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ROBERT FIELD: OK,
let's get started.
00:00:25.490 --> 00:00:34.730
So last time we talked about the
one-dimensional wave equation,
00:00:34.730 --> 00:00:40.710
which is a second-order
partial differential equation.
00:00:40.710 --> 00:00:44.220
This is not a math course.
00:00:44.220 --> 00:00:47.220
If you have a second-order
differential equation,
00:00:47.220 --> 00:00:52.940
there will be two linearly
independent solutions to it.
00:00:52.940 --> 00:00:56.100
And that's important
to remember.
00:00:56.100 --> 00:01:00.660
Now, there are
three steps that we
00:01:00.660 --> 00:01:03.727
use in approaching
a problem like this.
00:01:03.727 --> 00:01:06.060
Does anybody want to tell me
what those three steps are?
00:01:15.660 --> 00:01:16.415
Yes?
00:01:16.415 --> 00:01:20.750
AUDIENCE: [INAUDIBLE] it
has separate functions that
00:01:20.750 --> 00:01:22.590
take only one variable.
00:01:22.590 --> 00:01:25.770
ROBERT FIELD: OK,
that's most of it.
00:01:25.770 --> 00:01:28.530
You want to solve
the general equation,
00:01:28.530 --> 00:01:31.560
and one way to solve
the general equation
00:01:31.560 --> 00:01:33.960
is to try to separate variables.
00:01:33.960 --> 00:01:36.390
Always you want to
separate variables.
00:01:36.390 --> 00:01:39.750
Even if it's not
quite legal, you
00:01:39.750 --> 00:01:43.560
want to find a way to do that
because that breaks the problem
00:01:43.560 --> 00:01:45.670
down in a very useful way.
00:01:45.670 --> 00:01:53.270
So the first step is
the general solution,
00:01:53.270 --> 00:02:01.650
and it involves trying
something like this
00:02:01.650 --> 00:02:06.630
where we say u of x
and t is going to be
00:02:06.630 --> 00:02:09.060
treated in the separable form.
00:02:09.060 --> 00:02:10.949
If it doesn't work,
you're going to get 0.
00:02:10.949 --> 00:02:14.910
You're going to
get the solution.
00:02:14.910 --> 00:02:20.980
The only solution with separated
variables is nothing happening.
00:02:20.980 --> 00:02:23.100
And so that's unfortunate
if you do work
00:02:23.100 --> 00:02:26.400
and you get nothing for it,
but life is complicated.
00:02:26.400 --> 00:02:30.060
So then after you do the
general solution, what's next?
00:02:32.760 --> 00:02:33.260
Yes?
00:02:33.260 --> 00:02:34.706
AUDIENCE: You need to
set boundary conditions?
00:02:34.706 --> 00:02:35.456
ROBERT FIELD: Yes.
00:02:44.350 --> 00:02:50.230
Now if it's a second-order
differential equation,
00:02:50.230 --> 00:02:52.180
you're going to need
two boundary conditions.
00:02:54.800 --> 00:03:00.930
And when you impose two
boundary conditions,
00:03:00.930 --> 00:03:05.040
the second one gives some
sort of quantization.
00:03:05.040 --> 00:03:09.120
It makes the solutions discrete
or that there is a discrete set
00:03:09.120 --> 00:03:11.250
rather than a continuous set.
00:03:11.250 --> 00:03:16.650
The general solution is more or
less continuous, or continuous
00:03:16.650 --> 00:03:17.860
possibility.
00:03:17.860 --> 00:03:20.700
So we have now the
boundary conditions,
00:03:20.700 --> 00:03:26.910
and that gives us something
that we can start to visualize.
00:03:26.910 --> 00:03:29.540
So what are the important
things that, if you're
00:03:29.540 --> 00:03:34.040
going to be drawing pictures
rather than actually plotting
00:03:34.040 --> 00:03:37.600
some complicated mathematical
function, what are
00:03:37.600 --> 00:03:39.170
the first questions you ask?
00:03:44.260 --> 00:03:44.820
Yes?
00:03:44.820 --> 00:03:48.326
AUDIENCE: Is it
symmetric or asymmetric?
00:03:52.990 --> 00:03:56.680
ROBERT FIELD: Yes, if
the problem has symmetry,
00:03:56.680 --> 00:03:58.075
the solutions will
have symmetry.
00:04:01.940 --> 00:04:02.910
But there's more.
00:04:02.910 --> 00:04:04.860
I want another, my favorite.
00:04:04.860 --> 00:04:06.530
AUDIENCE: Where's it start?
00:04:06.530 --> 00:04:06.940
ROBERT FIELD: I'm sorry?
00:04:06.940 --> 00:04:08.190
AUDIENCE: Where does it start?
00:04:08.190 --> 00:04:11.730
So t equals 0 or x equals 0.
00:04:11.730 --> 00:04:14.940
ROBERT FIELD: The initial
condition is really
00:04:14.940 --> 00:04:18.423
the next thing I'm
going to ask you about,
00:04:18.423 --> 00:04:19.589
and that's called the pluck.
00:04:22.410 --> 00:04:23.870
You're right on target.
00:04:23.870 --> 00:04:26.310
But now if you're going
to draw a picture,
00:04:26.310 --> 00:04:28.650
the best thing you want
to do to draw a picture
00:04:28.650 --> 00:04:32.360
is have it not move.
00:04:32.360 --> 00:04:40.480
So you want to look at
the thing in position,
00:04:40.480 --> 00:04:45.640
and what are the things
about the position function
00:04:45.640 --> 00:04:50.460
that you can immediately
figure out and use
00:04:50.460 --> 00:04:51.855
in drawing a cartoon?
00:04:57.593 --> 00:04:58.590
AUDIENCE: Nodes.
00:04:58.590 --> 00:04:59.340
ROBERT FIELD: Yep.
00:05:04.320 --> 00:05:08.840
So how many nodes
and where are they?
00:05:08.840 --> 00:05:11.720
Are they equally spaced?
00:05:11.720 --> 00:05:16.710
And that's the most important
thing in drawing a picture,
00:05:16.710 --> 00:05:19.990
the number of zero crossings
that a wave function has,
00:05:19.990 --> 00:05:24.430
and how are they distributed?
00:05:24.430 --> 00:05:29.750
Their spacing of nodes
is half the wavelength,
00:05:29.750 --> 00:05:31.625
and the wavelength is
related to momentum.
00:05:34.460 --> 00:05:37.260
And so I'm jumping
into quantum mechanics,
00:05:37.260 --> 00:05:41.450
but it's still valid for
understanding the wave
00:05:41.450 --> 00:05:42.450
equation.
00:05:42.450 --> 00:05:50.270
So we want the number of nodes
for each specific solution
00:05:50.270 --> 00:05:54.610
satisfying the boundary
conditions and the spacing
00:05:54.610 --> 00:05:59.600
and the loops between nodes.
00:05:59.600 --> 00:06:02.010
Are they all identical?
00:06:02.010 --> 00:06:04.760
Is there some
systematic variation
00:06:04.760 --> 00:06:11.090
in the magnitude of each
of the loops between nodes?
00:06:11.090 --> 00:06:15.920
Because if you have just a
qualitative sense for how this
00:06:15.920 --> 00:06:19.740
works, you can draw
the wave function,
00:06:19.740 --> 00:06:23.950
and you can begin to make
conclusions about it.
00:06:23.950 --> 00:06:25.850
But it all starts with nodes.
00:06:25.850 --> 00:06:27.780
Nodes are really important.
00:06:27.780 --> 00:06:33.260
And quantum mechanics, the
wave functions have nodes.
00:06:33.260 --> 00:06:37.650
You can't do better
than focus on the nodes.
00:06:37.650 --> 00:06:42.240
And then I like
to call it pluck,
00:06:42.240 --> 00:06:52.270
but it's a superposition
of eigenstates.
00:06:52.270 --> 00:06:57.070
And those superpositions
for this problem can move.
00:07:00.510 --> 00:07:06.190
And so-- you're
missing a great--
00:07:06.190 --> 00:07:07.360
anyway.
00:07:07.360 --> 00:07:11.890
And so you want to know,
what is the kind of motion
00:07:11.890 --> 00:07:14.270
that this thing can have?
00:07:14.270 --> 00:07:17.020
And so one of the things--
00:07:17.020 --> 00:07:19.840
this is really the three
steps you go through
00:07:19.840 --> 00:07:22.540
in order to make a picture
in which you hang up
00:07:22.540 --> 00:07:24.070
your insights.
00:07:24.070 --> 00:07:35.900
And so if there's one
state or two states,
00:07:35.900 --> 00:07:40.370
one state is just going
to be standing waves.
00:07:40.370 --> 00:07:45.740
Two states is all the complexity
you're ever going to need.
00:07:45.740 --> 00:07:49.630
And if two states have
different frequencies,
00:07:49.630 --> 00:07:52.030
there will be motion,
and the motion
00:07:52.030 --> 00:07:57.550
can be side-to-side motion
or sort of breathing motion
00:07:57.550 --> 00:08:01.790
where amplitude moves in
from the turning-point region
00:08:01.790 --> 00:08:05.000
to the middle and
back out again.
00:08:05.000 --> 00:08:09.040
And so you can be
able to classify
00:08:09.040 --> 00:08:15.670
what you can understand and
to imagine doing experiments
00:08:15.670 --> 00:08:19.975
based on this simplified
version of the flux.
00:08:22.980 --> 00:08:25.200
In my opinion, the
most important thing
00:08:25.200 --> 00:08:30.480
you can do as a professional
quantum machinist
00:08:30.480 --> 00:08:33.299
and in preparation
for exams is to be
00:08:33.299 --> 00:08:37.595
able to draw these cartoons
quickly, really quickly.
00:08:37.595 --> 00:08:39.719
That means you have to
think about them in advance.
00:08:44.460 --> 00:08:48.570
And so this recipe
is how you're going
00:08:48.570 --> 00:08:52.880
to understand quantum
mechanical problems too.
00:08:52.880 --> 00:08:55.457
And this differential
equation is actually
00:08:55.457 --> 00:08:57.290
a little more complicated
than the first few
00:08:57.290 --> 00:09:01.220
that we're going to encounter
because the first few problems
00:09:01.220 --> 00:09:04.590
we're going to face
are not time dependent.
00:09:04.590 --> 00:09:08.590
There may still be a separation
of variables situation
00:09:08.590 --> 00:09:11.620
and imposing boundary
conditions and so on,
00:09:11.620 --> 00:09:15.880
but there is no motion.
00:09:15.880 --> 00:09:18.970
But eventually we'll get motion
because our real world has
00:09:18.970 --> 00:09:22.450
motion, and quantum mechanics
has to reproduce everything
00:09:22.450 --> 00:09:24.010
that our real world does.
00:09:26.680 --> 00:09:30.470
So, we're going to
begin quantum mechanics,
00:09:30.470 --> 00:09:33.860
and first of all I
will describe some
00:09:33.860 --> 00:09:37.020
of the rules we have
to obey in building
00:09:37.020 --> 00:09:39.440
a quantum-mechanical picture.
00:09:39.440 --> 00:09:42.470
And then I'll approach
two of the easiest
00:09:42.470 --> 00:09:46.310
problems, the free
particle and the particle
00:09:46.310 --> 00:09:47.230
in an infinite box.
00:10:05.730 --> 00:10:14.450
So we have the one-dimensional
Schrodinger equation,
00:10:14.450 --> 00:10:17.540
and the one-dimensional
Schrodinger equation looks
00:10:17.540 --> 00:10:19.340
like the wave equation.
00:10:19.340 --> 00:10:20.750
And why?
00:10:20.750 --> 00:10:25.700
Because waves interfere
with each other.
00:10:25.700 --> 00:10:28.880
We can have constructive and
destructive interference.
00:10:28.880 --> 00:10:33.620
Almost everything that is
wonderful about quantum
00:10:33.620 --> 00:10:38.390
mechanics is the solutions
to this Schrodinger equation
00:10:38.390 --> 00:10:42.780
also exhibit constructive
and destructive interference.
00:10:42.780 --> 00:10:46.370
And that's essential to our
understanding of how quantum
00:10:46.370 --> 00:10:50.150
mechanics describes the world.
00:10:50.150 --> 00:10:52.250
The next thing I want to
do is talk a little bit
00:10:52.250 --> 00:10:53.160
about postulates.
00:10:56.780 --> 00:11:00.410
Now I'm going to be introducing
the quantum-mechanic postulates
00:11:00.410 --> 00:11:02.750
as we need them
as opposed to just
00:11:02.750 --> 00:11:08.510
a dry lecture of these
strange and wonderful things
00:11:08.510 --> 00:11:11.630
before we're ready for them.
00:11:11.630 --> 00:11:15.520
But a postulate is something
that can't be proven right.
00:11:15.520 --> 00:11:18.140
It can be proven wrong.
00:11:18.140 --> 00:11:24.610
And we build a system of logic
based on these postulates.
00:11:24.610 --> 00:11:27.490
Now one of the great
experiences in my life
00:11:27.490 --> 00:11:33.760
was one time when I visited the
Exploratorium in San Francisco
00:11:33.760 --> 00:11:37.510
where there are rather crude,
or at least when I visited
00:11:37.510 --> 00:11:40.300
almost 50 years ago,
there were rather
00:11:40.300 --> 00:11:42.940
crude interactive
experiments where
00:11:42.940 --> 00:11:47.410
people can turn knobs and push
buttons and make things happen.
00:11:47.410 --> 00:11:52.300
And the most wonderful
thing was really young kids
00:11:52.300 --> 00:11:55.840
trying to break these exhibits.
00:11:55.840 --> 00:11:58.540
And what they did is
by trying to break them
00:11:58.540 --> 00:12:01.430
they discovered
patterns, some of them,
00:12:01.430 --> 00:12:02.920
and that's what
we're going to do.
00:12:02.920 --> 00:12:06.820
We're going to try to break or
think about breaking postulates
00:12:06.820 --> 00:12:12.040
and then see what we learn.
00:12:12.040 --> 00:12:14.140
So, let's begin.
00:12:17.630 --> 00:12:19.570
We have operators in
quantum mechanics.
00:12:23.010 --> 00:12:31.940
And we denote them either with
a hat or as a boldface object.
00:12:31.940 --> 00:12:33.870
We start using this
kind of notation when
00:12:33.870 --> 00:12:37.260
we do matrix mechanics,
which we will do,
00:12:37.260 --> 00:12:41.960
but this is just a general
symbol for an operator.
00:12:41.960 --> 00:12:46.460
And an operator
operates on a function
00:12:46.460 --> 00:12:50.050
and gives a different function.
00:12:50.050 --> 00:12:56.130
It operates to the
right, or at least we
00:12:56.130 --> 00:12:58.716
like to think about it as
operating to the right.
00:12:58.716 --> 00:13:00.090
If we let it
operate to the left,
00:13:00.090 --> 00:13:02.230
we have to figure out
what the rules are,
00:13:02.230 --> 00:13:04.260
and I'm not ready to
tell you about that.
00:13:04.260 --> 00:13:06.700
So this operator
has to be linear.
00:13:09.530 --> 00:13:14.590
And so if we have an
operator operating
00:13:14.590 --> 00:13:24.855
on a function, A f plus
bg, It has to do this.
00:13:30.010 --> 00:13:33.430
Now you'd think, well,
that's pretty simple.
00:13:33.430 --> 00:13:35.490
Anything should do that.
00:13:35.490 --> 00:13:37.460
So taking the
derivative does that.
00:13:37.460 --> 00:13:41.796
Doing an integral does that, but
taking the square root doesn't.
00:13:41.796 --> 00:13:44.220
So taking the square
root-- an operator says
00:13:44.220 --> 00:13:48.520
take the square root, well,
that's not a linear operator.
00:13:48.520 --> 00:13:50.700
Now the only operators
in quantum mechanics
00:13:50.700 --> 00:13:51.979
are linear operators.
00:13:58.400 --> 00:14:00.410
We have eigenvalue equations.
00:14:08.430 --> 00:14:13.090
So we have an operator
operating in some function.
00:14:13.090 --> 00:14:18.485
It gives a number and
the function back again.
00:14:18.485 --> 00:14:20.490
And this is called
the eigenvalues,
00:14:20.490 --> 00:14:21.740
and this is the eigenfunction.
00:14:21.740 --> 00:14:22.900
AUDIENCE: Dr. Field?
00:14:22.900 --> 00:14:23.730
ROBERT FIELD: Yes?
00:14:23.730 --> 00:14:25.140
AUDIENCE: In the
[INAUDIBLE] that
00:14:25.140 --> 00:14:28.228
should be b times
A hat [INAUDIBLE]..
00:14:31.210 --> 00:14:32.780
ROBERT FIELD: What did I do?
00:14:32.780 --> 00:14:35.490
Well, it should just be--
00:14:35.490 --> 00:14:38.060
sorry about that.
00:14:38.060 --> 00:14:41.910
I'm going to make
mistakes like this.
00:14:41.910 --> 00:14:43.870
The TAs are going
to catch me on it,
00:14:43.870 --> 00:14:45.150
and you're going to do it too.
00:14:48.230 --> 00:14:51.890
All right, so now here we
have an operator operating
00:14:51.890 --> 00:14:53.180
on some function.
00:14:53.180 --> 00:14:55.790
And this function is
special because when
00:14:55.790 --> 00:14:57.710
the operator operates
on it, it returns
00:14:57.710 --> 00:15:01.130
the function times a
number, the eigenvalue
00:15:01.130 --> 00:15:03.590
and the eigenfunction.
00:15:03.590 --> 00:15:04.520
We like these.
00:15:04.520 --> 00:15:07.460
Almost all quantum mechanics
is expressed in terms
00:15:07.460 --> 00:15:09.448
of eigenvalue equations.
00:15:17.420 --> 00:15:25.430
Operators in quantum mechanics--
00:15:25.430 --> 00:15:32.430
so for every physical quantity
in non-quantum-mechanical life
00:15:32.430 --> 00:15:35.620
there corresponds an operator
in quantum mechanics.
00:15:35.620 --> 00:15:40.960
So for the coordinate,
the operator
00:15:40.960 --> 00:15:43.530
is just the coordinate.
00:15:43.530 --> 00:15:46.550
For the momentum,
the operator is minus
00:15:46.550 --> 00:15:51.270
ih bar partial with respect to
x or derivative with respect
00:15:51.270 --> 00:15:52.650
to x.
00:15:52.650 --> 00:15:54.570
Now this is not too
surprising, but this
00:15:54.570 --> 00:16:00.610
is really puzzling because why
is there an imaginary number?
00:16:00.610 --> 00:16:04.680
This is the square root of
minus 1, which we call i.
00:16:04.680 --> 00:16:08.370
Why is that there and why
is the operator a derivative
00:16:08.370 --> 00:16:09.930
rather than just
some simple thing?
00:16:13.350 --> 00:16:16.620
Another operator is
the kinetic energy,
00:16:16.620 --> 00:16:20.070
and the kinetic energy
is p squared over 2m.
00:16:24.760 --> 00:16:30.570
And so that comes out to
be minus h bar squared
00:16:30.570 --> 00:16:36.570
over 2m second partial
with respect to x.
00:16:43.240 --> 00:16:45.800
Well, it's nice that I
don't have to memorize this
00:16:45.800 --> 00:16:49.940
because I can just square
this and this pops out,
00:16:49.940 --> 00:16:52.400
but you have to be
aware of how to operate
00:16:52.400 --> 00:16:56.060
with complex and
imaginary numbers.
00:16:56.060 --> 00:17:00.050
And there are so many
exercises on the problem set,
00:17:00.050 --> 00:17:03.440
so you should be
up to date on that.
00:17:03.440 --> 00:17:12.540
And now the potential
is just the potential.
00:17:12.540 --> 00:17:14.220
And now the most
important operator,
00:17:14.220 --> 00:17:16.680
at least when we start
out, is the Hamiltonian,
00:17:16.680 --> 00:17:20.040
which is the operator that
corresponds to energy,
00:17:20.040 --> 00:17:26.236
and that is kinetic energy
plus potential energy.
00:17:26.236 --> 00:17:28.369
And this is called
the Hamiltonian,
00:17:28.369 --> 00:17:32.010
and we're going to be
focusing a lot on that.
00:17:32.010 --> 00:17:35.650
So these are the operators
you're going to care about.
00:17:35.650 --> 00:17:42.470
The next thing we talk
about is commutation rules
00:17:42.470 --> 00:17:44.800
or commutators.
00:17:44.800 --> 00:17:48.520
And one really
important commutator
00:17:48.520 --> 00:17:50.830
is the commutator
of the coordinate
00:17:50.830 --> 00:17:53.470
with the conjugate
momentum, conjugate
00:17:53.470 --> 00:17:55.390
meaning in the same direction.
00:17:55.390 --> 00:18:02.350
And that is defined
as xp minus px.
00:18:02.350 --> 00:18:08.350
And the obvious thing is that
this commutator would be 0.
00:18:08.350 --> 00:18:11.490
Why does it matter which
order you write things?
00:18:11.490 --> 00:18:13.160
But it does matter.
00:18:13.160 --> 00:18:17.260
And, in fact, one approach
to quantum mechanics
00:18:17.260 --> 00:18:20.290
is to start not
with the postulates
00:18:20.290 --> 00:18:24.880
that you normally deal with
but a set of commutation rules,
00:18:24.880 --> 00:18:29.360
and everything can be derived
from the commutation rules.
00:18:29.360 --> 00:18:32.150
It's a much more
abstract approach,
00:18:32.150 --> 00:18:35.790
but it's a very
powerful approach.
00:18:35.790 --> 00:18:41.350
So this commutator is not zero.
00:18:41.350 --> 00:18:44.080
And how do you find out
what a commutator is?
00:18:44.080 --> 00:18:52.180
Well, you do xp minus px,
operate on some function,
00:18:52.180 --> 00:18:52.990
and you find out.
00:18:55.770 --> 00:18:58.140
And you could do that.
00:18:58.140 --> 00:18:59.310
I could do that.
00:18:59.310 --> 00:19:03.860
But the commutator is going
to be equal to ih bar.
00:19:07.850 --> 00:19:10.880
Now there is a little
bit of trickiness
00:19:10.880 --> 00:19:17.360
because the commutator xp is
ih bar and px is minus ih bar.
00:19:17.360 --> 00:19:20.870
And so I don't
recommend memorizing it.
00:19:20.870 --> 00:19:26.900
I recommend being able
to do this operation
00:19:26.900 --> 00:19:30.470
at the speed of light
so you know whether it's
00:19:30.470 --> 00:19:33.690
plus ih bar or minus
ih bar because you
00:19:33.690 --> 00:19:36.120
get into a whole lot of
trouble if you get it wrong.
00:19:39.390 --> 00:19:44.310
So this is really where
it all begins, and this
00:19:44.310 --> 00:19:48.480
is why you can't
make simultaneously
00:19:48.480 --> 00:19:53.160
precise measurements of
position and momentum,
00:19:53.160 --> 00:19:54.510
and lots of other good things.
00:19:57.250 --> 00:19:58.690
And then we have wave functions.
00:20:06.200 --> 00:20:16.480
So wave function for when the
time independent Hamiltonian is
00:20:16.480 --> 00:20:21.340
a function of one variable, and
it contains everything we could
00:20:21.340 --> 00:20:25.890
possibly know about the system.
00:20:25.890 --> 00:20:28.410
But this strange
and wonderful thing,
00:20:28.410 --> 00:20:32.680
which leads to all sorts
of philosophical debates,
00:20:32.680 --> 00:20:37.360
is that this guy, which contains
everything that we can know,
00:20:37.360 --> 00:20:40.920
can never be directly measured.
00:20:40.920 --> 00:20:43.350
You can only
measure what happens
00:20:43.350 --> 00:20:46.950
when you act on something
with a given wave function.
00:20:46.950 --> 00:20:49.320
You cannot observe
the wave function.
00:20:49.320 --> 00:20:57.490
And for a subject area where the
central thing is unobservable
00:20:57.490 --> 00:20:59.540
is rather spooky.
00:20:59.540 --> 00:21:02.310
And a lot of people
don't like that approach
00:21:02.310 --> 00:21:07.680
because it says we've got this
thing that we're relying on,
00:21:07.680 --> 00:21:08.790
but we can't observe it.
00:21:08.790 --> 00:21:13.260
We can only observe what
we do when we act on it.
00:21:13.260 --> 00:21:16.540
And usually the
action is destructive.
00:21:16.540 --> 00:21:18.690
It's destructive of the
state of the system.
00:21:18.690 --> 00:21:21.420
It causes the
state of the system
00:21:21.420 --> 00:21:26.580
to give you a set
of possible answers,
00:21:26.580 --> 00:21:30.680
and not the same
thing each time.
00:21:30.680 --> 00:21:31.880
So it's really weird.
00:21:34.780 --> 00:21:37.370
So we have wave functions.
00:21:37.370 --> 00:21:42.530
And we can use the
wave function to find
00:21:42.530 --> 00:21:51.640
the probability of the system
at x with a range of x, x
00:21:51.640 --> 00:21:53.730
to x plus dx.
00:21:53.730 --> 00:21:58.645
And that's psi
star x psi of x dx.
00:22:01.420 --> 00:22:04.270
So you notice we have two wave
functions, the product of two,
00:22:04.270 --> 00:22:07.550
and this star means takes
a complex conjugate.
00:22:07.550 --> 00:22:16.270
So if you have a complex number
z is equal to x plus iy--
00:22:16.270 --> 00:22:18.860
real part, imaginary part--
00:22:18.860 --> 00:22:22.900
and if we take z star,
that's x minus iy.
00:22:27.690 --> 00:22:32.340
So these wave functions
are complex functions
00:22:32.340 --> 00:22:35.110
of a real variable.
00:22:35.110 --> 00:22:39.170
And so we do things like
take the complex conjugate,
00:22:39.170 --> 00:22:43.220
and you have to become
familiar with that.
00:22:43.220 --> 00:22:56.620
Now we have what we call
the expectation value
00:22:56.620 --> 00:23:05.090
or the average value,
and we denote this as A.
00:23:05.090 --> 00:23:08.020
So for the state
function psi, we
00:23:08.020 --> 00:23:13.820
want the average value of the
operator A. Now in most life,
00:23:13.820 --> 00:23:20.960
that symbol is not included
just because people assume
00:23:20.960 --> 00:23:22.340
you know what you're doing.
00:23:22.340 --> 00:23:36.130
And this is psi star A hat
psi dx over psi star psi dx.
00:23:36.130 --> 00:23:39.640
And this is integral from
minus infinity to infinity.
00:23:42.330 --> 00:23:48.360
So this down here is a
normalization integral.
00:23:48.360 --> 00:23:53.690
Now we normally deal
with state functions
00:23:53.690 --> 00:23:56.840
which are normalized to
1, meaning the particle
00:23:56.840 --> 00:23:59.300
is somewhere.
00:23:59.300 --> 00:24:02.320
But if the particle
can go anywhere,
00:24:02.320 --> 00:24:08.600
then normalization to 1 means
it's approximately nowhere.
00:24:08.600 --> 00:24:11.380
And so we have to think
a little bit about what
00:24:11.380 --> 00:24:14.920
do we mean by
normalization, but this
00:24:14.920 --> 00:24:19.300
is how we define the average
value or the expectation
00:24:19.300 --> 00:24:23.120
value of the quantity
A for the state psi.
00:24:28.350 --> 00:24:30.510
So this is just a
little bit of a warning
00:24:30.510 --> 00:24:33.280
that, yeah, you would
think this is all you need,
00:24:33.280 --> 00:24:35.790
but you also need to at
least think about this.
00:24:39.549 --> 00:24:40.090
That's great.
00:24:40.090 --> 00:24:44.610
I'm at the top of the board
and we're now at the beginning.
00:24:44.610 --> 00:24:48.070
So the Schrodinger
equation is the last thing,
00:24:48.070 --> 00:24:51.850
and that's the Hamiltonian
operating on the function
00:24:51.850 --> 00:24:56.350
and gives an energy
times that function.
00:24:56.350 --> 00:24:57.820
And if it's an
eigenvalue, then we
00:24:57.820 --> 00:25:00.050
have this eigenvalue equation.
00:25:00.050 --> 00:25:03.110
We have these symbols here.
00:25:03.110 --> 00:25:07.820
So that's the energy associated
with the psi n function.
00:25:12.070 --> 00:25:15.910
So now we're ready to
start playing games
00:25:15.910 --> 00:25:18.710
with this strange new world.
00:25:18.710 --> 00:25:20.850
And so let's start out
with the free particle.
00:25:29.190 --> 00:25:33.510
Now because the free particle
has a complicated feature
00:25:33.510 --> 00:25:36.630
about how do we
normalize it, it really
00:25:36.630 --> 00:25:39.630
shouldn't be the first
thing we talk about.
00:25:39.630 --> 00:25:44.400
But it seems like the
simplest problem, so we will.
00:25:44.400 --> 00:25:47.314
So what's the Hamiltonian?
00:25:47.314 --> 00:25:52.460
The Hamiltonian is the kinetic
energy, minus h bar squared,
00:25:52.460 --> 00:25:58.830
or 2m second derivative
with respect to x
00:25:58.830 --> 00:26:02.346
plus the potential energy, V0.
00:26:02.346 --> 00:26:04.935
Free particle, the
potential is constant.
00:26:07.530 --> 00:26:11.520
We normally think of it
as the potential is zero,
00:26:11.520 --> 00:26:15.620
but there is no absolute
scale of a zero of energy,
00:26:15.620 --> 00:26:17.340
so we just need to specify this.
00:26:20.900 --> 00:26:24.770
And so we want to write
the Schrodinger equation,
00:26:24.770 --> 00:26:30.380
and we want to arrange it in
a form that is easy to solve.
00:26:30.380 --> 00:26:32.270
There is two steps
to the rearrangement,
00:26:32.270 --> 00:26:37.650
and I'll just write
the final thing.
00:26:37.650 --> 00:26:40.010
So the second
derivative of psi is
00:26:40.010 --> 00:26:51.380
equal to minus 2m over h bar
squared times E minus V0 psi.
00:26:51.380 --> 00:26:55.260
So this is the differential
equation that we have to solve.
00:26:55.260 --> 00:26:57.260
So there was a little bit
of rearrangement here,
00:26:57.260 --> 00:27:00.320
but you can do that.
00:27:00.320 --> 00:27:03.170
So the second derivative
of some function
00:27:03.170 --> 00:27:06.740
is equal to some constant
times that function.
00:27:06.740 --> 00:27:09.590
We've seen that problem before.
00:27:09.590 --> 00:27:12.260
It makes a lot of difference
whether that constant
00:27:12.260 --> 00:27:17.380
is positive or
negative, and it better,
00:27:17.380 --> 00:27:20.830
because if we have
a potential V0
00:27:20.830 --> 00:27:24.370
and we have an
energy up here, well,
00:27:24.370 --> 00:27:25.540
that's perfectly reasonable.
00:27:25.540 --> 00:27:28.210
The particle can be
there, classically.
00:27:28.210 --> 00:27:29.770
But suppose the
energy is down here.
00:27:32.420 --> 00:27:36.720
If the zero of energy is
here, you can't go below it.
00:27:39.940 --> 00:27:42.340
That's a classically
forbidden situation.
00:27:45.030 --> 00:27:48.930
And so for the classically
allowed situation,
00:27:48.930 --> 00:27:53.910
the quantity, this
constant, is negative.
00:27:53.910 --> 00:27:56.400
For the classically
forbidden situation,
00:27:56.400 --> 00:27:58.800
this constant is positive.
00:27:58.800 --> 00:28:00.630
You've already seen
the big difference
00:28:00.630 --> 00:28:06.090
in the way a second derivative,
this kind of equation,
00:28:06.090 --> 00:28:12.700
works when the constant
is positive or negative.
00:28:12.700 --> 00:28:17.900
When this constant is
negative, you get oscillation.
00:28:17.900 --> 00:28:22.130
When this constant is positive,
you'll get exponential.
00:28:27.780 --> 00:28:35.440
Now we're interested
in a free particle,
00:28:35.440 --> 00:28:44.980
so free particle
can be anywhere.
00:28:44.980 --> 00:28:48.790
And we insist that
the solution to
00:28:48.790 --> 00:28:55.390
our quantum-mechanical
problem, the wave function
00:28:55.390 --> 00:28:56.925
is what we say well behaved.
00:28:59.960 --> 00:29:02.200
So well behaved has many
meanings, but one of them
00:29:02.200 --> 00:29:04.170
is it never goes to infinity.
00:29:07.170 --> 00:29:09.400
Another is that when
you go to infinity,
00:29:09.400 --> 00:29:11.175
the wave function
should go to zero.
00:29:14.680 --> 00:29:20.130
But there's also things about
continuity and continuity
00:29:20.130 --> 00:29:22.200
of first derivatives
and continuity
00:29:22.200 --> 00:29:24.180
of second derivatives.
00:29:24.180 --> 00:29:28.410
We'll get into those,
but you know immediately
00:29:28.410 --> 00:29:32.380
that if this
constant is positive,
00:29:32.380 --> 00:29:37.300
you get an exponential
behavior, and you get the e
00:29:37.300 --> 00:29:40.120
to the ikx and e to the--
00:29:40.120 --> 00:29:44.980
not ik-- e to the kx
and e to the minus kx.
00:29:44.980 --> 00:29:50.590
And one of those blows up
at either positive infinity
00:29:50.590 --> 00:29:52.670
or negative infinity.
00:29:52.670 --> 00:29:57.400
So it's telling you
that in agreement
00:29:57.400 --> 00:30:02.240
with what you expect
for the classical world,
00:30:02.240 --> 00:30:07.740
an energy below the constant
potential is illegal.
00:30:10.480 --> 00:30:16.360
It's illegal when this
situation persists to infinity.
00:30:16.360 --> 00:30:19.570
But we'll discover
that it is legal
00:30:19.570 --> 00:30:26.680
if the range of coordinate
for which the energy is less
00:30:26.680 --> 00:30:28.570
than the potential is finite.
00:30:28.570 --> 00:30:31.120
And that's called
tunneling, and tunneling
00:30:31.120 --> 00:30:32.800
is a quantum-mechanical
phenomenon.
00:30:32.800 --> 00:30:36.090
We will encounter that.
00:30:36.090 --> 00:30:40.980
So we know from our experience
with this kind of differential
00:30:40.980 --> 00:30:49.350
equation that the solutions
will have the form sine
00:30:49.350 --> 00:30:52.890
kx and cosine kx.
00:30:55.570 --> 00:31:01.690
But we choose to use
instead e to the ikx
00:31:01.690 --> 00:31:12.480
and e to the minus ikx
because this cosine kx
00:31:12.480 --> 00:31:18.570
is 1/2 e to the ikx
plus e to the minus ikx.
00:31:18.570 --> 00:31:21.040
And so we can use
these functions
00:31:21.040 --> 00:31:23.750
because they're more
convenient, more memorable.
00:31:23.750 --> 00:31:28.270
All the integrals and
derivatives are trivial.
00:31:28.270 --> 00:31:31.350
And so we do that.
00:31:34.550 --> 00:31:39.488
So the differential equation--
00:31:58.940 --> 00:32:04.250
and we saw before that we
already have what k is.
00:32:04.250 --> 00:32:11.120
So minus k squared is minus
2m over h bar squared--
00:32:11.120 --> 00:32:16.260
minus 2m over h bar
squared E minus V0.
00:32:26.840 --> 00:32:29.375
We take the derivative
of this function.
00:32:35.180 --> 00:32:41.550
This is the function, and
this is the eigenvalue.
00:32:41.550 --> 00:32:43.740
We take the second
derivative with respect to x.
00:32:43.740 --> 00:32:47.190
We get an ik from
this term and then
00:32:47.190 --> 00:32:52.050
another ik, which
makes minus k squared.
00:32:52.050 --> 00:32:56.070
And we get a minus ik
and another minus ik,
00:32:56.070 --> 00:33:03.790
and that gives a
minus k squared.
00:33:03.790 --> 00:33:09.700
And so, in fact, this is
an eigenvalue equation.
00:33:09.700 --> 00:33:15.490
We have the form where this
equation is an eigenfunction.
00:33:15.490 --> 00:33:17.830
With this, we have everything.
00:33:17.830 --> 00:33:30.600
So the energies for the free
particle, h bar k over h bar k
00:33:30.600 --> 00:33:46.920
squared over 2m plus V0, so
this is an eigenfunction,
00:33:46.920 --> 00:33:50.760
and this is the eigenvalue
associated with that function.
00:33:53.412 --> 00:33:54.480
We're done.
00:33:54.480 --> 00:33:55.530
That was an easy problem.
00:33:55.530 --> 00:33:58.100
I skipped some steps
because it's an easy problem
00:33:58.100 --> 00:34:01.400
and I want you to go
over it and make sure
00:34:01.400 --> 00:34:05.060
that you understand
the logic and can
00:34:05.060 --> 00:34:06.363
come to the same solution.
00:34:11.170 --> 00:34:14.120
Let's take a little side issue.
00:34:14.120 --> 00:34:20.370
Suppose we have psi
of x is e to the ikx.
00:34:24.870 --> 00:34:32.719
Well, we're going to find that
this is an eigenfunction of p,
00:34:32.719 --> 00:34:38.570
and the eigenvalue or the
expectation value of p
00:34:38.570 --> 00:34:42.650
is h bar k.
00:34:42.650 --> 00:34:48.620
And if we had minus e to the
minus ikx, then what we'd get
00:34:48.620 --> 00:34:49.880
is minus h bar k.
00:34:52.750 --> 00:35:00.480
So we have this relationship
between p, expectation value,
00:35:00.480 --> 00:35:02.238
and h bar k.
00:35:04.830 --> 00:35:07.830
So this corresponds
to the particle going
00:35:07.830 --> 00:35:10.750
in the positive x direction,
and this corresponds
00:35:10.750 --> 00:35:15.180
to the particle going in
the negative x direction.
00:35:15.180 --> 00:35:18.180
Everything is
perfectly reasonable.
00:35:18.180 --> 00:35:21.420
We have solutions to
the Schrodinger equation
00:35:21.420 --> 00:35:23.160
for the free particle.
00:35:23.160 --> 00:35:25.590
The solutions to
the free particle
00:35:25.590 --> 00:35:30.600
are also solutions to
the eigenvalue equation
00:35:30.600 --> 00:35:32.270
for momentum.
00:35:32.270 --> 00:35:36.498
And the two possible
eigenvalues for a given k
00:35:36.498 --> 00:35:41.080
are plus h bar k minus h bar k.
00:35:41.080 --> 00:35:45.540
Now that's fine.
00:35:45.540 --> 00:35:48.000
So everything works out.
00:35:48.000 --> 00:35:50.250
We're getting
things, although we
00:35:50.250 --> 00:35:55.980
have the definition of the
momentum having a minus 1,
00:35:55.980 --> 00:36:00.510
an i factor, and a
derivative factor.
00:36:00.510 --> 00:36:02.540
Everything works.
00:36:02.540 --> 00:36:04.240
Everything is as
you would expect.
00:36:07.760 --> 00:36:16.060
And the general solution
to the Schrodinger equation
00:36:16.060 --> 00:36:20.980
can have two different values,
the superposition of these two.
00:36:24.750 --> 00:36:36.460
Right now, this wave function
is the localized overall space.
00:36:36.460 --> 00:36:39.020
Now if we want to
normalize it, we'd
00:36:39.020 --> 00:36:44.150
like to calculate integral
minus infinity to infinity
00:36:44.150 --> 00:36:52.334
of psi star x psi of x dx.
00:36:55.680 --> 00:37:00.390
This is why we
like this notation
00:37:00.390 --> 00:37:08.640
because suppose we have a
function like this, psi star--
00:37:08.640 --> 00:37:10.280
well, actually like this.
00:37:10.280 --> 00:37:28.250
Psi star is equal to a star e
to the minus ikx plus b star
00:37:28.250 --> 00:37:39.200
e to the ikx, and psi
is a e to the plus ikx.
00:37:39.200 --> 00:37:44.770
This would be e
to the minus ikx.
00:37:44.770 --> 00:37:47.620
And so when we write this
integral, what we get
00:37:47.620 --> 00:37:55.840
is integral of
psi star psi dx is
00:37:55.840 --> 00:38:06.200
equal to a squared integral
minus infinity to infinity
00:38:06.200 --> 00:38:12.720
of a squared plus b squared dx.
00:38:15.350 --> 00:38:18.200
So we have two constants
which are real numbers
00:38:18.200 --> 00:38:20.750
because they're square modulus.
00:38:20.750 --> 00:38:22.630
They're additive.
00:38:22.630 --> 00:38:27.040
And we're integrating this
constant from minus infinity
00:38:27.040 --> 00:38:29.770
to infinity.
00:38:29.770 --> 00:38:31.010
We'll get infinity.
00:38:31.010 --> 00:38:32.510
We can't make this equal to 1.
00:38:39.480 --> 00:38:41.330
So we have to put
this in our head
00:38:41.330 --> 00:38:43.400
and say, well,
there's a problem when
00:38:43.400 --> 00:38:46.640
you have a wave function
that extends over all space.
00:38:46.640 --> 00:38:49.070
It can't be normalized
to 1, but it
00:38:49.070 --> 00:38:55.740
can be normalized so that for
a given distance in real space,
00:38:55.740 --> 00:38:59.910
it's got a probability
of 1 in that distance.
00:38:59.910 --> 00:39:03.060
So we have a different
form of normalization.
00:39:03.060 --> 00:39:07.960
But when we actually
calculate expectation values,
00:39:07.960 --> 00:39:12.550
we can still use this naive idea
of the normalization interval
00:39:12.550 --> 00:39:14.980
and we get the
right answer, even
00:39:14.980 --> 00:39:18.400
though because both the
numerator and denominator
00:39:18.400 --> 00:39:21.560
go to infinity and
those infinities cancel
00:39:21.560 --> 00:39:22.850
and everything works out.
00:39:22.850 --> 00:39:27.450
This is why we don't
do this first usually
00:39:27.450 --> 00:39:28.950
because there's
all of these things
00:39:28.950 --> 00:39:30.840
that you have to
convince yourself are OK.
00:39:33.400 --> 00:39:36.140
And they are and you should.
00:39:36.140 --> 00:39:43.850
But now let's go to the
famous particle in a box.
00:39:43.850 --> 00:39:50.540
It's so famous that we
always use this notation.
00:39:50.540 --> 00:39:53.570
This is particle
in an infinite box,
00:39:53.570 --> 00:39:57.650
and that means the particle
is in a box like this
00:39:57.650 --> 00:40:00.660
where the walls go to infinity.
00:40:00.660 --> 00:40:06.620
And so we normally
locate this box
00:40:06.620 --> 00:40:09.550
at a place where this
is the x coordinate
00:40:09.550 --> 00:40:13.520
and this is the
potential energy,
00:40:13.520 --> 00:40:16.871
and the width of the box is a.
00:40:16.871 --> 00:40:22.790
And we normally put the
left edge of the box at zero
00:40:22.790 --> 00:40:24.950
because that problem
is a little easier
00:40:24.950 --> 00:40:27.740
to solve than the
more logical thing
00:40:27.740 --> 00:40:33.300
where you say, OK, this
box is centered about zero.
00:40:33.300 --> 00:40:35.660
And that should bother
you because anytime
00:40:35.660 --> 00:40:39.522
you're interested in asking
about the symmetry of things
00:40:39.522 --> 00:40:41.480
you'll want to choose a
coordinate system which
00:40:41.480 --> 00:40:44.360
reflects that.
00:40:44.360 --> 00:40:46.400
Don't worry.
00:40:46.400 --> 00:40:48.800
I am going to ask
you about symmetry,
00:40:48.800 --> 00:40:52.490
and it's a simple thing to take
the solution for this problem
00:40:52.490 --> 00:40:54.110
and move it to the
left by a over 2.
00:40:56.630 --> 00:41:05.630
So we have basically a
problem where the potential is
00:41:05.630 --> 00:41:14.910
equal to 0 for 0 is less
than or equal to x less than
00:41:14.910 --> 00:41:23.130
or equal to a, and it's equal to
infinity when x is less than 0
00:41:23.130 --> 00:41:24.060
or greater than a.
00:41:28.420 --> 00:41:34.540
So inside the box it looks
like a free particle,
00:41:34.540 --> 00:41:36.550
but it can't be a free
particle because there's
00:41:36.550 --> 00:41:40.800
got to be nodes at the walls.
00:41:40.800 --> 00:41:44.640
We know that outside the
box, the wave function
00:41:44.640 --> 00:41:50.280
has to be 0 everywhere because
it's classically forbidden,
00:41:50.280 --> 00:41:51.830
strongly forbidden.
00:41:56.060 --> 00:42:03.900
We know that the wave
function psi is continuous.
00:42:03.900 --> 00:42:08.610
So if it's at 0 outside, it's
going to be 0 at the wall.
00:42:08.610 --> 00:42:13.390
And so the wave functions
have boundary conditions
00:42:13.390 --> 00:42:17.140
where, at the wall, the
wave function goes to 0.
00:42:24.140 --> 00:42:28.110
So now we go and we
solve this problem.
00:42:28.110 --> 00:42:38.450
And so the Schrodinger equation
for the particle in the box
00:42:38.450 --> 00:42:41.120
where V of x is 0.
00:42:41.120 --> 00:42:42.410
Well, we don't need it.
00:42:42.410 --> 00:42:44.435
We just have the
kinetic-energy term,
00:42:44.435 --> 00:42:51.860
h bar squared over 2m second
derivative with respect to x.
00:42:51.860 --> 00:42:56.700
Psi is equal to e psi.
00:42:56.700 --> 00:42:58.980
Again, we rearrange it.
00:42:58.980 --> 00:43:03.090
And so we put the
derivative outside,
00:43:03.090 --> 00:43:10.547
and we have minus 2me
over h bar squared psi.
00:43:14.300 --> 00:43:16.670
And this is a number.
00:43:16.670 --> 00:43:21.760
And so we just call it
minus k squared psi.
00:43:24.390 --> 00:43:28.620
We know what that k is as long
as we know what the energy is,
00:43:28.620 --> 00:43:35.330
and k squared is equal to
2me over h bar squared.
00:43:39.430 --> 00:43:45.160
Now we have this thing which
is equal to a negative number
00:43:45.160 --> 00:43:47.140
times a wave function,
and we already
00:43:47.140 --> 00:43:49.285
know we have
exponential behavior.
00:43:51.790 --> 00:43:54.820
But in this case, we
use sines and cosines
00:43:54.820 --> 00:43:57.920
because it's more convenient.
00:43:57.920 --> 00:44:08.970
So psi of x is going to be
written as A sine kx plus B
00:44:08.970 --> 00:44:12.670
cosine kx.
00:44:12.670 --> 00:44:19.680
This is the general solution
for this differential equation
00:44:19.680 --> 00:44:25.700
where we have a negative
constant times the function.
00:44:25.700 --> 00:44:30.526
So the boundary
condition, psi of 0--
00:44:30.526 --> 00:44:35.180
well, psi of 0, this is
0, but this part is 1.
00:44:35.180 --> 00:44:41.150
So that means that
psi of 0 has to be 0,
00:44:41.150 --> 00:44:45.050
so B has to be equal to 0.
00:44:45.050 --> 00:44:48.530
And here, the other
boundary condition,
00:44:48.530 --> 00:44:57.290
that also has to be equal to 0,
and that has to be A sine ka.
00:44:57.290 --> 00:45:02.750
And ka has to be equal
to n times pi in order
00:45:02.750 --> 00:45:03.845
to satisfy this equation.
00:45:07.850 --> 00:45:14.943
Sine is 0 at 0, pi, 2
pi, 3 pi, et cetera.
00:45:20.870 --> 00:45:25.310
So k is equal to n pi over a.
00:45:28.214 --> 00:45:30.790
And so we have the solutions.
00:45:30.790 --> 00:45:43.340
Psi of x is equal to
A sine np a over x.
00:45:43.340 --> 00:45:47.609
And so we could put
a little n here.
00:45:47.609 --> 00:45:49.150
And this is starting
to make you feel
00:45:49.150 --> 00:45:55.470
really good because for
all positive integers
00:45:55.470 --> 00:45:57.890
there is a solution.
00:45:57.890 --> 00:46:00.930
There's an infinite
number of solutions.
00:46:00.930 --> 00:46:04.530
And their scaling with
quantum number is trivial.
00:46:04.530 --> 00:46:08.670
And it's really great
when you solve an equation
00:46:08.670 --> 00:46:11.640
and you are given an
infinite number of solutions.
00:46:15.317 --> 00:46:17.150
Well, there's one thing
more you have to do.
00:46:17.150 --> 00:46:20.510
You have to find out what the
normalization constant is,
00:46:20.510 --> 00:46:22.860
so you do the
normalization integral.
00:46:22.860 --> 00:46:25.880
And when you do
that, you discover
00:46:25.880 --> 00:46:37.880
that this is equal to 2 over a
square root sine n pi over a.
00:46:37.880 --> 00:46:41.660
So these are all the
solutions for the particle
00:46:41.660 --> 00:46:44.436
in an infinite box, all of them.
00:46:44.436 --> 00:46:54.300
And the energies you can write
as n squared times h squared
00:46:54.300 --> 00:47:02.590
over 8 m a squared or
n squared times E1.
00:47:02.590 --> 00:47:03.970
There's another thing.
00:47:03.970 --> 00:47:07.060
n equals 0.
00:47:07.060 --> 00:47:12.760
If n is equal to 0, the wave
function corresponding to n
00:47:12.760 --> 00:47:14.890
equals 0 is 0 everywhere.
00:47:14.890 --> 00:47:17.130
The particle isn't in the box.
00:47:17.130 --> 00:47:21.260
So n equals 0 is not a solution.
00:47:21.260 --> 00:47:22.960
So the solutions we have--
00:47:22.960 --> 00:47:27.580
n equals 1 2, et
cetera up to infinity,
00:47:27.580 --> 00:47:33.700
and the energies are
integer square multiples
00:47:33.700 --> 00:47:34.915
of a common factor.
00:47:39.350 --> 00:47:43.770
This is wonderful because
basically we have a problem.
00:47:43.770 --> 00:47:45.830
Maybe it's not that
interesting now
00:47:45.830 --> 00:47:49.820
because why do we have infinite
boxes and stuff like that?
00:47:49.820 --> 00:47:55.920
But if you ask, what
about the ideal gas law?
00:47:55.920 --> 00:47:59.740
We have particles that don't
interact with each other inside
00:47:59.740 --> 00:48:02.940
a container which
has infinite walls.
00:48:02.940 --> 00:48:05.520
And I can tell you
that in 5.62, there's
00:48:05.520 --> 00:48:07.500
a three-line derivation
of the ideal gas
00:48:07.500 --> 00:48:11.960
law based on solutions
to the particle in a box.
00:48:11.960 --> 00:48:14.690
Also, we often have
situations where
00:48:14.690 --> 00:48:17.690
you have molecules where
there's conjugation
00:48:17.690 --> 00:48:22.250
so that the molecule looks
like a not quite flat bottom
00:48:22.250 --> 00:48:25.000
box with walls.
00:48:25.000 --> 00:48:29.080
And this equation enables
you to learn something
00:48:29.080 --> 00:48:31.600
about the electronic
energy levels
00:48:31.600 --> 00:48:35.270
for linear conjugated molecules.
00:48:35.270 --> 00:48:37.730
And this leads to a lot
of qualitative insight
00:48:37.730 --> 00:48:42.870
into problems in photochemistry.
00:48:45.530 --> 00:48:53.900
Now the most important
thing, in my opinion,
00:48:53.900 --> 00:48:58.300
is being able to draw
cartoons, and these cartoons
00:48:58.300 --> 00:49:03.310
for the solutions the particle
in a box look like this.
00:49:03.310 --> 00:49:07.990
So what you frequently
do is draw the potential,
00:49:07.990 --> 00:49:11.600
and then you draw the energy
levels and wave functions.
00:49:17.710 --> 00:49:19.490
I have to cheat a little bit.
00:49:24.580 --> 00:49:27.430
So number of nodes--
00:49:27.430 --> 00:49:29.770
number of nodes, internal nodes.
00:49:29.770 --> 00:49:32.020
We don't count
zeros at the walls.
00:49:32.020 --> 00:49:35.560
The number of
nodes is n minus 1.
00:49:38.110 --> 00:49:46.270
The maximum of the wave
function is always 2 over a.
00:49:50.330 --> 00:49:53.780
So here 2 over a, here
2 over a, here minus 2
00:49:53.780 --> 00:49:57.320
over a i square
root of 2 over a.
00:49:57.320 --> 00:50:00.209
This slope is identical
to this slope.
00:50:00.209 --> 00:50:02.000
This slope is identical
to this slope which
00:50:02.000 --> 00:50:04.980
is identical to that slope.
00:50:04.980 --> 00:50:07.490
So there's a tremendous
amount that you
00:50:07.490 --> 00:50:12.576
can get by understanding
how the wave function looks
00:50:12.576 --> 00:50:13.700
and drawing these cartoons.
00:50:18.790 --> 00:50:27.250
And so now if instead we were
looking at problems where,
00:50:27.250 --> 00:50:30.700
instead of a particle
in a box like this,
00:50:30.700 --> 00:50:33.460
we have a little dimple
in the bottom of the box
00:50:33.460 --> 00:50:38.020
or we have something at the
bottom or the bottom of the box
00:50:38.020 --> 00:50:41.270
is slanted.
00:50:41.270 --> 00:50:45.290
You should be able to
intuit what these things do
00:50:45.290 --> 00:50:48.050
to the energy
levels, at least have
00:50:48.050 --> 00:50:52.290
the beginning of an intuition.
00:50:52.290 --> 00:50:59.730
So, we have an infinite number
of oscillating solutions.
00:50:59.730 --> 00:51:02.970
That means that we
could solve the problem
00:51:02.970 --> 00:51:07.740
for any kind of a box as long
as it has vertical walls,
00:51:07.740 --> 00:51:09.710
and that's called
a Fourier series.
00:51:12.890 --> 00:51:17.270
So for a finite range, we
can describe the solution
00:51:17.270 --> 00:51:20.240
to any quantum-mechanical
infinite
00:51:20.240 --> 00:51:24.140
box with a terrible
bottom, in principle,
00:51:24.140 --> 00:51:28.019
by a superposition of
our basis functions.
00:51:28.019 --> 00:51:29.060
That's what we call them.
00:51:32.090 --> 00:51:35.800
Now, there are several
methods for doing
00:51:35.800 --> 00:51:42.220
the solution of a problem
like this efficiently.
00:51:42.220 --> 00:51:46.620
And you're going to see
perturbation theory.
00:51:46.620 --> 00:51:48.120
And at the end of
the course, you're
00:51:48.120 --> 00:51:50.620
going to see something that
will really knock your socks off
00:51:50.620 --> 00:51:55.110
which is called the discrete
variable representation.
00:51:55.110 --> 00:51:58.080
And that enables you
to say, yeah, well,
00:51:58.080 --> 00:52:00.120
potential does terrible things.
00:52:00.120 --> 00:52:03.210
I solved the problem by not
doing any calculation at all
00:52:03.210 --> 00:52:06.630
because it's already been done.
00:52:06.630 --> 00:52:11.160
So these things are
fantastic that we
00:52:11.160 --> 00:52:15.360
have an infinite number of
solutions to a simple problem.
00:52:15.360 --> 00:52:20.880
We're always looking for a way
to describe a simple problem
00:52:20.880 --> 00:52:23.820
or maybe not so simple problem
with an infinite number
00:52:23.820 --> 00:52:28.230
of solutions where the energies
for the solutions and the wave
00:52:28.230 --> 00:52:32.771
functions behave in a simple
n-scaled, quantum-number-scaled
00:52:32.771 --> 00:52:33.270
way.
00:52:35.800 --> 00:52:40.450
And this provides us with
a way of looking at what
00:52:40.450 --> 00:52:43.730
these things do in real life.
00:52:43.730 --> 00:52:46.180
You do an experiment
on an eigenfunction
00:52:46.180 --> 00:52:50.470
of a box like this, and it will
have certain characteristics.
00:52:50.470 --> 00:52:52.960
And it tells you, oh,
if I measured the energy
00:52:52.960 --> 00:53:00.580
levels of a pathological box,
the quantum-number dependence
00:53:00.580 --> 00:53:03.590
of the energy levels
has a certain form,
00:53:03.590 --> 00:53:09.350
and each of the constants
in that special form
00:53:09.350 --> 00:53:14.530
sample a particular feature
of the pathological potential.
00:53:14.530 --> 00:53:16.360
And that's what we do
as spectroscopists.
00:53:16.360 --> 00:53:20.770
We find an efficient way to
fit the observables in order
00:53:20.770 --> 00:53:27.340
to characterize what's going on
inside something we can't see.
00:53:27.340 --> 00:53:29.440
And that's the game
in quantum mechanics.
00:53:29.440 --> 00:53:32.650
We can't see the
wave function ever.
00:53:32.650 --> 00:53:34.900
We know there are eigenstates.
00:53:34.900 --> 00:53:39.310
We can observe energy levels
and transition probabilities,
00:53:39.310 --> 00:53:45.100
and between those two things,
we can determine quantitatively
00:53:45.100 --> 00:53:50.470
all of the internal structure
of objects that we can't see.
00:53:50.470 --> 00:53:52.960
And this is what I do
as a spectroscopist.
00:53:52.960 --> 00:53:57.130
And I'm really a little
bit crazy about it
00:53:57.130 --> 00:54:02.880
because most people
instead of saying
00:54:02.880 --> 00:54:05.990
let's try to understand
based on something simple,
00:54:05.990 --> 00:54:08.180
they will just solve the
Schrodinger equations
00:54:08.180 --> 00:54:13.430
numerically and get a
bunch of small results
00:54:13.430 --> 00:54:16.730
and no intuition, no
cartoons, and no ability
00:54:16.730 --> 00:54:19.610
to do dynamics except
another picture
00:54:19.610 --> 00:54:23.950
where you have to work things
out in a complicated way.
00:54:23.950 --> 00:54:27.460
But I'm giving you the
standard problems from which
00:54:27.460 --> 00:54:30.620
you can solve almost anything.
00:54:30.620 --> 00:54:33.970
And this should sound
like fun, I hope.
00:54:33.970 --> 00:54:36.790
OK, so have a great weekend.