810 --> 00:13:04,000
We'll do it for the
harmonic oscillator.

205
00:13:04,000 --> 00:13:05,750
We'll see those infinite
number of states,

206
00:13:05,750 --> 00:13:09,110
but here we can't
prove it easily.

207
00:13:09,110 --> 00:13:11,120
Nevertheless, what
I want to argue

208
00:13:11,120 --> 00:13:15,510
for you is that these
states, as the first one,

209
00:13:15,510 --> 00:13:17,690
will have no nodes.

210
00:13:17,690 --> 00:13:20,490
The second state, the
first excited state,

211
00:13:20,490 --> 00:13:21,710
will have one node.

212
00:13:21,710 --> 00:13:24,290
Next will have two nodes,
three nodes, four nodes.

213
00:13:24,290 --> 00:13:29,050
We want to understand what
is this issue of the nodes.

214
00:13:29,050 --> 00:13:31,270
OK.

215
00:13:31,270 --> 00:13:34,000
You're not trying
to prove everything.

216
00:13:34,000 --> 00:13:36,360
But we're trying to
prove or understand

217
00:13:36,360 --> 00:13:39,120
something that is
very important.

218
00:13:39,120 --> 00:13:40,780
So how do we prove this?

219
00:13:40,780 --> 00:13:44,800
Or how do we understand that
the nodes-- so there will

220
00:13:44,800 --> 00:13:46,880
be an infinite number
of bound states,

221
00:13:46,880 --> 00:13:53,440
psi 0, psi 1, psi 2, up
to psi n, and it goes on.

222
00:13:53,440 --> 00:13:57,485
And psi n has n nodes.

223
00:14:01,410 --> 00:14:02,090
All right.

224
00:14:06,290 --> 00:14:11,030
So what I'm going to do, in
order to understand this,

225
00:14:11,030 --> 00:14:14,230
is I'm going to produce what we
will call screened potentials.

226
00:14:16,930 --> 00:14:19,620
Screened potentials.

227
00:14:23,370 --> 00:14:28,390
I'm going to select the
lowest point of the potential

228
00:14:28,390 --> 00:14:29,990
here for convenience.

229
00:14:29,990 --> 00:14:35,000
And I'm going to call it x
equals 0 is the lowest point.

230
00:14:40,770 --> 00:14:46,860
And the screen potential
will have a parameter a.

231
00:14:46,860 --> 00:14:52,280
It's a potential
which is equal to v

232
00:14:52,280 --> 00:14:57,470
of x if the absolute
value of x is less than a.

233
00:14:57,470 --> 00:15:04,580
And its infinity if the absolute
value of x is greater than a.

234
00:15:04,580 --> 00:15:07,810
So I come here,
and I want to see

235
00:15:07,810 --> 00:15:11,410
this is this
potential, v of x, what

236
00:15:11,410 --> 00:15:14,651
is the screened
potential for sum a?

237
00:15:14,651 --> 00:15:19,170
Well, I wanted colored
chalk, but I don't have it.

238
00:15:19,170 --> 00:15:23,190
I go mark a here minus a.

239
00:15:23,190 --> 00:15:26,600
Here are the points
between x and a.

240
00:15:26,600 --> 00:15:29,096
Absolute value of x less than a.

241
00:15:29,096 --> 00:15:32,290
Throughout this region
the screened potential

242
00:15:32,290 --> 00:15:35,540
is the potential that you have.

243
00:15:35,540 --> 00:15:39,160
Nevertheless, for the
rest, its infinite.

244
00:15:39,160 --> 00:15:42,820
So the screened
potential is this thing.

245
00:15:42,820 --> 00:15:45,750
Is infinite there, and
it's here this thing.

246
00:15:48,470 --> 00:15:52,410
So it's just some potential.

247
00:15:52,410 --> 00:15:55,150
You take it a screen
and you just see

248
00:15:55,150 --> 00:15:58,422
one part of the potential,
and let it go to infinity.

249
00:15:58,422 --> 00:15:59,630
So that's a screen potential.

250
00:16:02,190 --> 00:16:06,200
So now what I'm going
to do is that I'm

251
00:16:06,200 --> 00:16:08,550
going to try to
argue that you could

252
00:16:08,550 --> 00:16:13,590
try to find the bound state
of the screen potential.

253
00:16:13,590 --> 00:16:17,950
Unless you remove the
screen, you will find,

254
00:16:17,950 --> 00:16:20,830
as you let a go to
infinity, you will

255
00:16:20,830 --> 00:16:25,560
find the bound states of
the original potential.

256
00:16:25,560 --> 00:16:27,460
It's reasonable
that that's true,

257
00:16:27,460 --> 00:16:29,240
because as you
remove the screen,

258
00:16:29,240 --> 00:16:31,760
you're letting more of
the potential be exposed,

259
00:16:31,760 --> 00:16:33,730
and more of the
potential be exposed.

260
00:16:33,730 --> 00:16:37,440
And the wave functions
eventually die, so as the time

261
00:16:37,440 --> 00:16:40,570
that you're very far away,
you affect the wave functions

262
00:16:40,570 --> 00:16:41,425
less and less.

263
00:16:44,190 --> 00:16:45,680
So that's the argument.

264
00:16:45,680 --> 00:16:48,260
We're going to try
to argue that we're

265
00:16:48,260 --> 00:16:50,150
going to look at
the bound states

266
00:16:50,150 --> 00:16:54,550
of the screened potentials and
see what happened, whether they

267
00:16:54,550 --> 00:16:58,490
tell us about the bound
states the original potential.

268
00:16:58,490 --> 00:17:02,680
So for this, I'm going
to begin with a screen

269
00:17:02,680 --> 00:17:08,210
potential in which
a goes to 0 and say

270
00:17:08,210 --> 00:17:11,530
that a is equal to
epsilon, very small.

271
00:17:11,530 --> 00:17:13,839
So what potential so do I have?

272
00:17:13,839 --> 00:17:21,390
A very tiny potential here
from epsilon to minus epsilon.

273
00:17:21,390 --> 00:17:26,270
Now I chose the original point
down here to be the minimum.

274
00:17:26,270 --> 00:17:31,300
So actually, the bottom part of
the potential is really flat.

275
00:17:31,300 --> 00:17:35,240
And if you take
epsilon going to 0,

276
00:17:35,240 --> 00:17:38,220
well, the potential
might do this,

277
00:17:38,220 --> 00:17:42,390
but really at the bottom for
sufficiently small epsilon,

278
00:17:42,390 --> 00:17:48,500
this is an infinite square
well with psis to epsilon.

279
00:17:52,900 --> 00:17:57,520
I chose the minimum so that you
don't get something like this.

280
00:17:57,520 --> 00:17:59,310
If it would be a
point with a slope,

281
00:17:59,310 --> 00:18:00,710
this would be an ugly thing.

282
00:18:00,710 --> 00:18:02,460
So let's choose the minimum.

283
00:18:02,460 --> 00:18:05,770
And we have the screen
potential here, and that's it.

284
00:18:05,770 --> 00:18:07,070
Now look what we do.

285
00:18:07,070 --> 00:18:11,065
We say all right, here
there is a ground state.

286
00:18:13,680 --> 00:18:16,060
Very tiny.

287
00:18:16,060 --> 00:18:17,280
Goes like that.

288
00:18:17,280 --> 00:18:18,410
Vanishes here.

289
00:18:18,410 --> 00:18:21,090
Vanishes there.

290
00:18:21,090 --> 00:18:23,920
And has no nodes.

291
00:18:23,920 --> 00:18:24,720
Very tiny.

292
00:18:24,720 --> 00:18:28,190
You know the two 0s are
very close to each other.

293
00:18:28,190 --> 00:18:30,550
And now I'm going
to try to increase

294
00:18:30,550 --> 00:18:35,850
the value of the screen a.

295
00:18:35,850 --> 00:18:43,790
So suppose we've
increased the screen,

296
00:18:43,790 --> 00:18:49,040
and now the potential is here.

297
00:18:49,040 --> 00:18:53,050
And now we have a finite screen.

298
00:18:53,050 --> 00:18:55,060
Here is the potential.

299
00:18:55,060 --> 00:18:57,090
And I look at the wave function.

300
00:18:57,090 --> 00:18:58,520
How it looks.

301
00:18:58,520 --> 00:19:00,910
Here is psi 0.

302
00:19:00,910 --> 00:19:03,480
This ground state psi 0.

303
00:19:06,030 --> 00:19:13,410
Well, since this thing in here,
the potential becomes infinite,

304
00:19:13,410 --> 00:19:15,500
the wave function
still must vanish here

305
00:19:15,500 --> 00:19:18,150
and still must vanish here.

306
00:19:18,150 --> 00:19:23,830
Now just for your
imagination, think of this.

307
00:19:23,830 --> 00:19:28,900
At this stage, it still more
or less looks like this.

308
00:19:28,900 --> 00:19:30,830
Maybe.

309
00:19:30,830 --> 00:19:38,230
Now I'm going to ask, as I
increase, can I produce a node?

310
00:19:38,230 --> 00:19:40,550
And look what's going to happen.

311
00:19:40,550 --> 00:19:44,110
So suppose it might happen
that, as you increase,

312
00:19:44,110 --> 00:19:46,000
suddenly you produce a node.

313
00:19:46,000 --> 00:19:48,257
So here's what I'm saying here.

314
00:19:48,257 --> 00:19:49,340
I'm going to show it here.

315
00:19:49,340 --> 00:19:52,200
Suppose up to this
point, there is no node.

316
00:19:52,200 --> 00:19:56,000
But then when I double
it, when I increase it

317
00:19:56,000 --> 00:20:01,760
to twice the size, when I go
to screen potential like that,

318
00:20:01,760 --> 00:20:05,970
suddenly there is a
node in the middle.

319
00:20:05,970 --> 00:20:08,340
So if there is a
node in the middle,

320
00:20:08,340 --> 00:20:12,660
one thing that could have
happened is that you have this.

321
00:20:15,820 --> 00:20:19,920
And now look what must
have happened then.

322
00:20:19,920 --> 00:20:26,560
As I stretch this, this slope
must have been going down,

323
00:20:26,560 --> 00:20:31,080
and down, and down, until
it flips to the other side

324
00:20:31,080 --> 00:20:33,250
to produce a node here.

325
00:20:33,250 --> 00:20:34,960
It could have
happened on this side,

326
00:20:34,960 --> 00:20:36,810
but it's the same, so
the argument is just

327
00:20:36,810 --> 00:20:38,220
done with this side.

328
00:20:38,220 --> 00:20:41,780
To produce a node you
could have done somehow

329
00:20:41,780 --> 00:20:44,900
the slope here must
have changed sine.

330
00:20:44,900 --> 00:20:49,230
But for that to happen
continuously, at some point

331
00:20:49,230 --> 00:20:51,305
the this slope must have been 0.

332
00:20:54,660 --> 00:20:58,590
But you cannot have
a 0 and 0 slope.

333
00:20:58,590 --> 00:21:04,010
So this thing can't
flip, can't do this.

334
00:21:04,010 --> 00:21:05,810
Another thing that
could have happened

335
00:21:05,810 --> 00:21:09,060
is that when we
are here already,

336
00:21:09,060 --> 00:21:14,080
maybe the wave function
looks like that.

337
00:21:14,080 --> 00:21:17,120
It doesn't flip at the
edges, but produces something

338
00:21:17,120 --> 00:21:18,290
like that.

339
00:21:18,290 --> 00:21:21,590
But the only way this
can happen continuously,

340
00:21:21,590 --> 00:21:25,390
and this potential is
changing continuously,

341
00:21:25,390 --> 00:21:29,110
is for this thing at
some intermediate stage,

342
00:21:29,110 --> 00:21:34,310
as you keep stretching the
screen, this sort of starts

343
00:21:34,310 --> 00:21:37,560
to produce a depression here.

344
00:21:37,560 --> 00:21:42,000
And at some point, to get
here it has to do this.

345
00:21:42,000 --> 00:21:43,530
But it can't do this either.

346
00:21:43,530 --> 00:21:46,940
It cannot vanish and have
derivative like that.

347
00:21:46,940 --> 00:21:49,990
So actually, as you
stretch the screen,

348
00:21:49,990 --> 00:21:54,320
there's no way to
produce a node.

349
00:21:54,320 --> 00:21:57,360
That property forbids it.

350
00:21:57,360 --> 00:22:01,470
So by the time you go and
take the screen to infinity,

351
00:22:01,470 --> 00:22:04,255
this wave function has no nodes.

352
00:22:07,620 --> 00:22:12,100
So that proves it that the
ground state has no nodes.

353
00:22:12,100 --> 00:22:15,300
You could call this
a physicist proof,

354
00:22:15,300 --> 00:22:19,880
which means-- not in
the pejorative way.

355
00:22:19,880 --> 00:22:24,330
It means that it's
reasonable, it's intuitive,

356
00:22:24,330 --> 00:22:28,460
and a mathematician working
hard could make it rigorous.

357
00:22:32,300 --> 00:22:36,280
A bad physicist proof is
one that is a little sloppy

358
00:22:36,280 --> 00:22:39,990
and no mathematician could
fix it and make it work.

359
00:22:39,990 --> 00:22:44,370
So I think this is a good
physics proof in that sense.

360
00:22:44,370 --> 00:22:46,460
Probably you can
construct a real proof,

361
00:22:46,460 --> 00:22:50,410
or based on this, a
very precise proof.

362
00:22:50,410 --> 00:22:53,970
Now look at excited states.

363
00:22:53,970 --> 00:23:00,090
Suppose you take now here
this screen very little,

364
00:23:00,090 --> 00:23:05,375
and now consider the third
excited state, psi three.

365
00:23:08,250 --> 00:23:10,840
I'm sorry, we'll call this psi
2 because it has two nodes.

366
00:23:13,880 --> 00:23:15,565
Well, maybe I should do psi 1.

367
00:23:18,610 --> 00:23:19,710
Psi 1.

368
00:23:19,710 --> 00:23:21,670
One node.

369
00:23:21,670 --> 00:23:23,200
Same thing.

370
00:23:23,200 --> 00:23:27,750
As you increase
it, there's no way

371
00:23:27,750 --> 00:23:29,960
to create another
node continuously.

372
00:23:29,960 --> 00:23:32,620
Because again, you have
to flip at the edges,

373
00:23:32,620 --> 00:23:34,910
or you have to
depress in the middle.

374
00:23:34,910 --> 00:23:38,640
So this one will evolve
to a wave function that

375
00:23:38,640 --> 00:23:43,500
will have one node in
the whole big potential.

376
00:23:43,500 --> 00:23:49,680
Now stayed does that
state have more energy

377
00:23:49,680 --> 00:23:52,450
than the ground state?

378
00:23:52,450 --> 00:23:59,530
Well, it certainly begins with
a small screen with more energy,

379
00:23:59,530 --> 00:24:03,730
because in the square well
psi 1 has more energy.

380
00:24:03,730 --> 00:24:08,570
And that energy should
be clear that it's not

381
00:24:08,570 --> 00:24:12,440
going to go below the
energy of the ground state.

382
00:24:12,440 --> 00:24:13,640
Why?

383
00:24:13,640 --> 00:24:17,220
Because if it went below the
energy of the ground state

384
00:24:17,220 --> 00:24:21,500
slowly, at some point for
some value of the screen,

385
00:24:21,500 --> 00:24:25,160
it would have the same
energy as the ground state.

386
00:24:25,160 --> 00:24:29,960
But no degeneracy is possible
in one dimensional problems.

387
00:24:29,960 --> 00:24:31,750
So that can't happen.

388
00:24:31,750 --> 00:24:32,820
Cannot have that.

389
00:24:32,820 --> 00:24:35,980
So it will always
stay a little higher.

390
00:24:35,980 --> 00:24:39,810
And therefore with one node you
will be a little higher energy.

391
00:24:39,810 --> 00:24:43,390
With two nodes will
be higher and higher.

392
00:24:43,390 --> 00:24:44,570
And that's it.

393
00:24:44,570 --> 00:24:45,940
That's the argument.

394
00:24:45,940 --> 00:24:51,260
Now, we've argued by this
continuous deformation process

395
00:24:51,260 --> 00:24:54,760
that this potential not
only has these bound states,

396
00:24:54,760 --> 00:25:05,100
but this is n nodes and En is
greater than En prime for n

397
00:25:05,100 --> 00:25:08,890
greater than n prime.

398
00:25:08,890 --> 00:25:12,340
So the more nodes,
the more energy.

399
00:25:12,340 --> 00:25:15,920
Pretty nice result,
and that's really

400
00:25:15,920 --> 00:25:19,630
all I wanted to say
about this problem.

401
00:25:19,630 --> 00:25:21,210
Are there any questions?

402
00:25:21,210 --> 00:25:21,710
Any?

403
00:25:29,582 --> 00:25:31,070
OK.

404
00:25:31,070 --> 00:25:34,440
So what we do now is
the harmonic oscillator.

405
00:25:34,440 --> 00:25:39,570
That's going to keep us busy
for the rest of today's lecture.

406
00:25:39,570 --> 00:25:43,170
It's a very interesting problem.

407
00:25:43,170 --> 00:25:49,450
And it's a most famous quantum
mechanics problem in a sense,

408
00:25:49,450 --> 00:25:55,310
because it happens to be useful
in many, many applications.

409
00:25:55,310 --> 00:25:58,070
If you have any
potential-- so what

410
00:25:58,070 --> 00:26:02,010
is the characteristic of
the harmonic oscillator?

411
00:26:02,010 --> 00:26:03,245
Harmonic oscillator.

412
00:26:05,850 --> 00:26:08,720
Oscillator.

413
00:26:08,720 --> 00:26:15,580
Well, the energy operator is p
squared over 2m plus, we write,

414
00:26:15,580 --> 00:26:19,910
one half m omega
squared x squared

415
00:26:19,910 --> 00:26:23,180
where omega is this
omega that you always

416
00:26:23,180 --> 00:26:27,230
think of angular velocity,
or angular frequency.

417
00:26:27,230 --> 00:26:28,870
It's more like
angular frequency.

418
00:26:28,870 --> 00:26:32,670
Omega has units of 1 over time.

419
00:26:32,670 --> 00:26:36,880
It's actually put 2pi over
the period of an oscillation.

420
00:26:36,880 --> 00:26:40,090
And this you know from
classical mechanics.

421
00:26:40,090 --> 00:26:43,730
If you have a harmonic
oscillator of this form, yeah,

422
00:26:43,730 --> 00:26:47,620
it actually oscillates
with this frequency.

423
00:26:47,620 --> 00:26:49,680
And E is the energy
operator, and this

424
00:26:49,680 --> 00:26:53,020
is the energy of the oscillator.

425
00:26:53,020 --> 00:26:55,640
So what defines an oscillator?

426
00:26:55,640 --> 00:27:00,170
It's something in which the
potential energy, this term

427
00:27:00,170 --> 00:27:01,950
is v of x.

428
00:27:01,950 --> 00:27:04,880
v of x is quadratic in x.

429
00:27:04,880 --> 00:27:07,870
That is a harmonic oscillator.

430
00:27:07,870 --> 00:27:10,910
Then you arrange the
constants to make sense.

431
00:27:10,910 --> 00:27:13,260
This has units of
energy, because this

432
00:27:13,260 --> 00:27:16,700
has units of length squared.

433
00:27:16,700 --> 00:27:18,820
1 over time squared.

434
00:27:18,820 --> 00:27:21,210
Length over time
is velocity squared

435
00:27:21,210 --> 00:27:23,140
times mass is kinetic energy.

436
00:27:23,140 --> 00:27:26,680
So this term has
the units of energy.

437
00:27:26,680 --> 00:27:29,440
And you good with that.

438
00:27:29,440 --> 00:27:30,930
And why is this useful?

439
00:27:30,930 --> 00:27:39,190
Because actually in any sort
of arbitrary potential, rather

440
00:27:39,190 --> 00:27:42,030
general potential
at least, whenever

441
00:27:42,030 --> 00:27:47,190
you have a minimum where
the derivative vanishes,

442
00:27:47,190 --> 00:27:50,060
then the second derivative
need not vanish.

443
00:27:50,060 --> 00:27:53,790
Then it's a good
approximation to think

444
00:27:53,790 --> 00:27:57,810
of the potential at the minimum
as a quadratic potential.

445
00:27:57,810 --> 00:28:02,840
It fits the potential
nicely over a good region.

446
00:28:02,840 --> 00:28:06,190
And therefore when you have
two molecules with a bound

447
00:28:06,190 --> 00:28:09,250
or something oscillating,
there is a potential.

448
00:28:09,250 --> 00:28:11,780
It has a minimum at the
equilibrium position.

449
00:28:11,780 --> 00:28:15,060
And the oscillations
are governed

450
00:28:15,060 --> 00:28:18,110
by some harmonic oscillator.

451
00:28:18,110 --> 00:28:21,460
When you have photons
in space time traveling,

452
00:28:21,460 --> 00:28:23,810
there is a set of
harmonic oscillators

453
00:28:23,810 --> 00:28:27,420
that correspond to photons.

454
00:28:27,420 --> 00:28:28,830
Many, many applications.

455
00:28:28,830 --> 00:28:32,450
Endless amount of applications
for the harmonic oscillator.

456
00:28:32,450 --> 00:28:36,080
So we really want to
understand this system quantum

457
00:28:36,080 --> 00:28:37,570
mechanically.

458
00:28:37,570 --> 00:28:39,870
And what does that mean?

459
00:28:39,870 --> 00:28:43,790
Is that we really want
to calculate and solve

460
00:28:43,790 --> 00:28:45,790
the Schrodinger equation.

461
00:28:45,790 --> 00:28:49,570
This is our first step in
understanding the system.

462
00:28:49,570 --> 00:28:51,060
There's going to
be a lot of work

463
00:28:51,060 --> 00:28:54,320
to be done even once we have
the solutions of the Schrodinger

464
00:28:54,320 --> 00:28:54,940
equation.

465
00:28:54,940 --> 00:28:57,520
But the first thing
is to figure out

466
00:28:57,520 --> 00:29:01,530
what are the energy
eigenstates or the solutions

467
00:29:01,530 --> 00:29:04,500
of the Schrodinger
equation for this problem.

468
00:29:04,500 --> 00:29:08,750
So notice that here
in this problem

469
00:29:08,750 --> 00:29:12,910
there's an energy quantity.

470
00:29:16,160 --> 00:29:18,520
Remember, when you have
a harmontonian like that,

471
00:29:18,520 --> 00:29:22,000
and people say so what is
the ground state energy?

472
00:29:22,000 --> 00:29:25,390
Well, have to find the
ground state wave function.

473
00:29:25,390 --> 00:29:26,810
Have to do things.

474
00:29:26,810 --> 00:29:28,910
Give me an hour, I'll find it.

475
00:29:28,910 --> 00:29:29,900
And all that.

476
00:29:29,900 --> 00:29:33,170
But if you want an
approximate value,

477
00:29:33,170 --> 00:29:37,590
dimensional analysis will do it,
roughly what is it going to be.

478
00:29:37,590 --> 00:29:41,640
Well, with this constant how
do you produce an energy?

479
00:29:41,640 --> 00:29:44,730
Well, you remember
what Einstein did,

480
00:29:44,730 --> 00:29:51,020
and you know that h bar
omega has units of energy.

481
00:29:51,020 --> 00:29:55,320
So that's an energy associated
with Lagrangian energy

482
00:29:55,320 --> 00:29:57,150
like quantity.

483
00:29:57,150 --> 00:30:00,340
And we expect that
that energy is

484
00:30:00,340 --> 00:30:02,030
going to be the relevant energy.

485
00:30:02,030 --> 00:30:04,070
And in fact, we'll find
that the ground state

486
00:30:04,070 --> 00:30:06,010
energy is just one half of that.

487
00:30:08,870 --> 00:30:11,520
There's another quantity
that may be interesting.

488
00:30:11,520 --> 00:30:13,640
How about the length?

489
00:30:13,640 --> 00:30:16,205
How do you construct a
length from these quantities?

490
00:30:20,620 --> 00:30:26,070
Well, you can start
doing m omega h bar

491
00:30:26,070 --> 00:30:28,210
and put powers and struggle.

492
00:30:28,210 --> 00:30:31,100
I hate doing that.

493
00:30:31,100 --> 00:30:34,770
I always try to find some
way of doing it and avoiding

494
00:30:34,770 --> 00:30:35,780
that thing.

495
00:30:35,780 --> 00:30:42,030
So I know that
energies go like h

496
00:30:42,030 --> 00:30:47,240
over h squared over
m length squared.

497
00:30:47,240 --> 00:30:51,550
So I'm going to call
the length a quantity a.

498
00:30:51,550 --> 00:30:54,210
So ma squared.

499
00:30:54,210 --> 00:30:55,615
That has units of energy.

500
00:30:59,740 --> 00:31:01,917
And you should remember
that because energy

501
00:31:01,917 --> 00:31:07,790
is b squared over 2m,
and b by De Broglie

502
00:31:07,790 --> 00:31:10,910
is h bar over sub lamda.

503
00:31:10,910 --> 00:31:15,360
So h bar squared, lambda
squared, and m here,

504
00:31:15,360 --> 00:31:16,780
that's units of energy.

505
00:31:16,780 --> 00:31:18,570
So that's a length.

506
00:31:18,570 --> 00:31:20,970
On the other hand,
we have another way

507
00:31:20,970 --> 00:31:24,620
to construct an energy
is with this thing,

508
00:31:24,620 --> 00:31:27,190
m omega squared length squared.

509
00:31:27,190 --> 00:31:32,840
So that's also m omega
squared a squared.

510
00:31:32,840 --> 00:31:34,880
That's another energy.

511
00:31:34,880 --> 00:31:40,680
So from this equation I
find that a to the fourth

512
00:31:40,680 --> 00:31:46,160
is h squared over m
squared omega squared.

513
00:31:46,160 --> 00:31:49,390
And it's a little
complicated, so a squared

514
00:31:49,390 --> 00:31:52,495
is h bar over m omega.

515
00:31:55,360 --> 00:31:58,640
So that's a length.

516
00:31:58,640 --> 00:31:59,690
Length squared.

517
00:31:59,690 --> 00:32:01,480
I don't want to take
the square root.

518
00:32:01,480 --> 00:32:04,130
We can leave it
for a moment there.

519
00:32:04,130 --> 00:32:08,680
But that's important because
of that's a length scale.

520
00:32:08,680 --> 00:32:11,780
And if somebody would ask
you in the ground state,

521
00:32:11,780 --> 00:32:14,760
how far is this
particle oscillating,

522
00:32:14,760 --> 00:32:19,940
you would say probably
about a square root of this.

523
00:32:19,940 --> 00:32:27,710
Would be a natural answer
and probably about right.

524
00:32:27,710 --> 00:32:33,310
So OK, energy and
units is very important

525
00:32:33,310 --> 00:32:35,290
to begin your analysis.

526
00:32:35,290 --> 00:32:38,440
So what is the
Schrodinger equation?

527
00:32:38,440 --> 00:32:41,030
The Schrodinger
equation for this thing

528
00:32:41,030 --> 00:32:48,590
is going to be minus h
squared over 2m, d second psi,

529
00:32:48,590 --> 00:32:54,580
dx squared plus the potential,
one half m omega squared

530
00:32:54,580 --> 00:33:00,640
x squared psi is equal E psi.

531
00:33:00,640 --> 00:33:08,150
And the big problem is I don't
know psi and I don't know E.

532
00:33:08,150 --> 00:33:11,550
Now there's so many elegant
ways of solving the harmonic

533
00:33:11,550 --> 00:33:12,050
oscillator.

534
00:33:15,020 --> 00:33:18,222
You will see those next lecture.

535
00:33:18,222 --> 00:33:20,300
Allan Adams will be back here.

536
00:33:20,300 --> 00:33:25,850
But we all have to go
through once in your life

537
00:33:25,850 --> 00:33:30,670
through the direct, uninspired
method of solving it.

538
00:33:33,680 --> 00:33:38,390
Because most of the times
when you have a new problem,

539
00:33:38,390 --> 00:33:42,130
you will not come up with
a beautiful, elegant method

540
00:33:42,130 --> 00:33:44,830
to avoid solving the
differential equation.

541
00:33:44,830 --> 00:33:47,430
You will have to struggle with
the differential equation.

542
00:33:47,430 --> 00:33:51,220
So today we struggle with
the differential equation.

543
00:33:51,220 --> 00:33:53,080
We're going to just do it.

544
00:33:53,080 --> 00:33:56,600
And I'm going to do it slow
enough and in detail enough

545
00:33:56,600 --> 00:33:59,380
that I hope you
follow everything.

546
00:33:59,380 --> 00:34:01,280
I'll just keep a
couple of things,

547
00:34:01,280 --> 00:34:06,210
but it will be one line
computations that I will skip.

548
00:34:06,210 --> 00:34:10,949
So this equation is some sort
of fairly difficult thing.

549
00:34:10,949 --> 00:34:15,750
And it's complicated and
made fairly unpleasant

550
00:34:15,750 --> 00:34:18,960
by the presence of
all these constants.

551
00:34:18,960 --> 00:34:23,159
What kind of equation is that
with all these constants?

552
00:34:23,159 --> 00:34:26,719
They shouldn't be there,
all this constants, in fact.

553
00:34:26,719 --> 00:34:31,159
So this is the first step,
cleaning up the equation.

554
00:34:31,159 --> 00:34:32,510
We have to clean it up.

555
00:34:32,510 --> 00:34:33,010
Why?

556
00:34:33,010 --> 00:34:37,130
Because the nice
functions in life like y

557
00:34:37,130 --> 00:34:41,850
double prime is equal to
minus y have no units.

558
00:34:41,850 --> 00:34:44,909
The derivatives create no units.

559
00:34:44,909 --> 00:34:49,639
y has the same units of that,
and the solution is sine of x,

560
00:34:49,639 --> 00:34:52,400
where x must have no units,
because you cannot find

561
00:34:52,400 --> 00:34:55,960
the sine of one centimeter.

562
00:34:55,960 --> 00:34:59,400
So this thing, we should
have the same thing here.

563
00:34:59,400 --> 00:35:00,720
No units anywhere.

564
00:35:04,100 --> 00:35:05,830
So how can we do that?

565
00:35:05,830 --> 00:35:08,600
This is an absolutely
necessary first step.

566
00:35:08,600 --> 00:35:11,120
If you're going to be
carrying all these constants,

567
00:35:11,120 --> 00:35:13,610
you'll get nowhere.

568
00:35:13,610 --> 00:35:16,590
So we have to clean it up.

569
00:35:16,590 --> 00:35:18,520
So what I'm going
to try to see is

570
00:35:18,520 --> 00:35:22,260
that look, here is
psi, psi, and psi.

571
00:35:22,260 --> 00:35:26,750
So suppose I do the
following thing,

572
00:35:26,750 --> 00:35:30,050
that I will clean
up the right hand

573
00:35:30,050 --> 00:35:37,960
side by dividing by something
with units of energy.

574
00:35:37,960 --> 00:35:42,340
So I'm going to do
the following way.

575
00:35:42,340 --> 00:35:52,610
I'm going to divide all
by 1 over h bar omega.

576
00:35:52,610 --> 00:35:56,190
And this 2 I'm going
to multiply by 2.

577
00:35:56,190 --> 00:36:07,760
So multiply by 2
over h bar omega.

578
00:36:07,760 --> 00:36:11,190
So what do I achieve
with that first step?

579
00:36:11,190 --> 00:36:14,550
I achieve that
these 2s disappear.

580
00:36:14,550 --> 00:36:16,540
Well, that's not too bad.

581
00:36:16,540 --> 00:36:19,270
Not that great either, I think.

582
00:36:19,270 --> 00:36:23,050
But in the right hand side,
this has units of energy.

583
00:36:23,050 --> 00:36:26,490
And the right hand side will
not have units of energy.

584
00:36:26,490 --> 00:36:27,940
So what do we get here?

585
00:36:27,940 --> 00:36:29,510
So we get minus.

586
00:36:29,510 --> 00:36:39,280
The h becomes an h alone over--
the m disappears-- so m omega.

587
00:36:39,280 --> 00:36:44,000
The second psi the x squared.

588
00:36:44,000 --> 00:36:49,860
The 1/2 disappeared,
so m omega over h

589
00:36:49,860 --> 00:37:01,900
bar x squared psi equals
2 E over h bar omega psi.

590
00:37:01,900 --> 00:37:05,760
It looks actually
quite better already.

591
00:37:05,760 --> 00:37:07,070
Should agree with that.

592
00:37:07,070 --> 00:37:14,040
It looks a lot nicer Now
I can use a name for this.

593
00:37:14,040 --> 00:37:18,650
I want to call this
the dimensionless value

594
00:37:18,650 --> 00:37:19,750
of the energy.

595
00:37:19,750 --> 00:37:23,120
So a calligraphic e.

596
00:37:23,120 --> 00:37:25,510
It has no units.

597
00:37:25,510 --> 00:37:29,850
It's telling me if I find some
energy, that that energy really

598
00:37:29,850 --> 00:37:34,970
is this number, this pure
number is how many times bigger

599
00:37:34,970 --> 00:37:39,650
is e with respect
to h omega over 2.

600
00:37:39,650 --> 00:37:43,350
So I'll write this now as e psi.

601
00:37:43,350 --> 00:37:44,540
And look what I have.

602
00:37:44,540 --> 00:37:48,160
I have no units here.

603
00:37:48,160 --> 00:37:50,480
And I have a psi.

604
00:37:50,480 --> 00:37:52,140
And I have a psi.

605
00:37:52,140 --> 00:37:56,020
But things have
worked out already.

606
00:37:56,020 --> 00:38:00,000
Look, the same factor
here, h over m omega

607
00:38:00,000 --> 00:38:02,530
is upside down here.

608
00:38:02,530 --> 00:38:06,670
And this factor has
units of length squared.

609
00:38:06,670 --> 00:38:12,000
Length squared times d d
length squared has no units.

610
00:38:12,000 --> 00:38:14,320
And here's 1 over
length squared.

611
00:38:17,880 --> 00:38:20,620
1 over length squared
times length squared.

612
00:38:20,620 --> 00:38:22,660
So things have worked out.

613
00:38:22,660 --> 00:38:28,490
And we can now
simply say x is going

614
00:38:28,490 --> 00:38:32,560
to be equal to au,
a new variable.

615
00:38:32,560 --> 00:38:35,450
This is going to be your new
variable for your differential

616
00:38:35,450 --> 00:38:41,480
equation in which is this thing.

617
00:38:41,480 --> 00:38:49,570
And then this
differential equation

618
00:38:49,570 --> 00:38:53,025
really has cleaned
up perfectly well.

619
00:38:57,020 --> 00:38:59,450
So how does it look now?

620
00:38:59,450 --> 00:39:05,220
Well, it's all gone
actually, because if you

621
00:39:05,220 --> 00:39:14,050
have x equals au, d dx by
chain rule is 1 over a d du.

622
00:39:17,500 --> 00:39:19,440
And to derivatives
this with respect

623
00:39:19,440 --> 00:39:29,790
to x it's 1 over a squared
times the d second du squared.

624
00:39:29,790 --> 00:39:32,590
And this thing is a squared.

625
00:39:32,590 --> 00:39:36,470
So actually you
cancel this factor.

626
00:39:36,470 --> 00:39:39,360
And when I write x
equals to au, you

627
00:39:39,360 --> 00:39:41,205
get an a squared times this.

628
00:39:41,205 --> 00:39:43,950
And a squared times this is 1.

629
00:39:43,950 --> 00:39:46,190
So your differential
equations has

630
00:39:46,190 --> 00:39:53,230
become minus the
second psi du squared,

631
00:39:53,230 --> 00:39:56,740
where u is a dimensionless
quantity, because this has

632
00:39:56,740 --> 00:39:58,820
units of length, this
has units of length.

633
00:39:58,820 --> 00:40:01,640
No units here.

634
00:40:01,640 --> 00:40:02,720
You have no units.

635
00:40:02,720 --> 00:40:07,580
So minus d second
du squared plus u

636
00:40:07,580 --> 00:40:12,805
squared psi is equal to e psi.

637
00:40:18,320 --> 00:40:21,890
Much nicer.

638
00:40:21,890 --> 00:40:24,770
This is an equation
we can think about

639
00:40:24,770 --> 00:40:27,830
without being distracted
by this endless amount

640
00:40:27,830 --> 00:40:29,656
of little trivialities.

641
00:40:33,690 --> 00:40:38,760
But still we haven't
solved it, and how are we

642
00:40:38,760 --> 00:40:42,290
going to solve this equation?

643
00:40:42,290 --> 00:40:46,860
So let's again think
of what should happen.

644
00:40:46,860 --> 00:40:51,780
Somehow it should happen
that these e's get fixed.

645
00:40:51,780 --> 00:40:55,750
And there is some solution
just for some values of e's.

646
00:40:55,750 --> 00:41:00,090
It's not obvious at this stage
how that is going to happen.

647
00:41:00,090 --> 00:41:00,940
Yes?

648
00:41:00,940 --> 00:41:01,856
AUDIENCE: [INAUDIBLE].

649
00:41:06,290 --> 00:41:12,220
PROFESSOR: Here for example,
let me do this term.

650
00:41:12,220 --> 00:41:18,200
h bar over m omega is minus,
from that equation, a squared.

651
00:41:18,200 --> 00:41:26,640
But dx squared is 1 over
a squared d du squared.

652
00:41:26,640 --> 00:41:29,820
So a squared cancels.

653
00:41:29,820 --> 00:41:33,380
And here the x is equal
a squared times u,

654
00:41:33,380 --> 00:41:35,585
so again cancels.

655
00:41:39,060 --> 00:41:44,640
OK so what is the problem here?

656
00:41:44,640 --> 00:41:48,150
The problem is that most likely
what is going to go wrong

657
00:41:48,150 --> 00:41:51,760
is that this solution for
arbitrary values of e's is

658
00:41:51,760 --> 00:41:55,500
going to diverge at
infinity, and you're never

659
00:41:55,500 --> 00:41:57,810
going to be able
to normalize it.

660
00:41:57,810 --> 00:42:02,050
So let's try to understand
how the solution looks

661
00:42:02,050 --> 00:42:04,010
as we go to infinity.

662
00:42:04,010 --> 00:42:05,710
So this is the first
thing you should

663
00:42:05,710 --> 00:42:07,520
do with an equation like that.

664
00:42:07,520 --> 00:42:13,880
How does this solution
look as u goes to infinity?

665
00:42:13,880 --> 00:42:18,400
Now we may not be able to solve
it exactly in that case either,

666
00:42:18,400 --> 00:42:22,930
but we're going to gain
insight into what's happening.

667
00:42:22,930 --> 00:42:25,940
So here it is.

668
00:42:25,940 --> 00:42:30,950
When u goes to infinity,
this term, whatever psi is,

669
00:42:30,950 --> 00:42:33,190
this term is much
bigger than that,

670
00:42:33,190 --> 00:42:37,060
because we're presumably
working with some fixed energy

671
00:42:37,060 --> 00:42:38,810
that we still don't
know what it is,

672
00:42:38,810 --> 00:42:42,500
but it's a fixed number and,
for you, sufficiently large.

673
00:42:42,500 --> 00:42:44,430
This is going to dominate.

674
00:42:44,430 --> 00:42:46,620
So the equation
that we're trying

675
00:42:46,620 --> 00:42:53,840
to solve as u goes to infinity,
the equation sort of becomes

676
00:42:53,840 --> 00:42:58,240
psi double prime-- prime
is for two derivatives--

677
00:42:58,240 --> 00:43:02,330
is equal to u squared psi.

678
00:43:08,860 --> 00:43:13,270
OK, so how do we get
an idea what solves

679
00:43:13,270 --> 00:43:16,020
this is not all that obvious.

680
00:43:16,020 --> 00:43:19,720
It's certainly not a power of u,
because when you differentiate

681
00:43:19,720 --> 00:43:22,640
the power of u, you
lower the degree

682
00:43:22,640 --> 00:43:25,650
rather than increase the degree.

683
00:43:25,650 --> 00:43:29,200
So what function increases
degree as you differentiate?

684
00:43:29,200 --> 00:43:32,350
It's not the trivial function.

685
00:43:32,350 --> 00:43:34,420
Cannot be a polynomial.

686
00:43:34,420 --> 00:43:36,150
If it could be
even a polynomial,

687
00:43:36,150 --> 00:43:39,230
if you take two derivatives,
it kind cannot be equal to x

688
00:43:39,230 --> 00:43:40,440
squared times a polynomial.

689
00:43:40,440 --> 00:43:43,320
It's sort of upside down.

690
00:43:43,320 --> 00:43:46,390
So if you think about
it for a little while,

691
00:43:46,390 --> 00:43:48,180
you don't have an
exact solution,

692
00:43:48,180 --> 00:43:51,650
but you would imagine
that something like this

693
00:43:51,650 --> 00:43:55,780
would do it, an e
to the u squared.

694
00:43:55,780 --> 00:43:58,280
Because an e to
the u squared, when

695
00:43:58,280 --> 00:44:04,390
you differentiate with respect
to us, you produce a u down.

696
00:44:04,390 --> 00:44:05,830
When you one derivative.

697
00:44:05,830 --> 00:44:07,420
When you take
another derivative,

698
00:44:07,420 --> 00:44:09,650
well, it's more complicated,
but one term you

699
00:44:09,650 --> 00:44:11,980
will produce another u down.

700
00:44:11,980 --> 00:44:16,670
So that probably is quite good.

701
00:44:16,670 --> 00:44:18,290
So let's try that.

702
00:44:18,290 --> 00:44:21,310
Let's try to see if we
have something like that.

703
00:44:21,310 --> 00:44:22,890
So I will try something.

704
00:44:26,900 --> 00:44:33,290
I'll try psi equals 2.

705
00:44:37,420 --> 00:44:41,480
I'm going to try
the following thing.

706
00:44:41,480 --> 00:44:49,250
e to the alpha u squared over
2 where alpha is a number.

707
00:44:49,250 --> 00:44:50,590
I don't know how much it is.

708
00:44:50,590 --> 00:44:53,740
Alpha is some number.

709
00:44:53,740 --> 00:44:56,950
Now could try this
alone, but I actually

710
00:44:56,950 --> 00:45:01,860
want to emphasize
to you that if this

711
00:45:01,860 --> 00:45:05,090
is the behavior
near infinity, it

712
00:45:05,090 --> 00:45:07,430
won't make any difference
if you put here,

713
00:45:07,430 --> 00:45:12,800
for example, something
like u to the power k.

714
00:45:12,800 --> 00:45:17,250
It will also be
roughly a solution.

715
00:45:17,250 --> 00:45:20,660
So let's see that.

716
00:45:20,660 --> 00:45:22,475
So for that I have
to differentiate.

717
00:45:27,290 --> 00:45:31,460
And let's see what we get.

718
00:45:31,460 --> 00:45:37,850
So we're trying to see how the
function behaves far, far away.

719
00:45:37,850 --> 00:45:39,880
You might say well
look, probably

720
00:45:39,880 --> 00:45:41,850
that alpha should be negative.

721
00:45:41,850 --> 00:45:43,940
But let's see what
the equation tells us

722
00:45:43,940 --> 00:45:46,760
before we put anything in there.

723
00:45:46,760 --> 00:45:53,270
So if I do psi prime,
you would get what?

724
00:45:53,270 --> 00:45:54,900
You would get one
term that would

725
00:45:54,900 --> 00:46:02,170
be alpha u times this u to
the k into the alpha u squared

726
00:46:02,170 --> 00:46:02,760
over 2.

727
00:46:02,760 --> 00:46:06,580
I differentiated
the exponential.

728
00:46:06,580 --> 00:46:08,850
I differentiated
the exponential.

729
00:46:08,850 --> 00:46:10,640
And then you would
get a term where

730
00:46:10,640 --> 00:46:12,260
you differentiate the power.

731
00:46:12,260 --> 00:46:17,120
So you get ku to the k
minus 1 into the alpha u

732
00:46:17,120 --> 00:46:19,470
squared over 2.

733
00:46:19,470 --> 00:46:23,120
If you take a second
derivative, well, I

734
00:46:23,120 --> 00:46:25,620
can differentiate the
exponential again,

735
00:46:25,620 --> 00:46:28,840
so I will get alpha
u now squared,

736
00:46:28,840 --> 00:46:31,690
because each derivative
of this exponent

737
00:46:31,690 --> 00:46:35,120
produces a factor of alpha u.

738
00:46:35,120 --> 00:46:40,396
u to the k into the
alpha u squared over 2.

739
00:46:40,396 --> 00:46:50,240
And a couple more terms that
they all have less powers of u,

740
00:46:50,240 --> 00:46:56,020
because this term
has u to the k plus--

741
00:46:56,020 --> 00:46:58,270
already has u to the k plus 1.

742
00:46:58,270 --> 00:47:00,390
And this has u to the k minus 1.

743
00:47:00,390 --> 00:47:03,010
They differ by two powers of u.

744
00:47:03,010 --> 00:47:08,930
So for illustration,
please, if you want, do it.

745
00:47:08,930 --> 00:47:12,840
Three lines, you should skip
three lines in your notebook

746
00:47:12,840 --> 00:47:17,510
if you're taking notes
and get the following.

747
00:47:17,510 --> 00:47:21,080
No point in me doing
this algebra here.

748
00:47:21,080 --> 00:47:23,810
Alpha u squared over 2.

749
00:47:23,810 --> 00:47:28,550
Because actually it's
not all that important.

750
00:47:28,550 --> 00:47:37,840
Over alpha 1 over u squared
plus k minus 1 over alpha

751
00:47:37,840 --> 00:47:41,680
squared 1 over u to the fourth.

752
00:47:41,680 --> 00:47:42,840
That's all you get.

753
00:47:49,510 --> 00:47:56,510
Look, this is alpha
squared u squared

754
00:47:56,510 --> 00:48:01,260
times psi times these things.

755
00:48:01,260 --> 00:48:07,910
1 plus 2 k plus 1 over
alpha 1 over u squared.

756
00:48:07,910 --> 00:48:12,620
So when u goes to
infinity, your solution

757
00:48:12,620 --> 00:48:18,200
works, because these
thing's are negligible.

758
00:48:18,200 --> 00:48:21,400
So you get a number
times u squared.

759
00:48:21,400 --> 00:48:23,985
That is the equation you are
trying to solve up there.

760
00:48:27,020 --> 00:48:30,650
And therefore, you get
that the equation if alpha

761
00:48:30,650 --> 00:48:34,240
squared is equal to 1.

762
00:48:34,240 --> 00:48:40,050
And that means and really that
alpha can be plus minus 1.

763
00:48:40,050 --> 00:48:45,270
And roughly this
solution near infinity,

764
00:48:45,270 --> 00:48:48,990
probably there's two solutions.

765
00:48:51,820 --> 00:48:54,390
This is a second order
differential equation,

766
00:48:54,390 --> 00:48:57,910
so even near infinity there
should be two solutions.

767
00:48:57,910 --> 00:49:03,440
So we expect as u goes
to infinity psi of u

768
00:49:03,440 --> 00:49:09,290
will be some constant A
times u to the k times

769
00:49:09,290 --> 00:49:13,670
e to the minus u squared over 2.

770
00:49:13,670 --> 00:49:16,400
That's where alpha
equal minus 1.

771
00:49:16,400 --> 00:49:24,080
Plus Bu to the k into the
plus u squared over 2.

772
00:49:24,080 --> 00:49:26,340
And what is k?

773
00:49:26,340 --> 00:49:28,180
Well, we don't know what is k.

774
00:49:28,180 --> 00:49:30,700
It seems to work for all k.

775
00:49:30,700 --> 00:49:36,000
That may seem a little
confusing now, but don't worry.

776
00:49:36,000 --> 00:49:42,280
We'll see other things
happening here very soon.

777
00:49:42,280 --> 00:49:45,030
So look at what has happened.

778
00:49:45,030 --> 00:49:48,300
We've identified that most
likely your wave function

779
00:49:48,300 --> 00:49:50,920
is going to look like
this at infinity.

780
00:49:50,920 --> 00:49:54,540
So we're going to want to
this part not to be present.

781
00:49:54,540 --> 00:49:57,210
So presumably we're going
to want a solution that

782
00:49:57,210 --> 00:50:00,590
just has this, because
this is normalizable.

783
00:50:00,590 --> 00:50:05,930
The integral of any power times
a Gaussian is convergence.

784
00:50:05,930 --> 00:50:07,440
So this can be normalized.

785
00:50:07,440 --> 00:50:11,830
The Gaussian falls so
fast that any power can

786
00:50:11,830 --> 00:50:14,490
be integrated
against a Gaussian.

787
00:50:14,490 --> 00:50:17,660
Any power however big
doesn't grow big enough

788
00:50:17,660 --> 00:50:19,770
to compensate a Gaussian.

789
00:50:19,770 --> 00:50:21,655
It's impossible to
compensate a Gaussian.

790
00:50:25,210 --> 00:50:26,525
So we hope for this.

791
00:50:31,690 --> 00:50:34,700
But we want to
translate what we've

792
00:50:34,700 --> 00:50:38,630
learned into some
technical advantage

793
00:50:38,630 --> 00:50:40,430
in solving the
differential equation,

794
00:50:40,430 --> 00:50:44,960
because, after all, we wanted
be insight how it looks far way,

795
00:50:44,960 --> 00:50:48,100
but we wanted to solve
the differential equation.

796
00:50:48,100 --> 00:50:50,790
So how can we use
this insight we now

797
00:50:50,790 --> 00:50:57,560
have to simplify the solution
of the differential equation?

798
00:50:57,560 --> 00:51:03,000
The idea is to change
variables a little bit.

799
00:51:03,000 --> 00:51:14,660
So write psi of u to be
equal to h of u times e

800
00:51:14,660 --> 00:51:17,450
to the minus u squared over 2.

801
00:51:21,670 --> 00:51:23,780
Now you're going to say
wait, what are you doing?

802
00:51:23,780 --> 00:51:26,190
Are you making an
approximation now

803
00:51:26,190 --> 00:51:28,990
that this is what is
going to look far away?

804
00:51:28,990 --> 00:51:32,710
Or what are you putting there?

805
00:51:32,710 --> 00:51:35,050
I'm not making
any approximation.

806
00:51:35,050 --> 00:51:40,340
I'm just saying whatever
pis is, it can always

807
00:51:40,340 --> 00:51:42,100
be written in this way.

808
00:51:42,100 --> 00:51:42,600
Why?

809
00:51:42,600 --> 00:51:45,440
Because if you have
a psi of u, you

810
00:51:45,440 --> 00:51:51,840
can write it as psi of u
over e to the minus u squared

811
00:51:51,840 --> 00:51:55,940
over 2 times e minus
u squared over 2.

812
00:51:55,940 --> 00:51:59,210
Very trivially this
can always be done.

813
00:51:59,210 --> 00:52:02,640
As long as we say
that h is arbitrary,

814
00:52:02,640 --> 00:52:07,650
there's nothing,
no constraint here.

815
00:52:07,650 --> 00:52:10,040
I have not assume
anything, nothing.

816
00:52:10,040 --> 00:52:16,360
I'm just hoping that I have a
differential equation for psi.

817
00:52:16,360 --> 00:52:20,850
That because this is a very
clever factor, the differential

818
00:52:20,850 --> 00:52:24,500
equation for h will be simpler.

819
00:52:24,500 --> 00:52:30,010
Because part of the dependence
has been taken over.

820
00:52:30,010 --> 00:52:34,700
So maybe h, for example, could
be now a polynomial solution,

821
00:52:34,700 --> 00:52:38,520
because this product
has been taken care.

822
00:52:38,520 --> 00:52:43,580
So the hope is that by
writing this equation

823
00:52:43,580 --> 00:52:46,290
it will become an
equation for h of u,

824
00:52:46,290 --> 00:52:49,800
and that equation
will be simpler.

825
00:52:49,800 --> 00:52:52,070
So will it be simpler?

826
00:52:52,070 --> 00:52:56,340
Well, here again this
is not difficult.

827
00:52:56,340 --> 00:53:00,060
You're supposed to plug
into equation one--

828
00:53:00,060 --> 00:53:03,220
this is the equation
one-- plug into one.

829
00:53:11,900 --> 00:53:12,870
I won't do it.

830
00:53:12,870 --> 00:53:16,270
It's three lines of
algebra to plug into one

831
00:53:16,270 --> 00:53:18,580
and calculate the
equation for h of u.

832
00:53:18,580 --> 00:53:20,070
You should do it.

833
00:53:20,070 --> 00:53:24,185
It's the kind of thing that
one should do at least once.

834
00:53:26,830 --> 00:53:28,260
So please do it.

835
00:53:28,260 --> 00:53:30,890
It's three, four lines.

836
00:53:30,890 --> 00:53:32,520
It's not long.

837
00:53:32,520 --> 00:53:34,170
But I'll just write the answer.

838
00:53:37,710 --> 00:53:39,790
So by the time you
substitute, of course,

839
00:53:39,790 --> 00:53:41,980
the e to the minus
u squared over 2

840
00:53:41,980 --> 00:53:44,940
is going to cancel
from everywhere.

841
00:53:44,940 --> 00:53:46,210
It's very here.

842
00:53:46,210 --> 00:53:48,190
You just need to
take two derivatives,

843
00:53:48,190 --> 00:53:51,350
so it becomes a second
order differential equation.

844
00:53:51,350 --> 00:53:54,760
And indeed, it becomes
a tractable differential

845
00:53:54,760 --> 00:53:56,080
equation.

846
00:53:56,080 --> 00:54:08,070
The second h, du squared minus
2u dh du plus e minus 1 h

847
00:54:08,070 --> 00:54:08,880
equals 0.

848
00:54:15,740 --> 00:54:19,470
OK, that is our equation now.

849
00:54:19,470 --> 00:54:25,530
So now we face the problem
finally solving this equation.

850
00:54:25,530 --> 00:54:27,640
So before we start,
maybe there's

851
00:54:27,640 --> 00:54:31,441
some questions of what
we've done so far.

852
00:54:31,441 --> 00:54:31,940
Let's see.

853
00:54:31,940 --> 00:54:34,980
Any questions?

854
00:54:38,080 --> 00:54:38,970
Yes?

855
00:54:38,970 --> 00:54:42,130
AUDIENCE: Do you have
right there in the middle

856
00:54:42,130 --> 00:54:45,632
would be-- this equation
is linear, so can we just

857
00:54:45,632 --> 00:54:51,065
[INAUDIBLE] minus u squared over
2 and you stuck it to that u

858
00:54:51,065 --> 00:54:51,911
to the k.

859
00:54:51,911 --> 00:54:52,890
PROFESSOR: It's here?

860
00:54:52,890 --> 00:54:53,495
This thing?

861
00:54:53,495 --> 00:54:54,120
AUDIENCE: Yeah.

862
00:54:54,120 --> 00:54:57,900
Could you then just power series
what's going on at 0 with those

863
00:54:57,900 --> 00:55:01,120
u to the k terms [INAUDIBLE]?

864
00:55:01,120 --> 00:55:02,320
PROFESSOR: No.

865
00:55:02,320 --> 00:55:08,350
This is the behavior
as u goes to infinity.

866
00:55:08,350 --> 00:55:12,640
So I actually don't know
that the function near 0

867
00:55:12,640 --> 00:55:15,450
is going to behave
like u to the k.

868
00:55:15,450 --> 00:55:17,100
We really don't know.

869
00:55:17,100 --> 00:55:20,920
It suggest to you that
maybe the solution

870
00:55:20,920 --> 00:55:25,050
is going to be near 0 u to
the k times some polynomial

871
00:55:25,050 --> 00:55:26,060
or something like that.

872
00:55:26,060 --> 00:55:29,120
But it's not that, because
this analysis was just

873
00:55:29,120 --> 00:55:30,290
done at infinity.

874
00:55:30,290 --> 00:55:36,494
So we really have no information
still what's going on near 0.

875
00:55:36,494 --> 00:55:37,160
Other questions?

876
00:55:39,860 --> 00:55:40,958
Yes?

877
00:55:40,958 --> 00:55:45,369
AUDIENCE: So is k some arbitrary
number or is it an integer?

878
00:55:45,369 --> 00:55:47,660
PROFESSOR: At this moment,
actually, it doesn't matter.

879
00:55:47,660 --> 00:55:50,260
Is that right?

880
00:55:50,260 --> 00:55:51,320
Doesn't matter.

881
00:55:51,320 --> 00:55:56,420
The analysis that we did here
suggests it could be anything.

882
00:55:56,420 --> 00:56:00,640
That's why I just didn't
put it into h or u.

883
00:56:00,640 --> 00:56:03,520
I didn't put it because
would be strange to put here

884
00:56:03,520 --> 00:56:05,590
a u to the k.

885
00:56:05,590 --> 00:56:07,530
I wouldn't know
what to make of it.

886
00:56:07,530 --> 00:56:10,430
So at this moment,
the best thing to say

887
00:56:10,430 --> 00:56:14,050
is we don't know what it is,
and maybe we'll understand it.

888
00:56:14,050 --> 00:56:14,830
And we will.

889
00:56:14,830 --> 00:56:18,010
In a few seconds, we'll
sort of see what's going on.

890
00:56:20,570 --> 00:56:24,195
OK, so how does one
solve this equation?

891
00:56:27,900 --> 00:56:31,610
Well, it's not a
trivial equation, again.

892
00:56:31,610 --> 00:56:35,500
But it can be solved
by polynomials,

893
00:56:35,500 --> 00:56:36,670
and we'll see that.

894
00:56:36,670 --> 00:56:39,080
But the way we
solve this equation

895
00:56:39,080 --> 00:56:42,940
is by a power series expansion.

896
00:56:42,940 --> 00:56:45,670
Now you could do
it by hand first,

897
00:56:45,670 --> 00:56:48,370
and I did it when
I was preparing

898
00:56:48,370 --> 00:56:50,470
the lecture yesterday.

899
00:56:50,470 --> 00:56:57,060
I said I'm going to
just write h of u

900
00:56:57,060 --> 00:57:03,880
equal a constant a0 plus
a1u plus a2u squared

901
00:57:03,880 --> 00:57:07,360
plus a3u cubed.

902
00:57:07,360 --> 00:57:11,030
And I plugged it in here.

903
00:57:11,030 --> 00:57:13,220
And I just did the
first few terms

904
00:57:13,220 --> 00:57:15,490
and start to see what happened.

905
00:57:15,490 --> 00:57:17,850
And I found after
a little thinking

906
00:57:17,850 --> 00:57:24,410
that a2 is determined
by a0, and a3

907
00:57:24,410 --> 00:57:30,100
is determined by a1
once you substitute.

908
00:57:30,100 --> 00:57:34,090
It's not the obvious when you
look at this, but that happens.

909
00:57:34,090 --> 00:57:37,820
So when you face a
problem like that,

910
00:57:37,820 --> 00:57:41,060
don't go high power
to begin with.

911
00:57:41,060 --> 00:57:43,910
Just try a simple series
and see what happens.

912
00:57:43,910 --> 00:57:45,530
And you see a little pattern.

913
00:57:45,530 --> 00:57:48,670
And then you can do a more
sophisticated analysis.

914
00:57:48,670 --> 00:57:51,610
So what would be a more
sophisticated analysis?

915
00:57:51,610 --> 00:57:55,030
To write h of u
equal the sum from j

916
00:57:55,030 --> 00:58:00,262
equals 0 to infinity
aju to the j.

917
00:58:03,496 --> 00:58:06,370
Then if you take a
derivative, because we're

918
00:58:06,370 --> 00:58:11,780
going to need the
derivative, dh du

919
00:58:11,780 --> 00:58:18,860
would be the sum from
j equals 0 to infinity.

920
00:58:18,860 --> 00:58:24,500
j times aju to the j minus 1.

921
00:58:24,500 --> 00:58:27,360
You would say that
doesn't look very good

922
00:58:27,360 --> 00:58:31,000
because for j equals
0 you have 1 over u.

923
00:58:31,000 --> 00:58:31,990
That's crazy.

924
00:58:31,990 --> 00:58:35,650
But indeed for j equals 0,
the j here multiplies it

925
00:58:35,650 --> 00:58:37,090
and makes it 0.

926
00:58:37,090 --> 00:58:40,110
So this is OK.

927
00:58:40,110 --> 00:58:44,630
Now the term that we actually
need is minus 2u dh du.

928
00:58:44,630 --> 00:58:53,750
So here minus 2u dh du would
be equal to the sum from j

929
00:58:53,750 --> 00:59:02,745
equals 0 to infinity, and I
will have minus 2jaju to the j.

930
00:59:02,745 --> 00:59:07,990
The u makes this j minus 1 j,
and the constant went there.

931
00:59:07,990 --> 00:59:11,990
So here is so far h.

932
00:59:11,990 --> 00:59:14,250
Here is this other
term that we're

933
00:59:14,250 --> 00:59:18,582
going to need for the
differential equation.

934
00:59:18,582 --> 00:59:20,290
And then there's the
last term that we're

935
00:59:20,290 --> 00:59:22,125
going to need for the
differential equation,

936
00:59:22,125 --> 00:59:23,465
so I'm going to go here.

937
00:59:29,570 --> 00:59:32,110
So what do we get
for this last term.

938
00:59:32,110 --> 00:59:33,800
We'll have to take
a second derivative.

939
00:59:37,200 --> 00:59:40,290
So we'll take-- h
prime was there,

940
00:59:40,290 --> 00:59:46,790
so d second h du squared
will be the sum from j

941
00:59:46,790 --> 00:59:58,075
equals 0 of j times j minus
1 aju to the j minus 2.

942
01:00:03,980 --> 01:00:09,580
Now you have to rewrite this
in order to make it tractable.

943
01:00:09,580 --> 01:00:14,060
You want everything
to have u to the j's.

944
01:00:14,060 --> 01:00:17,960
You don't want actually to
have u to the j minus 2.

945
01:00:17,960 --> 01:00:20,130
So the first thing
that you notice

946
01:00:20,130 --> 01:00:28,690
is that this sum actually begins
with 2, because for 0 and 1

947
01:00:28,690 --> 01:00:29,370
it vanishes.

948
01:00:29,370 --> 01:00:37,600
So I can write j times j
minus 1 aj u to j minus 2.

949
01:00:37,600 --> 01:00:39,470
Like that.

950
01:00:39,470 --> 01:00:47,520
And then I can say let j
be equal to j prime plus 2.

951
01:00:52,270 --> 01:00:55,150
Look, j begins
with 2 in this sum.

952
01:00:55,150 --> 01:01:00,620
So if j is j prime plus 2,
j prime will begin with 0.

953
01:01:00,620 --> 01:01:06,490
So we've shifted the sum so it's
j prime equals 0 to infinity.

954
01:01:06,490 --> 01:01:09,850
And whenever I have a j I
must put j prime plus 2.

955
01:01:09,850 --> 01:01:13,210
So j prime plus 2.

956
01:01:13,210 --> 01:01:24,350
j prime plus 1 aj prime
plus 2 u to the j prime.

957
01:01:24,350 --> 01:01:28,670
Wherever I had j, I
put j prime plus 2.

958
01:01:28,670 --> 01:01:33,110
And finally you say j or
j prime is the same name,

959
01:01:33,110 --> 01:01:34,800
so let's call it j.

960
01:01:34,800 --> 01:01:36,830
j equals 0.

961
01:01:36,830 --> 01:01:38,610
j plus 2.

962
01:01:38,610 --> 01:01:41,480
j plus 1.

963
01:01:41,480 --> 01:01:43,997
aj plus 2 uj.

964
01:01:49,320 --> 01:01:52,730
So we got the series
expansion of everything,

965
01:01:52,730 --> 01:01:56,285
so we just plug into the
differential equation.

966
01:01:59,260 --> 01:02:01,170
So where is the
differential equation?

967
01:02:01,170 --> 01:02:03,150
It's here.

968
01:02:03,150 --> 01:02:04,990
So I'll plug it in.

969
01:02:04,990 --> 01:02:05,990
Let's see what we get.

970
01:02:08,590 --> 01:02:13,420
We'll get some from j
equals 0 to infinity.

971
01:02:13,420 --> 01:02:15,880
Let's see the second
derivative is here.

972
01:02:18,680 --> 01:02:35,510
j plus 2 times j plus 1 aj
plus 2 uj, so I'll put it here.

973
01:02:39,500 --> 01:02:41,830
So that's this second
derivative term.

974
01:02:41,830 --> 01:02:44,150
Now this one.

975
01:02:44,150 --> 01:02:44,870
It's easy.

976
01:02:44,870 --> 01:02:48,100
Minus 2j aj and the uj is there.

977
01:02:48,100 --> 01:02:49,390
So minus 2jaj.

978
01:02:55,050 --> 01:03:02,400
Last term is just e
minus 1, because it's

979
01:03:02,400 --> 01:03:05,880
the function this
times aj as well.

980
01:03:08,610 --> 01:03:10,890
That's h.

981
01:03:10,890 --> 01:03:13,870
And look, this whole
thing must be 0.

982
01:03:13,870 --> 01:03:16,970
So what you learn is that
this coefficient must

983
01:03:16,970 --> 01:03:19,920
be 0 for every value of j.

984
01:03:19,920 --> 01:03:23,710
Now it's possible to--
here is aj and aj,

985
01:03:23,710 --> 01:03:26,030
so it's actually
one single thing.

986
01:03:26,030 --> 01:03:27,720
Let me write it here.

987
01:03:27,720 --> 01:03:38,850
j plus 2 times j plus
1 aj plus 2 minus 2j

988
01:03:38,850 --> 01:03:46,870
plus 1 minus e aj uj.

989
01:03:46,870 --> 01:03:49,840
I think I got it right.

990
01:03:49,840 --> 01:03:51,130
Yes.

991
01:03:51,130 --> 01:03:52,300
And this is the same sum.

992
01:03:55,830 --> 01:04:01,370
And now, OK, it's a lot of
work, but we're getting there.

993
01:04:01,370 --> 01:04:03,330
This must be 0.

994
01:04:03,330 --> 01:04:08,230
So actually that solves for
aj plus 2 in terms of aj.

995
01:04:08,230 --> 01:04:11,760
What I had told you that you
can notice in two minutes

996
01:04:11,760 --> 01:04:12,880
if you try it a little.

997
01:04:12,880 --> 01:04:15,710
That a2 seems to be
determined by a0.

998
01:04:15,710 --> 01:04:18,740
And a3 seems to be
determined by a2.

999
01:04:18,740 --> 01:04:24,600
So this is saying
that aj plus 2 is

1000
01:04:24,600 --> 01:04:36,380
given by 2j plus 1 minus e
over j plus 2 j plus 1 aj.

1001
01:04:44,790 --> 01:04:47,460
A very nice recursive relation.

1002
01:04:47,460 --> 01:04:51,780
So indeed, if you
put the value of a0,

1003
01:04:51,780 --> 01:04:57,320
it will determine for you a2,
a4, a6, a8, all the even ones.

1004
01:04:57,320 --> 01:05:04,070
If you put the value of a1, it
will determine for you a3, a5.

1005
01:05:04,070 --> 01:05:10,070
So a solution is determined by
you telling me how much is a0,

1006
01:05:10,070 --> 01:05:12,600
and telling me how much is a1.

1007
01:05:12,600 --> 01:05:15,130
Two constants, two numbers.

1008
01:05:15,130 --> 01:05:17,830
That's what you expect from
a second order differential

1009
01:05:17,830 --> 01:05:18,660
equation.

1010
01:05:18,660 --> 01:05:20,370
The value of the
function at the point,

1011
01:05:20,370 --> 01:05:22,190
the derivative at a point.

1012
01:05:22,190 --> 01:05:27,240
In fact, you are
looking at a0 and a1

1013
01:05:27,240 --> 01:05:31,110
as the two constants that
will determine a solution.

1014
01:05:31,110 --> 01:05:34,300
And this is the value of h at 0.

1015
01:05:34,300 --> 01:05:37,590
This is the
derivative of h at 0.

1016
01:05:37,590 --> 01:05:51,830
So we can now write
the following facts

1017
01:05:51,830 --> 01:05:53,790
about the solution
that we have found.

1018
01:05:57,720 --> 01:05:59,090
So what do we know?

1019
01:05:59,090 --> 01:06:13,170
That solutions fixed
by giving a0 and a1.

1020
01:06:15,680 --> 01:06:19,245
That correspond to the
value of the function at 0

1021
01:06:19,245 --> 01:06:23,120
and the derivative
of the function at 0.

1022
01:06:23,120 --> 01:06:25,350
And this gives one solution.

1023
01:06:25,350 --> 01:06:29,780
Once you fix a0, you get a2, a4.

1024
01:06:29,780 --> 01:06:33,330
And this is an even
solution, because it

1025
01:06:33,330 --> 01:06:36,940
has only even powers.

1026
01:06:36,940 --> 01:06:43,360
And then from a1, you fixed
a3, a5, all the other ones

1027
01:06:43,360 --> 01:06:44,470
with an odd solution.

1028
01:06:50,055 --> 01:06:50,555
OK.

1029
01:06:53,680 --> 01:06:55,810
Well, we solve the
differential equation,

1030
01:06:55,810 --> 01:06:58,690
which is really,
in a sense, bad,

1031
01:06:58,690 --> 01:07:01,880
because we were expecting
that we can only solve it

1032
01:07:01,880 --> 01:07:03,185
for some values of the energy.

1033
01:07:05,690 --> 01:07:13,460
Moreover, you have a0,
you get a2, a4, a6, a8.

1034
01:07:13,460 --> 01:07:18,220
This will go on forever
and not terminate.

1035
01:07:18,220 --> 01:07:20,910
And then it will be an
infinite polynomial that

1036
01:07:20,910 --> 01:07:24,160
grows up and doesn't
ever decline,

1037
01:07:24,160 --> 01:07:28,070
which is sort of
contradictory with the idea

1038
01:07:28,070 --> 01:07:31,710
that we had before
that near infinity

1039
01:07:31,710 --> 01:07:37,160
the function was going to be
some power, some fixed power,

1040
01:07:37,160 --> 01:07:39,410
times this exponential.

1041
01:07:39,410 --> 01:07:44,950
So this is what we're looking
for, this h function now.

1042
01:07:44,950 --> 01:07:47,000
It doesn't look
like a fixed power.

1043
01:07:47,000 --> 01:07:50,240
It looks like it goes forever.

1044
01:07:50,240 --> 01:07:53,790
So let's see what
happens eventually

1045
01:07:53,790 --> 01:07:58,190
when the coefficient, the
value of the j index is large.

1046
01:07:58,190 --> 01:08:02,460
For large j.

1047
01:08:06,960 --> 01:08:13,500
aj plus 2 is roughly equal
to, for large a, whatever

1048
01:08:13,500 --> 01:08:17,580
the energy is, sufficiently
large, the most important here

1049
01:08:17,580 --> 01:08:20,565
is the 2j here, the j and the j.

1050
01:08:20,565 --> 01:08:23,600
So you get 2 over j aj.

1051
01:08:29,810 --> 01:08:37,649
So roughly for large j,
it behaves like that.

1052
01:08:37,649 --> 01:08:41,300
And now you have to ask
yourself the question,

1053
01:08:41,300 --> 01:08:45,140
if you have a power series
expansion whose coefficients

1054
01:08:45,140 --> 01:08:51,020
behave like that, how
badly is it at infinity?

1055
01:08:51,020 --> 01:08:52,130
How about is it?

1056
01:08:55,380 --> 01:08:57,800
You know it's the
power series expansion

1057
01:08:57,800 --> 01:09:01,770
because your h was all
these coefficients.

1058
01:09:01,770 --> 01:09:03,670
And suppose they
behave like that.

1059
01:09:03,670 --> 01:09:06,910
They grow in that way
or decay in this way,

1060
01:09:06,910 --> 01:09:09,470
because they're decaying.

1061
01:09:09,470 --> 01:09:11,770
Is this a solution
that's going to blow up?

1062
01:09:11,770 --> 01:09:15,050
Or is it not going to blow up?

1063
01:09:15,050 --> 01:09:19,569
And here comes an
important thing.

1064
01:09:19,569 --> 01:09:22,380
This is pretty bad
behavior, actually.

1065
01:09:22,380 --> 01:09:24,979
It's pretty awful behavior.

1066
01:09:24,979 --> 01:09:28,439
So let's see that.

1067
01:09:28,439 --> 01:09:29,290
That's pretty bad.

1068
01:09:34,810 --> 01:09:35,950
How do we see that?

1069
01:09:35,950 --> 01:09:39,210
Well you could do it
in different ways,

1070
01:09:39,210 --> 01:09:41,189
depending on whether
you want to derive

1071
01:09:41,189 --> 01:09:45,560
that this is a bad
behavior or guess it.

1072
01:09:45,560 --> 01:09:48,580
I'm going to guess something.

1073
01:09:48,580 --> 01:09:53,910
I'm going to look at how
does e to the u squared

1074
01:09:53,910 --> 01:09:56,430
behave as a power series.

1075
01:09:56,430 --> 01:09:59,030
Well, you know as a
power series exponential

1076
01:09:59,030 --> 01:10:04,750
is 1 over n u squared to the n.

1077
01:10:04,750 --> 01:10:06,080
Here's n factorial.

1078
01:10:06,080 --> 01:10:07,715
n equals 0 to infinity.

1079
01:10:10,950 --> 01:10:18,810
Now these two n's, u to the
2n, these are all even powers.

1080
01:10:18,810 --> 01:10:21,720
So I'm going to
change letters here,

1081
01:10:21,720 --> 01:10:28,400
and I'm going to work with j
from 0, 2, 4, over the evens.

1082
01:10:28,400 --> 01:10:32,980
So I will write u to the j here.

1083
01:10:32,980 --> 01:10:36,360
And that this correct, because
you produce u to the 0,

1084
01:10:36,360 --> 01:10:39,510
u to the 2, u to the
fourth, these things.

1085
01:10:39,510 --> 01:10:43,362
And j is really 2n,
so here you will

1086
01:10:43,362 --> 01:10:48,560
have one over j
over 2 factorial.

1087
01:10:48,560 --> 01:10:51,730
Now you might say, j over
2, isn't that a fraction?

1088
01:10:51,730 --> 01:10:54,170
No, it's not a fraction,
because j is even.

1089
01:10:54,170 --> 01:10:55,690
So this is a nice factorial.

1090
01:10:58,600 --> 01:11:06,056
Now this is the
coefficient, cj u to the j.

1091
01:11:06,056 --> 01:11:09,170
And let's see how this
coefficients vary.

1092
01:11:09,170 --> 01:11:14,490
So this cj is 1 over
j over 2 factorial.

1093
01:11:14,490 --> 01:11:20,800
What is cj plus 2 over cj?

1094
01:11:20,800 --> 01:11:23,560
Which is the analogue
of this thing.

1095
01:11:23,560 --> 01:11:30,940
Well, this would be 1 over
j plus 2 over 2 factorial.

1096
01:11:30,940 --> 01:11:36,340
And here is up there,
so j over 2 factorial.

1097
01:11:36,340 --> 01:11:40,310
Well, this has one more
factor in the denominator

1098
01:11:40,310 --> 01:11:41,970
than the numerator.

1099
01:11:41,970 --> 01:11:49,270
So this is roughly
one over j over 2

1100
01:11:49,270 --> 01:11:52,105
plus 1, the last value of this.

1101
01:11:52,105 --> 01:11:55,690
This integer is just
one bigger than that.

1102
01:11:55,690 --> 01:11:58,800
Now if j is large,
this is roughly 1

1103
01:11:58,800 --> 01:12:02,730
over j over 2,
which is 2 over j.

1104
01:12:02,730 --> 01:12:06,470
Oh, exactly that stuff.

1105
01:12:06,470 --> 01:12:09,145
So it's pretty bad.

1106
01:12:09,145 --> 01:12:13,380
If this series
goes on forever, it

1107
01:12:13,380 --> 01:12:18,080
will diverge like
e to the u squared.

1108
01:12:18,080 --> 01:12:24,160
And your h will be like e to the
u squared with e to the minus

1109
01:12:24,160 --> 01:12:27,780
u squared over 2 is going
to be like e to the plus.

1110
01:12:27,780 --> 01:12:32,160
u squared over 2 is going
to go and behave this one.

1111
01:12:32,160 --> 01:12:37,350
So it's going to do
exactly the wrong thing.

1112
01:12:37,350 --> 01:12:43,280
If this series
doesn't terminate,

1113
01:12:43,280 --> 01:12:46,290
we have not succeeded.

1114
01:12:46,290 --> 01:12:51,200
But happily, the
series may terminate,

1115
01:12:51,200 --> 01:12:54,420
because the j's are integers.

1116
01:12:54,420 --> 01:12:58,670
So maybe for some energies
that are integers,

1117
01:12:58,670 --> 01:13:01,720
it terminates, and
that's a solution.

1118
01:13:01,720 --> 01:13:05,090
The only way to get a solution
is if the series terminates.

1119
01:13:05,090 --> 01:13:08,360
The only way it can
terminate is that the e

1120
01:13:08,360 --> 01:13:14,020
is some odd number over here.

1121
01:13:14,020 --> 01:13:16,030
And that will solve the thing.

1122
01:13:16,030 --> 01:13:19,200
So we actually need to do this.

1123
01:13:19,200 --> 01:13:21,230
This shows the energy.

1124
01:13:21,230 --> 01:13:24,500
You found why it's quantized.

1125
01:13:24,500 --> 01:13:28,150
So let's do it then.

1126
01:13:28,150 --> 01:13:31,550
We're really done
with this in a sense.

1127
01:13:31,550 --> 01:13:36,460
This is the most important
point of the lecture,

1128
01:13:36,460 --> 01:13:39,480
is that the series
must terminate,

1129
01:13:39,480 --> 01:13:42,530
otherwise it will
blow up horrendously.

1130
01:13:42,530 --> 01:13:44,550
If it terminates
as a polynomial,

1131
01:13:44,550 --> 01:13:46,640
then everything is good.

1132
01:13:46,640 --> 01:13:58,900
So to terminate you can choose
2j plus 1 minus e to be 0.

1133
01:13:58,900 --> 01:14:05,010
This will make aj
plus 2 equal to 0.

1134
01:14:07,970 --> 01:14:13,930
And your solution, your
h of u, will begin.

1135
01:14:13,930 --> 01:14:17,800
aj will be the last
one that is non-zero,

1136
01:14:17,800 --> 01:14:25,020
so it will be aj
times u to the j,

1137
01:14:25,020 --> 01:14:33,500
and it will go down like aj
minus 2 u to the j minus 2.

1138
01:14:33,500 --> 01:14:37,380
It will go down in steps of
2, because this recursion

1139
01:14:37,380 --> 01:14:40,100
is always by steps of two.

1140
01:14:40,100 --> 01:14:41,232
So that's it.

1141
01:14:41,232 --> 01:14:44,350
That's going to be the solution
where these coefficients are

1142
01:14:44,350 --> 01:14:47,160
going to be fixed by
the recursive relation,

1143
01:14:47,160 --> 01:14:49,720
and we have this.

1144
01:14:49,720 --> 01:14:56,180
Now most people
here call j equal n.

1145
01:14:56,180 --> 01:14:58,080
So let's call it n.

1146
01:14:58,080 --> 01:15:04,420
And then we have 2n
plus 1 minus e equals 0.

1147
01:15:04,420 --> 01:15:12,580
And h of u would be an u to
the n plus all these things.

1148
01:15:12,580 --> 01:15:14,120
That's the h.

1149
01:15:14,120 --> 01:15:18,810
The full solution is h
times e to the minus u

1150
01:15:18,810 --> 01:15:21,290
squared over 2 as we will see.

1151
01:15:21,290 --> 01:15:25,250
But recall what e was.

1152
01:15:25,250 --> 01:15:28,480
e here is 2n plus 1.

1153
01:15:28,480 --> 01:15:36,620
But he was the true energy
divided by h omega over two.

1154
01:15:36,620 --> 01:15:37,935
That was long ago.

1155
01:15:41,310 --> 01:15:42,790
It's gone.

1156
01:15:42,790 --> 01:15:46,150
Long gone.

1157
01:15:46,150 --> 01:15:50,160
So what have you
found therefore?

1158
01:15:50,160 --> 01:15:59,258
That the energy,
that' we'll call en,

1159
01:15:59,258 --> 01:16:02,680
the energy of the
nth solution is

1160
01:16:02,680 --> 01:16:07,770
going to be h omega
over 2 2n plus 1.

1161
01:16:07,770 --> 01:16:13,890
So it's actually h omega, and
people write it n plus 1/2.

1162
01:16:13,890 --> 01:16:15,570
Very famous result.

1163
01:16:15,570 --> 01:16:20,440
The nth level of the harmonic
oscillator has this energy.

1164
01:16:20,440 --> 01:16:25,060
And moreover, these
objects, people

1165
01:16:25,060 --> 01:16:27,220
choose these-- you
see the constants are

1166
01:16:27,220 --> 01:16:28,720
related by steps of two.

1167
01:16:28,720 --> 01:16:32,770
So just like you could start
with a0, or a1 and go up,

1168
01:16:32,770 --> 01:16:34,640
you can go down.

1169
01:16:34,640 --> 01:16:38,990
People call these functions
Hermite functions.

1170
01:16:38,990 --> 01:16:44,230
And they fix the notation so
that this an is 2 to the n.

1171
01:16:44,230 --> 01:16:45,160
They like it.

1172
01:16:45,160 --> 01:16:46,880
It's a nice normalization.

1173
01:16:46,880 --> 01:16:52,410
So actually h of
n is what we call

1174
01:16:52,410 --> 01:16:56,880
the Hermite function of u sub n.

1175
01:16:56,880 --> 01:17:02,960
And it goes like 2 to the
n u to the n plus order u

1176
01:17:02,960 --> 01:17:06,170
to the n minus 2 plus
n minus 4, and it

1177
01:17:06,170 --> 01:17:07,855
goes on and on like that.

1178
01:17:13,180 --> 01:17:18,300
OK, a couple things
and we're done.

1179
01:17:18,300 --> 01:17:23,780
Just for reference,
the Hermite polynomial,

1180
01:17:23,780 --> 01:17:25,950
if you're interested
in it, is the one

1181
01:17:25,950 --> 01:17:28,230
that solves this equation.

1182
01:17:28,230 --> 01:17:30,470
And the Hermite
sub n corresponds

1183
01:17:30,470 --> 01:17:36,570
to e sub n, which is 2n plus 1.

1184
01:17:36,570 --> 01:17:39,330
So the Hermite solution
from that the equation

1185
01:17:39,330 --> 01:17:47,378
is that the Hermite polynomial
satisfies this minus 2u

1186
01:17:47,378 --> 01:17:52,480
d Hn du plus 2n.

1187
01:17:52,480 --> 01:17:55,050
Because en is 2n plus 1.

1188
01:17:55,050 --> 01:17:59,240
So it's 2n Hn equals 0.

1189
01:17:59,240 --> 01:18:02,130
That's the equation for
the Hermite polynomial,

1190
01:18:02,130 --> 01:18:05,530
and interesting thing to know.

1191
01:18:05,530 --> 01:18:08,100
Actually, if you want to
generate the efficiently

1192
01:18:08,100 --> 01:18:11,490
the Hermite polynomials, there's
something called the generating

1193
01:18:11,490 --> 01:18:13,450
function.

1194
01:18:13,450 --> 01:18:17,730
e to the minus z
squared plus 2zu.

1195
01:18:17,730 --> 01:18:23,440
If you expand it in
a power series of z,

1196
01:18:23,440 --> 01:18:27,660
it actually gives you
n equals 0 to infinity.

1197
01:18:27,660 --> 01:18:32,780
If it's a power series of z,
it will be some z to the n's.

1198
01:18:32,780 --> 01:18:37,605
You can put a factor here
n, and here is Hn of u.

1199
01:18:40,620 --> 01:18:43,740
So you can use your
mathematic program

1200
01:18:43,740 --> 01:18:46,590
and expand this in powers of z.

1201
01:18:46,590 --> 01:18:51,370
Collect the various powers of
u that appear with z to the n,

1202
01:18:51,370 --> 01:18:56,260
and that's Hn It's the most
efficient way of generating Hn

1203
01:18:56,260 --> 01:18:59,480
And moreover, if you want
to play in mathematics,

1204
01:18:59,480 --> 01:19:03,840
you can show that
such definition of Hn

1205
01:19:03,840 --> 01:19:05,950
satisfies this equation.

1206
01:19:05,950 --> 01:19:08,380
So it produces the solution.

1207
01:19:08,380 --> 01:19:10,810
So what have we found?

1208
01:19:10,810 --> 01:19:13,910
Our end result is the following.

1209
01:19:13,910 --> 01:19:17,080
Let me finish with that here.

1210
01:19:17,080 --> 01:19:25,070
We had this potential, and the
first energy level is called E0

1211
01:19:25,070 --> 01:19:27,620
and has energy h omega over 2.

1212
01:19:27,620 --> 01:19:29,940
The next energy is E1.

1213
01:19:29,940 --> 01:19:33,810
It has 3/2 h omega.

1214
01:19:33,810 --> 01:19:39,830
Next one is E2 5/2 h omega.

1215
01:19:39,830 --> 01:19:42,940
This polynomial is
nth degree polynomial.

1216
01:19:42,940 --> 01:19:46,860
So it has n zeros,
therefore n nodes.

1217
01:19:46,860 --> 01:19:50,470
So these wave functions will
have the right number of nodes.

1218
01:19:50,470 --> 01:19:54,030
E0, the psi 0,
will have no nodes.

1219
01:19:54,030 --> 01:19:59,300
When you have psi 0, the
Hn becomes a number for n

1220
01:19:59,300 --> 01:20:00,510
equals zero.

1221
01:20:00,510 --> 01:20:02,830
And the whole solution
is the exponential

1222
01:20:02,830 --> 01:20:05,610
of u squared over 2.

1223
01:20:05,610 --> 01:20:08,960
The whole solution, in
fact, is, as we wrote,

1224
01:20:08,960 --> 01:20:15,150
psi n Hn of u e to the
minus u squared over 2.

1225
01:20:15,150 --> 01:20:18,630
In plain English,
if you use an x,

1226
01:20:18,630 --> 01:20:24,890
it will be Hn u with x over
that constant a we had.

1227
01:20:24,890 --> 01:20:30,470
And you have minus x
squared over 2a squared.

1228
01:20:30,470 --> 01:20:32,810
Those are your eigenfunctions.

1229
01:20:32,810 --> 01:20:34,750
These are the solutions.

1230
01:20:34,750 --> 01:20:39,020
Discrete spectrum, evenly
spaced, the nicest spectrum

1231
01:20:39,020 --> 01:20:40,530
possible.

1232
01:20:40,530 --> 01:20:42,730
All the nodes are there.

1233
01:20:42,730 --> 01:20:46,620
You will solve this in a
more clever way next time.

1234
01:20:46,620 --> 01:20:48,170
[APPLAUSE]