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PROFESSOR: OK, so
I want to start out
00:00:25.030 --> 00:00:27.420
by reviewing a few
things and putting
00:00:27.420 --> 00:00:29.620
some machinery together.
00:00:29.620 --> 00:00:31.560
Unfortunately, this
thing is sort of stuck.
00:00:31.560 --> 00:00:35.300
We're going to need a
later, so I don't know.
00:00:35.300 --> 00:00:37.700
I'll put it up for now.
00:00:37.700 --> 00:00:40.410
So first just a bit of notation.
00:00:40.410 --> 00:00:42.250
This symbol, you
should think of it
00:00:42.250 --> 00:00:44.520
like the dot product,
or the inner product.
00:00:44.520 --> 00:00:49.160
It's just saying that
bracket f g is the integral.
00:00:49.160 --> 00:00:50.450
It's a number that you get.
00:00:50.450 --> 00:00:52.780
So this is a number that
you get from the function
00:00:52.780 --> 00:00:55.820
f and the function g
by taking f, taking
00:00:55.820 --> 00:00:58.840
its complex conjugate,
multiplying it by g,
00:00:58.840 --> 00:01:00.710
and then integrating
overall positions.
00:01:00.710 --> 00:01:01.490
All right?
00:01:01.490 --> 00:01:02.740
So it's a way to get a number.
00:01:02.740 --> 00:01:05.310
And you should think about it
as the analog for functions
00:01:05.310 --> 00:01:07.530
of the dot product for vectors.
00:01:07.530 --> 00:01:11.490
It's a way to get a
number out of two vectors.
00:01:11.490 --> 00:01:15.860
And so, for example, with
vectors we could do v dot w,
00:01:15.860 --> 00:01:16.940
and this is some number.
00:01:16.940 --> 00:01:20.712
And it has a nice
property that v dot v,
00:01:20.712 --> 00:01:23.170
we can think it as v squared,
it's something like a length.
00:01:23.170 --> 00:01:25.480
It's strictly positive,
and it's something
00:01:25.480 --> 00:01:26.760
like the length of a vector.
00:01:26.760 --> 00:01:31.320
Similarly, if I take f and
take its bracket with f,
00:01:31.320 --> 00:01:34.205
this is equal to the
integral dx of f squared,
00:01:34.205 --> 00:01:36.580
and in particular, f could be
complex, so f norm squared.
00:01:36.580 --> 00:01:38.000
This is strictly non-negative.
00:01:38.000 --> 00:01:41.550
It could vanish, but it's
not negative at a point,
00:01:41.550 --> 00:01:42.740
hence the norm squared.
00:01:42.740 --> 00:01:46.990
So this will be zero
if and only if what?
00:01:46.990 --> 00:01:49.010
f is 0, f is the
0 function, right.
00:01:49.010 --> 00:01:51.852
So the same way that
if you take a vector,
00:01:51.852 --> 00:01:54.310
and you take its dot product
with itself, take it the norm,
00:01:54.310 --> 00:01:56.470
it's 0 if an only
if the vector is 0.
00:01:56.470 --> 00:01:59.270
So this beast satisfies a lot
of the properties of a dot
00:01:59.270 --> 00:01:59.920
product.
00:01:59.920 --> 00:02:01.980
You should think about
it as morally equivalent.
00:02:01.980 --> 00:02:04.190
We'll talk about that
in more detail later.
00:02:04.190 --> 00:02:06.670
Second, basic postulate
of quantum mechanics,
00:02:06.670 --> 00:02:08.972
to every observable is
associated an operator,
00:02:08.972 --> 00:02:11.180
and it's an operator acting
on the space of functions
00:02:11.180 --> 00:02:14.340
or on the space
of wave functions.
00:02:14.340 --> 00:02:16.249
And to every operator
corresponding
00:02:16.249 --> 00:02:17.790
to an observable in
quantum mechanics
00:02:17.790 --> 00:02:20.060
are associated a special
set of functions called
00:02:20.060 --> 00:02:22.640
the eigenfunctions, such
that when the operator acts
00:02:22.640 --> 00:02:24.640
on that function, it gives
you the same function
00:02:24.640 --> 00:02:26.560
back times a constant.
00:02:26.560 --> 00:02:29.100
What these functions
mean, physically,
00:02:29.100 --> 00:02:32.860
is they are the wave functions
describing configurations
00:02:32.860 --> 00:02:35.930
with a definite value of the
corresponding observable.
00:02:35.930 --> 00:02:38.240
If I'm in an eigenfunction
of position with
00:02:38.240 --> 00:02:41.610
eigenvalue x naught, awesome.
00:02:41.610 --> 00:02:44.740
Thank you, AV person, thank you.
00:02:44.740 --> 00:02:48.742
So if your system is described
by a wave function which
00:02:48.742 --> 00:02:50.200
is an eigenfunction
of the position
00:02:50.200 --> 00:02:52.470
operator with
eigenvalue x naught,
00:02:52.470 --> 00:02:55.084
that means you can be
confident that the system is
00:02:55.084 --> 00:02:56.500
in the configuration
corresponding
00:02:56.500 --> 00:02:59.111
to having a definite
position x naught.
00:02:59.111 --> 00:02:59.610
Right?
00:02:59.610 --> 00:03:02.360
It's not a superposition
of different positions.
00:03:02.360 --> 00:03:04.980
It is at x naught.
00:03:04.980 --> 00:03:07.567
Similarly, momentum,
momentum has eigenfunctions,
00:03:07.567 --> 00:03:08.900
and we know what these guys are.
00:03:08.900 --> 00:03:12.300
These are the exponentials,
e to the iKX's.
00:03:12.300 --> 00:03:14.890
They're the eigenfunctions, and
those are the wave functions
00:03:14.890 --> 00:03:18.130
describing states with
definite value of the momentum,
00:03:18.130 --> 00:03:19.780
of the associated observable.
00:03:19.780 --> 00:03:26.640
Energy as an operator, energy is
described by an operator, which
00:03:26.640 --> 00:03:34.340
has eigenfunctions which I'll
call phi sub n, with energy
00:03:34.340 --> 00:03:37.910
as E sub n, those
are the eigenvalues.
00:03:37.910 --> 00:03:40.920
And if I tell you that your
wave function is the state phi
00:03:40.920 --> 00:03:43.330
sub 2, what that tells
you is that the system has
00:03:43.330 --> 00:03:46.540
a definite energy, E
sub 2, corresponding
00:03:46.540 --> 00:03:48.883
to that eigenvalue.
00:03:48.883 --> 00:03:51.166
Cool?
00:03:51.166 --> 00:03:53.040
And this is true for
any physical observable.
00:03:53.040 --> 00:03:54.350
But these are sort
of the basic ones
00:03:54.350 --> 00:03:56.849
that we'll keep focusing on,
position, momentum, and energy,
00:03:56.849 --> 00:03:58.930
for the next while.
00:03:58.930 --> 00:04:01.360
Now a nice property about
these eigenfunctions
00:04:01.360 --> 00:04:04.740
is that for different
eigenvalues,
00:04:04.740 --> 00:04:07.220
the associated wave functions
are different functions.
00:04:07.220 --> 00:04:08.320
And what I mean by
saying they're different
00:04:08.320 --> 00:04:09.736
functions is that
they're actually
00:04:09.736 --> 00:04:12.540
orthogonal functions in the
sense of this dot product.
00:04:12.540 --> 00:04:16.050
If I have a state
corresponding to be at x 0,
00:04:16.050 --> 00:04:17.579
definite position
x 0, that means
00:04:17.579 --> 00:04:20.371
they're in eigenfunction of
position with eigenvalue x 0,
00:04:20.371 --> 00:04:21.829
and I have another
that corresponds
00:04:21.829 --> 00:04:24.630
to being at x1, an eigenfunction
of the position operator
00:04:24.630 --> 00:04:28.240
or the eigenvalue x1, then
these wave functions are
00:04:28.240 --> 00:04:29.920
orthogonal to each other.
00:04:29.920 --> 00:04:32.960
And we get 0 if x 0
is not equal to x1.
00:04:32.960 --> 00:04:35.250
Everyone cool with that?
00:04:35.250 --> 00:04:37.520
Now, meanwhile not only
are they orthogonal
00:04:37.520 --> 00:04:39.837
but they're normalized
in a particular way.
00:04:39.837 --> 00:04:41.670
The inner product gives
me a delta function,
00:04:41.670 --> 00:04:44.490
which goes beep once, so that
if I integrate against it
00:04:44.490 --> 00:04:45.319
I get a 1.
00:04:45.319 --> 00:04:46.360
Same thing with momentum.
00:04:46.360 --> 00:04:48.734
And you do this, this you're
checking on the problem set.
00:04:48.734 --> 00:04:50.930
I don't remember if it
was last one or this one.
00:04:50.930 --> 00:04:53.410
And for the energies,
energy 1, if I
00:04:53.410 --> 00:04:58.436
know the system is in state
energy 1, and let's say e sub n
00:04:58.436 --> 00:05:00.625
and e sub m, those
are different states
00:05:00.625 --> 00:05:02.250
if n and m are not
equal to each other.
00:05:02.250 --> 00:05:04.867
And this inner product
is 0 if n and m are not
00:05:04.867 --> 00:05:06.450
equal to each other
and 1 if they are.
00:05:06.450 --> 00:05:08.220
Their properly normalized.
00:05:08.220 --> 00:05:10.240
Everyone cool with that?
00:05:10.240 --> 00:05:11.769
Yeah.
00:05:11.769 --> 00:05:13.768
AUDIENCE: Is it possible
that two eigenfunctions
00:05:13.768 --> 00:05:15.514
have the same eigenvalue?
00:05:15.514 --> 00:05:16.430
PROFESSOR: Absolutely.
00:05:16.430 --> 00:05:18.650
It is absolutely possible
for two eigenfunctions
00:05:18.650 --> 00:05:19.996
to have the same eigenvalue.
00:05:19.996 --> 00:05:21.120
That is certainly possible.
00:05:21.120 --> 00:05:23.210
AUDIENCE: [INAUDIBLE]
00:05:23.210 --> 00:05:25.257
PROFESSOR: Yeah, good.
00:05:25.257 --> 00:05:26.840
Thank you, this is
a good technicality
00:05:26.840 --> 00:05:28.970
that I didn't want to get into,
but I'll go and get into it.
00:05:28.970 --> 00:05:30.053
It's a very good question.
00:05:30.053 --> 00:05:31.900
So the question is,
is it possible for two
00:05:31.900 --> 00:05:34.620
different eigenfunctions to
have the same eigenvalue.
00:05:34.620 --> 00:05:36.640
Could there be two states
with the same energy ,
00:05:36.640 --> 00:05:38.550
different states, same energy?
00:05:38.550 --> 00:05:40.080
Yeah, that's
absolutely possible.
00:05:40.080 --> 00:05:41.280
And we'll run into that.
00:05:41.280 --> 00:05:43.550
And there's nice
physics encoded in it.
00:05:43.550 --> 00:05:45.200
But let's think about
what that means.
00:05:45.200 --> 00:05:47.400
The subsequent question is
well, if that's the case,
00:05:47.400 --> 00:05:49.910
are they really
still orthogonal?
00:05:49.910 --> 00:05:51.450
And here's the crucial thing.
00:05:51.450 --> 00:05:55.190
The crucial thing is, let's
say I take one function,
00:05:55.190 --> 00:05:58.750
I'll call the function phi 1,
consider the function phi 1.
00:05:58.750 --> 00:06:02.330
And let it have energy E1,
so that E acting on phi 1
00:06:02.330 --> 00:06:05.020
is equal to E1 phi 1.
00:06:05.020 --> 00:06:08.600
And let there be
another function, phi 2,
00:06:08.600 --> 00:06:10.810
such that the energy
operator acting on phi 2
00:06:10.810 --> 00:06:14.050
is also equal to E1 phi 2.
00:06:14.050 --> 00:06:16.700
These are said to be degenerate.
00:06:16.700 --> 00:06:18.950
Degenerate doesn't mean you
go out and trash your car,
00:06:18.950 --> 00:06:22.510
degenerate that the
energies are the same.
00:06:22.510 --> 00:06:23.997
So what does this tell me?
00:06:23.997 --> 00:06:25.080
This tells me a cool fact.
00:06:25.080 --> 00:06:30.390
If I take a wave function phi,
and I will call this phi star,
00:06:30.390 --> 00:06:32.560
in honor of Shri
Kulkarni, so I've
00:06:32.560 --> 00:06:35.770
got this phi star, which is
a linear combination alpha
00:06:35.770 --> 00:06:39.270
phi 1 plus beta phi 2,
a linear combination
00:06:39.270 --> 00:06:42.700
of them, a superposition
of those two states.
00:06:42.700 --> 00:06:46.060
Is this also an
energy eigenfunction?
00:06:46.060 --> 00:06:52.630
Yeah, because if I act on phi
star with E, then it's linear,
00:06:52.630 --> 00:06:55.960
so E acting on phi star is
E acting on alpha phi 1,
00:06:55.960 --> 00:06:57.370
alpha's a constant,
doesn't care.
00:06:57.370 --> 00:06:59.090
Phi 1 gives me an E1.
00:06:59.090 --> 00:07:02.060
Similarly, E acting on
phi 2 gives me an E1.
00:07:02.060 --> 00:07:06.490
So if I act with E
on this guy, this
00:07:06.490 --> 00:07:09.229
is equal to, from both of these
I get an overall factor of E1.
00:07:09.229 --> 00:07:11.270
So notice that we get the
same vector back, times
00:07:11.270 --> 00:07:16.260
a constant, a common constant.
00:07:16.260 --> 00:07:18.237
So when we have
degenerate eigenfunctions,
00:07:18.237 --> 00:07:20.320
we can take arbitrary
linear combinations to them,
00:07:20.320 --> 00:07:22.700
get another degenerate
eigenfunction.
00:07:22.700 --> 00:07:23.522
Cool?
00:07:23.522 --> 00:07:25.230
So this is like,
imagine I have a vector,
00:07:25.230 --> 00:07:26.460
and I have another vector.
00:07:26.460 --> 00:07:28.380
And they share the property
that they're both eigenfunctions
00:07:28.380 --> 00:07:29.240
of some operator.
00:07:29.240 --> 00:07:30.948
That means any linear
combination of them
00:07:30.948 --> 00:07:32.284
is also, right?
00:07:32.284 --> 00:07:33.950
So there's a whole
vector space, there's
00:07:33.950 --> 00:07:36.850
a whole space of
possible functions
00:07:36.850 --> 00:07:38.704
that all have the
same eigenvalue.
00:07:38.704 --> 00:07:40.870
So now you say, well, look,
are these two orthogonal
00:07:40.870 --> 00:07:41.360
to each other?
00:07:41.360 --> 00:07:41.640
No.
00:07:41.640 --> 00:07:41.820
These two?
00:07:41.820 --> 00:07:42.240
No.
00:07:42.240 --> 00:07:43.140
But here's the thing.
00:07:43.140 --> 00:07:45.306
If you have a vector space,
if you have a the space,
00:07:45.306 --> 00:07:48.030
you can always find
orthogonal guys and a basis
00:07:48.030 --> 00:07:50.190
for that space, yes?
00:07:50.190 --> 00:07:53.470
So while it's not true that
the eigenfunctions are always
00:07:53.470 --> 00:07:56.460
orthogonal, it is true--
00:07:56.460 --> 00:07:59.920
we will not prove this, but we
will discuss the proof of it
00:07:59.920 --> 00:08:01.770
later by pulling the
mathematician out
00:08:01.770 --> 00:08:03.930
of the closet--
00:08:03.930 --> 00:08:06.370
the proof will say that
it is possible to find
00:08:06.370 --> 00:08:09.620
a set of eigenfunctions which
are orthogonal in precisely
00:08:09.620 --> 00:08:12.380
this fashion, even if
there are degeneracies.
00:08:12.380 --> 00:08:12.880
OK?
00:08:12.880 --> 00:08:14.960
That theorem is called
the spectral theorem.
00:08:14.960 --> 00:08:16.724
And we'll discuss it later.
00:08:16.724 --> 00:08:18.140
So it is always
possible to do so.
00:08:18.140 --> 00:08:20.460
But you must be alert that
there may be degeneracies.
00:08:20.460 --> 00:08:22.140
There aren't always
degeneracies.
00:08:22.140 --> 00:08:24.040
In fact, degeneracies
are very special.
00:08:24.040 --> 00:08:27.840
Why should two numbers
happen to be the same?
00:08:27.840 --> 00:08:29.840
Something has to be forcing
them to be the same.
00:08:29.840 --> 00:08:31.715
That's going to be an
important theme for us.
00:08:31.715 --> 00:08:32.966
But it certainly is possible.
00:08:32.966 --> 00:08:33.549
Good question.
00:08:33.549 --> 00:08:36.085
Other questions?
00:08:36.085 --> 00:08:36.585
Yeah.
00:08:36.585 --> 00:08:42.340
AUDIENCE: [INAUDIBLE]
00:08:42.340 --> 00:08:46.110
PROFESSOR: Yeah, so using
the triangular brackets--
00:08:46.110 --> 00:08:49.140
so there's another notation for
the same thing, which is f g,
00:08:49.140 --> 00:08:51.351
but this carries some
slightly different weight.
00:08:51.351 --> 00:08:53.600
It mean something slightly--
you'll see this in books,
00:08:53.600 --> 00:08:55.500
and this means something
very similar to this.
00:08:55.500 --> 00:08:56.680
But I'm not going to
use this notation.
00:08:56.680 --> 00:08:58.010
It's called Dirac notation.
00:08:58.010 --> 00:08:59.360
We'll talk about it
later in the semester,
00:08:59.360 --> 00:09:00.640
but we're not going to
talk about it just yet.
00:09:00.640 --> 00:09:02.490
But when you see
this, effectively it
00:09:02.490 --> 00:09:05.565
means the same thing as this.
00:09:05.565 --> 00:09:07.260
This is sort of like dialect.
00:09:07.260 --> 00:09:10.590
You know, it's like
French and Quebecois.
00:09:10.590 --> 00:09:12.290
Other questions?
00:09:12.290 --> 00:09:14.190
My wife's Canadian.
00:09:14.190 --> 00:09:15.750
Other questions?
00:09:15.750 --> 00:09:17.760
OK.
00:09:17.760 --> 00:09:19.200
So given this fact,
given the fact
00:09:19.200 --> 00:09:20.574
that we can
associate observables
00:09:20.574 --> 00:09:23.210
to operators, operators
come with special functions,
00:09:23.210 --> 00:09:25.460
the eigenfunctions, those
eigenfunctions corresponding
00:09:25.460 --> 00:09:27.410
to have a definite
value of the observable,
00:09:27.410 --> 00:09:29.320
and they're orthonormal.
00:09:29.320 --> 00:09:30.940
This tells us,
and this is really
00:09:30.940 --> 00:09:32.523
the statement of the
spectral theorem,
00:09:32.523 --> 00:09:36.340
that any function can be
expanded in a basis of states
00:09:36.340 --> 00:09:38.550
with definite values
of some observable.
00:09:38.550 --> 00:09:40.200
So for example,
consider position.
00:09:40.200 --> 00:09:41.720
I claim that any
wave function can
00:09:41.720 --> 00:09:44.660
be expanded as a
superposition of states
00:09:44.660 --> 00:09:46.190
with definite position.
00:09:46.190 --> 00:09:48.510
So here's an arbitrary
function, here's
00:09:48.510 --> 00:09:52.140
this set of states with definite
position, the delta functions.
00:09:52.140 --> 00:09:55.290
And I can write any
function as a superposition
00:09:55.290 --> 00:09:57.960
with some coefficients of
states with definite position,
00:09:57.960 --> 00:10:01.814
integrating over all
possible positions, x0.
00:10:01.814 --> 00:10:03.480
And this is also sort
of trivially true,
00:10:03.480 --> 00:10:04.910
because what's this integral?
00:10:04.910 --> 00:10:07.650
Well, it's an integral, dx0
over all possible positions
00:10:07.650 --> 00:10:09.170
of this delta function.
00:10:09.170 --> 00:10:12.090
But we're evaluating
at x, so this is 0
00:10:12.090 --> 00:10:13.550
unless x is equal to x0.
00:10:13.550 --> 00:10:15.125
So I can just put
in x instead of x0,
00:10:15.125 --> 00:10:16.670
and that gives me psi of x.
00:10:16.670 --> 00:10:18.760
Sort of tautological We
can do the same thing
00:10:18.760 --> 00:10:20.442
for momentum eigenfunctions.
00:10:20.442 --> 00:10:22.150
I claim that any
function can be expanded
00:10:22.150 --> 00:10:24.340
in a superposition of
momentum eigenfunctions, where
00:10:24.340 --> 00:10:27.160
I sum over all possible
values in the momentum
00:10:27.160 --> 00:10:30.120
with some weight.
00:10:30.120 --> 00:10:31.670
This psi tilde of
K is just telling
00:10:31.670 --> 00:10:34.480
me how much amplitude there
is at that wave number.
00:10:34.480 --> 00:10:35.120
Cool?
00:10:35.120 --> 00:10:37.660
But this is the Fourier theorem,
it's a Fourier expansion.
00:10:37.660 --> 00:10:40.864
So purely mathematically,
we know that this is true.
00:10:40.864 --> 00:10:42.530
But there's also the
physical statement.
00:10:42.530 --> 00:10:45.050
Any state can be expressed
as a superposition of states
00:10:45.050 --> 00:10:47.950
with definite momentum.
00:10:47.950 --> 00:10:50.300
There's a math in here, but
there's also physics in it.
00:10:50.300 --> 00:10:52.370
Finally, this is less obvious
from a mathematical point
00:10:52.370 --> 00:10:54.036
of view, because I
haven't even told you
00:10:54.036 --> 00:10:56.560
what energy is, any wave
function can be expanded
00:10:56.560 --> 00:10:58.360
in states with definite energy.
00:10:58.360 --> 00:11:02.110
So this is a state, my state
En, with definite energy,
00:11:02.110 --> 00:11:05.200
with some coefficient summed
over all possible values
00:11:05.200 --> 00:11:07.790
of the energy.
00:11:07.790 --> 00:11:11.750
Given any physical observable,
any physical observable,
00:11:11.750 --> 00:11:13.860
momentum, position,
angular momentum,
00:11:13.860 --> 00:11:16.769
whatever, given any
physical observable,
00:11:16.769 --> 00:11:18.310
a given wave function
can be expanded
00:11:18.310 --> 00:11:21.920
as some superposition of
having definite values of that.
00:11:21.920 --> 00:11:25.950
Will it in general have definite
values of the observable?
00:11:25.950 --> 00:11:28.500
Well a general state be
an energy eigenfunction?
00:11:28.500 --> 00:11:29.420
No.
00:11:29.420 --> 00:11:35.210
But any state is a superposition
of energy eigenfunctions.
00:11:35.210 --> 00:11:38.240
Will a random state
have definite position?
00:11:38.240 --> 00:11:38.870
Certainly not.
00:11:38.870 --> 00:11:41.010
You could have
this wave function.
00:11:41.010 --> 00:11:43.110
Superposition.
00:11:43.110 --> 00:11:45.210
Yeah.
00:11:45.210 --> 00:11:50.710
AUDIENCE: Why is the
energy special such
00:11:50.710 --> 00:11:54.680
that you can make an arbitrary
state with a countable number
00:11:54.680 --> 00:11:56.954
of energy eigenfunctions
rather than having
00:11:56.954 --> 00:11:58.120
to do a continuous spectrum?
00:11:58.120 --> 00:11:58.890
PROFESSOR: Excellent question.
00:11:58.890 --> 00:12:00.340
So I'm going to phrase
that slightly differently.
00:12:00.340 --> 00:12:01.280
It's an excellent
question, and we'll
00:12:01.280 --> 00:12:03.460
come to that at the
end of today's lecture.
00:12:03.460 --> 00:12:06.190
So the question is,
those are integrals, that
00:12:06.190 --> 00:12:08.115
is a sum over discrete things.
00:12:08.115 --> 00:12:08.615
Why?
00:12:08.615 --> 00:12:12.012
Why is the possible values
of the position continuous,
00:12:12.012 --> 00:12:14.470
possible values of momentum
continuous, and possible values
00:12:14.470 --> 00:12:16.690
of energy discrete?
00:12:16.690 --> 00:12:20.050
The answer to this
will become apparent
00:12:20.050 --> 00:12:22.152
over the course of your
next few problem sets.
00:12:22.152 --> 00:12:23.610
You have to do some
problems to get
00:12:23.610 --> 00:12:25.190
your fingers dirty to
really understand this.
00:12:25.190 --> 00:12:26.648
But here's the
statement, and we'll
00:12:26.648 --> 00:12:30.700
see the first version of this
at the end of today's lecture.
00:12:30.700 --> 00:12:33.420
Sometimes the allowed energies
of a system, the energy
00:12:33.420 --> 00:12:35.240
eigenvalues, are discrete.
00:12:35.240 --> 00:12:37.170
Sometimes they are continuous.
00:12:37.170 --> 00:12:41.056
They will be discrete when you
have bound states, states that
00:12:41.056 --> 00:12:42.930
are trapped in some
region and aren't allowed
00:12:42.930 --> 00:12:44.410
to get arbitrarily far away.
00:12:44.410 --> 00:12:48.160
They'll be continuous
when you have states that
00:12:48.160 --> 00:12:50.720
can get arbitrarily far away.
00:12:50.720 --> 00:12:54.090
Sometimes the momentum will be
allowed to be discrete values,
00:12:54.090 --> 00:12:56.930
sometimes it will be allowed
to be continuous values.
00:12:56.930 --> 00:12:59.416
And we'll see exactly
why subsequently.
00:12:59.416 --> 00:13:00.790
But the thing I
want to emphasize
00:13:00.790 --> 00:13:03.720
is that I'm writing this to
emphasize that it's possible
00:13:03.720 --> 00:13:05.920
that each of these can be
discrete or continuous.
00:13:05.920 --> 00:13:09.120
The important thing is that once
you pick your physical system,
00:13:09.120 --> 00:13:11.572
you ask what are the allowed
values of position, what
00:13:11.572 --> 00:13:13.030
are the allowed
values of momentum,
00:13:13.030 --> 00:13:15.940
and what are the allowed
values of energy.
00:13:15.940 --> 00:13:18.290
And then you sum over
all possible values.
00:13:18.290 --> 00:13:21.450
Now, in the examples we looked
at yesterday, or last lecture,
00:13:21.450 --> 00:13:23.070
the energy could
have been discrete,
00:13:23.070 --> 00:13:25.070
as in the case of
the infinite well,
00:13:25.070 --> 00:13:28.680
or continuous, as in the
case of the free particle.
00:13:28.680 --> 00:13:30.180
In the case of a
continuous particle
00:13:30.180 --> 00:13:31.810
this would have
been an integral.
00:13:31.810 --> 00:13:35.890
In the case of the system
such as a free particle, where
00:13:35.890 --> 00:13:37.890
the energy could take any
of a continuous number
00:13:37.890 --> 00:13:40.424
of possible values, this would
be a continuous integral.
00:13:40.424 --> 00:13:41.840
To deal with that,
I'm often going
00:13:41.840 --> 00:13:45.660
to use the notation, just
shorthand, integral sum.
00:13:45.660 --> 00:13:47.500
Which I know is a
horrible bastardization
00:13:47.500 --> 00:13:50.120
of all that's good and
just, but on the other hand,
00:13:50.120 --> 00:13:53.000
emphasizes the fact
that in some systems
00:13:53.000 --> 00:13:55.200
you will get continuous,
in some systems discrete,
00:13:55.200 --> 00:13:57.520
and sometimes you'll have
both continuous and discrete.
00:13:57.520 --> 00:14:00.380
For example, in
hydrogen, in hydrogen
00:14:00.380 --> 00:14:02.310
we'll find that there
are bound states
00:14:02.310 --> 00:14:04.830
where the electron is stuck
to the hydrogen nucleus,
00:14:04.830 --> 00:14:05.920
to the proton.
00:14:05.920 --> 00:14:08.510
And there are discrete
allowed energy levels
00:14:08.510 --> 00:14:10.170
for that configuration.
00:14:10.170 --> 00:14:12.640
However, once you
ionize the hydrogen,
00:14:12.640 --> 00:14:15.290
the electron can add
any energy you want.
00:14:15.290 --> 00:14:16.170
It's no longer bound.
00:14:16.170 --> 00:14:18.090
It can just get
arbitrarily far away.
00:14:18.090 --> 00:14:21.360
And there are an uncountable
infinity, a continuous set
00:14:21.360 --> 00:14:22.744
of possible states.
00:14:22.744 --> 00:14:24.410
So in that situation,
we'll find that we
00:14:24.410 --> 00:14:27.230
have both the discrete
and continuous series
00:14:27.230 --> 00:14:28.582
of possible states.
00:14:28.582 --> 00:14:29.082
Yeah.
00:14:29.082 --> 00:14:32.420
AUDIENCE: [INAUDIBLE]
00:14:32.420 --> 00:14:34.620
PROFESSOR: Yeah, sure,
if you work on a lattice.
00:14:34.620 --> 00:14:36.870
So for example, consider the
following quantum system.
00:14:36.870 --> 00:14:38.550
I have an undergraduate.
00:14:38.550 --> 00:14:41.760
And that undergraduate has
been placed in 1 of 12 boxes.
00:14:41.760 --> 00:14:42.450
OK?
00:14:42.450 --> 00:14:44.309
Now, what's the state
of the undergraduate?
00:14:44.309 --> 00:14:44.850
I don't know.
00:14:44.850 --> 00:14:46.410
Is it a definite position state?
00:14:46.410 --> 00:14:47.290
It might be.
00:14:47.290 --> 00:14:49.660
But probably it's a
superposition, an arbitrary
00:14:49.660 --> 00:14:51.360
superposition, right?
00:14:51.360 --> 00:14:54.912
Very impressive
undergraduates at MIT.
00:14:54.912 --> 00:14:55.745
OK, other questions.
00:14:58.355 --> 00:14:59.491
Yeah.
00:14:59.491 --> 00:15:00.990
AUDIENCE: Do these
three [INAUDIBLE]
00:15:00.990 --> 00:15:03.580
hold even if the probability
changes over time?
00:15:03.580 --> 00:15:04.170
PROFESSOR: Excellent question.
00:15:04.170 --> 00:15:05.090
We'll come back to that.
00:15:05.090 --> 00:15:06.631
Very good question,
leading question.
00:15:06.631 --> 00:15:09.340
OK, so we have this.
00:15:09.340 --> 00:15:12.130
The next thing is that
energy eigenfunctions satisfy
00:15:12.130 --> 00:15:15.540
some very special properties.
00:15:15.540 --> 00:15:17.340
And in particular,
energy eigenfunctions
00:15:17.340 --> 00:15:20.785
have the property from the
Schrodinger equation i h
00:15:20.785 --> 00:15:26.150
bar d t on psi of x and t is
equal to the energy operator
00:15:26.150 --> 00:15:28.680
acting on psi of x and t.
00:15:28.680 --> 00:15:30.150
This tells us that
if we have psi
00:15:30.150 --> 00:15:37.000
x 0 time t 0 is equal to phi
n of x, as we saw last time,
00:15:37.000 --> 00:15:41.080
then the wave function,
psi at x at time t
00:15:41.080 --> 00:15:43.100
is equal to phi n of x.
00:15:43.100 --> 00:15:46.200
And it only changes by an
overall phase, e to the minus i
00:15:46.200 --> 00:15:50.510
En t over h bar.
00:15:50.510 --> 00:15:52.760
And this ratio En
upon h bar will often
00:15:52.760 --> 00:15:56.130
be written omega n is
equal to En over h bar.
00:15:56.130 --> 00:15:59.252
This is just the
Dupre relations.
00:15:59.252 --> 00:16:01.340
Everyone cool with that?
00:16:01.340 --> 00:16:02.600
So are energy eigenfunctions--
00:16:07.020 --> 00:16:07.570
how to say.
00:16:07.570 --> 00:16:12.800
No wave function is more
morally good than another.
00:16:12.800 --> 00:16:14.727
But some are
particularly convenient.
00:16:14.727 --> 00:16:16.560
Energy eigenfunctions
have the nice property
00:16:16.560 --> 00:16:17.990
that while they're not
in a definite position
00:16:17.990 --> 00:16:20.156
and they don't necessarily
have a definite momentum,
00:16:20.156 --> 00:16:22.950
they do evolve over time in
a particularly simple way.
00:16:22.950 --> 00:16:25.690
And that and the
superposition principle
00:16:25.690 --> 00:16:27.250
allow me to write the following.
00:16:27.250 --> 00:16:34.400
If I know that this is my wave
function at psi at x at time 0,
00:16:34.400 --> 00:16:38.230
so let's say in all these cases,
this is psi of x at time 0,
00:16:38.230 --> 00:16:40.590
how does this state
evolve forward in time?
00:16:44.909 --> 00:16:45.950
It's kind of complicated.
00:16:45.950 --> 00:16:49.060
How does this description,
how does psi tilde of k
00:16:49.060 --> 00:16:50.715
evolve forward in time?
00:16:50.715 --> 00:16:51.840
Again, kind of complicated.
00:16:51.840 --> 00:16:54.497
But when expressed in terms
of the energy eigenstates,
00:16:54.497 --> 00:16:56.330
the answer to how it
evolves forward in time
00:16:56.330 --> 00:16:58.460
is very simple, because
I know that this
00:16:58.460 --> 00:17:01.440
is a superposition, a
linear combination of states
00:17:01.440 --> 00:17:02.600
with definite energy.
00:17:02.600 --> 00:17:05.079
States with definite
energy evolve with a phase.
00:17:05.079 --> 00:17:07.010
And the Schrodinger
equation is linear,
00:17:07.010 --> 00:17:08.700
so solutions of the
Schrodinger equation
00:17:08.700 --> 00:17:11.680
evolve to become solutions
of the Schrodinger equation.
00:17:11.680 --> 00:17:13.950
So how does this state
evolve forward in time?
00:17:13.950 --> 00:17:18.476
It evolves forward with a phase,
e to the minus i omega n t.
00:17:18.476 --> 00:17:21.560
One for every different
terms in this sum.
00:17:21.560 --> 00:17:23.020
Cool?
00:17:23.020 --> 00:17:25.089
So we are going to harp
on energy functions,
00:17:25.089 --> 00:17:28.300
not because they're more moral,
or more just, or more good,
00:17:28.300 --> 00:17:31.040
but because they're more
convenient for solving the time
00:17:31.040 --> 00:17:33.680
evolution problem in
quantum mechanics.
00:17:33.680 --> 00:17:37.020
So most of today is going
to be about this expansion
00:17:37.020 --> 00:17:40.470
and qualitative features
of energy eigenfunctions.
00:17:40.470 --> 00:17:42.690
Cool?
00:17:42.690 --> 00:17:43.510
OK.
00:17:43.510 --> 00:17:45.670
And just to close
that out, I just
00:17:45.670 --> 00:17:47.420
want to remind you of
a couple of examples
00:17:47.420 --> 00:17:50.022
that we did last time,
just get them on board.
00:17:50.022 --> 00:17:51.355
So the first is a free particle.
00:17:54.440 --> 00:17:57.220
So for free particle, we have
that our wave functions--
00:18:00.380 --> 00:18:04.125
well, actually let me
not write that down.
00:18:04.125 --> 00:18:06.000
Actually, let me skip
over the free particle,
00:18:06.000 --> 00:18:06.999
because it's so trivial.
00:18:06.999 --> 00:18:10.417
Let me just talk about
the infinite well.
00:18:10.417 --> 00:18:12.000
So the potential is
infinite out here,
00:18:12.000 --> 00:18:13.750
and it's 0 inside
the well, and it
00:18:13.750 --> 00:18:18.790
goes from 0 to L. This is
just my choice of notation.
00:18:18.790 --> 00:18:22.300
And the energy
operator, as usual,
00:18:22.300 --> 00:18:26.165
is p squared upon
2m plus u of x.
00:18:26.165 --> 00:18:27.790
You might say, where
did I derive this,
00:18:27.790 --> 00:18:29.000
and the answer is I
didn't derive this.
00:18:29.000 --> 00:18:30.030
I just wrote it down.
00:18:30.030 --> 00:18:32.690
It's like force in
Newton's equations.
00:18:32.690 --> 00:18:34.730
You just declare some
force and you ask,
00:18:34.730 --> 00:18:36.190
what system does is model.
00:18:36.190 --> 00:18:38.120
So here's my system.
00:18:38.120 --> 00:18:41.630
It has what looks like a
classical kind of energy,
00:18:41.630 --> 00:18:43.450
except these are all operators.
00:18:43.450 --> 00:18:48.120
And the potential here is this
guy, it's 0 between 0 and L,
00:18:48.120 --> 00:18:49.730
and it's infinite elsewhere.
00:18:49.730 --> 00:18:51.740
And as we saw last
time, the solutions
00:18:51.740 --> 00:18:53.480
to the energy
eigenvalue equation
00:18:53.480 --> 00:18:55.190
are particularly simple.
00:18:55.190 --> 00:18:59.500
Phi sub n of x is
equal to root properly
00:18:59.500 --> 00:19:04.620
normalized 2 upon
L sine of Kn x,
00:19:04.620 --> 00:19:08.790
where kn is equal to
n plus 1 pi, where
00:19:08.790 --> 00:19:11.670
n is an integer upon L.
00:19:11.670 --> 00:19:14.070
And these were chosen to
satisfy our boundary conditions,
00:19:14.070 --> 00:19:16.403
that the wave function must
vanish here, hence the sine,
00:19:16.403 --> 00:19:19.690
and K was chosen so that it
turned over and just hit 0
00:19:19.690 --> 00:19:25.210
as we got to L. And that gave us
that the allowed energies were
00:19:25.210 --> 00:19:29.050
discrete, because the En, which
you can get by just plugging
00:19:29.050 --> 00:19:31.200
into the energy
eigenvalue equation,
00:19:31.200 --> 00:19:34.860
was equal to h bar squared
Kn squared upon 2m.
00:19:37.550 --> 00:19:39.370
So this tells us a nice thing.
00:19:39.370 --> 00:19:41.430
First off, in this system,
if I take a particle
00:19:41.430 --> 00:19:43.940
and I throw it in here
in some arbitrary state
00:19:43.940 --> 00:19:46.070
so that at time t
equals zero the wave
00:19:46.070 --> 00:19:54.170
function x 0 is equal to
sum over n phi n of x Cn.
00:19:54.170 --> 00:19:56.440
OK?
00:19:56.440 --> 00:19:57.220
Can I do this?
00:19:57.220 --> 00:19:58.845
Can I just pick some
arbitrary function
00:19:58.845 --> 00:20:01.320
which is a superposition
of energy eigenstates?
00:20:01.320 --> 00:20:02.956
Sure, because any function is.
00:20:02.956 --> 00:20:04.955
Any function can be
described as a superposition
00:20:04.955 --> 00:20:06.560
of energy eigenfunctions.
00:20:06.560 --> 00:20:09.350
And if I use the
energy eigenfunctions,
00:20:09.350 --> 00:20:11.600
it will automatically satisfy
the boundary conditions.
00:20:11.600 --> 00:20:13.190
All good things will happen.
00:20:13.190 --> 00:20:15.170
So this is perfectly
fine initial condition.
00:20:15.170 --> 00:20:16.950
What is the system at time t?
00:20:20.986 --> 00:20:22.360
Yeah, we just pick
up the phases.
00:20:22.360 --> 00:20:24.100
And what phase is this guy?
00:20:24.100 --> 00:20:28.420
It's this, e to the
minus i omega n t.
00:20:28.420 --> 00:20:31.150
And when I write omega n, let
me be more explicit about that,
00:20:31.150 --> 00:20:33.820
that's En over h bar.
00:20:33.820 --> 00:20:38.340
So that's h bar Kn
squared upon 2m t.
00:20:41.420 --> 00:20:42.550
Cool?
00:20:42.550 --> 00:20:48.740
So there is our solution for
arbitrary initial conditions
00:20:48.740 --> 00:20:54.069
to the infinite square well
problem in quantum mechanics.
00:20:54.069 --> 00:20:56.610
And you're going to study this
in some detail on your problem
00:20:56.610 --> 00:20:58.950
set.
00:20:58.950 --> 00:21:01.920
But just to start with a
little bit of intuition,
00:21:01.920 --> 00:21:04.390
let's look at the wave
functions and the probability
00:21:04.390 --> 00:21:06.177
distributions for the
lowest lying states.
00:21:06.177 --> 00:21:07.760
So for example, let's
look at the wave
00:21:07.760 --> 00:21:11.360
function for the ground state,
what I will call psi sub 0.
00:21:11.360 --> 00:21:14.959
And this is from 0 to L.
And I put these bars here
00:21:14.959 --> 00:21:16.750
not because we're
looking at the potential.
00:21:16.750 --> 00:21:23.954
I'm going to be plotting the
real part of the wave function.
00:21:23.954 --> 00:21:26.450
But I put these walls
here just to emphasize
00:21:26.450 --> 00:21:29.617
that that's where the walls are,
at x equals 0 and x equals L.
00:21:29.617 --> 00:21:30.700
So what does it look like?
00:21:30.700 --> 00:21:33.100
Well, the first one is
going to sine of Kn x.
00:21:33.100 --> 00:21:34.010
n is 0.
00:21:34.010 --> 00:21:38.804
Kn is going to be pi upon L.
So that's again just this guy.
00:21:38.804 --> 00:21:40.470
Now, what's the
probability distribution
00:21:40.470 --> 00:21:43.734
associated with psi 0?
00:21:43.734 --> 00:21:45.025
Where do you find the particle?
00:21:47.920 --> 00:21:51.850
So we know that it's just
the norm squared of this wave
00:21:51.850 --> 00:21:56.580
function and the norm
squared is here at 0, it's 0
00:21:56.580 --> 00:21:58.040
and it rises
linearly, because sine
00:21:58.040 --> 00:21:59.580
is linear for small values.
00:21:59.580 --> 00:22:02.980
That makes this
quadratic, and a maximum,
00:22:02.980 --> 00:22:04.600
and then quadratic again.
00:22:04.600 --> 00:22:06.710
So there's our
probability distribution.
00:22:06.710 --> 00:22:08.014
Now, here's a funny thing.
00:22:08.014 --> 00:22:09.930
Imagine I take a particle,
classical particle,
00:22:09.930 --> 00:22:11.607
and I put it in a box.
00:22:11.607 --> 00:22:13.690
And you put it in a box,
and you tell it, OK, it's
00:22:13.690 --> 00:22:14.390
got some energy.
00:22:14.390 --> 00:22:16.210
So classically it's
got some momentum.
00:22:16.210 --> 00:22:17.834
So it's sort of
bouncing back and forth
00:22:17.834 --> 00:22:20.000
and just bounces off the
arbitrarily hard walls
00:22:20.000 --> 00:22:20.880
and moves around.
00:22:20.880 --> 00:22:22.880
Where are you most likely
to find that particle?
00:22:27.850 --> 00:22:29.530
Where does it spend
most of its time?
00:22:33.052 --> 00:22:35.010
It spends the same amount
of time at any point.
00:22:35.010 --> 00:22:36.410
It's moving at
constant velocity.
00:22:36.410 --> 00:22:39.247
It goes boo, boo,
boo, boo, right?
00:22:39.247 --> 00:22:40.830
So what's the
probability distribution
00:22:40.830 --> 00:22:44.020
for finding it at any
point inside, classically?
00:22:44.020 --> 00:22:45.411
Constant.
00:22:45.411 --> 00:22:47.660
Classically, the probability
distribution is constant.
00:22:47.660 --> 00:22:49.576
You're just as likely
to find it near the wall
00:22:49.576 --> 00:22:51.615
as not near the wall.
00:22:51.615 --> 00:22:53.740
However, quantum mechanically,
for the lowest lying
00:22:53.740 --> 00:22:55.090
state that is clearly not true.
00:22:55.090 --> 00:22:58.310
You're really likely to
find it near the wall.
00:22:58.310 --> 00:23:01.060
What's up with that?
00:23:01.060 --> 00:23:03.630
So that's a question that
I want to put in your head
00:23:03.630 --> 00:23:05.850
and have you think about.
00:23:05.850 --> 00:23:08.309
You're going to see a similar
effect arising over and over.
00:23:08.309 --> 00:23:09.891
And we're going to
see at the very end
00:23:09.891 --> 00:23:12.460
that that is directly related,
the fact that this goes to 0,
00:23:12.460 --> 00:23:15.370
is directly related,
and I'm not kidding,
00:23:15.370 --> 00:23:16.940
to the transparency of diamond.
00:23:22.385 --> 00:23:24.000
OK, I think it was pretty cool.
00:23:26.720 --> 00:23:27.470
They're expensive.
00:23:30.399 --> 00:23:31.940
It's also related
to the transparency
00:23:31.940 --> 00:23:34.148
of cubic zirconium, which
I guess is less impressive.
00:23:37.020 --> 00:23:39.090
So the first state,
again, let's look
00:23:39.090 --> 00:23:43.790
at the real part of psi 1,
the first excited state.
00:23:43.790 --> 00:23:46.240
Well, this is now a
sine with one extra--
00:23:46.240 --> 00:23:50.940
with a 2 here, 2 pi,
so it goes through 0.
00:23:50.940 --> 00:23:53.740
So the probability distribution
associated with psi 1,
00:23:53.740 --> 00:23:56.774
and I should say write this
as a function of x, looks
00:23:56.774 --> 00:23:58.190
like, well, again,
it's quadratic.
00:23:58.190 --> 00:24:01.310
But it has a 0
again in the middle.
00:24:01.310 --> 00:24:04.250
So it's going to look like--
00:24:04.250 --> 00:24:07.670
oops, my bad art defeats me.
00:24:07.670 --> 00:24:08.580
OK, there we go.
00:24:08.580 --> 00:24:10.010
So now it's even worse.
00:24:10.010 --> 00:24:12.100
Not only is unlikely
to be out here,
00:24:12.100 --> 00:24:14.183
it's also very unlikely
to be found in the middle.
00:24:14.183 --> 00:24:16.750
In fact, there is 0 probability
you'll find it in the middle.
00:24:16.750 --> 00:24:18.700
That's sort of surprising.
00:24:18.700 --> 00:24:21.290
But you can quickly
guess what happens as you
00:24:21.290 --> 00:24:23.770
go to very high energies.
00:24:23.770 --> 00:24:30.010
The real part of psi n let's
say 10,000, 10 to the 4,
00:24:30.010 --> 00:24:31.610
what is that going to look like?
00:24:31.610 --> 00:24:33.620
Well, this had no
0s, this had one 0,
00:24:33.620 --> 00:24:35.300
and every time you
increase n by 1,
00:24:35.300 --> 00:24:37.570
you're just going to add
one more 0 to the sign.
00:24:37.570 --> 00:24:39.570
That's an interesting
and suggestive fact.
00:24:39.570 --> 00:24:42.650
So if it's size of
10,000, how many nodes
00:24:42.650 --> 00:24:45.411
are there going to be in
the middle of the domain?
00:24:45.411 --> 00:24:45.910
10,000.
00:24:45.910 --> 00:24:48.690
And the amplitude is
going to be the same.
00:24:48.690 --> 00:24:51.220
I'm not to be able to do
this, but you get the idea.
00:24:51.220 --> 00:24:53.520
And now if I construct the
probability distribution,
00:24:53.520 --> 00:24:55.780
what's the probability
distribution going to be?
00:24:55.780 --> 00:25:02.030
Probability of the 10,000th
psi sub 10 to the 4 of x.
00:25:02.030 --> 00:25:05.830
Well, it's again going
to be strictly positive.
00:25:05.830 --> 00:25:10.040
And if you are not able to make
measurements on the scale of L
00:25:10.040 --> 00:25:14.340
upon 10,000, but just say like
L over 3, because you have
00:25:14.340 --> 00:25:17.331
a thumb and you don't have an
infinitely accurate meter, what
00:25:17.331 --> 00:25:17.830
do you see?
00:25:17.830 --> 00:25:20.384
You see effectively a constant
probability distribution.
00:25:20.384 --> 00:25:22.050
And actually, I
shouldn't draw it there.
00:25:22.050 --> 00:25:23.834
I should draw it
through the half,
00:25:23.834 --> 00:25:27.170
because sine squared over
2 averages to one half,
00:25:27.170 --> 00:25:30.500
or, sorry, sine squared averages
to one half over many periods.
00:25:30.500 --> 00:25:33.660
So what we see is that
the classical probability
00:25:33.660 --> 00:25:38.380
distribution constant
does arise when we look
00:25:38.380 --> 00:25:41.590
at very high energy states.
00:25:41.590 --> 00:25:42.880
Cool?
00:25:42.880 --> 00:25:46.084
But it is manifestly
not a good description.
00:25:46.084 --> 00:25:48.250
The classical description
is not a good description.
00:25:48.250 --> 00:25:51.070
Your intuition is
crappy at low energies,
00:25:51.070 --> 00:25:53.570
near the ground state, where
quantum effects are dominating,
00:25:53.570 --> 00:25:55.980
because indeed, classically
there was no minimum energy.
00:25:55.980 --> 00:25:57.930
Quantum effects have
to be dominating there.
00:25:57.930 --> 00:25:59.780
And here we see that even the
probability distribution's
00:25:59.780 --> 00:26:01.542
radically different
than our intuition.
00:26:01.542 --> 00:26:02.042
Yeah.
00:26:02.042 --> 00:26:12.650
AUDIENCE: [INAUDIBLE]
00:26:12.650 --> 00:26:13.900
PROFESSOR: Keep working on it.
00:26:13.900 --> 00:26:18.820
So I want you all to
think about what--
00:26:18.820 --> 00:26:21.250
you're not, I promise
you, unless you've already
00:26:21.250 --> 00:26:22.230
seen some quantum
mechanics, you're
00:26:22.230 --> 00:26:24.271
not going to be able to
answer this question now.
00:26:24.271 --> 00:26:27.010
But I want you to have it as
an uncomfortable little piece
00:26:27.010 --> 00:26:31.420
of sand in the back
of your oyster mind--
00:26:31.420 --> 00:26:38.020
no offense-- what
is causing that 0?
00:26:38.020 --> 00:26:39.300
Why are we getting 0?
00:26:39.300 --> 00:26:40.575
And I'll give you a hint.
00:26:40.575 --> 00:26:42.960
In quantum mechanics,
anytime something interesting
00:26:42.960 --> 00:26:45.985
happens it's because of
superposition and interference.
00:26:49.220 --> 00:26:50.850
All right.
00:26:50.850 --> 00:26:55.044
So with all that said, so any
questions now over this story
00:26:55.044 --> 00:26:56.585
about energy
eigenfunctions expanding
00:26:56.585 --> 00:27:01.136
in a basis, et cetera,
before we get moving?
00:27:01.136 --> 00:27:03.370
No, OK.
00:27:03.370 --> 00:27:05.880
In that case, get
out your clickers.
00:27:05.880 --> 00:27:07.620
We're going to test
your knowledge.
00:27:17.720 --> 00:27:20.836
Channel 41, for those of
you who have to adjust it.
00:27:20.836 --> 00:27:32.380
[CHATTER]
00:27:32.380 --> 00:27:32.880
Wow.
00:27:37.100 --> 00:27:38.100
That's kind of worrying.
00:27:42.980 --> 00:27:43.480
Aha.
00:27:58.400 --> 00:28:01.780
OK, ready?
00:28:01.780 --> 00:28:07.010
OK, channel 41, and here we go.
00:28:18.300 --> 00:28:19.789
So go ahead and start now.
00:28:19.789 --> 00:28:21.830
Sorry, there was a little
technical glitch there.
00:28:21.830 --> 00:28:24.640
So psi 1 and psi
2 are eigenstates.
00:28:24.640 --> 00:28:27.590
They're non-degenerate, meaning
the energies are different.
00:28:27.590 --> 00:28:30.020
Is a superposition psi 1 plus
psi 2 also an eigenstate?
00:28:43.398 --> 00:28:45.580
All right, four more seconds.
00:28:48.610 --> 00:28:49.560
All right.
00:28:49.560 --> 00:28:52.910
I want everyone to turn
to the person next to you
00:28:52.910 --> 00:28:54.050
and discuss this.
00:28:54.050 --> 00:28:56.538
You've got about 30 seconds
to discuss, or a minute.
00:28:56.538 --> 00:29:39.780
[CHATTER]
00:29:39.780 --> 00:29:41.940
All right.
00:29:41.940 --> 00:29:45.230
I want everyone, now that you've
got an answer, click again,
00:29:45.230 --> 00:29:47.300
put in your current best guess.
00:29:50.564 --> 00:29:51.230
Oh, wait, sorry.
00:29:51.230 --> 00:29:54.180
For some reason I have
to start over again.
00:29:54.180 --> 00:29:56.080
OK, now click.
00:30:02.690 --> 00:30:05.090
This is the best.
00:30:05.090 --> 00:30:08.180
I'm such a convert to clickers,
this is just fantastic.
00:30:08.180 --> 00:30:11.750
So you guys went from,
so roughly you all
00:30:11.750 --> 00:30:20.480
went from about 30, 60,
10, to what are we now?
00:30:20.480 --> 00:30:26.990
8, 82, and 10.
00:30:26.990 --> 00:30:29.230
So it sounds like you guys
are predicting answer b.
00:30:29.230 --> 00:30:30.200
And the answer is--
00:30:33.587 --> 00:30:34.420
I like the suspense.
00:30:34.420 --> 00:30:37.170
There we go.
00:30:37.170 --> 00:30:38.150
B, good.
00:30:38.150 --> 00:30:40.160
So here's a quick question.
00:30:42.890 --> 00:30:44.020
So why?
00:30:44.020 --> 00:30:49.910
And the reason why is that if
we have E on psi 1 plus psi 2,
00:30:49.910 --> 00:30:56.540
this is equal to E on psi 1 plus
E on psi 2, operator, operator,
00:30:56.540 --> 00:30:59.200
operator, but this
is equal to E 1
00:30:59.200 --> 00:31:05.380
psi 1 E 2 psi 2, which if
E1 and E2 are not equal,
00:31:05.380 --> 00:31:09.070
which is not equal to E
times psi 1 plus psi 2.
00:31:09.070 --> 00:31:10.350
Right?
00:31:10.350 --> 00:31:14.740
Not equal to E anything
times psi 1 plus psi 2.
00:31:14.740 --> 00:31:16.280
And it needs to
be, in order to be
00:31:16.280 --> 00:31:18.300
an eigenfunction, an
eigenfunction of the energy
00:31:18.300 --> 00:31:20.960
operator.
00:31:20.960 --> 00:31:22.538
Yeah.
00:31:22.538 --> 00:31:24.162
AUDIENCE: So I was
thinking about this,
00:31:24.162 --> 00:31:26.037
if this was kind of a
silly random case where
00:31:26.037 --> 00:31:27.140
one of the energies is 0.
00:31:27.140 --> 00:31:29.640
Does this only happen if you
have something that's infinite?
00:31:29.640 --> 00:31:31.360
PROFESSOR: Yeah, that's
a really good question.
00:31:31.360 --> 00:31:33.340
So first off, how do
you measure an energy?
00:31:36.240 --> 00:31:38.790
Do you ever measure an energy?
00:31:38.790 --> 00:31:41.365
Do you ever measure a
voltage, the actual value
00:31:41.365 --> 00:31:43.490
of the scalar potential,
the electromagnetic scalar
00:31:43.490 --> 00:31:43.670
potential?
00:31:43.670 --> 00:31:44.169
No.
00:31:44.169 --> 00:31:46.259
You measure a difference.
00:31:46.259 --> 00:31:47.550
Do you ever measure the energy?
00:31:47.550 --> 00:31:49.296
No, you measure a
difference in energy.
00:31:49.296 --> 00:31:51.670
So the absolute value of energy
is sort of a silly thing.
00:31:51.670 --> 00:31:54.440
But we always talk
about it as if it's not.
00:31:54.440 --> 00:31:55.680
We say, that's got energy 14.
00:31:55.680 --> 00:31:58.200
It's a little bit suspicious.
00:31:58.200 --> 00:32:00.480
So to answer your
question, there's
00:32:00.480 --> 00:32:02.290
nothing hallowed
about the number 0,
00:32:02.290 --> 00:32:04.282
although we will often
refer to zero energy
00:32:04.282 --> 00:32:05.490
with a very specific meaning.
00:32:05.490 --> 00:32:06.980
What we really mean
in that case is
00:32:06.980 --> 00:32:09.420
the value of the potential
energy at infinity.
00:32:09.420 --> 00:32:11.220
So when I say energy,
usually what I mean
00:32:11.220 --> 00:32:12.790
is relative to the
value at infinity.
00:32:12.790 --> 00:32:14.415
So then let me ask
your question again.
00:32:14.415 --> 00:32:16.510
Your question is it
possible to have energy 0?
00:32:16.510 --> 00:32:17.802
Absolutely, and we'll see that.
00:32:17.802 --> 00:32:20.093
And it's actually going to
be really interesting what's
00:32:20.093 --> 00:32:21.910
true of states with
energy 0 in that sense.
00:32:21.910 --> 00:32:23.451
Second part of your
question, though,
00:32:23.451 --> 00:32:25.690
is how does energy
being 0 fit into this?
00:32:25.690 --> 00:32:26.970
Well, does that save us?
00:32:26.970 --> 00:32:29.210
Suppose one of
the energies is 0.
00:32:29.210 --> 00:32:31.690
Then that says E on psi
1 plus psi 2 is equal to,
00:32:31.690 --> 00:32:34.790
let's say E2 is 0.
00:32:34.790 --> 00:32:35.950
Well, that term is gone.
00:32:35.950 --> 00:32:37.074
So there's just the one E1.
00:32:37.074 --> 00:32:38.657
Are we in energy eigenstate?
00:32:38.657 --> 00:32:40.240
No, because it's
still not of the form
00:32:40.240 --> 00:32:42.290
E times psi 1 plus psi 2.
00:32:42.290 --> 00:32:44.740
So it doesn't save us, but
it's an interesting question
00:32:44.740 --> 00:32:46.200
for the future.
00:32:46.200 --> 00:32:48.290
All right.
00:32:48.290 --> 00:32:50.935
Next question, four parts.
00:33:07.100 --> 00:33:10.250
So the question says x
and p commute to i h bar.
00:33:10.250 --> 00:33:11.470
We've shown this.
00:33:11.470 --> 00:33:14.580
Is p x equal to i h
bar, and is ip plus cx
00:33:14.580 --> 00:33:15.910
the same as cx plus ip?
00:33:29.920 --> 00:33:33.740
If you're really unsure you
can ask the person next to you,
00:33:33.740 --> 00:33:37.170
but you don't have to.
00:33:37.170 --> 00:33:41.290
OK, so this is looking good.
00:33:41.290 --> 00:33:44.060
Everyone have an answer in?
00:33:44.060 --> 00:33:44.560
No?
00:33:49.770 --> 00:33:55.310
Five, four, three,
two, one, OK, good.
00:33:55.310 --> 00:34:03.242
So the answer is C, which most
of you got, but not everyone.
00:34:03.242 --> 00:34:05.200
A bunch of you put D. So
let's talk through it.
00:34:05.200 --> 00:34:10.639
So remember what the definition
of the commutator is.
00:34:10.639 --> 00:34:15.219
x with p by definition
is equal to xp minus px.
00:34:15.219 --> 00:34:18.500
If we change the
order here, px is
00:34:18.500 --> 00:34:22.630
equal to minus
this, px minus xp.
00:34:22.630 --> 00:34:24.620
It's just the definition
of the commutator.
00:34:24.620 --> 00:34:27.800
So on the other hand, if you
add things, does 7 plus 6
00:34:27.800 --> 00:34:28.940
equal 6 plus 7?
00:34:28.940 --> 00:34:29.440
Yeah.
00:34:29.440 --> 00:34:31.116
Well, of course 6
times 7 is 7 times 6.
00:34:31.116 --> 00:34:32.699
So that's not a
terribly good analogy.
00:34:36.880 --> 00:34:39.880
Does the order of addition
of operators matter?
00:34:39.880 --> 00:34:42.540
No.
00:34:42.540 --> 00:34:44.639
Yeah.
00:34:44.639 --> 00:34:45.630
Yeah, exactly.
00:34:45.630 --> 00:34:46.130
Exactly.
00:34:46.130 --> 00:34:47.130
So it's slightly sneaky.
00:34:47.130 --> 00:34:49.840
OK, next question.
00:34:49.840 --> 00:34:51.170
OK, this one has five.
00:34:54.860 --> 00:34:56.670
f and g are both wave functions.
00:34:56.670 --> 00:34:58.090
c is a constant.
00:34:58.090 --> 00:35:03.150
Then if we take the inner
product c times f with g,
00:35:03.150 --> 00:35:05.160
this is equal to what?
00:35:21.000 --> 00:35:24.240
Three, two, one, OK.
00:35:24.240 --> 00:35:25.720
So the answer is--
00:35:25.720 --> 00:35:27.610
so this one definitely discuss.
00:35:27.610 --> 00:35:29.778
Discuss with the
person next to you.
00:35:29.778 --> 00:35:55.520
[CHATTER]
00:35:55.520 --> 00:35:56.280
All right.
00:36:01.440 --> 00:36:05.910
OK, go ahead and enter
your guess again,
00:36:05.910 --> 00:36:08.490
or your answer again,
let it not be a guess.
00:36:11.220 --> 00:36:12.855
OK, 10 seconds.
00:36:15.450 --> 00:36:15.950
Wow.
00:36:15.950 --> 00:36:16.630
OK, fantastic.
00:36:16.630 --> 00:36:17.630
That works like a champ.
00:36:17.630 --> 00:36:19.040
So what's the answer?
00:36:19.040 --> 00:36:20.970
Yes, complex conjugation.
00:36:20.970 --> 00:36:21.970
Don't screw that one up.
00:36:21.970 --> 00:36:24.110
It's very easy to forget,
but it matters a lot.
00:36:28.297 --> 00:36:29.380
Cursor keeps disappearing.
00:36:29.380 --> 00:36:32.360
OK, next one.
00:36:32.360 --> 00:36:40.120
A wave function
has been expressed
00:36:40.120 --> 00:36:41.570
as a sum of energy
eigenfunctions.
00:36:41.570 --> 00:36:45.445
Here I'm calling them mu rather
than phi, but same thing.
00:36:45.445 --> 00:36:47.070
Compared to the
original wave function,
00:36:47.070 --> 00:36:49.920
the set of coefficients, given
that we're using the energy
00:36:49.920 --> 00:36:52.730
basis, the set of coefficients
contains more or less
00:36:52.730 --> 00:36:55.390
the same information, or
it can't be determined.
00:37:00.650 --> 00:37:02.360
OK, five seconds.
00:37:06.990 --> 00:37:07.780
All right.
00:37:07.780 --> 00:37:11.660
And the answer is C, great.
00:37:11.660 --> 00:37:13.390
OK, next one.
00:37:16.429 --> 00:37:17.720
So right now we're normalizing.
00:37:22.540 --> 00:37:23.470
OK.
00:37:23.470 --> 00:37:25.820
All stationary states, or
all energy eigenstates,
00:37:25.820 --> 00:37:28.570
have the form that spatial
and time dependence
00:37:28.570 --> 00:37:30.760
is the spatial dependence,
the energy eigenfunction,
00:37:30.760 --> 00:37:35.110
times a phase, so that the norm
squared is time independent.
00:37:35.110 --> 00:37:37.290
Consider the sum of two
non-degenerate energy
00:37:37.290 --> 00:37:40.280
eigenstates psi 1 and psi 2.
00:37:40.280 --> 00:37:42.990
Non-degenerate means they
have different energy.
00:37:45.540 --> 00:37:47.140
Is the wave function stationary?
00:37:47.140 --> 00:37:48.840
Is the probability
distribution time
00:37:48.840 --> 00:37:50.420
independent or is
it time dependent?
00:38:00.600 --> 00:38:02.980
This one's not trivial.
00:38:02.980 --> 00:38:03.530
Oh, shoot.
00:38:03.530 --> 00:38:04.730
I forgot to get it started.
00:38:04.730 --> 00:38:06.399
Sorry.
00:38:06.399 --> 00:38:08.940
It's particularly non-trivial
if you can't enter your answer.
00:38:08.940 --> 00:38:09.210
Right.
00:38:09.210 --> 00:38:10.626
So go ahead and
enter your answer.
00:38:17.438 --> 00:38:19.420
Whoo, yeah.
00:38:19.420 --> 00:38:20.715
This one always kills people.
00:38:31.315 --> 00:38:32.190
No chatting just yet.
00:38:32.190 --> 00:38:33.565
Test yourself,
not your neighbor.
00:38:38.427 --> 00:38:40.010
It's fine to look
deep into your soul,
00:38:40.010 --> 00:38:41.980
but don't look deep into the
soul of the person sitting next
00:38:41.980 --> 00:38:42.480
to you.
00:38:46.660 --> 00:38:47.590
All right.
00:38:47.590 --> 00:38:51.840
So at this point, chat
with your neighbor.
00:38:51.840 --> 00:38:54.086
Let me just give
you some presage.
00:38:54.086 --> 00:38:57.720
The parallel strategy's probably
not so good, because about half
00:38:57.720 --> 00:39:00.273
of you got it right, and about
half of you got it wrong.
00:39:00.273 --> 00:40:07.070
[CHATTER]
00:40:07.070 --> 00:40:07.950
All right.
00:40:07.950 --> 00:40:09.375
Let's vote again.
00:40:11.980 --> 00:40:15.720
And hold on, starting now.
00:40:15.720 --> 00:40:16.480
OK, vote again.
00:40:16.480 --> 00:40:18.063
You've got 10 seconds
to enter a vote.
00:40:24.520 --> 00:40:25.710
Wow.
00:40:25.710 --> 00:40:28.110
OK, two seconds.
00:40:28.110 --> 00:40:29.020
Good.
00:40:29.020 --> 00:40:36.940
So the distribution on
this one went from 30, 50,
00:40:36.940 --> 00:40:47.290
20 initially, to now
it is 10, 80, and 10.
00:40:47.290 --> 00:40:51.030
Amazingly, you guys got worse.
00:40:51.030 --> 00:40:55.620
The answer is C. And I want
you to discuss with each other
00:40:55.620 --> 00:40:57.174
why it's C.
00:40:57.174 --> 00:41:27.840
[CHATTER]
00:41:27.840 --> 00:41:29.420
All right.
00:41:29.420 --> 00:41:32.060
OK.
00:41:32.060 --> 00:41:33.830
So let me talk you through it.
00:41:33.830 --> 00:41:36.230
So the wave function,
we've said psi of x and t
00:41:36.230 --> 00:41:45.865
is equal to phi 1 at x, e to
the minus i omega 1 t plus phi
00:41:45.865 --> 00:41:50.940
2 of x e to the
minus i omega 2 t.
00:41:50.940 --> 00:41:52.670
So great, we take
the norm squared.
00:41:52.670 --> 00:41:55.342
What's the probability to
find it at x at time t.
00:41:55.342 --> 00:41:56.800
The probability
density is the norm
00:41:56.800 --> 00:41:59.260
squared of this guy,
psi squared, which
00:41:59.260 --> 00:42:04.590
is equal to phi 1 complex
conjugate e to the plus i omega
00:42:04.590 --> 00:42:12.200
1 t plus phi 2 complex conjugate
e to the plus i omega 2t times
00:42:12.200 --> 00:42:16.210
the thing itself phi 1
of x e to the minus i
00:42:16.210 --> 00:42:23.320
omega 1 t plus phi 2 of x e to
the minus i omega 2t, right?
00:42:23.320 --> 00:42:25.870
So this has four terms.
00:42:25.870 --> 00:42:28.470
The first term is
psi 1 norm squared.
00:42:28.470 --> 00:42:30.650
The phases cancel, right?
00:42:30.650 --> 00:42:33.340
You're going to see this
happen a billion times in 804.
00:42:33.340 --> 00:42:35.850
The first term is going
to be phi 1 norm squared.
00:42:35.850 --> 00:42:37.970
There's another term, which
is phi 2 norm squared.
00:42:37.970 --> 00:42:40.345
Again the phases exactly
cancel, even the minus i omega 2
00:42:40.345 --> 00:42:42.080
t to the plus i omega 2 t.
00:42:42.080 --> 00:42:44.470
Plus phi 2 squared.
00:42:44.470 --> 00:42:47.280
But then there are two cross
terms, the interference terms.
00:42:47.280 --> 00:42:55.180
Plus phi 1 complex conjugate
phi 2 e to the i omega 1 t
00:42:55.180 --> 00:43:00.670
e to the plus i omega 1 t, i
omega 1 t, and e to the minus
00:43:00.670 --> 00:43:01.869
i omega 2t, minus omega 2.
00:43:01.869 --> 00:43:04.160
So we have a cross-term which
depends on the difference
00:43:04.160 --> 00:43:05.194
in frequencies.
00:43:05.194 --> 00:43:07.110
Frequencies are like
energies modulo on h-bar,
00:43:07.110 --> 00:43:09.240
so it's a difference
in energies.
00:43:09.240 --> 00:43:10.740
And then there's
another term, which
00:43:10.740 --> 00:43:12.670
is the complex
conjugate of this guy,
00:43:12.670 --> 00:43:15.950
phi 2 star times phi 1 phi
2 complex conjugate phi 1
00:43:15.950 --> 00:43:17.825
and the phases are also
the complex conjugate
00:43:17.825 --> 00:43:28.350
e to the minus i omega 1 minus
omega 2 t of x of x of x of x.
00:43:28.350 --> 00:43:31.880
So is there time dependence
in this, in principle?
00:43:31.880 --> 00:43:33.890
Absolutely, from the
interference terms.
00:43:33.890 --> 00:43:35.500
Were we not in
the superposition,
00:43:35.500 --> 00:43:37.470
we would not have
interference terms.
00:43:37.470 --> 00:43:40.330
Time dependence comes from
interference, when we expand
00:43:40.330 --> 00:43:41.600
in energy eigenfunctions.
00:43:41.600 --> 00:43:43.187
Cool?
00:43:43.187 --> 00:43:44.270
However, can these vanish?
00:43:44.270 --> 00:43:44.769
When?
00:43:48.450 --> 00:43:50.460
Sorry, say again?
00:43:50.460 --> 00:43:54.577
Great, so when omega 1
equals omega 2, what happens?
00:43:54.577 --> 00:43:55.660
Time dependence goes away.
00:43:55.660 --> 00:44:00.750
But omega 1 is e 1 over h bar,
omega 2 is e 2 over h bar,
00:44:00.750 --> 00:44:03.595
and we started out by saying
these are non-degenerate.
00:44:03.595 --> 00:44:05.970
So if they're non-degenerate,
the energies are different,
00:44:05.970 --> 00:44:08.361
the frequencies are different,
so that doesn't help us.
00:44:08.361 --> 00:44:09.860
How do we kill this
time dependence?
00:44:12.860 --> 00:44:13.399
Yes.
00:44:13.399 --> 00:44:15.190
If the two functions
aren't just orthogonal
00:44:15.190 --> 00:44:17.150
in a functional sense, but
if we have the following.
00:44:17.150 --> 00:44:18.410
Suppose phi 1 is like this.
00:44:18.410 --> 00:44:21.430
It's 0 everywhere except for
in some lump that's phi 1,
00:44:21.430 --> 00:44:25.230
and phi 2 is 0
everywhere except here.
00:44:25.230 --> 00:44:27.670
Then anywhere that phi 1
is non-zero, phi 2 is zero.
00:44:27.670 --> 00:44:30.790
And anywhere where phi 2
is non-zero, phi 1 is zero.
00:44:30.790 --> 00:44:33.940
So this can point-wise vanish.
00:44:33.940 --> 00:44:37.250
Do you expect this to
happen generically?
00:44:37.250 --> 00:44:39.850
Does it happen for the
energy eigenfunctions
00:44:39.850 --> 00:44:41.270
in the infinite square well?
00:44:44.710 --> 00:44:45.360
Sine waves?
00:44:47.981 --> 00:44:48.480
No.
00:44:48.480 --> 00:44:50.100
They have zero at
isolated points,
00:44:50.100 --> 00:44:53.019
but they're non-zero
generically.
00:44:53.019 --> 00:44:54.310
Yeah, so it doesn't work there.
00:44:54.310 --> 00:44:55.875
What about for
the free particle?
00:44:55.875 --> 00:44:57.250
Well, those are
just plain waves.
00:44:57.250 --> 00:44:58.680
Does that ever happen?
00:44:58.680 --> 00:44:59.610
No.
00:44:59.610 --> 00:45:02.540
OK, so this is an
incredibly special case.
00:45:02.540 --> 00:45:04.040
We'll actually see
it in one problem
00:45:04.040 --> 00:45:05.485
on a problem set later on.
00:45:05.485 --> 00:45:07.130
It's a very special case.
00:45:07.130 --> 00:45:09.935
So technically, the
answer is C. And I
00:45:09.935 --> 00:45:11.570
want you guys to
keep your minds open
00:45:11.570 --> 00:45:15.310
on these sorts of questions,
when does a spatial dependence
00:45:15.310 --> 00:45:17.370
matter and when are
there interference terms.
00:45:17.370 --> 00:45:18.260
Those are two
different questions,
00:45:18.260 --> 00:45:19.718
and I want you to
tease them apart.
00:45:19.718 --> 00:45:21.490
OK?
00:45:21.490 --> 00:45:22.590
Cool?
00:45:22.590 --> 00:45:23.784
Yeah?
00:45:23.784 --> 00:45:26.462
AUDIENCE: Is a valid
way to think about this
00:45:26.462 --> 00:45:34.594
to think that you're fixing
the initial [INAUDIBLE]
00:45:34.594 --> 00:45:36.760
PROFESSOR: That's a very
good way to think about it.
00:45:36.760 --> 00:45:39.150
That's exactly right.
00:45:39.150 --> 00:45:41.082
That's a very,
very good question.
00:45:41.082 --> 00:45:43.040
Let me say that subtly
differently, and tell me
00:45:43.040 --> 00:45:44.956
if this agrees with what
you were just saying.
00:45:44.956 --> 00:45:47.806
So I can look at
this wave function,
00:45:47.806 --> 00:45:49.930
and I already know that
the overall phase of a wave
00:45:49.930 --> 00:45:51.150
function doesn't matter.
00:45:51.150 --> 00:45:52.960
That's what it is to say a
stationary state is stationary.
00:45:52.960 --> 00:45:54.960
It's got an overall phase
that's the only thing,
00:45:54.960 --> 00:45:56.540
norm squared it goes away.
00:45:56.540 --> 00:46:01.950
So I can write this as e to
the minus i omega 1 t times phi
00:46:01.950 --> 00:46:08.750
1 of x plus phi 2 of x
e to the minus i omega
00:46:08.750 --> 00:46:12.506
2 minus omega 1 t.
00:46:12.506 --> 00:46:13.920
Is that what you mean?
00:46:13.920 --> 00:46:15.170
So that's one way to do this.
00:46:15.170 --> 00:46:16.503
We could also do something else.
00:46:16.503 --> 00:46:22.752
We could do e to the minus i
omega 1 plus omega 2 upon 2 t.
00:46:22.752 --> 00:46:24.210
And this is more,
I think, what you
00:46:24.210 --> 00:46:26.085
were thinking of, a sort
of average frequency
00:46:26.085 --> 00:46:28.070
and then a relative
frequency, and then
00:46:28.070 --> 00:46:31.050
the change in the frequencies
on these two terms.
00:46:31.050 --> 00:46:31.760
Absolutely.
00:46:31.760 --> 00:46:36.720
So you can organize
this in many, many ways.
00:46:36.720 --> 00:46:39.670
But your question gets at
a very important point,
00:46:39.670 --> 00:46:41.670
which is that the overall
phase doesn't matter.
00:46:41.670 --> 00:46:46.400
But relative phases in a
superposition do matter.
00:46:46.400 --> 00:46:48.727
So when does a phase
matter in a wave function?
00:46:48.727 --> 00:46:50.560
It does not matter if
it's an overall phase.
00:46:50.560 --> 00:46:53.170
But it does matter if it's a
relative phase between terms
00:46:53.170 --> 00:46:54.790
in a superposition.
00:46:54.790 --> 00:46:55.870
Cool?
00:46:55.870 --> 00:46:57.510
Very good question.
00:46:57.510 --> 00:47:00.330
Other questions?
00:47:00.330 --> 00:47:03.550
If not, then I have some.
00:47:03.550 --> 00:47:07.930
So, consider a system
which is in the state--
00:47:07.930 --> 00:47:10.227
so I should give you five--
00:47:10.227 --> 00:47:12.810
system is in a state which is a
linear combination of n equals
00:47:12.810 --> 00:47:14.280
1 and n equals 2 eigenstates.
00:47:16.956 --> 00:47:18.580
What's the probability
that measurement
00:47:18.580 --> 00:47:20.082
will give us energy E1?
00:47:20.082 --> 00:47:21.373
And it's in this superposition.
00:47:28.390 --> 00:47:29.215
OK, five seconds.
00:47:33.780 --> 00:47:34.660
OK, fantastic.
00:47:34.660 --> 00:47:36.450
What's the answer?
00:47:36.450 --> 00:47:37.497
Yes, C, great.
00:47:37.497 --> 00:47:38.580
OK, everyone got that one.
00:47:38.580 --> 00:47:42.100
So one's a slightly more
interesting question.
00:47:45.580 --> 00:47:55.210
Suppose I have an infinite
well with width L.
00:47:55.210 --> 00:47:57.150
How does the energy,
the ground state energy,
00:47:57.150 --> 00:48:01.175
compare to that of a
system with a wider well?
00:48:16.070 --> 00:48:21.335
So L versus a larger
L. OK, four seconds.
00:48:25.140 --> 00:48:28.940
OK, quickly discuss amongst
yourselves, like 10 seconds.
00:48:28.940 --> 00:48:45.160
[CHATTER]
00:48:45.160 --> 00:48:47.300
All right.
00:48:47.300 --> 00:48:48.185
Now click again.
00:48:51.493 --> 00:48:51.993
Yeah.
00:48:56.080 --> 00:48:57.520
All right.
00:48:57.520 --> 00:48:58.260
Five seconds.
00:48:58.260 --> 00:49:02.270
One, two, three,
four, five, great.
00:49:02.270 --> 00:49:08.470
OK, the answer is A. OK,
great, because the energy
00:49:08.470 --> 00:49:11.530
of the infinite well
goes like K squared.
00:49:11.530 --> 00:49:18.974
K goes like 1 over L. So the
energy is, if we make it wider,
00:49:18.974 --> 00:49:21.140
the energy if we make it
wider is going to be lower.
00:49:27.260 --> 00:49:28.725
And last couple of questions.
00:49:31.510 --> 00:49:34.900
OK, so t equals 0.
00:49:34.900 --> 00:49:38.350
Could the wave function for an
electron in an infinite square
00:49:38.350 --> 00:49:44.880
well of width a, rather than
L, be A sine squared of pi x
00:49:44.880 --> 00:49:48.680
upon a, where A is suitably
chosen to be normalized?
00:50:10.460 --> 00:50:13.870
All right, you've got
about five seconds left.
00:50:17.550 --> 00:50:21.420
And OK, we are at chance.
00:50:21.420 --> 00:50:24.710
We are at even
odds, and the answer
00:50:24.710 --> 00:50:28.380
is not a superposition
of A and B,
00:50:28.380 --> 00:50:31.130
so I encourage you to discuss
with the people around you.
00:50:31.130 --> 00:50:34.567
[CHATTER]
00:50:51.731 --> 00:50:52.230
Great.
00:50:52.230 --> 00:50:54.400
What properties had it
better satisfy in order
00:50:54.400 --> 00:50:56.774
to be a viable wave function?
00:50:56.774 --> 00:50:58.440
What properties should
the wave function
00:50:58.440 --> 00:51:00.391
have so that it's reasonable?
00:51:00.391 --> 00:51:00.890
Yeah.
00:51:00.890 --> 00:51:02.080
Is it zero at the ends?
00:51:02.080 --> 00:51:03.001
Yeah.
00:51:03.001 --> 00:51:03.500
Good.
00:51:03.500 --> 00:51:04.841
Is it smooth?
00:51:04.841 --> 00:51:07.100
Yeah.
00:51:07.100 --> 00:51:07.830
Exactly.
00:51:07.830 --> 00:51:09.691
And so you can write
it as a superposition.
00:51:09.691 --> 00:51:10.190
Excellent.
00:51:10.190 --> 00:51:11.692
So the answer is?
00:51:11.692 --> 00:51:12.192
Yeah.
00:51:17.516 --> 00:51:18.015
All right.
00:51:20.780 --> 00:51:21.750
Vote again.
00:51:25.170 --> 00:51:26.794
OK, I might have
missed a few people.
00:51:26.794 --> 00:51:27.710
So go ahead and start.
00:51:31.800 --> 00:51:33.185
OK, five more seconds.
00:51:37.230 --> 00:51:37.950
All right.
00:51:37.950 --> 00:51:43.820
So we went from 50-50 to 77-23.
00:51:43.820 --> 00:51:44.820
That's pretty good.
00:51:44.820 --> 00:51:47.260
What's the answer?
00:51:47.260 --> 00:51:47.890
A. Why?
00:51:51.710 --> 00:51:53.892
Is this an energy eigenstate?
00:51:53.892 --> 00:51:55.160
No.
00:51:55.160 --> 00:51:56.960
Does that matter?
00:51:56.960 --> 00:51:57.620
No.
00:51:57.620 --> 00:51:59.710
What properties had this
wave function better
00:51:59.710 --> 00:52:05.460
satisfy to be a reasonable wave
function in this potential?
00:52:05.460 --> 00:52:06.660
Say again?
00:52:06.660 --> 00:52:08.020
It's got to vanish at the walls.
00:52:08.020 --> 00:52:09.853
It's got to satisfy the
boundary conditions.
00:52:09.853 --> 00:52:12.290
What else must be true
of this wave function?
00:52:12.290 --> 00:52:13.010
Normalizable.
00:52:13.010 --> 00:52:14.240
Is it normalizable?
00:52:14.240 --> 00:52:14.880
Yeah.
00:52:14.880 --> 00:52:16.400
What else?
00:52:16.400 --> 00:52:16.900
Continuous.
00:52:16.900 --> 00:52:18.524
It better not have
any discontinuities.
00:52:18.524 --> 00:52:19.471
Is it continuous?
00:52:19.471 --> 00:52:19.970
Great.
00:52:19.970 --> 00:52:20.470
OK.
00:52:20.470 --> 00:52:23.691
Is there any reason that this
is a stupid wave function?
00:52:23.691 --> 00:52:24.190
No.
00:52:24.190 --> 00:52:25.500
It's perfectly reasonable.
00:52:25.500 --> 00:52:29.930
It's not an energy
eigenfunction, but--
00:52:29.930 --> 00:52:31.330
Yeah, cool?
00:52:31.330 --> 00:52:32.150
Yeah.
00:52:32.150 --> 00:52:34.252
AUDIENCE: This is sort
of like a math question.
00:52:34.252 --> 00:52:35.768
So to write that
at a superposition,
00:52:35.768 --> 00:52:38.636
you have to write it like
basically a Fourier sign
00:52:38.636 --> 00:52:39.592
series?
00:52:39.592 --> 00:52:42.940
Isn't the [INAUDIBLE]
function even, though?
00:52:42.940 --> 00:52:46.426
PROFESSOR: On this domain,
that and the sines are even.
00:52:46.426 --> 00:52:48.925
So this is actually odd, but
we're only looking at it from 0
00:52:48.925 --> 00:52:53.480
to L. So, I mean
that half of it.
00:52:53.480 --> 00:52:56.080
The sines are odd, but we're
only looking at the first peak.
00:52:56.080 --> 00:52:56.780
So you could just
as well have written
00:52:56.780 --> 00:52:59.420
that as cosine of the
midpoint plus the distance
00:52:59.420 --> 00:53:01.595
from the midpoint.
00:53:01.595 --> 00:53:03.470
Actually, let me say
that again, because it's
00:53:03.470 --> 00:53:05.511
a much better question I
just give it shrift for.
00:53:10.050 --> 00:53:11.010
So here's the question.
00:53:11.010 --> 00:53:15.359
The question is, look, so
sine is an odd function,
00:53:15.359 --> 00:53:16.900
but sine squared is
an even function.
00:53:16.900 --> 00:53:19.670
So how can you expand sine
squared, an even function,
00:53:19.670 --> 00:53:21.780
in terms of sines,
an odd function?
00:53:21.780 --> 00:53:23.860
But think about this physically.
00:53:23.860 --> 00:53:30.570
Here's sine squared in our
domain, and here's sine.
00:53:30.570 --> 00:53:31.810
Now what do you mean by even?
00:53:31.810 --> 00:53:33.905
Usually by even we mean
reflection around zero.
00:53:33.905 --> 00:53:35.780
But I could just as well
have said reflection
00:53:35.780 --> 00:53:36.770
around the origin.
00:53:36.770 --> 00:53:39.030
This potential is symmetric.
00:53:39.030 --> 00:53:41.745
And the energy eigenfunctions
are symmetric about the origin.
00:53:41.745 --> 00:53:44.120
They're not symmetric about
reflection around this point.
00:53:44.120 --> 00:53:46.380
But they are symmetric about
reflection around this point.
00:53:46.380 --> 00:53:48.210
That's a particularly
natural place to call it 0.
00:53:48.210 --> 00:53:49.930
So I was calling them sine
because I was calling this 0,
00:53:49.930 --> 00:53:51.304
but I could have
called it cosine
00:53:51.304 --> 00:53:55.470
if I called this
0, for the same Kx.
00:53:55.470 --> 00:53:58.570
And indeed, can we
expand this sine
00:53:58.570 --> 00:54:01.570
squared function in terms
of a basis of these sines
00:54:01.570 --> 00:54:02.610
on the domain 0 to L?
00:54:02.610 --> 00:54:04.030
Absolutely.
00:54:04.030 --> 00:54:07.220
Very good question.
00:54:07.220 --> 00:54:09.355
And lastly, last
clicker question.
00:54:12.700 --> 00:54:13.460
Oops.
00:54:13.460 --> 00:54:16.030
Whatever.
00:54:16.030 --> 00:54:17.210
OK.
00:54:17.210 --> 00:54:20.600
At t equals 0, a particle is
described by the wave function
00:54:20.600 --> 00:54:21.130
we just saw.
00:54:23.461 --> 00:54:25.710
Which of the following is
true about the wave function
00:54:25.710 --> 00:54:26.600
at subsequent times?
00:54:37.650 --> 00:54:38.210
5 seconds.
00:54:42.776 --> 00:54:43.730
Whew.
00:54:43.730 --> 00:54:44.650
Oh, OK.
00:54:44.650 --> 00:54:47.360
In the last few seconds we had
an explosive burst for A, B,
00:54:47.360 --> 00:54:49.870
and C. So our
current distribution
00:54:49.870 --> 00:54:54.960
is 8, 16, 10, and 67,
sounds like 67 is popular.
00:54:54.960 --> 00:54:58.630
Discuss quickly, very quickly,
with the person next to you.
00:54:58.630 --> 00:55:02.510
[CHATTER]
00:55:13.670 --> 00:55:17.093
OK, and vote again.
00:55:21.580 --> 00:55:22.980
OK, five seconds.
00:55:22.980 --> 00:55:24.180
Get your last vote in.
00:55:28.240 --> 00:55:29.300
All right.
00:55:29.300 --> 00:55:33.472
And the answer is D. Yay.
00:55:37.289 --> 00:55:38.580
So let's think about the logic.
00:55:38.580 --> 00:55:39.950
Let's go through the logic here.
00:55:39.950 --> 00:55:42.360
So as was pointed out by
a student up here earlier,
00:55:42.360 --> 00:55:46.070
the wave function
sine squared of pi x
00:55:46.070 --> 00:55:49.100
can be expanded in terms of
the energy eigenfunction.
00:55:49.100 --> 00:55:51.150
Any reasonable function
can be expanded
00:55:51.150 --> 00:55:53.800
in terms of a superposition
of definite energy states
00:55:53.800 --> 00:55:55.600
of energy eigenfunctions.
00:55:55.600 --> 00:55:58.490
So that means we can
write psi at some time
00:55:58.490 --> 00:56:04.060
as a superposition Cn sine of n
pi x upon a e to the minus i e
00:56:04.060 --> 00:56:09.450
n t upon h bar, since those are,
in fact, the eigenfunctions.
00:56:09.450 --> 00:56:10.200
So we can do that.
00:56:10.200 --> 00:56:11.877
Now, when we look at
the time evolution,
00:56:11.877 --> 00:56:13.710
we know that each term
in that superposition
00:56:13.710 --> 00:56:15.450
evolves with a phase.
00:56:15.450 --> 00:56:19.220
The overall wave function
does not evolve with a phase.
00:56:19.220 --> 00:56:21.360
It is not an energy eigenstate.
00:56:21.360 --> 00:56:23.460
There are going to be
interference terms due
00:56:23.460 --> 00:56:26.207
to the fact that
it's a superposition.
00:56:26.207 --> 00:56:28.540
So its probability distribution
is not time-independent.
00:56:28.540 --> 00:56:29.640
It is a superposition.
00:56:29.640 --> 00:56:33.430
And so the wave function doesn't
rotate by an overall phase.
00:56:33.430 --> 00:56:37.030
However, we can solve
the Schrodinger equation,
00:56:37.030 --> 00:56:37.810
as we did before.
00:56:37.810 --> 00:56:40.440
The wave function is
expanded at time 0
00:56:40.440 --> 00:56:43.544
as the energy eigenfunctions
times some set of coefficients.
00:56:43.544 --> 00:56:44.960
And the time
evolution corresponds
00:56:44.960 --> 00:56:48.230
to adding two each independent
term in the superposition
00:56:48.230 --> 00:56:52.120
the appropriate phase for
that energy eigenstate.
00:56:52.120 --> 00:56:54.040
Cool?
00:56:54.040 --> 00:56:54.540
All right.
00:56:54.540 --> 00:57:03.220
So the answer is D. And that's
it for the clicker questions.
00:57:03.220 --> 00:57:07.050
OK, so any questions on the
clicker questions so far?
00:57:07.050 --> 00:57:09.050
OK, those are going to
be posted on the web site
00:57:09.050 --> 00:57:12.190
so you can go over them.
00:57:12.190 --> 00:57:14.250
And now back to
energy eigenfunctions.
00:57:24.770 --> 00:57:27.730
So what I want to talk about
now is the qualitative behavior
00:57:27.730 --> 00:57:29.140
of energy eigenfunctions.
00:57:29.140 --> 00:57:31.460
Suppose I know I have
an energy eigenfunction.
00:57:31.460 --> 00:57:33.955
What can I say generally
about its structure?
00:57:38.870 --> 00:57:41.175
So let me ask the question,
qualitative behavior.
00:57:47.270 --> 00:57:49.670
So suppose someone hands
you a potential U of x.
00:57:49.670 --> 00:57:51.990
Someone hands you some
potential, U of x,
00:57:51.990 --> 00:57:54.900
and says, look, I've
got this potential.
00:57:54.900 --> 00:57:57.580
Maybe I'll draw it for you.
00:57:57.580 --> 00:57:59.850
It's got some wiggles,
and then a big wiggle,
00:57:59.850 --> 00:58:01.880
and then it's got a
big wiggle, and then--
00:58:01.880 --> 00:58:03.195
do I want to do that?
00:58:03.195 --> 00:58:04.275
Yeah, let's do that.
00:58:04.275 --> 00:58:08.089
Then a big wiggle, and
something like this.
00:58:08.089 --> 00:58:09.630
And someone shows
you this potential.
00:58:09.630 --> 00:58:11.921
And they say, look, what are
the energy eigenfunctions?
00:58:13.770 --> 00:58:17.510
Well, OK, free
particle was easy.
00:58:17.510 --> 00:58:19.070
The infinite square
well was easy.
00:58:19.070 --> 00:58:21.207
We could solve
that analytically.
00:58:21.207 --> 00:58:23.290
The next involved solving
a differential equation.
00:58:23.290 --> 00:58:25.360
So what differential equation
is this going to lead us to?
00:58:25.360 --> 00:58:27.570
Well, we know that the
energy eigenvalue equation
00:58:27.570 --> 00:58:32.070
is minus h bar
squared upon 2 m phi
00:58:32.070 --> 00:58:38.570
prime prime of x plus U of x
phi x, so that's the energy
00:58:38.570 --> 00:58:41.620
operator acting on
phi, is equal to,
00:58:41.620 --> 00:58:47.949
saying that it's an energy
eigenfunction, phi sub E,
00:58:47.949 --> 00:58:49.740
says that it's equal
to the energy operator
00:58:49.740 --> 00:58:52.670
acting on this eigenfunction
is just a constant E phi sub
00:58:52.670 --> 00:58:54.870
E of x.
00:58:54.870 --> 00:58:56.680
And I'm going to work
at moment in time,
00:58:56.680 --> 00:59:00.420
so we're going to drop all the
t dependence for the moment.
00:59:00.420 --> 00:59:01.920
So this is the
differential equation
00:59:01.920 --> 00:59:07.072
we need to solve where U of
x is this god-awful function.
00:59:07.072 --> 00:59:08.780
Do you think it's very
likely that you're
00:59:08.780 --> 00:59:11.670
going to be able to
solve this analytically?
00:59:11.670 --> 00:59:13.700
Probably not.
00:59:13.700 --> 00:59:18.761
However, some basic
ideas will help
00:59:18.761 --> 00:59:21.010
you get an intuition for
what the wave function should
00:59:21.010 --> 00:59:22.240
look like.
00:59:22.240 --> 00:59:25.910
And I cannot overstate the
importance of being able
00:59:25.910 --> 00:59:29.560
to eyeball a system and guess
the qualitative features of its
00:59:29.560 --> 00:59:32.030
wave functions,
because that intuition,
00:59:32.030 --> 00:59:34.590
that ability to estimate, is
going to contain an awful lot
00:59:34.590 --> 00:59:35.340
of physics.
00:59:35.340 --> 00:59:36.790
So let's try to extract it.
00:59:36.790 --> 00:59:38.930
So in order to do
so, I want to start
00:59:38.930 --> 00:59:40.945
by massaging this
equation into a form which
00:59:40.945 --> 00:59:42.070
is particularly convenient.
00:59:44.790 --> 00:59:47.910
So in particular, I'm going to
write this equation as phi sub
00:59:47.910 --> 00:59:49.449
E prime prime.
00:59:49.449 --> 00:59:51.740
So what I'm going to do is
I'm going to take this term,
00:59:51.740 --> 00:59:53.615
I'm going to notice this
has two derivatives,
00:59:53.615 --> 00:59:56.026
this has no derivatives,
this has no derivatives.
00:59:56.026 --> 00:59:59.130
And I'm going to move this
term over here and combine
00:59:59.130 --> 01:00:01.600
these terms into E
minus U of x, and I'm
01:00:01.600 --> 01:00:05.320
going to divide each side by 2m
upon h bar squared with a minus
01:00:05.320 --> 01:00:08.260
sign, giving me that
phi prime prime of E
01:00:08.260 --> 01:00:14.420
of x upon phi E of x dividing
through by this phi E
01:00:14.420 --> 01:00:18.250
is equal to minus 2m
over h bar squared.
01:00:21.682 --> 01:00:23.140
And let's just get
our signs right.
01:00:23.140 --> 01:00:24.560
We've got the minus
from here, so this
01:00:24.560 --> 01:00:25.960
is going to be E minus U of x.
01:00:33.931 --> 01:00:35.680
So you might look at
that and think, well,
01:00:35.680 --> 01:00:40.010
why is that any better than
what I've just written down.
01:00:40.010 --> 01:00:42.724
But what is the second
derivative of function?
01:00:42.724 --> 01:00:44.890
It's telling you not its
slope, but it's telling you
01:00:44.890 --> 01:00:46.550
how the slope changes.
01:00:46.550 --> 01:00:49.588
It's telling about the
curvature of the function.
01:00:49.588 --> 01:00:52.046
And what this is telling me is
something very, very useful.
01:00:54.179 --> 01:00:55.970
So for example, let's
look at the function.
01:00:55.970 --> 01:00:57.360
Let's assume that
the function is real,
01:00:57.360 --> 01:00:58.901
although we know in
general it's not.
01:00:58.901 --> 01:01:02.100
Let's assume that the function
is real for simplicity.
01:01:02.100 --> 01:01:05.910
So we're going to plot
the real part of phi
01:01:05.910 --> 01:01:07.210
in the vertical axis.
01:01:07.210 --> 01:01:08.630
And this is x.
01:01:08.630 --> 01:01:15.130
Suppose the real part of phi
is positive at some point.
01:01:15.130 --> 01:01:17.070
Phi prime prime,
if it's positive,
01:01:17.070 --> 01:01:19.297
tells us that not only
is the slope positive,
01:01:19.297 --> 01:01:20.130
but it's increasing.
01:01:20.130 --> 01:01:21.600
Or it doesn't tell us
anything about the slope,
01:01:21.600 --> 01:01:23.080
but it tells us that whatever
the slope, it's increasing.
01:01:23.080 --> 01:01:25.650
If it's negative, the slope is
increasing as we increase x.
01:01:25.650 --> 01:01:28.210
If it's positive, it's
increasing as we increase x.
01:01:28.210 --> 01:01:31.450
So it's telling us that the
wave function looks like this,
01:01:31.450 --> 01:01:34.880
locally, something like that.
01:01:34.880 --> 01:01:39.040
If phi is negative,
if phi is negative,
01:01:39.040 --> 01:01:41.254
then if this
quantity is positive,
01:01:41.254 --> 01:01:42.920
then phi prime prime
has to be negative.
01:01:42.920 --> 01:01:45.150
But negative is curving down.
01:01:50.210 --> 01:01:54.780
So if this quantity, which
I will call the curvature,
01:01:54.780 --> 01:02:01.260
if this quantity is positive,
it curves away from the axis.
01:02:01.260 --> 01:02:07.100
So this is phi prime prime
over phi greater than 0.
01:02:07.100 --> 01:02:09.700
If this quantity is
positive, the function
01:02:09.700 --> 01:02:11.050
curves away from the axis.
01:02:11.050 --> 01:02:12.440
Cool?
01:02:12.440 --> 01:02:16.270
If this quantity is negative,
phi prime prime upon
01:02:16.270 --> 01:02:19.490
phi less than 0,
exactly the opposite.
01:02:19.490 --> 01:02:20.920
This has to be negative.
01:02:20.920 --> 01:02:23.480
If phi is positive, then phi
prime prime has to be negative.
01:02:23.480 --> 01:02:24.907
It has to be curving down.
01:02:27.710 --> 01:02:30.290
And similarly, if
phi is negative,
01:02:30.290 --> 01:02:33.170
then phi prime prime
has to be positive,
01:02:33.170 --> 01:02:34.500
and it has to curve up.
01:02:34.500 --> 01:02:37.840
So if this quantity is positive,
if the curvature is positive,
01:02:37.840 --> 01:02:39.440
it curves away from the axis.
01:02:39.440 --> 01:02:41.950
If the curvature is negative,
if this quantity is negative,
01:02:41.950 --> 01:02:44.120
it curves towards the axis.
01:02:44.120 --> 01:02:47.590
So what does that tell you about
solutions when the curvature is
01:02:47.590 --> 01:02:49.206
positive or negative?
01:02:49.206 --> 01:02:50.330
It tells you the following.
01:02:52.930 --> 01:02:54.980
It tells you that,
imagine we have
01:02:54.980 --> 01:02:59.450
a function where phi prime
prime over phi is constant.
01:02:59.450 --> 01:03:03.050
And in particular, let's
let phi prime prime over phi
01:03:03.050 --> 01:03:06.760
be a constant,
which is positive.
01:03:06.760 --> 01:03:10.671
And I'll call that positive
constant kappa squared.
01:03:10.671 --> 01:03:12.170
And to emphasize
that it's positive,
01:03:12.170 --> 01:03:14.174
I'm going to call
it kappa squared.
01:03:14.174 --> 01:03:15.090
It's a positive thing.
01:03:15.090 --> 01:03:16.980
It's a real number squared.
01:03:16.980 --> 01:03:18.635
What does the
solution look like?
01:03:22.510 --> 01:03:24.120
Well, this quantity is positive.
01:03:24.120 --> 01:03:25.661
It's always going
to be curving away.
01:03:25.661 --> 01:03:28.287
So we have solutions that
look like this or solutions
01:03:28.287 --> 01:03:29.120
that look like this.
01:03:29.120 --> 01:03:29.990
Can it ever be 0?
01:03:32.907 --> 01:03:34.740
Yeah, sure, it could
be an inflection point.
01:03:34.740 --> 01:03:36.698
So for example, here the
curvature is positive,
01:03:36.698 --> 01:03:40.170
but at this point the curvature
has to switch to be like this.
01:03:40.170 --> 01:03:41.960
What functions are of this form?
01:03:44.752 --> 01:03:45.960
Let me give you another hint.
01:03:45.960 --> 01:03:46.550
Here's one.
01:03:46.550 --> 01:03:49.400
Is this curvature positive?
01:03:49.400 --> 01:03:50.057
Yes.
01:03:50.057 --> 01:03:50.890
What about this one?
01:03:53.470 --> 01:03:53.970
Yup.
01:03:53.970 --> 01:03:56.220
Those are all
positive curvature.
01:03:56.220 --> 01:03:57.910
And these are exponentials.
01:03:57.910 --> 01:03:59.930
And the solution to this
differential equation
01:03:59.930 --> 01:04:06.670
is e to the plus kappa x
or e to the minus kappa x.
01:04:06.670 --> 01:04:08.940
And an arbitrary
solution of this equation
01:04:08.940 --> 01:04:12.130
is a superposition A e
to the kappa x plus B
01:04:12.130 --> 01:04:14.346
e to the minus kappa x.
01:04:14.346 --> 01:04:16.244
Everyone cool with that?
01:04:16.244 --> 01:04:17.660
When this quantity
is positive, we
01:04:17.660 --> 01:04:19.390
get growing and
collapsing exponentials.
01:04:21.890 --> 01:04:22.390
Yeah?
01:04:25.610 --> 01:04:29.990
On the other hand, if
phi prime prime over phi
01:04:29.990 --> 01:04:34.010
is a negative number, i.e.
minus what I'll call k
01:04:34.010 --> 01:04:44.010
squared, then the curvature
has to be negative.
01:04:44.010 --> 01:04:47.460
And what functions have
everywhere negative curvature?
01:04:47.460 --> 01:04:49.169
Sinusoidals.
01:04:49.169 --> 01:04:49.669
Cool?
01:04:53.540 --> 01:05:00.445
And the general solution
is A e to the i K x plus B
01:05:00.445 --> 01:05:03.226
e to the minus i K x.
01:05:03.226 --> 01:05:06.710
So that differential equation,
also known as sine and cosine.
01:05:09.170 --> 01:05:09.670
Cool?
01:05:12.540 --> 01:05:19.765
So putting that together
with our original function,
01:05:19.765 --> 01:05:21.140
let's bring this up.
01:05:24.110 --> 01:05:26.680
So we want to think about
the wave functions here.
01:05:26.680 --> 01:05:29.014
But in order to think about
the energy eigenstates,
01:05:29.014 --> 01:05:30.305
we need to decide on an energy.
01:05:32.850 --> 01:05:35.794
We need to pick an
energy, because you
01:05:35.794 --> 01:05:37.960
can't find the solution
without figuring the energy.
01:05:37.960 --> 01:05:38.950
But notice something nice here.
01:05:38.950 --> 01:05:40.260
So suppose the energy is e.
01:05:40.260 --> 01:05:42.360
And let me just draw
E. This is a constant.
01:05:42.360 --> 01:05:43.187
The energy is this.
01:05:43.187 --> 01:05:45.520
So this is the value of E.
Here we're drawing potential.
01:05:45.520 --> 01:05:48.489
But this is the value of the
energy, which is a constant.
01:05:48.489 --> 01:05:49.280
It's just a number.
01:05:53.680 --> 01:05:57.200
If you had a classical particle
moving in this potential,
01:05:57.200 --> 01:05:58.760
what would happen?
01:05:58.760 --> 01:05:59.650
It would roll around.
01:05:59.650 --> 01:06:01.360
So for example, let's say
you gave it this energy
01:06:01.360 --> 01:06:02.210
by putting it here.
01:06:02.210 --> 01:06:04.168
And think of this as a
gravitational potential.
01:06:04.168 --> 01:06:06.157
You put it here, you let
go, and it falls down.
01:06:06.157 --> 01:06:07.990
And it'll keep rolling
until it gets up here
01:06:07.990 --> 01:06:10.230
to the classical turning point.
01:06:10.230 --> 01:06:12.080
And at that point,
its kinetic energy
01:06:12.080 --> 01:06:13.710
must be 0, because
its potential energy
01:06:13.710 --> 01:06:15.410
is its total energy,
at which point
01:06:15.410 --> 01:06:17.670
it will turn around
and fall back.
01:06:17.670 --> 01:06:18.359
Yes?
01:06:18.359 --> 01:06:20.150
If you take your ball,
and you put it here,
01:06:20.150 --> 01:06:21.525
and you let it
roll, does it ever
01:06:21.525 --> 01:06:23.960
get here, to this position?
01:06:23.960 --> 01:06:25.990
No, because it doesn't
have enough energy.
01:06:25.990 --> 01:06:29.620
Classically, this is
a forbidden position.
01:06:29.620 --> 01:06:31.790
So given an energy
and given a potential,
01:06:31.790 --> 01:06:38.200
we can break the system up
into classically allowed zones
01:06:38.200 --> 01:06:39.760
and classically forbidden zones.
01:06:45.170 --> 01:06:47.120
Cool?
01:06:47.120 --> 01:06:49.210
Now, in a classically
allowed zone,
01:06:49.210 --> 01:06:53.670
the energy is greater
than the potential.
01:06:53.670 --> 01:06:55.170
And in a classically
forbidden zone,
01:06:55.170 --> 01:06:57.300
the energy is less
than the potential.
01:07:00.530 --> 01:07:03.230
Everyone cool with that?
01:07:03.230 --> 01:07:06.080
But this tells us
something really nice.
01:07:06.080 --> 01:07:09.429
If the energy is greater
than the potential,
01:07:09.429 --> 01:07:10.970
what do you know
about the curvature?
01:07:15.050 --> 01:07:15.550
Yeah.
01:07:15.550 --> 01:07:17.175
If we're in a
classically allowed zone,
01:07:17.175 --> 01:07:20.150
so the energy is greater
than the potential,
01:07:20.150 --> 01:07:22.860
then this quantity is positive,
there's a minus sign here,
01:07:22.860 --> 01:07:23.920
so this is negative.
01:07:23.920 --> 01:07:25.760
So the curvature is negative.
01:07:28.920 --> 01:07:31.680
Remember, curvature
is negative means
01:07:31.680 --> 01:07:35.000
that we curve towards the axis.
01:07:35.000 --> 01:07:37.015
So in a classically
allowed region,
01:07:37.015 --> 01:07:38.640
the wave function
should be sinusoidal.
01:07:43.840 --> 01:07:46.050
What about in the classically
forbidden regions?
01:07:46.050 --> 01:07:47.280
In the classically
forbidden regions,
01:07:47.280 --> 01:07:48.890
the energy is less
than the potential.
01:07:48.890 --> 01:07:50.848
That means in magnitude
this is less than this,
01:07:50.848 --> 01:07:53.270
this is a negative
number, minus sign,
01:07:53.270 --> 01:07:55.365
the curvature is
going to be minus
01:07:55.365 --> 01:07:59.050
times a minus is a positive,
so the curvature's positive.
01:08:02.260 --> 01:08:05.190
So the solutions are
either growing exponentials
01:08:05.190 --> 01:08:10.000
or shrinking exponentials
or superpositions of them.
01:08:10.000 --> 01:08:11.810
Everyone cool with that?
01:08:11.810 --> 01:08:14.340
So let's think about
a simple example.
01:08:14.340 --> 01:08:16.452
Let's work through this
in a simple example.
01:08:16.452 --> 01:08:18.575
And let me give you a
little bit more board space.
01:08:22.430 --> 01:08:26.122
Simple example would be a
potential that looks like this.
01:08:26.122 --> 01:08:27.580
And let's just
suppose that we want
01:08:27.580 --> 01:08:33.760
to find an energy eigenfunction
with energy that's E. Well,
01:08:33.760 --> 01:08:36.250
this is a classically
allowed zone,
01:08:36.250 --> 01:08:39.939
and these are the classically
forbidden regions.
01:08:39.939 --> 01:08:43.370
Now I want to ask, what does
the wave function look like?
01:08:43.370 --> 01:08:46.630
And I don't want to draw it
on top of the energy diagram,
01:08:46.630 --> 01:08:49.037
because wave function
is not an energy.
01:08:49.037 --> 01:08:50.620
Wave function is a
different quantity,
01:08:50.620 --> 01:08:52.370
because it's got
different axes and I want
01:08:52.370 --> 01:08:54.279
it drawn on a different plot.
01:08:54.279 --> 01:08:57.960
So but as a function of x--
01:08:57.960 --> 01:09:00.935
so just to get the
positions straight,
01:09:00.935 --> 01:09:02.560
these are the bounds
of the classically
01:09:02.560 --> 01:09:04.899
allowed and forbidden regions.
01:09:04.899 --> 01:09:07.140
What do we expect?
01:09:07.140 --> 01:09:09.830
Well, we expect that it's
going to be sinusoidal in here.
01:09:13.160 --> 01:09:15.770
We expect that it's going
to be exponential growing
01:09:15.770 --> 01:09:20.399
or converging out here, exp.
01:09:20.399 --> 01:09:25.100
But one last important thing is
that not only is the curvature
01:09:25.100 --> 01:09:28.270
negative in here in these
classically allowed regions,
01:09:28.270 --> 01:09:30.080
but the magnitude
of the curvature,
01:09:30.080 --> 01:09:32.870
how rapidly it's turning over,
how big that second derivative
01:09:32.870 --> 01:09:35.040
is, depends on the
difference between the energy
01:09:35.040 --> 01:09:35.865
and the potential.
01:09:35.865 --> 01:09:38.240
The greater the difference,
the more rapid the curvature,
01:09:38.240 --> 01:09:40.500
the more rapid the turning
over and fluctuation.
01:09:40.500 --> 01:09:43.450
If the differences between the
potential and the true energy,
01:09:43.450 --> 01:09:46.720
the total energy, is small, then
the curvature is very small.
01:09:46.720 --> 01:09:49.990
So the derivative
changes very gradually.
01:09:49.990 --> 01:09:51.660
What does that tell us?
01:09:51.660 --> 01:09:54.194
That tells us that in here the
wave function is oscillating
01:09:54.194 --> 01:09:56.110
rapidly, because the
curvature, the difference
01:09:56.110 --> 01:09:58.320
between the energy and
the potential is large,
01:09:58.320 --> 01:10:01.690
and so the wave function
is oscillating rapidly.
01:10:01.690 --> 01:10:04.200
As we get out towards the
classical turning points,
01:10:04.200 --> 01:10:07.640
the wave function will be
oscillating less rapidly.
01:10:07.640 --> 01:10:09.865
The slope will be
changing more gradually.
01:10:09.865 --> 01:10:11.810
And as a consequence,
two things happen.
01:10:11.810 --> 01:10:13.768
Let me actually draw this
slightly differently.
01:10:17.647 --> 01:10:19.230
So as a consequence
two things happen.
01:10:19.230 --> 01:10:21.130
One is the wavelength
gets longer,
01:10:21.130 --> 01:10:22.847
because the
curvature is smaller.
01:10:22.847 --> 01:10:24.680
And the second is the
amplitude gets larger,
01:10:24.680 --> 01:10:27.280
because it keeps on having
a positive slope for longer
01:10:27.280 --> 01:10:31.520
and longer, and it takes
longer to curve back down.
01:10:31.520 --> 01:10:34.260
So here we have
rapid oscillations.
01:10:34.260 --> 01:10:41.430
And then the oscillations get
longer and longer wavelength,
01:10:41.430 --> 01:10:43.892
until we get out to the
classical turning point.
01:10:43.892 --> 01:10:45.225
And at this point, what happens?
01:10:47.704 --> 01:10:49.120
Yeah, it's got to
be [INAUDIBLE]..
01:10:49.120 --> 01:10:51.860
Now, here we have some sine,
and some superpositions
01:10:51.860 --> 01:10:53.647
of sine and cosines,
exponentials.
01:10:53.647 --> 01:10:55.730
And in particular, it
arrives here with some slope
01:10:55.730 --> 01:10:57.490
and with some value.
01:10:57.490 --> 01:11:00.140
We know this side we've
got to get exponentials.
01:11:00.140 --> 01:11:02.330
And so this sum of sines
and cosines at this point
01:11:02.330 --> 01:11:06.400
must match the sum
of exponentials.
01:11:06.400 --> 01:11:08.282
How must it do so?
01:11:08.282 --> 01:11:10.490
What must be true of the
wave function at this point?
01:11:13.880 --> 01:11:16.090
Can it be discontinuous?
01:11:16.090 --> 01:11:17.850
Can its derivative
be discontinuous?
01:11:17.850 --> 01:11:18.350
No.
01:11:18.350 --> 01:11:21.800
So the value and the
derivative must be continuous.
01:11:21.800 --> 01:11:24.690
So that tells us precisely
which linear combination
01:11:24.690 --> 01:11:29.089
of positive growing and
shrinking exponentials we get.
01:11:29.089 --> 01:11:30.630
So we'll get some
linear combination,
01:11:30.630 --> 01:11:32.360
which may do this for awhile.
01:11:32.360 --> 01:11:33.860
But since it's got
some contribution
01:11:33.860 --> 01:11:36.490
of positive exponential, it'll
just grow exponentially off
01:11:36.490 --> 01:11:37.982
to infinity.
01:11:37.982 --> 01:11:39.940
And as the energy gets
further and further away
01:11:39.940 --> 01:11:42.780
from the potential, now in
their negative sine, what
01:11:42.780 --> 01:11:45.444
happens to the rate of growth?
01:11:45.444 --> 01:11:46.610
It gets more and more rapid.
01:11:46.610 --> 01:11:48.540
So this just diverges
more and more rapidly.
01:11:48.540 --> 01:11:51.554
Similarly, out here we
have to match the slope.
01:11:51.554 --> 01:11:52.970
And we know that
the curvature has
01:11:52.970 --> 01:11:55.530
to be now positive,
so it has to do this.
01:11:59.120 --> 01:12:00.040
So two questions.
01:12:00.040 --> 01:12:03.820
First off, is this sketch
of the wave function
01:12:03.820 --> 01:12:05.560
a reasonable sketch,
given what we
01:12:05.560 --> 01:12:08.570
know about curvature and this
potential of a wave function
01:12:08.570 --> 01:12:09.280
with that energy?
01:12:13.007 --> 01:12:14.840
Are there ways in which
it's a bad estimate?
01:12:17.798 --> 01:12:19.590
AUDIENCE: [INAUDIBLE]
01:12:19.590 --> 01:12:21.685
PROFESSOR: OK, excellent.
01:12:21.685 --> 01:12:24.600
AUDIENCE: On the right side,
could it have crossed zero?
01:12:24.600 --> 01:12:25.720
PROFESSOR: Absolutely, it
could have crossed zero.
01:12:25.720 --> 01:12:26.920
So I may have drawn this badly.
01:12:26.920 --> 01:12:28.060
It turned out it
was a little subtle.
01:12:28.060 --> 01:12:28.740
It's not obvious.
01:12:28.740 --> 01:12:30.340
Maybe it actually punched
all the way through zero,
01:12:30.340 --> 01:12:31.660
and then it diverged
down negative.
01:12:31.660 --> 01:12:32.785
That's absolutely positive.
01:12:32.785 --> 01:12:35.075
So that was one of the
quibbles you could have.
01:12:35.075 --> 01:12:36.450
Another quibble
you could have is
01:12:36.450 --> 01:12:39.200
that it looks like I have
constant wavelength in here.
01:12:39.200 --> 01:12:41.010
But the potential's
actually changing.
01:12:41.010 --> 01:12:43.070
And what you should
chalk this up to,
01:12:43.070 --> 01:12:46.299
if you'll pardon the pun, is
my artistic skills are limited.
01:12:46.299 --> 01:12:48.340
So this is always going
to be sort of inescapable
01:12:48.340 --> 01:12:50.505
when you qualitatively
draw something.
01:12:50.505 --> 01:12:53.130
On a test, I'm not going to bag
you points on things like that.
01:12:53.130 --> 01:12:54.584
That's what I want to emphasize.
01:12:54.584 --> 01:12:56.250
But the second thing,
is there something
01:12:56.250 --> 01:12:57.622
bad about this wave function?
01:12:57.622 --> 01:12:58.830
Yes, you've already named it.
01:12:58.830 --> 01:13:01.090
What's bad about
this wave function?
01:13:01.090 --> 01:13:03.150
It's badly non-normalizable.
01:13:03.150 --> 01:13:06.820
It diverges off to infinity
out here and out here.
01:13:06.820 --> 01:13:10.110
What does that tell you?
01:13:10.110 --> 01:13:10.860
It's not physical.
01:13:10.860 --> 01:13:11.170
Good.
01:13:11.170 --> 01:13:13.045
What else does it tell
you about this system?
01:13:16.110 --> 01:13:18.220
Sorry?
01:13:18.220 --> 01:13:18.870
Excellent.
01:13:18.870 --> 01:13:21.670
Is this an allowable energy?
01:13:21.670 --> 01:13:22.840
No.
01:13:22.840 --> 01:13:25.130
If the wave function
has this energy,
01:13:25.130 --> 01:13:28.340
it is impossible to
make it continuous,
01:13:28.340 --> 01:13:30.010
assuming that I
drew it correctly,
01:13:30.010 --> 01:13:31.530
and have it converge.
01:13:31.530 --> 01:13:33.640
Is this wave function allowable?
01:13:33.640 --> 01:13:36.180
No, because it does not satisfy
our boundary conditions.
01:13:36.180 --> 01:13:38.763
Our boundary conditions are that
the wave function must vanish
01:13:38.763 --> 01:13:41.710
out here and it must
vanish out here at infinity
01:13:41.710 --> 01:13:43.785
in order to be normalizable.
01:13:43.785 --> 01:13:44.410
Here we failed.
01:13:47.590 --> 01:13:49.610
Now, you can imagine
that-- so let's decrease
01:13:49.610 --> 01:13:50.610
the energy a little bit.
01:13:50.610 --> 01:13:53.760
If we decrease the energy,
our trial energy just a little
01:13:53.760 --> 01:13:55.200
tiny bit, what happens?
01:13:55.200 --> 01:13:58.570
Well, that's going to decrease
the curvature in here.
01:13:58.570 --> 01:14:01.260
We decrease, we bring the energy
in just a little tiny bit.
01:14:01.260 --> 01:14:03.550
That means this is a
little bit smaller.
01:14:03.550 --> 01:14:05.130
The potential stays the same.
01:14:05.130 --> 01:14:06.760
So the curvature in
the allowed region
01:14:06.760 --> 01:14:09.010
is just a little
tiny bit smaller.
01:14:09.010 --> 01:14:10.600
And meanwhile,
the allowed region
01:14:10.600 --> 01:14:12.649
has got just a
little bit thinner.
01:14:12.649 --> 01:14:14.940
And what that will do is the
curvature's a little less,
01:14:14.940 --> 01:14:17.500
the region's a little
less, so now we have--
01:14:28.130 --> 01:14:30.885
Sorry, I get excited.
01:14:30.885 --> 01:14:33.010
And if we tweak the energy,
what's going to happen?
01:14:33.010 --> 01:14:35.176
Well, it's going to arrive
here a little bit sooner.
01:14:38.220 --> 01:14:40.490
And let's imagine
something like this.
01:14:40.490 --> 01:14:45.250
And if we chose the
energy just right,
01:14:45.250 --> 01:14:48.120
we would get it to match
to a linear combination
01:14:48.120 --> 01:14:50.560
of collapsing and
growing exponentials,
01:14:50.560 --> 01:14:53.085
where the contribution from
the growing exponential
01:14:53.085 --> 01:14:54.210
in this direction vanishes.
01:14:57.180 --> 01:14:59.320
There's precisely one
value of the energy
01:14:59.320 --> 01:15:01.992
that lets me do that with
this number of wiggles.
01:15:01.992 --> 01:15:03.950
And so then it goes
through and does its thing.
01:15:07.860 --> 01:15:11.570
And we need it to
happen on both sides.
01:15:11.570 --> 01:15:14.510
Now if I take that solution,
so that it achieves convergence
01:15:14.510 --> 01:15:18.740
out here, and it achieves
convergence out here,
01:15:18.740 --> 01:15:21.630
and I take that energy and
I increase it by epsilon,
01:15:21.630 --> 01:15:24.450
by just the tiniest little bit,
what will happen to this wave
01:15:24.450 --> 01:15:26.057
function?
01:15:26.057 --> 01:15:26.640
It'll diverge.
01:15:26.640 --> 01:15:28.670
It will no longer
be normalizable.
01:15:28.670 --> 01:15:31.070
When you have classically
forbidden regions,
01:15:31.070 --> 01:15:36.010
are the allowed energies
continuous or discrete?
01:15:36.010 --> 01:15:38.820
And that answers a question
from earlier in the class.
01:15:38.820 --> 01:15:40.570
And it also is going
to be the beginning
01:15:40.570 --> 01:15:43.153
of the answer to the question,
why is the spectrum of hydrogen
01:15:43.153 --> 01:15:44.000
discrete.
01:15:44.000 --> 01:15:45.960
See you next time.