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PROFESSOR: So,
finally, before I get
00:00:23.720 --> 00:00:26.230
started on the new
stuff, questions
00:00:26.230 --> 00:00:27.500
from the previous lectures?
00:00:34.820 --> 00:00:37.052
No questions?
00:00:37.052 --> 00:00:37.552
Yeah.
00:00:37.552 --> 00:00:38.025
AUDIENCE: I have a question.
00:00:38.025 --> 00:00:39.483
You might have said
this last time,
00:00:39.483 --> 00:00:41.069
but when is the first exam?
00:00:41.069 --> 00:00:42.110
PROFESSOR: Ah, excellent.
00:00:42.110 --> 00:00:47.150
Those will be posted on the
Stellar page later today.
00:00:47.150 --> 00:00:48.221
Yeah.
00:00:48.221 --> 00:00:50.145
AUDIENCE: OK, so we're
associating operators
00:00:50.145 --> 00:00:51.444
with observables, right?
00:00:51.444 --> 00:00:52.069
PROFESSOR: Yes.
00:00:52.069 --> 00:00:53.568
AUDIENCE: And
Professor [? Zugoff ?]
00:00:53.568 --> 00:00:55.676
mentioned that whenever
we have done a wave
00:00:55.676 --> 00:01:00.082
function with an
operator, it collapses.
00:01:00.082 --> 00:01:02.040
PROFESSOR: OK, so let me
rephrase the question.
00:01:02.040 --> 00:01:04.640
This is a very valuable
question to talk through.
00:01:04.640 --> 00:01:05.720
So, thanks for asking it.
00:01:05.720 --> 00:01:09.486
So, we've previously observed
that observables are associated
00:01:09.486 --> 00:01:10.860
with operators--
and we'll review
00:01:10.860 --> 00:01:12.930
that in more detail
in a second--
00:01:12.930 --> 00:01:16.100
and the statement was
then made, does that
00:01:16.100 --> 00:01:21.210
mean that acting on a wave
function with an operator
00:01:21.210 --> 00:01:22.660
is like measuring
the observable?
00:01:22.660 --> 00:01:25.540
And it's absolutely
essential that you understand
00:01:25.540 --> 00:01:29.960
that acting on a wave
function with an operator
00:01:29.960 --> 00:01:34.380
has nothing whatsoever to do
with measuring that associated
00:01:34.380 --> 00:01:35.240
observable.
00:01:35.240 --> 00:01:36.360
Nothing.
00:01:36.360 --> 00:01:36.860
OK?
00:01:36.860 --> 00:01:37.895
And we'll talk about
the relationship
00:01:37.895 --> 00:01:39.160
and what those things mean.
00:01:39.160 --> 00:01:41.400
But here's a very
tempting thing to think.
00:01:41.400 --> 00:01:42.540
I have a wave function.
00:01:42.540 --> 00:01:43.890
I want to know the momentum.
00:01:43.890 --> 00:01:46.160
I will thus operate with
the momentum operator.
00:01:46.160 --> 00:01:47.400
Completely wrong.
00:01:47.400 --> 00:01:50.430
So, before I even tell you
what the right statement is,
00:01:50.430 --> 00:01:52.030
let me just get that
out of your head,
00:01:52.030 --> 00:01:54.384
and then we'll talk through
that in much more detail
00:01:54.384 --> 00:01:55.300
over the next lecture.
00:01:55.300 --> 00:01:55.970
Yeah.
00:01:55.970 --> 00:01:58.395
AUDIENCE: Why doesn't it
collapse by special relativity?
00:01:58.395 --> 00:01:59.770
PROFESSOR: We're
doing everything
00:01:59.770 --> 00:02:00.730
non-relativistically.
00:02:00.730 --> 00:02:02.620
Quantum Mechanics
for 804 is going
00:02:02.620 --> 00:02:05.600
to be a universe in which
there is no relativity.
00:02:05.600 --> 00:02:08.530
If you ask me that more
precisely in my office hours,
00:02:08.530 --> 00:02:10.970
I will tell you a
relativistic story.
00:02:10.970 --> 00:02:13.130
But it doesn't violate
anything relativistic.
00:02:13.130 --> 00:02:13.700
At all.
00:02:13.700 --> 00:02:16.690
We'll talk about that-- just
to be a little more detailed--
00:02:16.690 --> 00:02:19.380
that will be a very
important question that we'll
00:02:19.380 --> 00:02:22.140
deal with in the last two
lectures of the course,
00:02:22.140 --> 00:02:24.894
when we come back to Bell's
inequality and locality.
00:02:24.894 --> 00:02:25.560
Other questions?
00:02:29.580 --> 00:02:31.844
OK.
00:02:31.844 --> 00:02:32.760
So, let's get started.
00:02:32.760 --> 00:02:35.010
So, just to review where we are.
00:02:35.010 --> 00:02:38.292
In Quantum Mechanics
according to 804,
00:02:38.292 --> 00:02:40.500
our first pass at the
definition of quantum mechanics
00:02:40.500 --> 00:02:43.290
is that the configuration of
any system-- and in particular,
00:02:43.290 --> 00:02:45.206
think about a single
point particle--
00:02:45.206 --> 00:02:46.580
the configuration
of our particle
00:02:46.580 --> 00:02:49.696
is specified by giving
a wave function, which
00:02:49.696 --> 00:02:51.320
is a function which
may depend on time,
00:02:51.320 --> 00:02:52.445
but a function of position.
00:02:55.310 --> 00:02:57.956
Observables-- and this is
a complete specification
00:02:57.956 --> 00:02:59.080
of the state of the system.
00:02:59.080 --> 00:03:01.540
If I know the wave
function, I neither
00:03:01.540 --> 00:03:05.140
needed nor have access to
any further information
00:03:05.140 --> 00:03:05.870
about the system.
00:03:05.870 --> 00:03:08.650
All the information specifying
the configuration system
00:03:08.650 --> 00:03:13.420
is completely contained
in the wave function.
00:03:13.420 --> 00:03:16.090
Secondly, observables
in quantum mechanics
00:03:16.090 --> 00:03:17.840
are associated with operators.
00:03:17.840 --> 00:03:19.750
Something you can
build an experiment
00:03:19.750 --> 00:03:24.920
to observe or to measure is
associated with an operator.
00:03:24.920 --> 00:03:27.990
And by an operator, I
mean a rule or a map,
00:03:27.990 --> 00:03:30.170
something that tells you
if you give me a function,
00:03:30.170 --> 00:03:32.091
I will give you a
different function back.
00:03:32.091 --> 00:03:32.590
OK?
00:03:32.590 --> 00:03:34.631
An operator is just a
thing which eats a function
00:03:34.631 --> 00:03:37.280
and spits out another function.
00:03:37.280 --> 00:03:39.840
Now, operators-- which I
will denote with a hat,
00:03:39.840 --> 00:03:43.730
as long as I can remember
to do so-- operators
00:03:43.730 --> 00:03:46.230
come-- and in particular, the
kinds of operators we're going
00:03:46.230 --> 00:03:48.521
to care about, linear operators,
which you talked about
00:03:48.521 --> 00:03:53.300
in detail last lecture--
linear operators come endowed
00:03:53.300 --> 00:03:56.220
with a natural set
of special functions
00:03:56.220 --> 00:03:59.930
called Eigenfunctions with
the following property.
00:03:59.930 --> 00:04:03.930
Your operator, acting
on its Eigenfunction,
00:04:03.930 --> 00:04:07.030
gives you that same function
back times a constant.
00:04:09.920 --> 00:04:12.395
So, that's a very
special generically.
00:04:12.395 --> 00:04:14.270
An operator will take
a function and give you
00:04:14.270 --> 00:04:15.630
some other random
function that doesn't
00:04:15.630 --> 00:04:17.200
look all like the
original function.
00:04:17.200 --> 00:04:19.491
It's a very special thing to
give you the same function
00:04:19.491 --> 00:04:21.180
back times a constant.
00:04:21.180 --> 00:04:23.070
So, a useful thing
to think about here
00:04:23.070 --> 00:04:24.940
is just in the case
of vector spaces.
00:04:24.940 --> 00:04:27.730
So, I'm going to consider
the operation corresponding
00:04:27.730 --> 00:04:30.210
to rotation around the
z-axis by a small angle.
00:04:30.210 --> 00:04:31.130
OK?
00:04:31.130 --> 00:04:34.810
So, under rotation around
the z-axis by a small angle,
00:04:34.810 --> 00:04:37.287
I take an arbitrary vector
to some other stupid vector.
00:04:37.287 --> 00:04:39.370
Which vector is completely
determined by the rule?
00:04:39.370 --> 00:04:40.970
I rotate by a small
amount, right?
00:04:40.970 --> 00:04:43.110
I take this vector and
it gives me this one.
00:04:43.110 --> 00:04:45.750
I take that vector,
it gives me this one.
00:04:45.750 --> 00:04:46.970
Everyone agree with that?
00:04:46.970 --> 00:04:50.350
What are the Eigenvectors
of the rotation
00:04:50.350 --> 00:04:52.006
by a small angle
around the z-axis?
00:04:52.006 --> 00:04:53.470
AUDIENCE: [INAUDIBLE]
00:04:53.470 --> 00:04:55.520
PROFESSOR: Yeah, it's
got to be a vector that
00:04:55.520 --> 00:04:57.040
doesn't change its direction.
00:04:57.040 --> 00:04:58.720
It just changes by magnitude.
00:04:58.720 --> 00:05:00.660
So there's one, right?
00:05:00.660 --> 00:05:01.270
I rotate.
00:05:01.270 --> 00:05:02.897
And what's its Eigenvalue?
00:05:02.897 --> 00:05:03.480
AUDIENCE: One.
00:05:03.480 --> 00:05:05.490
PROFESSOR: One, because
nothing changed, right?
00:05:05.490 --> 00:05:07.480
Now, let's consider the
following operation.
00:05:07.480 --> 00:05:10.640
Rotate by small angle
and double its length.
00:05:10.640 --> 00:05:12.020
OK, that's a different operator.
00:05:12.020 --> 00:05:13.430
I rotate and I
double the length.
00:05:13.430 --> 00:05:15.550
I rotate and I
double the length.
00:05:15.550 --> 00:05:17.930
I rotate and I
double the length.
00:05:17.930 --> 00:05:20.360
Yeah, so what's the Eigenvalue
under that operator?
00:05:20.360 --> 00:05:21.185
AUDIENCE: Two.
00:05:21.185 --> 00:05:21.810
PROFESSOR: Two.
00:05:21.810 --> 00:05:22.650
Right, exactly.
00:05:22.650 --> 00:05:25.482
So these are a very
special set of functions.
00:05:25.482 --> 00:05:27.690
This is the same idea, but
instead of having vectors,
00:05:27.690 --> 00:05:29.340
we have functions.
00:05:29.340 --> 00:05:30.722
Questions?
00:05:30.722 --> 00:05:32.580
I thought I saw a hand pop up.
00:05:32.580 --> 00:05:33.080
No?
00:05:33.080 --> 00:05:35.800
OK, cool.
00:05:35.800 --> 00:05:37.580
Third, superposition.
00:05:37.580 --> 00:05:41.620
Given any two viable
wave functions
00:05:41.620 --> 00:05:43.120
that could describe
our system, that
00:05:43.120 --> 00:05:46.620
could specify states or
configurations of our system,
00:05:46.620 --> 00:05:50.130
an arbitrary superposition of
them-- arbitrary linear sum--
00:05:50.130 --> 00:05:52.900
could also be a valid
physical configuration.
00:05:52.900 --> 00:05:55.172
There is also a
state corresponding
00:05:55.172 --> 00:05:56.380
to being in an arbitrary sum.
00:05:56.380 --> 00:05:59.230
For example, if we know that
the electron could be black
00:05:59.230 --> 00:06:00.980
and it could be
white, it could also
00:06:00.980 --> 00:06:03.810
be in an arbitrary superposition
of being black and white.
00:06:03.810 --> 00:06:06.975
And that is a statement in
which the electron is not black.
00:06:06.975 --> 00:06:08.490
The electron is not white.
00:06:08.490 --> 00:06:10.530
It is in the
superposition of the two.
00:06:10.530 --> 00:06:13.040
It does not have
a definite color.
00:06:13.040 --> 00:06:15.680
And that is exactly
the configuration
00:06:15.680 --> 00:06:19.740
we found inside our apparatus
in the first lecture.
00:06:19.740 --> 00:06:20.800
Yeah.
00:06:20.800 --> 00:06:23.420
AUDIENCE: Are those Phi-A
arbitrary functions,
00:06:23.420 --> 00:06:25.170
or are they supposed
to be Eigenfunctions?
00:06:25.170 --> 00:06:25.670
PROFESSOR: Excellent.
00:06:25.670 --> 00:06:27.600
So, in general the
superposition thank you.
00:06:27.600 --> 00:06:28.725
It's an excellent question.
00:06:28.725 --> 00:06:31.082
The question was are these
Phi-As arbitrary functions,
00:06:31.082 --> 00:06:32.540
or are they specific
Eigenfunctions
00:06:32.540 --> 00:06:33.704
of some operator?
00:06:33.704 --> 00:06:35.370
So, the superposition
principle actually
00:06:35.370 --> 00:06:36.453
says a very general thing.
00:06:36.453 --> 00:06:39.860
It says, given any two
viable wave functions,
00:06:39.860 --> 00:06:42.630
an arbitrary sum, an
arbitrary linear combination,
00:06:42.630 --> 00:06:43.962
is also a viable wave function.
00:06:43.962 --> 00:06:46.170
But here I want to mark
something slightly different.
00:06:46.170 --> 00:06:49.220
And this is why I chose
the notation I did.
00:06:49.220 --> 00:06:51.310
Given an operator
A, it comes endowed
00:06:51.310 --> 00:06:54.870
with a special set of functions,
its Eigenfunctions, right?
00:06:54.870 --> 00:06:57.410
We saw the last time.
00:06:57.410 --> 00:07:00.480
And I claimed the following.
00:07:00.480 --> 00:07:03.050
Beyond just the usual
superposition principle,
00:07:03.050 --> 00:07:05.670
the set of Eigenfunctions
of operators
00:07:05.670 --> 00:07:07.810
corresponding to physical
observables-- so, pick
00:07:07.810 --> 00:07:10.460
your observable, like momentum.
00:07:10.460 --> 00:07:12.050
That corresponds to an operator.
00:07:12.050 --> 00:07:14.632
Consider the
Eigenfunctions of momentum.
00:07:14.632 --> 00:07:15.840
Those we know what those are.
00:07:15.840 --> 00:07:17.970
They're plane waves with
definite wavelength,
00:07:17.970 --> 00:07:18.803
right? u to the ikx.
00:07:21.678 --> 00:07:25.240
Any function can be
expressed as a superposition
00:07:25.240 --> 00:07:29.170
of those Eigenfunctions of
your physical observable.
00:07:29.170 --> 00:07:32.280
We'll go over this in
more detail in a minute.
00:07:32.280 --> 00:07:35.880
But here I want to emphasize
that the Eigenfunctions have
00:07:35.880 --> 00:07:38.461
a special property that-- for
observables, for operators
00:07:38.461 --> 00:07:40.710
corresponding to observables--
the Eigenfunctions form
00:07:40.710 --> 00:07:42.160
a basis.
00:07:42.160 --> 00:07:46.320
Any function can be expanded
as some linear combination
00:07:46.320 --> 00:07:48.660
of these basis functions,
the classic example
00:07:48.660 --> 00:07:51.070
being the Fourier expansion.
00:07:51.070 --> 00:07:53.370
Any function, any
periodic function,
00:07:53.370 --> 00:07:55.890
can be expanded as a sum
of sines and cosines,
00:07:55.890 --> 00:07:57.850
and any function
on the real line
00:07:57.850 --> 00:08:02.010
can be expanded as a sum of
exponentials, e to the ikx.
00:08:02.010 --> 00:08:03.210
This is the same statement.
00:08:03.210 --> 00:08:05.270
The Eigenfunctions
of momentum are what?
00:08:05.270 --> 00:08:07.230
e to the ikx.
00:08:07.230 --> 00:08:09.320
So, this is the same that
an arbitrary function--
00:08:09.320 --> 00:08:11.304
when the observable
is the momentum,
00:08:11.304 --> 00:08:13.470
this is the statement that
an arbitrary function can
00:08:13.470 --> 00:08:17.550
be expanded as a superposition,
or a sum of exponentials,
00:08:17.550 --> 00:08:19.480
and that's the Fourier theorem.
00:08:19.480 --> 00:08:20.360
Cool?
00:08:20.360 --> 00:08:21.806
Was there a question?
00:08:21.806 --> 00:08:22.770
AUDIENCE: [INAUDIBLE]
00:08:22.770 --> 00:08:24.920
PROFESSOR: OK, good.
00:08:24.920 --> 00:08:28.890
Other questions on these points?
00:08:28.890 --> 00:08:31.800
So, these should not yet be
trivial and obvious to you.
00:08:31.800 --> 00:08:34.690
If they are, then that's
great, but if they're not,
00:08:34.690 --> 00:08:36.440
we're going to be
working through examples
00:08:36.440 --> 00:08:39.289
for the next several
lectures and problem sets.
00:08:39.289 --> 00:08:43.010
The point now is to give you
a grounding on which to stand.
00:08:43.010 --> 00:08:44.250
Fourth postulate.
00:08:44.250 --> 00:08:47.524
What these expansion
coefficients mean.
00:08:47.524 --> 00:08:48.940
And this is also
an interpretation
00:08:48.940 --> 00:08:50.439
of the meaning of
the wave function.
00:08:50.439 --> 00:08:52.050
What these expansion
coefficients mean
00:08:52.050 --> 00:08:55.550
is that the probability that
I measure the observable
00:08:55.550 --> 00:08:58.810
to be a particular Eigenvalue
is the norm squared
00:08:58.810 --> 00:09:00.635
of the expansion coefficient.
00:09:00.635 --> 00:09:01.910
OK?
00:09:01.910 --> 00:09:03.510
So, I tell you that
any function can
00:09:03.510 --> 00:09:06.360
be expanded as a superposition
of plane waves-- waves
00:09:06.360 --> 00:09:09.082
with definite momentum--
with some coefficients.
00:09:09.082 --> 00:09:11.040
And those coefficients
depend on which function
00:09:11.040 --> 00:09:12.210
I'm talking about.
00:09:12.210 --> 00:09:14.442
What these coefficients
tell me is the probability
00:09:14.442 --> 00:09:16.650
that I will measure the
momentum to be the associated
00:09:16.650 --> 00:09:18.655
value, the Eigenvalue.
00:09:18.655 --> 00:09:19.290
OK?
00:09:19.290 --> 00:09:21.164
Take that coefficient,
take its norm squared,
00:09:21.164 --> 00:09:23.790
that gives me the probability.
00:09:23.790 --> 00:09:26.902
How do we compute these
expansion coefficients?
00:09:26.902 --> 00:09:29.110
I think Barton didn't
introduce to you this notation,
00:09:29.110 --> 00:09:30.402
but he certainly told you this.
00:09:30.402 --> 00:09:32.943
So let me introduce to you this
notation which I particularly
00:09:32.943 --> 00:09:33.780
like.
00:09:33.780 --> 00:09:36.460
We can extract the
expansion coefficient
00:09:36.460 --> 00:09:42.030
if we know the wave function
by taking this integral,
00:09:42.030 --> 00:09:43.690
taking the wave
function, multiplying
00:09:43.690 --> 00:09:46.859
by the complex conjugate of
the associated Eigenfunction,
00:09:46.859 --> 00:09:47.650
doing the integral.
00:09:47.650 --> 00:09:52.630
And that notation is this
round brackets with Phi A
00:09:52.630 --> 00:09:56.880
and Psi is my notation
for this integral.
00:09:59.390 --> 00:10:02.200
And again, we'll still see
this in more detail later on.
00:10:02.200 --> 00:10:04.650
And finally we have
collapse, the statement that,
00:10:04.650 --> 00:10:08.630
if we go about measuring
some observable A,
00:10:08.630 --> 00:10:13.100
then we will always, always
observe precisely one
00:10:13.100 --> 00:10:15.140
of the Eigenvalues
of that operator.
00:10:15.140 --> 00:10:17.360
We will never measure
anything else.
00:10:17.360 --> 00:10:20.110
If the Eigenvalues are one,
two, three, four, and five,
00:10:20.110 --> 00:10:24.380
you will never measure
half, 13 halves.
00:10:24.380 --> 00:10:27.280
You will always
measure an Eigenvalue.
00:10:27.280 --> 00:10:30.060
And upon measuring
that Eigenvalue,
00:10:30.060 --> 00:10:33.720
you can be confident that that's
the actual value of the system.
00:10:33.720 --> 00:10:36.180
I observe that it's
a white electron,
00:10:36.180 --> 00:10:39.950
then it will remain white if I
subsequently measure its color.
00:10:39.950 --> 00:10:43.230
What that's telling you is
it's no longer a superposition
00:10:43.230 --> 00:10:45.570
of white and black,
but it's wave function
00:10:45.570 --> 00:10:48.800
is that corresponding
to a definite value
00:10:48.800 --> 00:10:50.790
of the observable.
00:10:50.790 --> 00:10:53.090
So, somehow the process
of measurement-- and this
00:10:53.090 --> 00:10:56.740
is a disturbing statement, to
which we'll return-- somehow
00:10:56.740 --> 00:10:59.780
the process of measuring
the observable changes
00:10:59.780 --> 00:11:02.980
the wave function from our
arbitrary superposition
00:11:02.980 --> 00:11:06.937
to a specific Eigenfunction,
one particular Eigenfunction
00:11:06.937 --> 00:11:08.270
of the operator we're measuring.
00:11:10.972 --> 00:11:13.180
And this is called the
collapse of the wave function.
00:11:13.180 --> 00:11:15.060
It collapses from
being a superposition
00:11:15.060 --> 00:11:18.050
over possible states to
being in a definite state
00:11:18.050 --> 00:11:18.900
upon measurement.
00:11:18.900 --> 00:11:20.680
And the definite
state is that state
00:11:20.680 --> 00:11:25.020
corresponding to the value
we observed or measured.
00:11:25.020 --> 00:11:25.727
Yeah.
00:11:25.727 --> 00:11:27.685
AUDIENCE: So, when the
wave function collapses,
00:11:27.685 --> 00:11:29.790
does it instantaneously not
become a function of time
00:11:29.790 --> 00:11:30.290
anymore?
00:11:30.290 --> 00:11:32.510
Because originally
we had Psi of (x,t).
00:11:32.510 --> 00:11:34.468
PROFESSOR: Yeah, that's
a really good question.
00:11:34.468 --> 00:11:38.234
So I wrote this only
in terms of position,
00:11:38.234 --> 00:11:39.650
but I should more
precisely write.
00:11:39.650 --> 00:11:42.390
So, the question was, does this
happen instantaneously, or more
00:11:42.390 --> 00:11:45.250
precisely, does it cease
to be a function of time?
00:11:45.250 --> 00:11:45.750
Thank you.
00:11:45.750 --> 00:11:46.750
It's very good question.
00:11:46.750 --> 00:11:48.833
So, no, it doesn't cease
to be a function of time.
00:11:48.833 --> 00:11:50.460
It just says that
Psi at x-- what
00:11:50.460 --> 00:11:51.960
you know upon doing
this measurement
00:11:51.960 --> 00:11:55.360
is that Psi, as a function
of x, at the time which I'll
00:11:55.360 --> 00:11:58.460
call T star, at what
you've done the measurement
00:11:58.460 --> 00:12:00.067
is equal to this wave function.
00:12:00.067 --> 00:12:02.400
And so that leaves us with
the following question, which
00:12:02.400 --> 00:12:04.608
is another way of asking
the question you just asked.
00:12:04.608 --> 00:12:05.810
What happens next?
00:12:05.810 --> 00:12:08.550
How does the system
evolve subsequently?
00:12:08.550 --> 00:12:10.450
And at the very end
of the last lecture,
00:12:10.450 --> 00:12:13.720
we answered that--
or rather, Barton
00:12:13.720 --> 00:12:16.850
answered that-- by introducing
the Schrodinger equation.
00:12:16.850 --> 00:12:19.760
And the Schrodinger equation,
we don't derive, we just posit.
00:12:19.760 --> 00:12:21.650
Much like Newton
posits f equals ma.
00:12:21.650 --> 00:12:25.110
You can motivate it,
but you can't derive it.
00:12:25.110 --> 00:12:28.900
It's just what we mean by
the quantum mechanical model.
00:12:28.900 --> 00:12:32.550
And Schrodinger's equation
says, given a wave function,
00:12:32.550 --> 00:12:34.510
I can determine the time
derivative, the time
00:12:34.510 --> 00:12:36.530
rate of changes of
that wave function,
00:12:36.530 --> 00:12:38.950
and determine its
time evolution,
00:12:38.950 --> 00:12:42.070
and its time derivative,
its slope-- its velocity,
00:12:42.070 --> 00:12:48.090
if you will-- is one upon I h
bar, the energy operator acting
00:12:48.090 --> 00:12:49.460
on that wave function.
00:12:49.460 --> 00:12:52.090
So, suppose we measure
that our observable capital
00:12:52.090 --> 00:12:53.580
A takes the value
of little a, one
00:12:53.580 --> 00:12:55.870
of the Eigenvalues of
the associated operators.
00:12:55.870 --> 00:12:58.550
Suppose we measure
that A equals little a
00:12:58.550 --> 00:13:00.602
at some particular
moment T start.
00:13:00.602 --> 00:13:02.060
Then we know that
the wave function
00:13:02.060 --> 00:13:04.735
is Psi of x at that
moment in time.
00:13:04.735 --> 00:13:06.360
We can then compute
the time derivative
00:13:06.360 --> 00:13:08.151
of the wave function
at that moment in time
00:13:08.151 --> 00:13:11.150
by acting on this wave
function with the operator e
00:13:11.150 --> 00:13:12.690
hat, the energy operator.
00:13:12.690 --> 00:13:15.190
And we can then integrate that
differential equation forward
00:13:15.190 --> 00:13:19.100
in time and determine how
the wave function evolves.
00:13:19.100 --> 00:13:21.370
The point of today's
lecture is going
00:13:21.370 --> 00:13:24.980
to be to study how time
evolution works in quantum
00:13:24.980 --> 00:13:27.950
mechanics, and to look
at some basic examples
00:13:27.950 --> 00:13:31.060
and basic strategies for
solving the time evolution
00:13:31.060 --> 00:13:33.540
problem in quantum mechanics.
00:13:33.540 --> 00:13:35.590
One of the great surprises
in quantum mechanics--
00:13:35.590 --> 00:13:38.048
hold on just one sec-- one of
the real surprises in quantum
00:13:38.048 --> 00:13:39.770
mechanics is that
time evolution is
00:13:39.770 --> 00:13:43.070
in a very specific sense
trivial in quantum mechanics.
00:13:43.070 --> 00:13:44.680
It's preposterously simple.
00:13:44.680 --> 00:13:49.010
In particular, time evolution is
governed by a linear equation.
00:13:49.010 --> 00:13:52.760
How many of you have studied
a classical mechanical system
00:13:52.760 --> 00:13:57.181
where the time evolution is
governed by a linear equation?
00:13:57.181 --> 00:13:57.680
Right.
00:13:57.680 --> 00:13:58.305
OK, all of you.
00:13:58.305 --> 00:13:59.740
The harmonic oscillator.
00:13:59.740 --> 00:14:02.600
But otherwise, not at all.
00:14:02.600 --> 00:14:04.890
Otherwise, the equations
in classical mechanics
00:14:04.890 --> 00:14:06.860
are generically
highly nonlinear.
00:14:06.860 --> 00:14:09.762
The time rate of change
of position of a particle
00:14:09.762 --> 00:14:12.095
is the gradient of the force,
and the force is generally
00:14:12.095 --> 00:14:13.940
some complicated
function of position.
00:14:13.940 --> 00:14:16.050
You've got some capacitors
over here, and maybe
00:14:16.050 --> 00:14:16.980
some magnetic field.
00:14:16.980 --> 00:14:18.062
It's very nonlinear.
00:14:18.062 --> 00:14:19.770
Evolution in quantum
mechanics is linear,
00:14:19.770 --> 00:14:21.921
and this is going
to be surprising.
00:14:21.921 --> 00:14:24.170
It's going to lead to some
surprising simplifications.
00:14:24.170 --> 00:14:25.350
And we'll turn
back to that, but I
00:14:25.350 --> 00:14:27.319
want to put that your
mind like a little hook,
00:14:27.319 --> 00:14:28.860
that that's something
you should mark
00:14:28.860 --> 00:14:30.724
on to as different from
classical mechanics.
00:14:30.724 --> 00:14:31.890
And we'll come back to that.
00:14:31.890 --> 00:14:32.380
Yeah.
00:14:32.380 --> 00:14:34.005
AUDIENCE: If a particle
is continuously
00:14:34.005 --> 00:14:35.600
observed as a not
evolving particle?
00:14:35.600 --> 00:14:37.183
PROFESSOR: That's
an awesome question.
00:14:37.183 --> 00:14:39.780
The question is, look,
imagine I observe--
00:14:39.780 --> 00:14:42.030
I'm going to paraphrase--
imagine I observe a particle
00:14:42.030 --> 00:14:43.320
and I observe that it's here.
00:14:43.320 --> 00:14:43.820
OK?
00:14:43.820 --> 00:14:45.710
Subsequently, its wave function
will evolve in some way--
00:14:45.710 --> 00:14:47.995
and we'll actually study that
later today-- its wave function
00:14:47.995 --> 00:14:49.780
will evolve in some
way, and it'll change.
00:14:49.780 --> 00:14:51.780
It won't necessarily be
definitely here anymore.
00:14:51.780 --> 00:14:54.640
But if I just keep measuring it
over and over and over again,
00:14:54.640 --> 00:14:56.348
I just keep measure
it to be right there.
00:14:56.348 --> 00:14:58.290
It can't possibly evolve.
00:14:58.290 --> 00:15:00.180
And that's actually
true, and it's
00:15:00.180 --> 00:15:02.210
called the Quantum Zeno problem.
00:15:02.210 --> 00:15:04.660
So, it's the observation
that if you continuously
00:15:04.660 --> 00:15:06.480
measure a thing,
you can't possibly
00:15:06.480 --> 00:15:09.200
have its wave function
evolve significantly.
00:15:09.200 --> 00:15:11.780
And not only is it a cute
idea, but it's something people
00:15:11.780 --> 00:15:13.340
do in the laboratory.
00:15:13.340 --> 00:15:15.850
So, Martin-- well, OK.
00:15:15.850 --> 00:15:18.040
People do it in a
laboratory and it's cool.
00:15:18.040 --> 00:15:19.240
Come ask me and I'll tell
you about the experiments.
00:15:19.240 --> 00:15:19.580
Other questions?
00:15:19.580 --> 00:15:20.040
There were a bunch.
00:15:20.040 --> 00:15:20.450
Yeah.
00:15:20.450 --> 00:15:22.334
AUDIENCE: So after you measure,
the Schrodinger equation
00:15:22.334 --> 00:15:24.540
also gives you the
evolution backwards in time?
00:15:24.540 --> 00:15:25.810
PROFESSOR: Oh, crap!
00:15:25.810 --> 00:15:26.390
Yes.
00:15:26.390 --> 00:15:27.590
That's such a good question.
00:15:27.590 --> 00:15:28.160
OK.
00:15:28.160 --> 00:15:29.670
I hate it when people
ask that at this point,
00:15:29.670 --> 00:15:31.211
because I had to
then say more words.
00:15:31.211 --> 00:15:32.920
That's a very good question.
00:15:32.920 --> 00:15:34.390
So the question goes like this.
00:15:34.390 --> 00:15:37.300
So this was going to be a
punchline later on in the
00:15:37.300 --> 00:15:41.910
in the lecture but you stole
my thunder, so that's awesome.
00:15:41.910 --> 00:15:43.520
So, here's the deal.
00:15:43.520 --> 00:15:46.960
We have a rule for time
evolution of a wave function,
00:15:46.960 --> 00:15:48.840
and it has some
lovely properties.
00:15:48.840 --> 00:15:55.550
In particular-- let me talk
through this-- in particular,
00:15:55.550 --> 00:15:59.167
this equation is linear.
00:15:59.167 --> 00:16:00.500
So what properties does it have?
00:16:00.500 --> 00:16:01.680
Let me just-- I'm
going to come back
00:16:01.680 --> 00:16:02.800
to your question
in just a second,
00:16:02.800 --> 00:16:05.280
but first I want to set it up
so we have a little more meat
00:16:05.280 --> 00:16:07.460
to answer your
question precisely.
00:16:07.460 --> 00:16:09.730
So we note some properties
of this equation, this time
00:16:09.730 --> 00:16:11.860
evolution equation.
00:16:11.860 --> 00:16:15.330
The first is that it's
a linear equation.
00:16:15.330 --> 00:16:17.660
The derivative of
a sum of function
00:16:17.660 --> 00:16:19.224
is a sum of the derivatives.
00:16:19.224 --> 00:16:20.890
The energy operator's
a linear operator,
00:16:20.890 --> 00:16:23.370
meaning the energy operator
acting on a sum of functions
00:16:23.370 --> 00:16:26.370
is a sum of the energy operator
acting on each function.
00:16:26.370 --> 00:16:28.620
You guys studied linear
operators in your problem set,
00:16:28.620 --> 00:16:29.786
right?
00:16:29.786 --> 00:16:30.660
So, these are linear.
00:16:30.660 --> 00:16:34.780
What that tells you
is if Psi 1 of x and t
00:16:34.780 --> 00:16:38.580
solves the Schrodinger equation,
and Psi 2 of x and t-- two
00:16:38.580 --> 00:16:40.870
different functions of
position in time-- both
00:16:40.870 --> 00:16:46.000
solve the Schrodinger equation,
then any combination of them--
00:16:46.000 --> 00:16:51.630
alpha Psi 1 plus Beta Psi 2--
also solves-- which I will call
00:16:51.630 --> 00:16:56.130
Psi, and I'll make it
a capital Psi for fun--
00:16:56.130 --> 00:17:01.040
solves the Schrodinger
equation automatically.
00:17:01.040 --> 00:17:03.329
Given two solutions of
the Schrodinger equation,
00:17:03.329 --> 00:17:05.579
a superposition of them--
an arbitrary superposition--
00:17:05.579 --> 00:17:07.660
also solves the
Schrodinger equation.
00:17:07.660 --> 00:17:10.390
This is linearity.
00:17:10.390 --> 00:17:12.500
Cool?
00:17:12.500 --> 00:17:14.450
Next property.
00:17:14.450 --> 00:17:15.869
It's unitary.
00:17:15.869 --> 00:17:18.800
What I mean by unitary is this.
00:17:18.800 --> 00:17:21.089
It concerns probability.
00:17:21.089 --> 00:17:23.579
And you'll give a
precise derivation
00:17:23.579 --> 00:17:25.859
of what I mean by
unitary and you'll
00:17:25.859 --> 00:17:27.859
demonstrate that, in fact,
Schrodinger evolution
00:17:27.859 --> 00:17:30.075
is unitary on your
next problem set.
00:17:30.075 --> 00:17:31.570
It's not on the current one.
00:17:31.570 --> 00:17:33.995
But what I mean by unitary is
that conserves probability.
00:17:33.995 --> 00:17:35.160
Whoops, that's an o.
00:17:35.160 --> 00:17:39.120
Conserves probability.
00:17:39.120 --> 00:17:42.180
IE, if there's an
electron here, or if we
00:17:42.180 --> 00:17:43.910
have an object, a
piece of chalk-- which
00:17:43.910 --> 00:17:45.365
I'm treating as a quantum
mechanical point particle--
00:17:45.365 --> 00:17:47.430
it's described by
the wave function.
00:17:47.430 --> 00:17:50.530
The integral, the probability
distribution over all the
00:17:50.530 --> 00:17:52.550
places it could possibly
be had better be one,
00:17:52.550 --> 00:17:57.410
because it had better be
somewhere with probability one.
00:17:57.410 --> 00:18:00.230
That had better
not change in time.
00:18:00.230 --> 00:18:02.705
If I solve the Schrodinger
equation evolve the system
00:18:02.705 --> 00:18:05.007
forward for half an
hour, it had better not
00:18:05.007 --> 00:18:06.590
be the case that the
total probability
00:18:06.590 --> 00:18:08.610
of finding the
particle is one half.
00:18:08.610 --> 00:18:10.570
That means things
disappear in the universe.
00:18:10.570 --> 00:18:12.240
And much as my socks
would seem to be
00:18:12.240 --> 00:18:15.690
a counter example of that,
things don't disappear, right?
00:18:15.690 --> 00:18:17.020
It just doesn't happen.
00:18:17.020 --> 00:18:20.180
So, quantum mechanics
is demonstrably-- well,
00:18:20.180 --> 00:18:22.330
quantum mechanics
is unitary, and this
00:18:22.330 --> 00:18:26.050
is a demonstrably good
description of the real world.
00:18:26.050 --> 00:18:28.750
It fits all the observations
we've ever made.
00:18:28.750 --> 00:18:32.050
No one's ever discovered
an experimental violation
00:18:32.050 --> 00:18:33.850
of unitarity of
quantum mechanics.
00:18:33.850 --> 00:18:37.490
I will note that there is
a theoretical violation
00:18:37.490 --> 00:18:39.710
of unitarity in quantum
mechanics, which
00:18:39.710 --> 00:18:40.750
is dear to my heart.
00:18:40.750 --> 00:18:43.670
It's called the Hawking Effect,
and it's an observation that,
00:18:43.670 --> 00:18:47.520
due quantum mechanics, black
holes in general relativity--
00:18:47.520 --> 00:18:52.520
places from which light
cannot escape-- evaporate.
00:18:52.520 --> 00:18:54.500
So you throw stuff and
you form a black hole.
00:18:54.500 --> 00:18:55.310
It's got a horizon.
00:18:55.310 --> 00:18:56.684
If you fall through
that horizon,
00:18:56.684 --> 00:18:57.730
we never see you again.
00:18:57.730 --> 00:19:00.290
Surprisingly, a black hole's
a hot object like an iron,
00:19:00.290 --> 00:19:02.667
and it sends off radiation.
00:19:02.667 --> 00:19:04.750
As it sends off radiation,
it's losing its energy.
00:19:04.750 --> 00:19:05.760
It's shrinking.
00:19:05.760 --> 00:19:07.760
And eventually it will,
like the classical atom,
00:19:07.760 --> 00:19:09.160
collapse to nothing.
00:19:09.160 --> 00:19:10.660
There's a quibble
going on right now
00:19:10.660 --> 00:19:12.040
over whether it really
collapses to nothing,
00:19:12.040 --> 00:19:13.655
or whether there's
a little granule
00:19:13.655 --> 00:19:15.577
nugget of quantum goodness.
00:19:15.577 --> 00:19:16.710
[LAUGHTER]
00:19:16.710 --> 00:19:18.220
We argue about this.
00:19:18.220 --> 00:19:19.930
We get paid to argue about this.
00:19:19.930 --> 00:19:20.430
[LAUGHTER]
00:19:20.430 --> 00:19:22.440
So, but here's the funny thing.
00:19:22.440 --> 00:19:26.107
If you threw in a dictionary and
then the black hole evaporates,
00:19:26.107 --> 00:19:28.440
where did the information
about what made the black hole
00:19:28.440 --> 00:19:30.660
go if it's just thermal
radiation coming out?
00:19:30.660 --> 00:19:32.480
So, this is a
classic calculation,
00:19:32.480 --> 00:19:34.350
which to a theorist says, ah ha!
00:19:34.350 --> 00:19:35.810
Maybe unitarity isn't conserved.
00:19:35.810 --> 00:19:37.960
But, look.
00:19:37.960 --> 00:19:39.610
Black holes, theorists.
00:19:39.610 --> 00:19:41.680
There's no
experimental violation
00:19:41.680 --> 00:19:43.420
of unitarity anywhere.
00:19:43.420 --> 00:19:45.390
And if anyone ever did
find such a violation,
00:19:45.390 --> 00:19:48.210
it would shatter the basic
tenets of quantum mechanics,
00:19:48.210 --> 00:19:50.450
in particular the
Schrodinger equation.
00:19:50.450 --> 00:19:52.925
So that's something we would
love to see but never have.
00:19:52.925 --> 00:19:54.300
It depends on your
point of view.
00:19:54.300 --> 00:19:55.469
You might hate to see it.
00:19:55.469 --> 00:19:57.010
And the third-- and
this is, I think,
00:19:57.010 --> 00:20:00.904
the most important-- is that
the Schrodinger evolution, this
00:20:00.904 --> 00:20:02.730
is a time derivative.
00:20:02.730 --> 00:20:04.150
It's a differential equation.
00:20:04.150 --> 00:20:05.567
If you know the
initial condition,
00:20:05.567 --> 00:20:07.941
and you know the derivative,
you can integrate it forward
00:20:07.941 --> 00:20:08.470
in time.
00:20:08.470 --> 00:20:11.836
And they're existence and
uniqueness theorems for this.
00:20:11.836 --> 00:20:14.740
The system is deterministic.
00:20:14.740 --> 00:20:19.600
What that means
is that if I have
00:20:19.600 --> 00:20:23.795
complete knowledge of the
system at some moment in time,
00:20:23.795 --> 00:20:25.920
if I know the wave function
at some moment in time,
00:20:25.920 --> 00:20:27.720
I can determine
unambiguously the wave
00:20:27.720 --> 00:20:29.840
function in all subsequent
moments of time.
00:20:29.840 --> 00:20:30.720
Unambiguously.
00:20:30.720 --> 00:20:34.870
There's no probability, there's
no likelihood, it's determined.
00:20:34.870 --> 00:20:36.150
Completely determined.
00:20:36.150 --> 00:20:39.680
Given full knowledge now, I
will have full knowledge later.
00:20:39.680 --> 00:20:41.450
Does everyone agree
that this equation
00:20:41.450 --> 00:20:45.380
is a deterministic
equation in that sense?
00:20:45.380 --> 00:20:46.110
Question.
00:20:46.110 --> 00:20:47.193
AUDIENCE: It's also local?
00:20:47.193 --> 00:20:48.510
PROFESSOR: It's all-- well, OK.
00:20:48.510 --> 00:20:51.350
This one happens
to be-- you need
00:20:51.350 --> 00:20:54.250
to give me a better
definition of local.
00:20:54.250 --> 00:20:56.926
So give me a definition
of local that you want.
00:20:56.926 --> 00:20:59.854
AUDIENCE: The time evolution
of the wave function
00:20:59.854 --> 00:21:03.270
happens only at a
point that depends only
00:21:03.270 --> 00:21:06.055
on the value of the derivatives
of the wave function
00:21:06.055 --> 00:21:07.680
and its potential
energy at that point.
00:21:07.680 --> 00:21:08.240
PROFESSOR: No.
00:21:08.240 --> 00:21:09.420
Unfortunately,
that's not the case.
00:21:09.420 --> 00:21:11.050
We'll see counter
examples of that.
00:21:11.050 --> 00:21:13.165
The wave function--
the energy operator.
00:21:13.165 --> 00:21:15.040
So let's think about
what this equation says.
00:21:15.040 --> 00:21:16.748
What this says is the
time rate of change
00:21:16.748 --> 00:21:19.110
of the value of the wave
function at some position
00:21:19.110 --> 00:21:23.875
and some moment in time is the
energy operator acting on Psi
00:21:23.875 --> 00:21:24.927
at x of t.
00:21:24.927 --> 00:21:27.010
But I didn't tell you what
the energy operator is.
00:21:27.010 --> 00:21:29.057
The energy operator
just has to be linear.
00:21:29.057 --> 00:21:31.390
But it doesn't have to be--
it could know about the wave
00:21:31.390 --> 00:21:32.223
function everywhere.
00:21:32.223 --> 00:21:35.850
The energy operator's a map
that takes the wave function
00:21:35.850 --> 00:21:38.410
and tells you what
it should be later.
00:21:38.410 --> 00:21:41.550
And so, at this level there's
nothing about locality built
00:21:41.550 --> 00:21:43.110
in to the energy
operator, and we'll
00:21:43.110 --> 00:21:45.530
see just how bad that can be.
00:21:45.530 --> 00:21:47.100
So, this is related
to your question
00:21:47.100 --> 00:21:50.760
about special relativity, and
so those are deeply intertwined.
00:21:50.760 --> 00:21:52.900
We don't have that
property here yet.
00:21:52.900 --> 00:21:55.259
But keep that in your
mind, and ask questions
00:21:55.259 --> 00:21:56.300
when it seems to come up.
00:21:56.300 --> 00:21:57.870
Because it's a very,
very, very important
00:21:57.870 --> 00:21:59.610
question when we talk
about relativity.
00:21:59.610 --> 00:22:00.385
Yeah.
00:22:00.385 --> 00:22:02.385
AUDIENCE: Are postulates
six and three redundant
00:22:02.385 --> 00:22:05.077
if the Schrodinger equation
has superposition in it?
00:22:05.077 --> 00:22:05.660
PROFESSOR: No.
00:22:05.660 --> 00:22:06.624
Excellent question.
00:22:06.624 --> 00:22:07.790
That's a very good question.
00:22:07.790 --> 00:22:10.849
The question is, look, there's
postulate three, which says,
00:22:10.849 --> 00:22:12.890
given any two wave functions
that are viable wave
00:22:12.890 --> 00:22:15.340
functions of the
system, then there's
00:22:15.340 --> 00:22:16.930
another state which
is a viable wave
00:22:16.930 --> 00:22:19.570
function at some moment in time,
which is also a viable wave
00:22:19.570 --> 00:22:21.120
function.
00:22:21.120 --> 00:22:25.711
But number six, the Schrodinger
equation-- or sorry,
00:22:25.711 --> 00:22:27.710
really the linearity
property of the Schrodinger
00:22:27.710 --> 00:22:31.236
equation-- so it needs to be
the case for the Schrodinger
00:22:31.236 --> 00:22:33.360
question, but it says
something slightly different.
00:22:33.360 --> 00:22:42.070
It doesn't just say that any
any plausible or viable wave
00:22:42.070 --> 00:22:45.360
function and another
can be superposed.
00:22:45.360 --> 00:22:48.210
It says that, specifically,
any solution of the Schrodinger
00:22:48.210 --> 00:22:51.150
equation plus any other solution
of the Schrodinger equation
00:22:51.150 --> 00:22:52.760
is again the
Schrodinger operation.
00:22:52.760 --> 00:22:54.690
So, it's a slightly
more specific thing
00:22:54.690 --> 00:22:55.970
than postulate three.
00:22:55.970 --> 00:23:00.180
However, your question is
excellent because could it
00:23:00.180 --> 00:23:02.230
have been that the
Schrodinger evolution didn't
00:23:02.230 --> 00:23:05.680
respect superposition?
00:23:05.680 --> 00:23:07.410
Well, you could imagine
something, sure.
00:23:07.410 --> 00:23:09.160
We could've done a
differ equation, right?
00:23:09.160 --> 00:23:10.110
It might not have been linear.
00:23:10.110 --> 00:23:12.068
We could have had that
Schrodinger equation was
00:23:12.068 --> 00:23:12.970
equal to dt Psi.
00:23:12.970 --> 00:23:14.240
So imagine this equation.
00:23:14.240 --> 00:23:17.230
How do we have blown linearity
while preserving determinism?
00:23:17.230 --> 00:23:25.500
So we could have added plus, I
don't know, PSI squared of x.
00:23:25.500 --> 00:23:27.250
So that would now be
a nonlinear equation.
00:23:27.250 --> 00:23:29.820
It's actually refer to as the
nonlinear Schrodinger equation.
00:23:32.425 --> 00:23:34.050
Well, people mean
many different things
00:23:34.050 --> 00:23:35.410
by the nonlinear
Schrodinger equation,
00:23:35.410 --> 00:23:37.390
but that's a nonlinear
Schrodinger equation.
00:23:37.390 --> 00:23:40.380
So you could certainly
write this down.
00:23:40.380 --> 00:23:43.120
It's not linear.
00:23:43.120 --> 00:23:46.340
Does it violate
the statement three
00:23:46.340 --> 00:23:49.540
that any two states
of the system
00:23:49.540 --> 00:23:52.400
could be superposed to
give another viable state
00:23:52.400 --> 00:23:54.080
at a moment in time?
00:23:54.080 --> 00:23:54.914
No, right?
00:23:54.914 --> 00:23:56.080
It doesn't directly violate.
00:23:56.080 --> 00:23:58.272
It violates the spirit of it.
00:23:58.272 --> 00:23:59.730
And as we'll see
later, it actually
00:23:59.730 --> 00:24:01.090
would cause dramatic problems.
00:24:01.090 --> 00:24:03.896
It's something we don't
usually emphasize-- something
00:24:03.896 --> 00:24:05.770
I don't usually emphasize
in lectures of 804,
00:24:05.770 --> 00:24:07.410
but I will make
a specific effort
00:24:07.410 --> 00:24:11.740
to mark the places where
this would cause disasters.
00:24:11.740 --> 00:24:14.090
But, so this is actually
a logically independent,
00:24:14.090 --> 00:24:16.400
although morally--
and in some sense
00:24:16.400 --> 00:24:19.410
is a technically related point
to the superposition principle
00:24:19.410 --> 00:24:20.710
number three.
00:24:20.710 --> 00:24:21.896
Yeah.
00:24:21.896 --> 00:24:28.175
AUDIENCE: For postulate three,
can that sum be infinite sum?
00:24:28.175 --> 00:24:29.624
PROFESSOR: Absolutely.
00:24:29.624 --> 00:24:31.556
AUDIENCE: Can you
do bad things, then,
00:24:31.556 --> 00:24:33.488
like creating discontinuous
wave functions?
00:24:33.488 --> 00:24:34.454
PROFESSOR: Oh yes.
00:24:34.454 --> 00:24:36.150
Oh, yes you can.
00:24:36.150 --> 00:24:37.090
So here's the thing.
00:24:37.090 --> 00:24:40.310
Look, if you have two functions
and you add them together--
00:24:40.310 --> 00:24:41.950
like two smooth
continuous functions,
00:24:41.950 --> 00:24:43.616
you add them together--
what do you get?
00:24:43.616 --> 00:24:45.970
You get another smooth
continuous function, right?
00:24:45.970 --> 00:24:47.390
Take seven.
00:24:47.390 --> 00:24:49.270
You get another.
00:24:49.270 --> 00:24:51.960
But if you take an
infinite number-- look,
00:24:51.960 --> 00:24:53.230
mathematicians are sneaky.
00:24:53.230 --> 00:24:55.170
There's a reason we keep
them down that hall,
00:24:55.170 --> 00:24:56.043
far away from us.
00:24:56.043 --> 00:24:56.850
[LAUGHTER]
00:24:56.850 --> 00:24:57.690
They're very sneaky.
00:24:57.690 --> 00:25:02.680
And if you give them an infinite
number of continuous functions,
00:25:02.680 --> 00:25:06.035
they'll build for you a
discontinuous function, right?
00:25:06.035 --> 00:25:06.535
Sneaky.
00:25:09.810 --> 00:25:13.170
Does that seem
terribly physical?
00:25:13.170 --> 00:25:13.670
No.
00:25:13.670 --> 00:25:16.003
It's what happens when you
give a mathematician too much
00:25:16.003 --> 00:25:19.010
paper and time, right?
00:25:19.010 --> 00:25:21.760
So, I mean this less
flippantly than I'm saying it,
00:25:21.760 --> 00:25:24.910
but it's worth being a
little flippant here.
00:25:24.910 --> 00:25:27.290
In a physical
setting, we will often
00:25:27.290 --> 00:25:30.050
find that there are
effectively an infinite number
00:25:30.050 --> 00:25:32.010
of possible things
that could happen.
00:25:32.010 --> 00:25:33.970
So, for example in this room,
where is this piece of chalk?
00:25:33.970 --> 00:25:35.430
It's described by a
continuous variable.
00:25:35.430 --> 00:25:37.700
That's an uncountable
infinite number of positions.
00:25:37.700 --> 00:25:38.800
Now, in practice,
you can't really
00:25:38.800 --> 00:25:40.220
build an experiment
that does that,
00:25:40.220 --> 00:25:42.136
but it is in principle
an uncountable infinity
00:25:42.136 --> 00:25:44.100
of possible positions, right?
00:25:44.100 --> 00:25:46.560
You will never get a
discontinuous wave function
00:25:46.560 --> 00:25:47.590
for this guy, because
it would correspond
00:25:47.590 --> 00:25:49.000
to divergent
amounts of momentum,
00:25:49.000 --> 00:25:51.790
as you showed on the
previous problem set.
00:25:51.790 --> 00:25:56.000
So, in general, we will often
be in a situation as physicists
00:25:56.000 --> 00:25:58.690
where there's the
possibility of using
00:25:58.690 --> 00:26:00.640
the machinery-- the
mathematical machinery--
00:26:00.640 --> 00:26:02.790
to create pathological examples.
00:26:02.790 --> 00:26:05.380
And yes, that is a risk.
00:26:05.380 --> 00:26:07.260
But physically it never happens.
00:26:07.260 --> 00:26:09.120
Physically it's
extraordinarily rare
00:26:09.120 --> 00:26:12.910
that such infinite
divergences could matter.
00:26:12.910 --> 00:26:15.080
Now, I'm not saying
that they never do.
00:26:15.080 --> 00:26:18.060
But we're going to be very
carefree and casual in 804
00:26:18.060 --> 00:26:20.910
and just assume that when
problems can arise from,
00:26:20.910 --> 00:26:23.244
say, superposing an infinite
number of smooth functions,
00:26:23.244 --> 00:26:24.826
leading potentially
to discontinuities
00:26:24.826 --> 00:26:27.790
or singularities, that they will
either not happen for us-- not
00:26:27.790 --> 00:26:30.480
be relevant-- or they will
happen because they're
00:26:30.480 --> 00:26:32.310
forced too, so for
physical reasons
00:26:32.310 --> 00:26:34.402
we'll be able to identify.
00:26:34.402 --> 00:26:35.860
So, this is a very
important point.
00:26:35.860 --> 00:26:37.526
We're not proving
mathematical theorems.
00:26:37.526 --> 00:26:39.060
We're not trying to be rigorous.
00:26:39.060 --> 00:26:40.590
To prove a mathematical
theorem you
00:26:40.590 --> 00:26:42.530
have to look at all
the exceptional cases
00:26:42.530 --> 00:26:44.700
and say, those
exceptional cases,
00:26:44.700 --> 00:26:46.600
we can deal with
them mathematically.
00:26:46.600 --> 00:26:48.939
To a physicist, exceptional
cases are exceptional.
00:26:48.939 --> 00:26:49.730
They're irrelevant.
00:26:49.730 --> 00:26:50.480
They don't happen.
00:26:50.480 --> 00:26:51.600
It doesn't matter.
00:26:51.600 --> 00:26:52.126
OK?
00:26:52.126 --> 00:26:53.500
And it doesn't
mean that we don't
00:26:53.500 --> 00:26:55.375
care about the mathematical
precision, right?
00:26:55.375 --> 00:26:57.290
I mean, I publish
papers in math journals,
00:26:57.290 --> 00:26:59.800
so I have a deep love
for these questions.
00:26:59.800 --> 00:27:02.780
But they're not salient for
most of the physical questions
00:27:02.780 --> 00:27:03.580
we care about.
00:27:03.580 --> 00:27:08.440
So, do your best to try not
to let those special cases get
00:27:08.440 --> 00:27:11.022
in the way of your understanding
of the general case.
00:27:11.022 --> 00:27:12.730
I don't want you to
not think about them,
00:27:12.730 --> 00:27:16.700
I just want you not
let them stop you, OK?
00:27:16.700 --> 00:27:17.561
Yeah.
00:27:17.561 --> 00:27:19.590
AUDIENCE: So, in
postulate five, you
00:27:19.590 --> 00:27:21.256
mentioned that
[? functions ?] in effect
00:27:21.256 --> 00:27:24.669
was a experiment that
more or less proves
00:27:24.669 --> 00:27:26.100
this collapse [INAUDIBLE]
00:27:26.100 --> 00:27:30.410
But, so I read that it
is not [? complicit. ?]
00:27:30.410 --> 00:27:33.050
PROFESSOR: Yeah, so as with many
things in quantum mechanics--
00:27:33.050 --> 00:27:34.008
that's a fair question.
00:27:34.008 --> 00:27:37.170
So, let me make a slightly
more general statement
00:27:37.170 --> 00:27:40.540
than answering that
question directly.
00:27:40.540 --> 00:27:50.460
Many things will-- how to
say-- so, we will not prove--
00:27:50.460 --> 00:27:53.150
and experimentally you almost
never prove a positive thing.
00:27:53.150 --> 00:27:56.504
You can show that a prediction
is violated by experiment.
00:27:56.504 --> 00:27:58.420
So there's always going
to be some uncertainty
00:27:58.420 --> 00:27:59.166
in your measurements,
there's always
00:27:59.166 --> 00:28:01.124
going to be some uncertainty
in your arguments.
00:28:01.124 --> 00:28:04.070
However, in the absence
of a compelling alternate
00:28:04.070 --> 00:28:05.680
theoretical
description, you cling
00:28:05.680 --> 00:28:07.930
on to what you've got it as
long as it fits your data,
00:28:07.930 --> 00:28:09.790
and this fits the
data like a champ.
00:28:09.790 --> 00:28:10.290
Right?
00:28:10.290 --> 00:28:11.630
So, does it prove?
00:28:11.630 --> 00:28:12.939
No.
00:28:12.939 --> 00:28:14.480
It fits pretty well,
and nothing else
00:28:14.480 --> 00:28:15.870
comes even within the ballpark.
00:28:15.870 --> 00:28:17.260
And there's no
explicit violation
00:28:17.260 --> 00:28:19.510
that's better than our
experimental uncertainties.
00:28:19.510 --> 00:28:21.590
So, I don't know if
I'd say, well, we
00:28:21.590 --> 00:28:25.550
could prove such a
thing, but it fits.
00:28:25.550 --> 00:28:26.610
And I'm a physicist.
00:28:26.610 --> 00:28:28.400
I'm looking for things that fit.
00:28:28.400 --> 00:28:29.750
I'm not a metaphysicist.
00:28:29.750 --> 00:28:33.320
I'm not trying to give you
some ontological commitment
00:28:33.320 --> 00:28:35.800
about what things are true
and exist in the world, right?
00:28:35.800 --> 00:28:36.610
That's not my job.
00:28:39.910 --> 00:28:42.462
OK.
00:28:42.462 --> 00:28:43.420
So much for our review.
00:28:43.420 --> 00:28:45.010
But let me finally
come back to-- now
00:28:45.010 --> 00:28:46.280
that we've observed
that it's determinist,
00:28:46.280 --> 00:28:47.821
let me come back to
the question that
00:28:47.821 --> 00:28:50.960
was asked a few minutes
ago, which is, look,
00:28:50.960 --> 00:28:54.637
suppose we take
our superposition.
00:28:54.637 --> 00:28:56.970
We evolve it forward for some
time using the Schrodinger
00:28:56.970 --> 00:28:58.180
evolution.
00:28:58.180 --> 00:28:59.805
Notice that it's time reversal.
00:28:59.805 --> 00:29:01.180
If we know it's
time reverted, we
00:29:01.180 --> 00:29:02.430
could run it
backwards just as well
00:29:02.430 --> 00:29:03.620
as we could run it
forwards, right?
00:29:03.620 --> 00:29:04.780
We could integrate
that in time back,
00:29:04.780 --> 00:29:06.280
or we could integrate
that in time forward.
00:29:06.280 --> 00:29:07.730
So, if we know the wave
function at some moment in time,
00:29:07.730 --> 00:29:09.670
we can integrate it forward,
and we can integrate it back
00:29:09.670 --> 00:29:10.490
in time.
00:29:10.490 --> 00:29:13.360
But, If at some
point we measure,
00:29:13.360 --> 00:29:16.720
then the wave
function collapses.
00:29:16.720 --> 00:29:18.900
And subsequently, the
system evolves according
00:29:18.900 --> 00:29:20.441
to the Schrodinger
equation, but with
00:29:20.441 --> 00:29:22.100
this new initial condition.
00:29:22.100 --> 00:29:24.400
So now we seem to
have a problem.
00:29:24.400 --> 00:29:25.901
We seem to have--
and I believe this
00:29:25.901 --> 00:29:27.233
was the question that was asked.
00:29:27.233 --> 00:29:28.570
I don't remember who asked it.
00:29:28.570 --> 00:29:29.111
Who asked it?
00:29:31.847 --> 00:29:32.680
So someone asked it.
00:29:32.680 --> 00:29:33.990
It was a good question.
00:29:33.990 --> 00:29:35.580
We have this problem
that there seem
00:29:35.580 --> 00:29:39.300
to be two definitions of time
evolution in quantum mechanics.
00:29:39.300 --> 00:29:41.064
One is the Schrodinger
equation, which
00:29:41.064 --> 00:29:42.480
says that things
deterministically
00:29:42.480 --> 00:29:44.970
evolve forward in time.
00:29:44.970 --> 00:29:48.530
And the second is collapse,
that if you do a measurement,
00:29:48.530 --> 00:29:52.610
things non-deterministically
by probabilities collapse
00:29:52.610 --> 00:29:56.020
to some possible state.
00:29:56.020 --> 00:29:56.520
Yeah?
00:29:56.520 --> 00:29:57.895
And the probability
is determined
00:29:57.895 --> 00:30:01.190
by which wave function you have.
00:30:01.190 --> 00:30:04.010
How can these
things both be true?
00:30:04.010 --> 00:30:07.770
How can you have two different
definitions of time evolution?
00:30:07.770 --> 00:30:09.700
So, this sort of
frustration lies
00:30:09.700 --> 00:30:12.180
at the heart of much
of the sort of spiel
00:30:12.180 --> 00:30:14.247
about the interpretation
of quantum mechanics.
00:30:14.247 --> 00:30:15.830
On the one hand, we
want to say, well,
00:30:15.830 --> 00:30:18.520
the world is inescapably
probabilistic.
00:30:18.520 --> 00:30:21.909
Measurement comes with
probabilistic outcomes
00:30:21.909 --> 00:30:23.700
and leads to collapse
of the wave function.
00:30:23.700 --> 00:30:25.491
On the other hand, when
you're not looking,
00:30:25.491 --> 00:30:27.340
the system evolves
deterministically.
00:30:27.340 --> 00:30:29.167
And this sounds horrible.
00:30:29.167 --> 00:30:31.000
It sounds horrible to
a classical physicist.
00:30:31.000 --> 00:30:32.830
It sounds horrible to me.
00:30:32.830 --> 00:30:34.790
It just sounds awful.
00:30:34.790 --> 00:30:35.920
It sounds arbitrary.
00:30:35.920 --> 00:30:38.030
Meanwhile, it makes it
sound like the world cares.
00:30:38.030 --> 00:30:39.930
It evolves differently depending
on whether you're looking
00:30:39.930 --> 00:30:40.700
or not.
00:30:40.700 --> 00:30:42.000
And that-- come on.
00:30:42.000 --> 00:30:45.560
I mean, I think we can all
agree that that's just crazy.
00:30:45.560 --> 00:30:46.920
So what's going on?
00:30:46.920 --> 00:30:51.940
So for a long time, physicists
in practice-- and still
00:30:51.940 --> 00:30:54.550
in practice-- for a
long time physicists
00:30:54.550 --> 00:30:56.490
almost exclusively
looked at this problem
00:30:56.490 --> 00:30:59.710
and said, look,
don't worry about.
00:30:59.710 --> 00:31:00.970
It fits the data.
00:31:00.970 --> 00:31:03.820
It makes good predictions.
00:31:03.820 --> 00:31:04.950
Work with me here.
00:31:04.950 --> 00:31:05.590
Right?
00:31:05.590 --> 00:31:09.210
And it's really hard to
argue against that attitude.
00:31:09.210 --> 00:31:10.290
You have a set of rules.
00:31:10.290 --> 00:31:11.350
It allows you to compute things.
00:31:11.350 --> 00:31:12.058
You compute them.
00:31:12.058 --> 00:31:13.300
They fit the data.
00:31:13.300 --> 00:31:13.900
Done.
00:31:13.900 --> 00:31:15.140
That is triumph.
00:31:15.140 --> 00:31:17.870
But it's deeply disconcerting.
00:31:17.870 --> 00:31:21.900
So, over the last, I
don't know, in the second
00:31:21.900 --> 00:31:24.670
or the last quarter,
roughly, the last third
00:31:24.670 --> 00:31:27.080
of the 20th century,
various people
00:31:27.080 --> 00:31:29.530
started getting more
upset about this.
00:31:29.530 --> 00:31:33.530
So, this notion of just
shut up and calculate,
00:31:33.530 --> 00:31:36.870
which has been enshrined
in the physics literature,
00:31:36.870 --> 00:31:39.880
goes under the name of the
Copenhagen interpretation,
00:31:39.880 --> 00:31:42.540
which roughly says,
look, just do this.
00:31:42.540 --> 00:31:43.270
Don't ask.
00:31:43.270 --> 00:31:47.100
Compute the numbers,
and get what you will.
00:31:47.100 --> 00:31:55.200
And people have questioned the
sanity or wisdom of doing that.
00:31:55.200 --> 00:31:57.061
And in particular,
there's an idea--
00:31:57.061 --> 00:31:59.560
so I refer to the Copenhagen
interpretation with my students
00:31:59.560 --> 00:32:03.000
as the cop out,
because it's basically
00:32:03.000 --> 00:32:04.240
disavowal of responsibility.
00:32:04.240 --> 00:32:05.620
Look, it doesn't make
sense, but I'm not
00:32:05.620 --> 00:32:06.545
responsible for making sense.
00:32:06.545 --> 00:32:08.378
I'm just responsible
for making predictions.
00:32:08.378 --> 00:32:09.050
Come on.
00:32:09.050 --> 00:32:14.680
So, more recently has come
the theory of decoherence.
00:32:14.680 --> 00:32:17.650
And we're not going to
talk about it in any detail
00:32:17.650 --> 00:32:19.530
until the last couple
lectures of 804.
00:32:19.530 --> 00:32:22.060
Decoherence.
00:32:22.060 --> 00:32:23.310
I can't spell to save my life.
00:32:23.310 --> 00:32:24.740
So, the theory of decoherence.
00:32:24.740 --> 00:32:26.580
And here's roughly
what the theory says.
00:32:26.580 --> 00:32:28.670
The theory says,
look, the reason
00:32:28.670 --> 00:32:33.530
you have this problem between
on the one hand, Schrodinger
00:32:33.530 --> 00:32:35.850
evolution of a quantum
system, and on the other hand,
00:32:35.850 --> 00:32:37.420
measurement leading
to collapse, is
00:32:37.420 --> 00:32:39.760
that in the case of measurement
meaning to collapse,
00:32:39.760 --> 00:32:42.480
you're not really studying the
evolution of a quantum system.
00:32:42.480 --> 00:32:45.360
You're studying the evolution
of a quantum system-- ie
00:32:45.360 --> 00:32:47.977
a little thing that you're
measuring-- interacting
00:32:47.977 --> 00:32:50.060
with your experimental
apparatus, which is made up
00:32:50.060 --> 00:32:53.600
of 10 to the 27th particles,
and you made up of 10
00:32:53.600 --> 00:32:55.320
to the 28 particles.
00:32:55.320 --> 00:32:55.820
Whatever.
00:32:55.820 --> 00:32:56.865
It's a large number.
00:32:56.865 --> 00:32:58.750
OK, a lot more than that.
00:32:58.750 --> 00:33:04.272
You, a macroscopic object,
where classical dynamics
00:33:04.272 --> 00:33:05.230
are a good description.
00:33:05.230 --> 00:33:06.605
In particular,
what that means is
00:33:06.605 --> 00:33:08.600
that the quantum effects
are being washed out.
00:33:08.600 --> 00:33:10.110
You're washing out
the interference
00:33:10.110 --> 00:33:12.400
of fringes, which is why
I can catch this thing
00:33:12.400 --> 00:33:14.910
and not have it split into
many different possible wave
00:33:14.910 --> 00:33:16.940
functions and where it went.
00:33:16.940 --> 00:33:20.694
So, dealing with that is hard,
because now if you really
00:33:20.694 --> 00:33:22.860
want to treat the system
with Schrodinger evolution,
00:33:22.860 --> 00:33:24.859
you have to study the
trajectory and the motion,
00:33:24.859 --> 00:33:27.640
the dynamics, of every particle
in the system, every degree
00:33:27.640 --> 00:33:28.930
of freedom in the system.
00:33:28.930 --> 00:33:31.831
So here's the question that
decoherence is trying to ask.
00:33:31.831 --> 00:33:34.080
If you take a system where
you have one little quantum
00:33:34.080 --> 00:33:35.760
subsystem that you're
trying to measure,
00:33:35.760 --> 00:33:37.990
and then again a gagillion
other degrees of freedom,
00:33:37.990 --> 00:33:39.850
some of which you care
about-- they're made of you--
00:33:39.850 --> 00:33:40.880
some of which you don't,
like the particles
00:33:40.880 --> 00:33:42.760
of gas in the room,
the environment.
00:33:42.760 --> 00:33:46.700
If you take that whole
system, does Schrodinger
00:33:46.700 --> 00:33:51.180
evolution in the end
boil down to collapse
00:33:51.180 --> 00:33:53.650
for that single
quantum microsystem?
00:33:53.650 --> 00:33:55.460
And the answer is yes.
00:33:55.460 --> 00:33:57.720
Showing that take
some work, and we'll
00:33:57.720 --> 00:33:59.300
touch on it at the end of 804.
00:33:59.300 --> 00:34:01.010
But I want to mark
right here that this
00:34:01.010 --> 00:34:03.690
is one of the most deeply
unsatisfying points
00:34:03.690 --> 00:34:06.070
in the basic story
of quantum mechanics,
00:34:06.070 --> 00:34:08.239
and that it's deeply
unsatisfying because of the way
00:34:08.239 --> 00:34:09.750
that we're presenting it.
00:34:09.750 --> 00:34:11.550
And there's a much
more satisfying--
00:34:11.550 --> 00:34:14.540
although still you
never escape the fact
00:34:14.540 --> 00:34:16.909
that quantum mechanics
violates your intuition.
00:34:16.909 --> 00:34:17.889
That's inescapable.
00:34:17.889 --> 00:34:19.750
But at least it's not illogical.
00:34:19.750 --> 00:34:21.659
it doesn't directly
contradict itself.
00:34:21.659 --> 00:34:25.500
So that story is the
story of decoherence.
00:34:25.500 --> 00:34:27.139
And if we're very
lucky, I think we'll
00:34:27.139 --> 00:34:31.679
try to get one of my friends
who's a quantum computing
00:34:31.679 --> 00:34:33.610
guy to talk about it.
00:34:33.610 --> 00:34:34.540
Yeah.
00:34:34.540 --> 00:34:37.040
AUDIENCE: [INAUDIBLE]
Is it possible
00:34:37.040 --> 00:34:39.639
that we get two
different results?
00:34:39.639 --> 00:34:40.560
PROFESSOR: No.
00:34:40.560 --> 00:34:41.060
No.
00:34:41.060 --> 00:34:41.560
No.
00:34:41.560 --> 00:34:43.889
There's never any ambiguity
about what result you got.
00:34:43.889 --> 00:34:46.139
You never end up in a state
of-- and this is also something
00:34:46.139 --> 00:34:47.805
that decoherence is
supposed to explain.
00:34:47.805 --> 00:34:50.850
You never end up in a situation
where you go like, wait, wait.
00:34:50.850 --> 00:34:51.710
I don't know.
00:34:51.710 --> 00:34:53.293
Maybe it was here,
maybe it was there.
00:34:53.293 --> 00:34:54.320
I'm really confused.
00:34:54.320 --> 00:34:55.986
I mean, you can get
up in that situation
00:34:55.986 --> 00:34:57.550
because you did a
bad job, but you
00:34:57.550 --> 00:34:58.900
don't end up in that
situation because you're
00:34:58.900 --> 00:35:00.220
in a superposition state.
00:35:00.220 --> 00:35:02.442
You always end up when you're
a classical beast doing
00:35:02.442 --> 00:35:03.900
a classical
measurement, you always
00:35:03.900 --> 00:35:06.070
end up in some definite state.
00:35:06.070 --> 00:35:08.060
Now, what wave
function describes
00:35:08.060 --> 00:35:09.520
you doesn't
necessarily correspond
00:35:09.520 --> 00:35:10.400
to you being in a simple state.
00:35:10.400 --> 00:35:12.400
You might be in a superposition
of thinking this and thinking
00:35:12.400 --> 00:35:13.080
that.
00:35:13.080 --> 00:35:15.690
But, when you think this,
that's in fact what happened.
00:35:15.690 --> 00:35:18.651
And when you think that,
that's in fact what happened.
00:35:18.651 --> 00:35:19.150
OK.
00:35:19.150 --> 00:35:21.190
So I'm going to leave
this alone for the moment,
00:35:21.190 --> 00:35:23.350
but I just wanted to mark
that as an important part
00:35:23.350 --> 00:35:26.120
of the quantum mechanical story.
00:35:26.120 --> 00:35:28.430
OK.
00:35:28.430 --> 00:35:32.771
So let's go on to solving
the Schrodinger equation.
00:35:32.771 --> 00:35:34.520
So what I want to do
for the rest of today
00:35:34.520 --> 00:35:36.478
is talk about solving
the Schrodinger equation.
00:35:43.740 --> 00:35:46.115
So when we set about solving
the Schrodinger equation,
00:35:46.115 --> 00:35:47.490
the first thing
we should realize
00:35:47.490 --> 00:35:49.490
is that at the end of the
day, the Schrodinger equation
00:35:49.490 --> 00:35:50.890
is just some
differential equation.
00:35:50.890 --> 00:35:53.390
And in fact, it's a particularly
easy differential equation.
00:35:53.390 --> 00:35:56.460
It's a first order linear
differential equation.
00:35:56.460 --> 00:35:56.960
Right?
00:35:56.960 --> 00:35:59.750
We know how to solve those.
00:35:59.750 --> 00:36:02.104
But, while it's
first order in time,
00:36:02.104 --> 00:36:04.270
we have to think about what
this energy operator is.
00:36:04.270 --> 00:36:06.370
So, just like the Newton
equation f equals ma,
00:36:06.370 --> 00:36:07.995
we have to specify
the energy operative
00:36:07.995 --> 00:36:10.330
before we can actually solve
the dynamics of the system.
00:36:10.330 --> 00:36:11.788
In f equals ma, we
have to tell you
00:36:11.788 --> 00:36:14.280
what the force is before
we can solve for p,
00:36:14.280 --> 00:36:17.554
from p is equal to f.
00:36:17.554 --> 00:36:18.220
So, for example.
00:36:20.880 --> 00:36:23.930
So one strategy to solve
the Schrodinger equation
00:36:23.930 --> 00:36:26.605
is to say, look, it's just
a differential equation,
00:36:26.605 --> 00:36:28.480
and I'll solve it using
differential equation
00:36:28.480 --> 00:36:29.510
techniques.
00:36:29.510 --> 00:36:34.690
So let me specify, for
example, the energy operator.
00:36:34.690 --> 00:36:38.190
What's an easy energy operator?
00:36:38.190 --> 00:36:41.230
Well, imagine you had a harmonic
oscillator, which, you know,
00:36:41.230 --> 00:36:42.930
physicists, that's your go-to.
00:36:42.930 --> 00:36:45.380
So, harmonic
oscillator has energy p
00:36:45.380 --> 00:36:49.840
squared over 2m plus M Omega
squared upon 2x squared.
00:36:49.840 --> 00:36:51.480
But we're going
quantum mechanics,
00:36:51.480 --> 00:36:53.882
so we replace these
guys by operators.
00:36:53.882 --> 00:36:55.090
So that's an energy operator.
00:36:55.090 --> 00:36:58.520
It's a perfectly
viable operator.
00:36:58.520 --> 00:37:01.240
And what is the differential
equation that this leads to?
00:37:01.240 --> 00:37:03.050
What's the Schrodinger
equation leads to?
00:37:03.050 --> 00:37:05.216
Well, I'm going to put the
ih bar on the other side.
00:37:05.216 --> 00:37:10.300
ih bar derivative with respect
to time of Psi of x and t
00:37:10.300 --> 00:37:12.210
is equal to p squared.
00:37:12.210 --> 00:37:14.200
Well, we remember that
p is equal to h bar
00:37:14.200 --> 00:37:17.160
upon i, derivative
with respect to x.
00:37:17.160 --> 00:37:20.120
So p squared is minus h bar
squared derivative with respect
00:37:20.120 --> 00:37:24.270
to x squared upon 2m, or
minus h bar squared upon 2m.
00:37:27.042 --> 00:37:28.425
Psi prime prime.
00:37:28.425 --> 00:37:31.200
Let me write this as dx squared.
00:37:31.200 --> 00:37:37.570
Two spatial derivatives acting
on Psi of x and t plus m
00:37:37.570 --> 00:37:44.720
omega squared upon 2x
squared Psi of x and t.
00:37:44.720 --> 00:37:47.160
So here's a
differential equation.
00:37:47.160 --> 00:37:51.241
And if we want to know how
does a system evolve in time,
00:37:51.241 --> 00:37:53.240
ie given some initial
wave function, how does it
00:37:53.240 --> 00:37:55.531
evolve in time, we just take
this differential equation
00:37:55.531 --> 00:37:56.280
and we solve it.
00:37:56.280 --> 00:37:58.540
And there are many tools to
solve this partial differential
00:37:58.540 --> 00:37:59.040
equation.
00:37:59.040 --> 00:38:01.300
For example, you could
put it on Mathematica
00:38:01.300 --> 00:38:03.175
and just use NDSolve, right?
00:38:03.175 --> 00:38:04.550
This wasn't
available, of course,
00:38:04.550 --> 00:38:06.480
to the physicists at
the turn of the century,
00:38:06.480 --> 00:38:09.467
but they were less timid
about differential equations
00:38:09.467 --> 00:38:11.550
than we are, because they
didn't have Mathematica.
00:38:11.550 --> 00:38:13.130
So, this is a very
straightforward differential
00:38:13.130 --> 00:38:13.950
equation to solve,
and we're going
00:38:13.950 --> 00:38:15.510
to solve it in a
couple of lectures.
00:38:15.510 --> 00:38:17.801
We're going to study the
harmonic oscillator in detail.
00:38:17.801 --> 00:38:20.430
What I want to emphasize for
you is that any system has have
00:38:20.430 --> 00:38:23.540
some specified energy operator,
just like any classical system,
00:38:23.540 --> 00:38:26.980
has some definite
force function.
00:38:26.980 --> 00:38:28.780
And given that energy
operator, that's
00:38:28.780 --> 00:38:30.100
going to lead to a
differential equation.
00:38:30.100 --> 00:38:31.975
So one way to solve the
differential equation
00:38:31.975 --> 00:38:35.040
is just to go ahead and
brute force solve it.
00:38:35.040 --> 00:38:36.770
But, at the end of
the day, solving
00:38:36.770 --> 00:38:38.520
the Schrodinger equation
is always, always
00:38:38.520 --> 00:38:40.180
going to boil down
to some version
00:38:40.180 --> 00:38:43.900
morally of solve this
differential equation.
00:38:43.900 --> 00:38:46.119
Questions about that?
00:38:46.119 --> 00:38:48.900
OK.
00:38:48.900 --> 00:38:51.129
But when we actually look
at a differential equation
00:38:51.129 --> 00:38:53.420
like this-- so, say we have
this differential equation.
00:38:53.420 --> 00:38:55.211
It's got a derivative
with respect to time,
00:38:55.211 --> 00:38:57.210
so we have to specify
some initial condition.
00:38:57.210 --> 00:38:59.010
There are many ways to solve it.
00:38:59.010 --> 00:39:06.250
So given E, given
some specific E,
00:39:06.250 --> 00:39:08.190
given some specific
energy operator,
00:39:08.190 --> 00:39:09.420
there are many ways to solve.
00:39:13.570 --> 00:39:16.444
The resulting
differential equation.
00:39:16.444 --> 00:39:18.485
And I'm just going to mark
that, in general, it's
00:39:18.485 --> 00:39:20.100
a PDE, because it's got
derivatives with respect
00:39:20.100 --> 00:39:22.016
to time and derivatives
with respect to space.
00:39:25.810 --> 00:39:27.950
And roughly speaking,
all these techniques
00:39:27.950 --> 00:39:29.300
fall into three camps.
00:39:29.300 --> 00:39:30.700
The first is just brute force.
00:39:33.440 --> 00:39:37.230
That means some analog of
throw it on Mathematica,
00:39:37.230 --> 00:39:39.262
go to the closet and pull
out your mathematician
00:39:39.262 --> 00:39:41.560
and tie them to the
chalkboard until they're done,
00:39:41.560 --> 00:39:42.520
and then put them back.
00:39:42.520 --> 00:39:46.600
But some version of a
brute force, which is just
00:39:46.600 --> 00:39:48.700
use, by hook or by
crook, some technique
00:39:48.700 --> 00:39:51.710
that allows you to solve
the differential equation.
00:39:51.710 --> 00:39:52.500
OK.
00:39:52.500 --> 00:39:55.810
The second is
extreme cleverness.
00:39:55.810 --> 00:39:57.990
And you'd be amazed how
often this comes in handy.
00:39:57.990 --> 00:39:59.490
So, extreme
cleverness-- which we'll
00:39:59.490 --> 00:40:02.610
see both of these
techniques used
00:40:02.610 --> 00:40:05.860
for the harmonic oscillator.
00:40:05.860 --> 00:40:07.200
That's what we'll do next week.
00:40:07.200 --> 00:40:08.790
First, the brute
force, and secondly,
00:40:08.790 --> 00:40:10.909
the clever way of solving
the harmonic oscillator.
00:40:10.909 --> 00:40:12.950
When I say extreme
cleverness, what I really mean
00:40:12.950 --> 00:40:15.310
is a more elegant use
of your mathematician.
00:40:15.310 --> 00:40:17.890
You know something
about the structure,
00:40:17.890 --> 00:40:21.430
the mathematical structure of
your differential equation.
00:40:21.430 --> 00:40:23.690
And you're going to
use that structure
00:40:23.690 --> 00:40:27.240
to figure out a good way to
organize the differential
00:40:27.240 --> 00:40:29.500
equation, the good way
to organize the problem.
00:40:29.500 --> 00:40:30.977
And that will teach you physics.
00:40:30.977 --> 00:40:33.310
And the reason I distinguish
brute force from cleverness
00:40:33.310 --> 00:40:34.850
in this sense is
that brute force,
00:40:34.850 --> 00:40:35.890
you just get a list of numbers.
00:40:35.890 --> 00:40:37.764
Cleverness, you learn
something about the way
00:40:37.764 --> 00:40:39.750
the physics of the
system operates.
00:40:39.750 --> 00:40:43.264
We'll see this at work
in the next two lectures.
00:40:43.264 --> 00:40:45.555
And see, I really should
separate this out numerically.
00:40:48.760 --> 00:40:52.100
And here I don't just mean
sticking it into MATLAB.
00:40:52.100 --> 00:40:53.850
Numerically, it can
be enormously valuable
00:40:53.850 --> 00:40:54.690
for a bunch of reasons.
00:40:54.690 --> 00:40:56.130
First off, there
are often situations
00:40:56.130 --> 00:40:58.330
where no classic technique
in differential equations
00:40:58.330 --> 00:41:00.760
or no simple mathematical
structure that would just
00:41:00.760 --> 00:41:02.780
leap to the imagination
comes to use.
00:41:02.780 --> 00:41:04.780
And you have some horrible
differential you just
00:41:04.780 --> 00:41:07.100
have to solve, and you
can solve it numerically.
00:41:07.100 --> 00:41:09.220
Very useful lesson, and
a reason to not even--
00:41:09.220 --> 00:41:11.720
how many of y'all are thinking
about being theorists of some
00:41:11.720 --> 00:41:13.540
stripe or other?
00:41:13.540 --> 00:41:14.040
OK.
00:41:14.040 --> 00:41:15.080
And how many of y'all
are thinking about being
00:41:15.080 --> 00:41:17.260
experimentalists of
some stripe or another?
00:41:17.260 --> 00:41:17.990
OK, cool.
00:41:17.990 --> 00:41:21.380
So, look, there's this
deep, deep prejudice
00:41:21.380 --> 00:41:27.030
in theory against numerical
solutions of problems.
00:41:27.030 --> 00:41:28.640
It's myopia.
00:41:28.640 --> 00:41:31.140
It's a terrible attitude,
and here's the reason.
00:41:31.140 --> 00:41:34.220
Computers are stupid.
00:41:34.220 --> 00:41:36.050
Computers are
breathtakingly dumb.
00:41:36.050 --> 00:41:37.800
They will do whatever
you tell them to do,
00:41:37.800 --> 00:41:40.091
but they will not tell you
that was a dumb thing to do.
00:41:40.091 --> 00:41:40.940
They have no idea.
00:41:40.940 --> 00:41:43.830
So, in order to solve an
interesting physical problem,
00:41:43.830 --> 00:41:46.480
you have to first
extract all the physics
00:41:46.480 --> 00:41:48.770
and organize the
problem in such a way
00:41:48.770 --> 00:41:52.510
that a stupid computer
can do the solution.
00:41:52.510 --> 00:41:55.430
As a consequence, you learn
the physics about the problem.
00:41:55.430 --> 00:41:58.019
It's extremely valuable to
learn how to solve problems
00:41:58.019 --> 00:42:00.310
numerically, and we're going
to have problem sets later
00:42:00.310 --> 00:42:01.100
in the course in
which you're going
00:42:01.100 --> 00:42:03.010
to be required to
numerically solve
00:42:03.010 --> 00:42:05.020
some of these
differential equations.
00:42:05.020 --> 00:42:07.340
But it's useful because
you get numbers,
00:42:07.340 --> 00:42:09.810
and you can check
against data, but also it
00:42:09.810 --> 00:42:11.640
lets you in the process
of understanding
00:42:11.640 --> 00:42:12.681
how to solve the problem.
00:42:12.681 --> 00:42:17.360
You learn things
about the problem.
00:42:17.360 --> 00:42:20.640
So I want to mark that as a
separate logical way to do it.
00:42:20.640 --> 00:42:24.690
So today, I want to
start our analysis
00:42:24.690 --> 00:42:27.070
by looking at a
couple of examples
00:42:27.070 --> 00:42:33.430
of solving the
Schrodinger equation.
00:42:33.430 --> 00:42:38.740
And I want to start by looking
at energy Eigenfunctions.
00:42:44.540 --> 00:42:54.780
And then once we understand how
a single energy Eigenfunction
00:42:54.780 --> 00:42:56.975
evolves in time, once we
understand that solution
00:42:56.975 --> 00:42:58.570
to the Schrodinger
equation, we're
00:42:58.570 --> 00:43:01.660
going to use the linearity
of the Schrodinger equation
00:43:01.660 --> 00:43:06.380
to write down a general solution
of the Schrodinger equation.
00:43:06.380 --> 00:43:07.940
OK.
00:43:07.940 --> 00:43:10.840
So, first.
00:43:10.840 --> 00:43:13.090
What happens if we have a
single energy Eigenfunction?
00:43:13.090 --> 00:43:16.820
So, suppose our wave function
as a function of x at time t
00:43:16.820 --> 00:43:20.570
equals zero is in a known
configuration, which
00:43:20.570 --> 00:43:24.480
is an energy Eigenfunction
Phi sub E of x.
00:43:24.480 --> 00:43:28.270
What I mean by Phi sub E of x is
if I take the energy operator,
00:43:28.270 --> 00:43:32.050
and I act on Phi sub E
of x, this gives me back
00:43:32.050 --> 00:43:34.620
the number E Phi sub E of x.
00:43:37.994 --> 00:43:38.920
OK?
00:43:38.920 --> 00:43:41.830
So it's an Eigenfunction of the
energy operator, the Eigenvalue
00:43:41.830 --> 00:43:42.460
E.
00:43:42.460 --> 00:43:44.560
So, suppose our
initial condition is
00:43:44.560 --> 00:43:48.090
that our system
began life at time t
00:43:48.090 --> 00:43:51.306
equals zero in this state with
definite energy E. Everyone
00:43:51.306 --> 00:43:52.360
cool with that?
00:43:52.360 --> 00:43:53.810
First off, question.
00:43:53.810 --> 00:43:56.400
Suppose I immediately
at time zero
00:43:56.400 --> 00:43:59.210
measure the energy
of this system.
00:43:59.210 --> 00:44:00.310
What will I get?
00:44:00.310 --> 00:44:01.726
AUDIENCE: E.
00:44:01.726 --> 00:44:03.100
PROFESSOR: With
what probability?
00:44:03.100 --> 00:44:04.000
AUDIENCE: 100%
00:44:04.000 --> 00:44:06.920
PROFESSOR: 100%, because this
is, in fact, of this form,
00:44:06.920 --> 00:44:09.890
it's a superposition
of energy Eigenstates,
00:44:09.890 --> 00:44:11.350
except there's only one term.
00:44:11.350 --> 00:44:12.850
And the coefficient
of that one term
00:44:12.850 --> 00:44:16.170
is one, and the probability
that I measure the energy
00:44:16.170 --> 00:44:19.170
to be equal to that value is
the coefficient norm squared,
00:44:19.170 --> 00:44:21.040
and that's one norm squared.
00:44:21.040 --> 00:44:22.499
Everyone cool with that?
00:44:22.499 --> 00:44:25.040
Consider on the other hand, if
I had taken this wave function
00:44:25.040 --> 00:44:28.390
and I had multiplied it
by phase E to the i Alpha.
00:44:28.390 --> 00:44:31.370
What now is the probability
where alpha is just a number?
00:44:31.370 --> 00:44:34.380
What now is the probability
that I measured the state
00:44:34.380 --> 00:44:35.710
to have energy E?
00:44:35.710 --> 00:44:36.455
AUDIENCE: One.
00:44:36.455 --> 00:44:38.580
PROFESSOR: It's still one,
because the norm squared
00:44:38.580 --> 00:44:40.430
of a phase is one.
00:44:40.430 --> 00:44:41.266
Right?
00:44:41.266 --> 00:44:42.180
OK.
00:44:42.180 --> 00:44:46.530
The overall phase
does not matter.
00:44:46.530 --> 00:44:48.540
So, suppose I have this
as my initial condition.
00:44:48.540 --> 00:44:49.630
Let's take away
the overall phase
00:44:49.630 --> 00:44:50.921
because my life will be easier.
00:44:53.094 --> 00:44:54.260
So here's the wave function.
00:44:54.260 --> 00:44:55.635
What is the
Schrodinger equation?
00:44:55.635 --> 00:44:57.330
Well, the Schrodinger
equation says
00:44:57.330 --> 00:45:01.550
that ih bar time
derivative of Psi
00:45:01.550 --> 00:45:04.409
is equal to the energy
operator acting on Psi.
00:45:04.409 --> 00:45:05.450
And I should be specific.
00:45:05.450 --> 00:45:11.740
This is Psi at x at time t,
Eigenvalued at this time zero
00:45:11.740 --> 00:45:15.610
is equal to the energy operator
acting on this wave function.
00:45:15.610 --> 00:45:19.880
But what's the energy operator
acting on this wave function?
00:45:19.880 --> 00:45:20.380
AUDIENCE: E.
00:45:20.380 --> 00:45:22.420
PROFESSOR: E. E on
Psi is equal to E
00:45:22.420 --> 00:45:25.750
on Phi sub E, which
is just E the number.
00:45:25.750 --> 00:45:28.720
This is the number E,
the Eigenvalue E times
00:45:28.720 --> 00:45:31.867
Psi at x zero.
00:45:31.867 --> 00:45:34.200
And now, instead of having
an operator on the right hand
00:45:34.200 --> 00:45:35.814
side, we just have a number.
00:45:35.814 --> 00:45:38.230
So, I'm going to write this
differential equation slightly
00:45:38.230 --> 00:45:44.470
differently, ie time
derivative of Psi
00:45:44.470 --> 00:45:54.450
is equal to E upon ih bar,
or minus ie over h bar Psi.
00:45:58.941 --> 00:45:59.440
Yeah?
00:45:59.440 --> 00:46:00.439
Everyone cool with that?
00:46:00.439 --> 00:46:04.050
This is the easiest differential
equation in the world to solve.
00:46:04.050 --> 00:46:06.700
So, the time derivative
is a constant.
00:46:06.700 --> 00:46:08.090
Well, times itself.
00:46:08.090 --> 00:46:13.660
That means that
therefore Psi at x and t
00:46:13.660 --> 00:46:23.329
is equal to E to the minus i
ET over h bar Psi at x zero.
00:46:23.329 --> 00:46:25.620
Where I've imposed the initial
condition that at time t
00:46:25.620 --> 00:46:27.310
equals zero, the
wave function is
00:46:27.310 --> 00:46:28.980
just equal to Psi of x at zero.
00:46:32.340 --> 00:46:35.250
And in particular, I know
what Psi of x and zero is.
00:46:35.250 --> 00:46:37.360
It's Phi E of x.
00:46:37.360 --> 00:46:39.836
So I can simply write
this as Phi E of x.
00:46:44.970 --> 00:46:47.850
Are we cool with that?
00:46:47.850 --> 00:46:51.420
So, what this tells me is
that under time evolution,
00:46:51.420 --> 00:46:55.284
a state which is initially
in an energy Eigenstate
00:46:55.284 --> 00:46:57.450
remains in an energy
Eigenstate with the same energy
00:46:57.450 --> 00:46:57.975
Eigenvalue.
00:46:57.975 --> 00:47:00.100
The only thing that changes
about the wave function
00:47:00.100 --> 00:47:02.320
is that its phase
changes, and its phase
00:47:02.320 --> 00:47:05.300
changes by rotating with
a constant velocity.
00:47:05.300 --> 00:47:08.270
E to the minus i, the
energy Eigenvalue,
00:47:08.270 --> 00:47:10.674
times time upon h bar.
00:47:10.674 --> 00:47:12.840
Now, first off, before we
do anything else as usual,
00:47:12.840 --> 00:47:14.923
we should first check the
dimensions of our result
00:47:14.923 --> 00:47:16.680
to make sure we
didn't make a goof.
00:47:16.680 --> 00:47:20.860
So, does this make
sense dimensionally?
00:47:20.860 --> 00:47:22.806
Let's quickly check.
00:47:22.806 --> 00:47:23.449
Yeah, it does.
00:47:23.449 --> 00:47:24.490
Let's just quickly check.
00:47:24.490 --> 00:47:30.791
So we have that the exponent
there is Et over h bar.
00:47:30.791 --> 00:47:31.290
OK?
00:47:31.290 --> 00:47:33.650
And this should have dimensions
of what in order to make sense?
00:47:33.650 --> 00:47:34.630
AUDIENCE: Nothing.
00:47:34.630 --> 00:47:35.220
PROFESSOR: Nothing, exactly.
00:47:35.220 --> 00:47:36.410
It should be dimensionless.
00:47:36.410 --> 00:47:38.554
So what are the
dimensions of h bar?
00:47:38.554 --> 00:47:39.490
AUDIENCE: [INAUDIBLE]
00:47:39.490 --> 00:47:40.800
PROFESSOR: Oh, no, the
dimensions, guys, not
00:47:40.800 --> 00:47:41.240
the units.
00:47:41.240 --> 00:47:42.239
What are the dimensions?
00:47:42.239 --> 00:47:44.205
AUDIENCE: [INAUDIBLE]
00:47:44.205 --> 00:47:46.330
PROFESSOR: It's an action,
which is energy of time.
00:47:46.330 --> 00:47:49.146
So the units of
the dimensions of h
00:47:49.146 --> 00:47:51.480
are an energy times
a time, also known
00:47:51.480 --> 00:47:54.651
as a momentum times a position.
00:47:54.651 --> 00:47:55.150
OK?
00:48:03.350 --> 00:48:06.280
So, this has dimensions of
action or energy times time,
00:48:06.280 --> 00:48:07.780
and then upstairs
we have dimensions
00:48:07.780 --> 00:48:09.590
of energy times time.
00:48:09.590 --> 00:48:11.510
So that's consistent.
00:48:11.510 --> 00:48:14.510
So this in fact is dimensionally
sensible, which is good.
00:48:14.510 --> 00:48:16.310
Now, this tells you a
very important thing.
00:48:16.310 --> 00:48:17.590
In fact, we just
answered this equation.
00:48:17.590 --> 00:48:19.100
At time t equals
zero, what will we
00:48:19.100 --> 00:48:22.290
get if we measure the energy?
00:48:22.290 --> 00:48:26.480
E. At time t prime-- some
subsequent time-- what energy
00:48:26.480 --> 00:48:27.180
will we measure?
00:48:27.180 --> 00:48:27.874
AUDIENCE: E.
00:48:27.874 --> 00:48:28.540
PROFESSOR: Yeah.
00:48:28.540 --> 00:48:31.810
Does the energy
change over time?
00:48:31.810 --> 00:48:32.440
No.
00:48:32.440 --> 00:48:34.530
When I say that, what I
mean is, does the energy
00:48:34.530 --> 00:48:36.410
that you expect to
measure change over time?
00:48:36.410 --> 00:48:36.730
No.
00:48:36.730 --> 00:48:38.645
Does the probability that
you measure energy E change?
00:48:38.645 --> 00:48:41.228
No, because it's just a phase,
and the norm squared of a phase
00:48:41.228 --> 00:48:42.400
is one.
00:48:42.400 --> 00:48:43.440
Yeah?
00:48:43.440 --> 00:48:45.160
Everyone cool with that?
00:48:45.160 --> 00:48:48.109
Questions at this point.
00:48:48.109 --> 00:48:49.650
This is very simple
example, but it's
00:48:49.650 --> 00:48:50.270
going to have a lot of power.
00:48:50.270 --> 00:48:51.061
Oh, yeah, question.
00:48:51.061 --> 00:48:51.850
Thank you.
00:48:51.850 --> 00:48:54.224
AUDIENCE: Are we going to deal
with energy operators that
00:48:54.224 --> 00:48:56.100
change over time?
00:48:56.100 --> 00:48:57.350
PROFESSOR: Excellent question.
00:48:57.350 --> 00:48:59.470
We will later, but not in 804.
00:48:59.470 --> 00:49:02.020
In 805, you'll discuss
it in more detail.
00:49:02.020 --> 00:49:04.650
Nothing dramatic
happens, but you just
00:49:04.650 --> 00:49:06.405
have to add more symbols.
00:49:06.405 --> 00:49:07.700
There's nothing deep about it.
00:49:07.700 --> 00:49:08.370
It's a very good question.
00:49:08.370 --> 00:49:09.953
The question was,
are we going to deal
00:49:09.953 --> 00:49:12.877
with energy operators
that change in time?
00:49:12.877 --> 00:49:14.960
My answer was no, not in
804, but you will in 805.
00:49:14.960 --> 00:49:17.550
And what you'll find is
that it's not a big deal.
00:49:17.550 --> 00:49:19.200
Nothing particularly
dramatic happens.
00:49:19.200 --> 00:49:23.530
We will deal with systems where
the energy operator changes
00:49:23.530 --> 00:49:24.455
instantaneously.
00:49:24.455 --> 00:49:26.080
So not a continuous
function, but we're
00:49:26.080 --> 00:49:27.830
at some of them you turn
on the electric field,
00:49:27.830 --> 00:49:28.890
or something like that.
00:49:28.890 --> 00:49:30.719
So we'll deal with
that later on.
00:49:30.719 --> 00:49:32.760
But we won't develop a
theory of energy operators
00:49:32.760 --> 00:49:35.490
that depend on time.
00:49:35.490 --> 00:49:38.110
But you could do it,
and you will do in 805.
00:49:38.110 --> 00:49:39.610
There's nothing
mysterious about it.
00:49:39.610 --> 00:49:42.400
Other questions?
00:49:42.400 --> 00:49:43.760
OK.
00:49:43.760 --> 00:49:48.770
So, these states-- a
state Psi of x and t,
00:49:48.770 --> 00:49:52.690
which is of the form e
to the minus i Omega t,
00:49:52.690 --> 00:49:56.117
where Omega is equal
to E over h bar.
00:49:56.117 --> 00:49:57.200
This should look familiar.
00:49:57.200 --> 00:50:00.770
It's the de Broglie relation,
[INAUDIBLE] relation, whatever.
00:50:00.770 --> 00:50:04.370
Times some Phi E
of x, where this
00:50:04.370 --> 00:50:07.290
is an energy Eigenfunction.
00:50:07.290 --> 00:50:10.390
These states are called
stationary states.
00:50:19.459 --> 00:50:20.750
And what's the reason for that?
00:50:20.750 --> 00:50:22.333
Why are they called
stationary states?
00:50:22.333 --> 00:50:24.160
I'm going to erase this.
00:50:24.160 --> 00:50:27.990
Well, suppose this is my wave
function as a function of time.
00:50:27.990 --> 00:50:31.030
What is the probability
that at time t
00:50:31.030 --> 00:50:33.080
I will measure the particle
to be at position x,
00:50:33.080 --> 00:50:34.205
or the probability density?
00:50:36.980 --> 00:50:40.580
Well, the probability density
we know from our postulates,
00:50:40.580 --> 00:50:42.580
it's just the norm squared
of the wave function.
00:50:42.580 --> 00:50:47.090
This is Psi at x t norm squared.
00:50:47.090 --> 00:50:49.500
But this is equal to
the norm squared of e
00:50:49.500 --> 00:50:53.024
to the minus Psi Omega t Phi
E by the Schrodinger equation.
00:50:53.024 --> 00:50:54.440
But when we take
the norm squared,
00:50:54.440 --> 00:50:56.720
this phase cancels
out, as we already saw.
00:50:56.720 --> 00:51:01.600
So this is just equal to
Phi E of x norm squared,
00:51:01.600 --> 00:51:04.820
the energy Eigenfunction norm
squared independent of time.
00:51:11.700 --> 00:51:14.100
So, if we happen to know that
our state is in an energy
00:51:14.100 --> 00:51:17.100
Eigenfunction, then
the probability density
00:51:17.100 --> 00:51:19.180
for finding the particle
at any given position
00:51:19.180 --> 00:51:20.550
does not change in time.
00:51:20.550 --> 00:51:22.520
It remains invariant.
00:51:22.520 --> 00:51:24.620
The wave function rotates
by an overall phase,
00:51:24.620 --> 00:51:26.620
but the probability density
is the norm squared.
00:51:26.620 --> 00:51:28.070
It's insensitive to
that overall phase,
00:51:28.070 --> 00:51:29.650
and so the probability
density just
00:51:29.650 --> 00:51:31.358
remains constant in
whatever shape it is.
00:51:33.439 --> 00:51:34.980
Hence it's called
a stationary state.
00:51:34.980 --> 00:51:37.470
Notice its consequence.
00:51:37.470 --> 00:51:39.440
What can you say about
the expectation value
00:51:39.440 --> 00:51:41.030
of the position as
a function of time?
00:51:44.090 --> 00:51:47.360
Well, this is equal
to the integral dx
00:51:47.360 --> 00:51:50.702
in the state Psi of x and t.
00:51:50.702 --> 00:51:52.740
And I'll call this Psi
sub E just to emphasize.
00:51:52.740 --> 00:51:54.990
It's the integral of the x,
integral over all possible
00:51:54.990 --> 00:51:57.520
positions of the
probability distribution,
00:51:57.520 --> 00:52:00.570
probability of x
at time t times x.
00:52:00.570 --> 00:52:03.610
But this is equal
to the integral dx
00:52:03.610 --> 00:52:08.760
of Phi E of x squared x.
00:52:08.760 --> 00:52:13.680
But that's equal to expectation
value of x at any time,
00:52:13.680 --> 00:52:15.640
or time zero. t equals zero.
00:52:15.640 --> 00:52:20.030
And maybe the best way to write
this is as a function of time.
00:52:20.030 --> 00:52:22.940
So, the expectation value
of x doesn't change.
00:52:22.940 --> 00:52:26.010
In a stationary state,
expected positions,
00:52:26.010 --> 00:52:28.840
energy-- these
things don't change.
00:52:28.840 --> 00:52:30.430
Everyone cool with that?
00:52:30.430 --> 00:52:32.240
And it's because
of this basic fact
00:52:32.240 --> 00:52:34.260
that the wave
function only rotates
00:52:34.260 --> 00:52:36.512
by a phase under
time evolution when
00:52:36.512 --> 00:52:37.970
the system is an
energy Eigenstate.
00:52:42.010 --> 00:52:44.320
Questions?
00:52:44.320 --> 00:52:46.200
OK.
00:52:46.200 --> 00:52:49.960
So, here's a couple of
questions for you guys.
00:52:54.320 --> 00:52:57.740
Are all systems always
in energy Eigenstates?
00:53:00.950 --> 00:53:02.320
Am I in an energy Eigenstate?
00:53:05.530 --> 00:53:06.490
AUDIENCE: No.
00:53:06.490 --> 00:53:07.690
PROFESSOR: No, right?
00:53:07.690 --> 00:53:09.950
OK, expected position of my
hand is changing in time.
00:53:09.950 --> 00:53:14.240
I am not in-- so obviously,
things change in time.
00:53:14.240 --> 00:53:16.267
Energies change in time.
00:53:16.267 --> 00:53:18.600
Positions-- expected typical
positions-- change in time.
00:53:18.600 --> 00:53:21.740
We are not in
energy Eigenstates.
00:53:21.740 --> 00:53:23.590
That's a highly
non-generic state.
00:53:23.590 --> 00:53:25.160
So here's another question.
00:53:25.160 --> 00:53:28.790
Are any states ever truly
in energy Eigenstates?
00:53:32.670 --> 00:53:34.790
Can you imagine an
object in the world
00:53:34.790 --> 00:53:36.680
that is truly
described precisely
00:53:36.680 --> 00:53:41.717
by an energy Eigenstate
in the real world?
00:53:41.717 --> 00:53:42.258
AUDIENCE: No.
00:53:44.962 --> 00:53:46.670
PROFESSOR: Ok, there
have been a few nos.
00:53:46.670 --> 00:53:47.410
Why?
00:53:47.410 --> 00:53:49.050
Why not?
00:53:49.050 --> 00:53:53.360
Does anything really
remain invariant in time?
00:53:53.360 --> 00:53:54.370
No, right?
00:53:54.370 --> 00:53:56.060
Everything is getting
buffeted around
00:53:56.060 --> 00:53:57.480
by the rest of the universe.
00:53:57.480 --> 00:54:01.030
So, not only are these
not typical states,
00:54:01.030 --> 00:54:03.050
not only are stationary
states not typical,
00:54:03.050 --> 00:54:05.970
but they actually never
exist in the real world.
00:54:05.970 --> 00:54:07.930
So why am I talking
about them at all?
00:54:11.030 --> 00:54:13.600
So here's why.
00:54:13.600 --> 00:54:16.350
And actually I'm
going to do this here.
00:54:16.350 --> 00:54:16.950
So here's why.
00:54:16.950 --> 00:54:20.810
The reason is this guy, the
superposition principle,
00:54:20.810 --> 00:54:25.940
which tells me that if
I have possible states,
00:54:25.940 --> 00:54:27.557
I can build
superpositions of them.
00:54:27.557 --> 00:54:29.182
And this statement--
and in particular,
00:54:29.182 --> 00:54:31.180
linearity-- which says
that given any two
00:54:31.180 --> 00:54:34.250
solutions of the
Schrodinger equation,
00:54:34.250 --> 00:54:36.210
I can take a
superposition and build
00:54:36.210 --> 00:54:38.170
a new solution of the
Schrodinger equation.
00:54:38.170 --> 00:54:39.220
So, let me build it.
00:54:39.220 --> 00:54:40.720
So, in particular,
I want to exploit
00:54:40.720 --> 00:54:46.770
the linearity of the Schrodinger
equation to do the following.
00:54:49.470 --> 00:54:52.850
Suppose Psi.
00:54:52.850 --> 00:54:54.400
And I'm going to
label these by n.
00:54:54.400 --> 00:55:02.590
Psi n of x and t is equal
to e to the minus i Omega nt
00:55:02.590 --> 00:55:09.632
Phi sub En of x, where En
is equal to h bar Omega n.
00:55:09.632 --> 00:55:11.840
n labels the various different
energy Eigenfunctions.
00:55:11.840 --> 00:55:15.290
So, consider all the energy
Eigenfunctions Phi sub En.
00:55:15.290 --> 00:55:18.690
n is a number which labels them.
00:55:18.690 --> 00:55:21.540
And this is the solution to
the Schrodinger equation, which
00:55:21.540 --> 00:55:24.310
at time zero is just
equal to the energy
00:55:24.310 --> 00:55:25.440
Eigenfunction of interest.
00:55:25.440 --> 00:55:26.345
Cool?
00:55:26.345 --> 00:55:30.060
So, consider these guys.
00:55:30.060 --> 00:55:34.280
So, suppose we have
these guys such that they
00:55:34.280 --> 00:55:40.160
solve the Schrodinger equation.
00:55:40.160 --> 00:55:42.127
Solve the Schrodinger equation.
00:55:42.127 --> 00:55:44.210
Suppose these guys solve
the Schrodinger equation.
00:55:44.210 --> 00:55:48.790
Then, by linearity, we
can take Psi of x and t
00:55:48.790 --> 00:55:54.950
to be an arbitrary superposition
sum over n, c sub n, Psi sub
00:55:54.950 --> 00:55:57.480
n of x and t.
00:55:57.480 --> 00:56:01.460
And this will automatically
solve the Schrodinger equation
00:56:01.460 --> 00:56:03.660
by linearity of the
Schrodinger equation.
00:56:03.660 --> 00:56:04.194
Yeah.
00:56:04.194 --> 00:56:05.569
AUDIENCE: But
can't we just get n
00:56:05.569 --> 00:56:07.902
as the sum of the
energy Eigenstate
00:56:07.902 --> 00:56:10.840
by just applying that and
by just measuring that?
00:56:10.840 --> 00:56:13.080
PROFESSOR: Excellent.
00:56:13.080 --> 00:56:14.080
So, here's the question.
00:56:14.080 --> 00:56:15.660
The question is,
look, a minute ago
00:56:15.660 --> 00:56:21.010
you said no system is truly in
an energy Eigenstate, right?
00:56:21.010 --> 00:56:23.435
But can't we put a system
in an energy Eigenstate
00:56:23.435 --> 00:56:26.630
by just measuring the energy?
00:56:26.630 --> 00:56:27.130
Right?
00:56:27.130 --> 00:56:30.720
Isn't that exactly what the
collapse postulate says?
00:56:30.720 --> 00:56:31.806
So here's my question.
00:56:31.806 --> 00:56:33.430
How confident are
you that you actually
00:56:33.430 --> 00:56:36.370
measure the energy precisely?
00:56:36.370 --> 00:56:38.930
With what accuracy can
we measure the energy?
00:56:38.930 --> 00:56:41.430
So here's the unfortunate
truth, the unfortunate practical
00:56:41.430 --> 00:56:41.585
truth.
00:56:41.585 --> 00:56:42.610
And I'm not talking about
in principle things.
00:56:42.610 --> 00:56:45.151
I'm talking about it in practice
things in the real universe.
00:56:45.151 --> 00:56:47.840
When you measure the energy of
something, you've got a box,
00:56:47.840 --> 00:56:50.430
and the box has a dial,
and the dial has a needle,
00:56:50.430 --> 00:56:52.570
it has a finite width,
and your current meter
00:56:52.570 --> 00:56:54.800
has a finite sensitivity
to the current.
00:56:54.800 --> 00:56:57.040
So you never truly measure
the energy exactly.
00:56:57.040 --> 00:56:59.440
You measure it to
within some tolerance.
00:56:59.440 --> 00:56:59.940
And
00:56:59.940 --> 00:57:02.180
In fact, there's a
fundamental bound--
00:57:02.180 --> 00:57:06.140
there's a fundamental bound on
the accuracy with which you can
00:57:06.140 --> 00:57:08.425
make a measurement, which
is just the following.
00:57:08.425 --> 00:57:10.550
And this is the analog of
the uncertainty equation.
00:57:10.550 --> 00:57:11.760
We'll talk about
this more later,
00:57:11.760 --> 00:57:13.470
but let me just jump
ahead a little bit.
00:57:13.470 --> 00:57:17.600
Suppose I want to
measure frequency.
00:57:17.600 --> 00:57:20.180
So I have some
signal, and I look
00:57:20.180 --> 00:57:22.610
at that signal for 10 minutes.
00:57:22.610 --> 00:57:23.200
OK?
00:57:23.200 --> 00:57:26.480
Can I be absolutely confident
that this signal is in fact
00:57:26.480 --> 00:57:29.210
a plane wave with the given
frequency that I just did?
00:57:29.210 --> 00:57:30.390
No, because it could
change outside that.
00:57:30.390 --> 00:57:31.806
But more to the
point, there might
00:57:31.806 --> 00:57:33.300
have been small
variations inside.
00:57:33.300 --> 00:57:35.510
There could've been
a wavelength that
00:57:35.510 --> 00:57:38.140
could change on a time
scale longer than the time
00:57:38.140 --> 00:57:38.900
that I measured.
00:57:38.900 --> 00:57:41.180
So, to know that the
system doesn't change
00:57:41.180 --> 00:57:43.930
on a arbitrary-- that
it's strictly fixed Omega,
00:57:43.930 --> 00:57:46.640
I have to wait a very long time.
00:57:46.640 --> 00:57:49.470
And in particular, how confident
you can be of the frequency
00:57:49.470 --> 00:57:55.430
is bounded by the time over
which-- so your confidence,
00:57:55.430 --> 00:57:57.290
your uncertainty
in the frequency,
00:57:57.290 --> 00:58:01.460
is bounded in the
following fashion.
00:58:01.460 --> 00:58:04.180
Delta Omega, Delta t is always
greater than or equal to one,
00:58:04.180 --> 00:58:05.904
approximately.
00:58:05.904 --> 00:58:07.320
What this says is
that if you want
00:58:07.320 --> 00:58:09.440
to be absolute confident
of the frequency,
00:58:09.440 --> 00:58:11.817
you have to wait an
arbitrarily long time.
00:58:11.817 --> 00:58:13.650
Now if I multiply this
whole thing by h bar,
00:58:13.650 --> 00:58:14.640
I get the following.
00:58:14.640 --> 00:58:17.010
Delta E-- so this is
a classic equation
00:58:17.010 --> 00:58:20.480
that signals analysis--
Delta E, Delta t
00:58:20.480 --> 00:58:22.817
is greater than or
approximately equal to h bar.
00:58:22.817 --> 00:58:25.025
This is a hallowed time-
energy uncertainty relation,
00:58:25.025 --> 00:58:27.580
which we haven't talked about.
00:58:27.580 --> 00:58:30.400
So, in fact, it is
possible to make
00:58:30.400 --> 00:58:32.510
an arbitrarily precise
measurement of the energy.
00:58:32.510 --> 00:58:34.240
What do I have to do?
00:58:34.240 --> 00:58:36.070
I have to wait forever.
00:58:36.070 --> 00:58:38.450
How patient are you, right?
00:58:38.450 --> 00:58:39.400
So, that's the issue.
00:58:39.400 --> 00:58:41.630
In the real world, we can't make
arbitrarily long measurements,
00:58:41.630 --> 00:58:43.800
and we can't isolate systems
for an arbitrarily long amount
00:58:43.800 --> 00:58:44.379
of time.
00:58:44.379 --> 00:58:46.670
So, we can't put things in
a definite energy Eigenstate
00:58:46.670 --> 00:58:48.017
by measurement.
00:58:48.017 --> 00:58:49.100
That answer your question?
00:58:49.100 --> 00:58:49.683
AUDIENCE: Yes.
00:58:49.683 --> 00:58:50.815
PROFESSOR: Great.
00:58:50.815 --> 00:58:52.190
How many people
have seen signals
00:58:52.190 --> 00:58:55.520
in this expression, the
bound on the frequency?
00:58:55.520 --> 00:58:56.150
Oh, good.
00:58:56.150 --> 00:58:59.240
So we'll talk about that
later in the course.
00:58:59.240 --> 00:59:02.030
OK, so coming back to this.
00:59:02.030 --> 00:59:06.120
So, we have our solutions
of the Schrodinger equation
00:59:06.120 --> 00:59:07.786
that are initially
energy Eigenstates.
00:59:07.786 --> 00:59:10.035
I claim I can take an arbitrary
superposition of them,
00:59:10.035 --> 00:59:14.705
and by linearity derive
that this is also
00:59:14.705 --> 00:59:16.330
a solution to the
Schrodinger equation.
00:59:18.840 --> 00:59:29.300
And in particular, what
that tells me is-- well,
00:59:29.300 --> 00:59:31.000
another way to say
this is that if I
00:59:31.000 --> 00:59:37.660
know that Psi of x times zero
is equal to sum over n-- so
00:59:37.660 --> 00:59:40.706
if sum Psi of x--
if the wave function
00:59:40.706 --> 00:59:42.080
at some particular
moment in time
00:59:42.080 --> 00:59:50.064
can be expanded as sum over
n Cn Phi E of x, if this is
00:59:50.064 --> 00:59:51.980
my initial condition,
my initial wave function
00:59:51.980 --> 00:59:56.007
is some superposition, then I
know what the wave function is
00:59:56.007 --> 00:59:56.840
at subsequent times.
00:59:56.840 --> 01:00:00.530
The wave function by
superposition Psi of x and t
01:00:00.530 --> 01:00:05.400
is equal to sum over
n Cn e to the minus i
01:00:05.400 --> 01:00:12.830
Omega nt Phi n-- sorry, this
should've been Phi sub n-- Phi
01:00:12.830 --> 01:00:13.750
n of x.
01:00:21.109 --> 01:00:22.900
And I know this has to
be true because this
01:00:22.900 --> 01:00:25.710
is a solution to the Schrodinger
equation by construction,
01:00:25.710 --> 01:00:28.475
and at time t equals
zero, it reduces to this.
01:00:28.475 --> 01:00:30.600
So, this is a solution to
the Schrodinger equation,
01:00:30.600 --> 01:00:34.950
satisfying this condition at
the initial time t equals zero.
01:00:34.950 --> 01:00:36.520
Don't even have to
do a calculation.
01:00:36.520 --> 01:00:38.646
So, having solved the
Schrodinger equation once
01:00:38.646 --> 01:00:40.020
for energy,
Eigenstates allows me
01:00:40.020 --> 01:00:41.860
to solve it for
general superposition.
01:00:41.860 --> 01:00:43.950
However, what I just
said isn't quite enough.
01:00:43.950 --> 01:00:46.120
I need one more argument.
01:00:46.120 --> 01:00:52.260
And that one more argument is
really the stronger version
01:00:52.260 --> 01:00:54.600
of three that we talked
about before, which
01:00:54.600 --> 01:01:05.090
is that, given an
energy operator E,
01:01:05.090 --> 01:01:07.720
we find the set of
wave functions Phi sub
01:01:07.720 --> 01:01:10.610
E, the Eigenfunctions
of the energy operator,
01:01:10.610 --> 01:01:14.680
with Eigenvalue E.
01:01:14.680 --> 01:01:17.620
So, given the energy operator,
we find its Eigenfunctions.
01:01:17.620 --> 01:01:22.300
Any wave function Psi at
x-- we'll say at time zero--
01:01:22.300 --> 01:01:25.930
any function of x can
be expanded as a sum.
01:01:25.930 --> 01:01:34.530
Specific superposition sum
over n Cn Phi E sub n of x.
01:01:38.010 --> 01:01:39.860
And if any function
can be expanded
01:01:39.860 --> 01:01:42.140
as a superposition of
energy Eigenfunctions,
01:01:42.140 --> 01:01:43.985
and we know how to
take a superposition,
01:01:43.985 --> 01:01:45.620
an arbitrary
superposition of energy
01:01:45.620 --> 01:01:47.974
Eigenfunctions, and find
the corresponding solution
01:01:47.974 --> 01:01:49.140
to the Schrodinger equation.
01:01:49.140 --> 01:01:52.540
What this means is, we can take
an arbitrary initial condition
01:01:52.540 --> 01:01:55.990
and compute the full solution
of the Schrodinger equation.
01:01:55.990 --> 01:02:00.770
All we have to do is figure out
what these coefficients Cn are.
01:02:00.770 --> 01:02:02.300
Everyone cool with that?
01:02:02.300 --> 01:02:04.950
So, we have thus, using
superposition and energy
01:02:04.950 --> 01:02:08.280
Eigenvalues, totally solved
the Schrodinger equation,
01:02:08.280 --> 01:02:10.860
and reduced it to the problem
of finding these expansion
01:02:10.860 --> 01:02:12.530
coefficients.
01:02:12.530 --> 01:02:14.850
Meanwhile, these expansion
coefficients have a meaning.
01:02:14.850 --> 01:02:16.510
They correspond
to the probability
01:02:16.510 --> 01:02:18.240
that we measure the
energy to be equal
01:02:18.240 --> 01:02:21.740
to the corresponding
energy E sub n.
01:02:21.740 --> 01:02:23.960
And it's just the norm
squared of that coefficient.
01:02:23.960 --> 01:02:28.380
So those coefficients
mean something.
01:02:28.380 --> 01:02:31.790
And they allow us to
solve the problem.
01:02:31.790 --> 01:02:32.850
Cool?
01:02:32.850 --> 01:02:34.940
So this is fairly abstract.
01:02:34.940 --> 01:02:37.540
So let's make it concrete
by looking at some examples.
01:02:37.540 --> 01:02:39.660
So, just as a quick aside.
01:02:39.660 --> 01:02:42.400
This should sound an awful
lot like the Fourier theorem.
01:02:42.400 --> 01:02:44.750
And let me comment on that.
01:02:44.750 --> 01:02:47.245
This statement originally was
about a general observable
01:02:47.245 --> 01:02:48.120
and general operator.
01:02:48.120 --> 01:02:49.700
Here I'm talking
about the energy.
01:02:49.700 --> 01:02:52.559
But let's think about a
slightly more special example,
01:02:52.559 --> 01:02:53.600
or more familiar example.
01:02:53.600 --> 01:02:54.766
Let's consider the momentum.
01:02:54.766 --> 01:02:57.390
Given the momentum, we can
find a set of Eigenstates.
01:02:57.390 --> 01:02:59.510
What are the set of
good, properly normalized
01:02:59.510 --> 01:03:00.750
Eigenfunctions of momentum?
01:03:03.382 --> 01:03:05.590
What are the Eigenfunctions
of the momentum operator?
01:03:05.590 --> 01:03:06.841
AUDIENCE: E to the ikx.
01:03:06.841 --> 01:03:07.930
PROFESSOR: E to the ikx.
01:03:07.930 --> 01:03:08.430
Exactly.
01:03:08.430 --> 01:03:11.940
In particular, one
over 2 pi e to the ikx.
01:03:16.160 --> 01:03:18.160
So I claim that, for every
different value of k,
01:03:18.160 --> 01:03:20.650
I get a different value of p,
and the Eigenvalue associated
01:03:20.650 --> 01:03:24.450
to this guy is p is
equal to h bar k.
01:03:24.450 --> 01:03:25.477
That's the Eigenvalue.
01:03:25.477 --> 01:03:27.560
And we get that by acting
with the momentum, which
01:03:27.560 --> 01:03:31.600
is h bar upon i, h bar times
derivative with respect to x.
01:03:31.600 --> 01:03:33.225
Derivative with
respect to x pulls down
01:03:33.225 --> 01:03:35.230
an ik times the same thing.
01:03:35.230 --> 01:03:38.200
H bar multiplies the
k over i, kills the i,
01:03:38.200 --> 01:03:40.470
and leaves us with an overall
coefficient of h bar k.
01:03:40.470 --> 01:03:42.330
This is an Eigenfunction
of the momentum
01:03:42.330 --> 01:03:45.740
operator with
Eigenvalue h bar k.
01:03:45.740 --> 01:03:48.600
And that statement
three is the statement
01:03:48.600 --> 01:03:51.000
that an arbitrary
function f of x
01:03:51.000 --> 01:03:52.940
can be expanded
as a superposition
01:03:52.940 --> 01:03:54.790
of all possible
energy Eigenvalues.
01:03:54.790 --> 01:03:58.200
But k is continuously
valued and the momentum,
01:03:58.200 --> 01:04:03.850
so that's an integral
dk one over 2 pi,
01:04:03.850 --> 01:04:06.150
e to the ikx times
some coefficients.
01:04:06.150 --> 01:04:08.042
And those coefficients
are labeled by k,
01:04:08.042 --> 01:04:10.500
but since k is continuous, I'm
going to call it a function.
01:04:10.500 --> 01:04:12.791
And just to give it a name,
instead of calling C sub k,
01:04:12.791 --> 01:04:15.530
I'll call it f tilde of k.
01:04:15.530 --> 01:04:17.180
This is of exactly
the same form.
01:04:17.180 --> 01:04:22.110
Here is the expansion--
there's the Eigenfunction, here
01:04:22.110 --> 01:04:24.940
is the Eigenfunction, here
is the expansion coefficient,
01:04:24.940 --> 01:04:26.250
here is expansion coefficient.
01:04:26.250 --> 01:04:27.490
And this has a familiar name.
01:04:27.490 --> 01:04:30.280
It's the Fourier theorem.
01:04:30.280 --> 01:04:31.740
So, we see that
the Fourier theorem
01:04:31.740 --> 01:04:33.948
is this statement, statement
three, the superposition
01:04:33.948 --> 01:04:37.700
principal, for the
momentum operator.
01:04:37.700 --> 01:04:40.502
We also see that it's true
for the energy operator.
01:04:40.502 --> 01:04:42.210
And what we're claiming
here is that it's
01:04:42.210 --> 01:04:43.320
true for any observable.
01:04:43.320 --> 01:04:47.260
Given any observable, you
can find its Eigenfunctions,
01:04:47.260 --> 01:04:50.410
and they form a basis on the
space of all good functions,
01:04:50.410 --> 01:04:54.310
and an arbitrary function can
be expanded in that basis.
01:04:54.310 --> 01:04:58.785
So, as a last example,
consider the following.
01:04:58.785 --> 01:04:59.535
We've done energy.
01:04:59.535 --> 01:05:00.368
We've done momentum.
01:05:00.368 --> 01:05:02.240
What's another
operator we care about?
01:05:02.240 --> 01:05:04.620
What about position?
01:05:04.620 --> 01:05:07.661
What are the
Eigenfunctions of position?
01:05:07.661 --> 01:05:12.640
Well, x hat on
Delta of x minus y
01:05:12.640 --> 01:05:18.547
is equal to y Delta x minus y.
01:05:18.547 --> 01:05:20.880
So, these are the states with
definite value of position
01:05:20.880 --> 01:05:22.600
x is equal to y.
01:05:22.600 --> 01:05:26.020
And the reason this is true
is that when x is equal to y,
01:05:26.020 --> 01:05:28.320
x is the operator that
multiplies by the variable x.
01:05:28.320 --> 01:05:31.610
But it's zero, except
at x is equal to y,
01:05:31.610 --> 01:05:35.100
so we might as well
replace x by y.
01:05:35.100 --> 01:05:36.500
So, there are the
Eigenfunctions.
01:05:36.500 --> 01:05:38.000
And this statement
is a statement
01:05:38.000 --> 01:05:39.875
that we can represent
an arbitrary function f
01:05:39.875 --> 01:05:43.515
of x in a superposition of
these states of definite x.
01:05:43.515 --> 01:05:47.170
f of x is equal to the integral
over all possible expansion
01:05:47.170 --> 01:05:51.920
coefficients dy delta x
minus y times some expansion
01:05:51.920 --> 01:05:52.420
coefficient.
01:05:52.420 --> 01:05:55.069
And what's the
expansion coefficient?
01:05:55.069 --> 01:05:56.360
It's got to be a function of y.
01:05:56.360 --> 01:05:57.870
And what function
of y must it be?
01:06:01.430 --> 01:06:03.000
Just f of y.
01:06:03.000 --> 01:06:05.240
Because this integral
against this delta function
01:06:05.240 --> 01:06:06.537
had better give me f of x.
01:06:06.537 --> 01:06:08.411
And that will only be
true if this is f of x.
01:06:11.060 --> 01:06:12.994
So here we see, in some
sense, the definition
01:06:12.994 --> 01:06:13.910
of the delta function.
01:06:13.910 --> 01:06:16.702
But really, this is a statement
of the superposition principle,
01:06:16.702 --> 01:06:18.660
the statement that any
function can be expanded
01:06:18.660 --> 01:06:21.210
as a superposition of
Eigenfunctions of the position
01:06:21.210 --> 01:06:22.300
operator.
01:06:22.300 --> 01:06:24.620
Any function can be
expanded as a superposition
01:06:24.620 --> 01:06:26.830
of Eigenfunctions of momentum.
01:06:26.830 --> 01:06:29.630
Any function can be
expanded as a superposition
01:06:29.630 --> 01:06:31.780
of Eigenfunctions of energy.
01:06:31.780 --> 01:06:34.910
Any function can be
expanded as a superposition
01:06:34.910 --> 01:06:40.110
of Eigenfunctions of any
operator of your choice.
01:06:40.110 --> 01:06:40.950
OK?
01:06:40.950 --> 01:06:44.650
The special cases-- the Fourier
theorem, the general cases,
01:06:44.650 --> 01:06:47.170
the superposition postulate.
01:06:47.170 --> 01:06:48.660
Cool?
01:06:48.660 --> 01:06:49.930
Powerful tool.
01:06:49.930 --> 01:06:51.640
And we've used
this powerful tool
01:06:51.640 --> 01:06:54.550
to write down a general
expression for a solution
01:06:54.550 --> 01:06:56.050
to the Schrodinger equation.
01:06:56.050 --> 01:06:57.060
That's good.
01:06:57.060 --> 01:06:58.156
That's progress.
01:06:58.156 --> 01:06:59.780
So let's look at some
examples of this.
01:07:02.957 --> 01:07:03.790
I can leave this up.
01:07:03.790 --> 01:07:07.900
So, our first example is going
to be for the free particle.
01:07:07.900 --> 01:07:12.720
So, a particle whose
energy operator
01:07:12.720 --> 01:07:14.870
has no potential whatsoever.
01:07:14.870 --> 01:07:16.960
So the energy operator
is going to be
01:07:16.960 --> 01:07:19.340
just equal to p squared upon 2m.
01:07:22.060 --> 01:07:22.880
Kinetic energy.
01:07:22.880 --> 01:07:25.100
Yeah.
01:07:25.100 --> 01:07:28.030
AUDIENCE: When you say
any wave function can
01:07:28.030 --> 01:07:29.925
be expanded in terms of--
01:07:29.925 --> 01:07:31.300
PROFESSOR: Energy
Eigenfunctions,
01:07:31.300 --> 01:07:33.430
position Eigenfunctions,
momentum Eigenfunctions--
01:07:33.430 --> 01:07:35.350
AUDIENCE: Eigenbasis,
does the Eigenbasis
01:07:35.350 --> 01:07:38.565
have to come from an operator
corresponding to an observable?
01:07:38.565 --> 01:07:39.190
PROFESSOR: Yes.
01:07:39.190 --> 01:07:39.690
Absolutely.
01:07:39.690 --> 01:07:41.106
I'm starting with
that assumption.
01:07:41.106 --> 01:07:42.420
AUDIENCE: OK.
01:07:42.420 --> 01:07:44.590
PROFESSOR: So, again,
this is a first pass
01:07:44.590 --> 01:07:46.720
of the axioms of
quantum mechanics.
01:07:46.720 --> 01:07:50.492
We'll make this more precise,
and we'll make it more general,
01:07:50.492 --> 01:07:52.950
later on in the course, as we
go through a second iteration
01:07:52.950 --> 01:07:53.660
of this.
01:07:53.660 --> 01:07:55.280
And there we'll talk about
exactly what we need,
01:07:55.280 --> 01:07:57.160
and what operators are
appropriate operators.
01:07:57.160 --> 01:07:59.201
But for the moment, the
sufficient and physically
01:07:59.201 --> 01:08:00.850
correct answer is,
operators correspond
01:08:00.850 --> 01:08:02.070
to each observable values.
01:08:02.070 --> 01:08:02.569
Yeah.
01:08:02.569 --> 01:08:05.150
AUDIENCE: So are the set of
all reasonable wave functions
01:08:05.150 --> 01:08:07.320
in the vector space
that is the same
01:08:07.320 --> 01:08:10.674
as the one with
the Eigenfunctions?
01:08:10.674 --> 01:08:12.340
PROFESSOR: That's an
excellent question.
01:08:12.340 --> 01:08:14.322
In general, no.
01:08:14.322 --> 01:08:15.280
So here's the question.
01:08:15.280 --> 01:08:17.470
The question is,
look, if this is true,
01:08:17.470 --> 01:08:20.100
shouldn't it be that the
Eigenfunctions, since they're
01:08:20.100 --> 01:08:21.660
our basis for the
good functions,
01:08:21.660 --> 01:08:24.430
are inside the space of
reasonable functions,
01:08:24.430 --> 01:08:26.430
they should also be
reasonable functions, right?
01:08:26.430 --> 01:08:28.710
Because if you're going
to expand-- for example,
01:08:28.710 --> 01:08:29.819
consider two dimensional
vector space.
01:08:29.819 --> 01:08:31.235
And you want to
say any vector can
01:08:31.235 --> 01:08:34.990
be expanded in a basis of pairs
of vectors in two dimensions,
01:08:34.990 --> 01:08:36.176
like x and y.
01:08:36.176 --> 01:08:38.300
You really want to make
sure that those vectors are
01:08:38.300 --> 01:08:39.341
inside your vector space.
01:08:39.341 --> 01:08:41.250
But if you say this
vector in this space
01:08:41.250 --> 01:08:43.580
can be expanded in terms
of two vectors, this vector
01:08:43.580 --> 01:08:46.400
and that vector, you're
in trouble, right?
01:08:46.400 --> 01:08:47.774
That's not going
to work so well.
01:08:47.774 --> 01:08:50.024
So you want to make sure
that your vectors, your basis
01:08:50.024 --> 01:08:51.819
vectors, are in the space.
01:08:51.819 --> 01:08:55.069
For position, the basis
vector's a delta function.
01:08:55.069 --> 01:08:58.130
Is that a smooth, continuous
normalizable function?
01:08:58.130 --> 01:08:58.930
No.
01:08:58.930 --> 01:09:01.029
For momentum, the
basis functions
01:09:01.029 --> 01:09:03.200
are plane waves that
extend off to infinity
01:09:03.200 --> 01:09:05.399
and have support everywhere.
01:09:05.399 --> 01:09:08.020
Is that a normalizable
reasonable function?
01:09:08.020 --> 01:09:08.520
No.
01:09:08.520 --> 01:09:10.634
So, both of these
sets are really bad.
01:09:10.634 --> 01:09:13.300
So, at that point you might say,
look, this is clearly nonsense.
01:09:13.300 --> 01:09:14.550
But here's an important thing.
01:09:17.029 --> 01:09:18.695
So this is a totally
mathematical aside,
01:09:18.695 --> 01:09:19.840
and for those of you who
don't care about the math,
01:09:19.840 --> 01:09:20.689
don't worry about it.
01:09:20.689 --> 01:09:22.050
Well, these guys
don't technically
01:09:22.050 --> 01:09:23.841
live in the space of
non-stupid functions--
01:09:23.841 --> 01:09:27.800
reasonable, smooth,
normalizable functions.
01:09:27.800 --> 01:09:31.050
What you can show is that
they exist in the closure,
01:09:31.050 --> 01:09:33.700
in the completion of that space.
01:09:33.700 --> 01:09:34.550
OK?
01:09:34.550 --> 01:09:38.439
So, you can find a
sequence of wave functions
01:09:38.439 --> 01:09:40.970
that are good wave functions,
an infinite sequence,
01:09:40.970 --> 01:09:43.238
that eventually that
infinite sequence converges
01:09:43.238 --> 01:09:45.029
to these guys, even
though these are silly.
01:09:45.029 --> 01:09:47.090
So, for example, for the
position Eigenstates,
01:09:47.090 --> 01:09:50.398
the delta function is not a
continuous smooth function.
01:09:50.398 --> 01:09:51.439
It's not even a function.
01:09:51.439 --> 01:09:53.540
Really, it's some god- awful
thing called a distribution.
01:09:53.540 --> 01:09:54.360
It's some horrible thing.
01:09:54.360 --> 01:09:56.485
It's the thing that tells
you, give it an integral,
01:09:56.485 --> 01:09:57.800
it'll give you a number.
01:09:57.800 --> 01:09:59.722
Or a function.
01:09:59.722 --> 01:10:01.180
But how do we build
this as a limit
01:10:01.180 --> 01:10:02.513
of totally reasonable functions?
01:10:02.513 --> 01:10:03.960
We've already done that.
01:10:03.960 --> 01:10:07.670
Take this function with
area one, and if you want,
01:10:07.670 --> 01:10:10.230
you can round this out by
making it hyperbolic tangents.
01:10:10.230 --> 01:10:11.190
OK?
01:10:11.190 --> 01:10:13.360
We did it on one of
the problem sets.
01:10:13.360 --> 01:10:15.360
And then just make it
more narrow and more tall.
01:10:15.360 --> 01:10:17.901
And keep making it more narrow
and more tall, and more narrow
01:10:17.901 --> 01:10:19.930
and more tall, keeping
its area to be one.
01:10:19.930 --> 01:10:24.190
And I claim that eventually
that series, that sequence
01:10:24.190 --> 01:10:26.340
of functions, converges
to the delta function.
01:10:26.340 --> 01:10:29.430
So, while this function is
not technically in our space,
01:10:29.430 --> 01:10:31.600
it's in the completion
of our space,
01:10:31.600 --> 01:10:34.290
in the sense that we take a
series and they converge to it.
01:10:34.290 --> 01:10:36.715
And that's what you need for
this theorem to work out.
01:10:36.715 --> 01:10:38.590
That's what you need
for the Fourier theorem.
01:10:38.590 --> 01:10:40.470
And in some sense,
that observation
01:10:40.470 --> 01:10:42.070
was really the
genius of Fourier,
01:10:42.070 --> 01:10:44.571
understanding that
that could be done.
01:10:44.571 --> 01:10:46.070
That was totally
mathematical aside.
01:10:46.070 --> 01:10:47.180
But that answer your question?
01:10:47.180 --> 01:10:47.480
AUDIENCE: Yes.
01:10:47.480 --> 01:10:48.330
PROFESSOR: OK.
01:10:48.330 --> 01:10:49.891
Every once in a
while I can't resist
01:10:49.891 --> 01:10:51.390
talking about these
sort of details,
01:10:51.390 --> 01:10:52.515
because I really like them.
01:10:52.515 --> 01:10:56.350
But it's good to know that
stupid things like this
01:10:56.350 --> 01:10:58.930
can't matter for
us, and they don't.
01:10:58.930 --> 01:11:01.820
But it's a very good question.
01:11:01.820 --> 01:11:03.920
If you're confused about
some mathematical detail,
01:11:03.920 --> 01:11:05.970
no matter how elementary, ask.
01:11:05.970 --> 01:11:09.410
If you're confused, someone
else the room is also confused.
01:11:09.410 --> 01:11:11.350
So please don't hesitate.
01:11:11.350 --> 01:11:14.990
OK, so our first example's
going to be the free particle.
01:11:14.990 --> 01:11:17.820
And this operator can be
written in a nice way.
01:11:17.820 --> 01:11:21.980
We can write it as minus--
so p is h bar upon iddx,
01:11:21.980 --> 01:11:23.550
so this line is
minus h bar squared
01:11:23.550 --> 01:11:27.434
upon 2m to the derivative
with respect to x.
01:11:27.434 --> 01:11:28.600
There's the energy operator.
01:11:31.550 --> 01:11:35.750
So, we want to solve
for the wave functions.
01:11:35.750 --> 01:11:37.670
So let's solve it
using an expansion
01:11:37.670 --> 01:11:38.660
in terms of energy
Eigenfunctions.
01:11:38.660 --> 01:11:40.243
So what are the
energy Eigenfunctions?
01:11:40.243 --> 01:11:45.280
We want to find the
functions E on Phi sub E
01:11:45.280 --> 01:11:47.007
such that this is
equal to-- whoops.
01:11:47.007 --> 01:11:47.840
That's not a vector.
01:11:47.840 --> 01:11:54.510
That's a hat-- such as this is
equal to a number E Phi sub E.
01:11:54.510 --> 01:11:56.290
But given this
energy operator, this
01:11:56.290 --> 01:12:00.950
says that minus h bar squared
over 2m-- whoops, that's a 2.
01:12:00.950 --> 01:12:10.060
2m-- Phi prime prime of
x is equal to E Phi of x.
01:12:10.060 --> 01:12:18.240
Or equivalently, Phi prime
prime of x plus 2me over h bar
01:12:18.240 --> 01:12:21.720
squared Phi of x
is equal to zero.
01:12:27.140 --> 01:12:30.280
So I'm just going to
call 2me-- because it's
01:12:30.280 --> 01:12:32.030
annoying to write it
over and over again--
01:12:32.030 --> 01:12:32.460
over h bar squared.
01:12:32.460 --> 01:12:34.170
Well, first off,
what are its units?
01:12:34.170 --> 01:12:35.810
What are the units
of this coefficient?
01:12:35.810 --> 01:12:36.770
Well, you could do it two ways.
01:12:36.770 --> 01:12:38.770
You could either do dimensional
analysis of each thing here,
01:12:38.770 --> 01:12:41.228
or you could just know that we
started with a dimensionally
01:12:41.228 --> 01:12:42.910
sensible equation,
and this has units
01:12:42.910 --> 01:12:46.160
of this divided by length twice.
01:12:46.160 --> 01:12:48.182
So this must have to
whatever length squared.
01:12:48.182 --> 01:12:49.640
So I'm going to
call this something
01:12:49.640 --> 01:12:51.130
like k squared,
something that has
01:12:51.130 --> 01:12:52.504
units of one over
length squared.
01:12:54.940 --> 01:12:56.410
And the general
solution of this is
01:12:56.410 --> 01:13:02.750
that phi E E of x-- well, this
is a second order differential
01:13:02.750 --> 01:13:05.240
equation that will have two
solutions with two expansion
01:13:05.240 --> 01:13:15.960
coefficients-- A e to the ikx
plus B e to the minus iks.
01:13:15.960 --> 01:13:19.070
A state with definite momentum
and definite negative momentum
01:13:19.070 --> 01:13:26.736
where such that E is equal to h
bar squared k squared upon 2m.
01:13:26.736 --> 01:13:30.720
And we get that just from this.
01:13:30.720 --> 01:13:36.386
So, this is the solution of the
energy Eigenfunction equation.
01:13:36.386 --> 01:13:37.510
Just a note of terminology.
01:13:37.510 --> 01:13:38.950
People sometimes
call the equation
01:13:38.950 --> 01:13:40.984
determining an energy
Eigenfunction-- the energy
01:13:40.984 --> 01:13:42.400
Eigenfunction
equation-- sometimes
01:13:42.400 --> 01:13:45.110
that's refer to as the
Schrodinger equation.
01:13:45.110 --> 01:13:48.060
That's a sort of cruel
thing to do to language,
01:13:48.060 --> 01:13:50.380
because the Schrodinger's
equation is about time
01:13:50.380 --> 01:13:54.500
evolution, and this equation
is about energy Eigenfunctions.
01:13:54.500 --> 01:13:57.670
Now, it's true that energy
Eigenfunctions evolve
01:13:57.670 --> 01:14:00.151
in a particularly simple
way under time evolution,
01:14:00.151 --> 01:14:01.400
but it's a different equation.
01:14:01.400 --> 01:14:03.890
This is telling you about
energy Eigenstates, OK?
01:14:07.370 --> 01:14:09.170
And then more
discussion of this is
01:14:09.170 --> 01:14:12.100
done in the notes, which I will
leave aside for the moment.
01:14:12.100 --> 01:14:15.250
But I want to do one more
example before we take off.
01:14:15.250 --> 01:14:16.384
Wow.
01:14:16.384 --> 01:14:18.300
We got through a lot
less than expected today.
01:14:21.410 --> 01:14:23.190
And the one last example
is the following.
01:14:23.190 --> 01:14:25.085
It's a particle
in a box And this
01:14:25.085 --> 01:14:26.500
is going to be important
for your problem sets,
01:14:26.500 --> 01:14:28.833
so I'm going to go ahead and
get this one out of the way
01:14:28.833 --> 01:14:30.620
as quickly as possible.
01:14:30.620 --> 01:14:31.680
So, example two.
01:14:35.130 --> 01:14:36.290
Particle in a box.
01:14:40.979 --> 01:14:42.520
So, what I mean by
particle in a box.
01:14:42.520 --> 01:14:45.551
I'm going to take a system
that has a deep well.
01:14:45.551 --> 01:14:47.550
So what I'm drawing here
is the potential energy
01:14:47.550 --> 01:14:55.130
U of x, where this is
some energy E naught,
01:14:55.130 --> 01:14:57.270
and this is the
energy zero, and this
01:14:57.270 --> 01:14:59.330
is the position x
equals zero, and this
01:14:59.330 --> 01:15:00.724
is position x equals l.
01:15:00.724 --> 01:15:02.640
And I'm going to idealize
this by saying look,
01:15:02.640 --> 01:15:05.300
I'm going to be interested
in low energy physics,
01:15:05.300 --> 01:15:08.030
so I'm going to just treat
this as infinitely deep.
01:15:08.030 --> 01:15:09.770
And meanwhile, my
life is easier if I
01:15:09.770 --> 01:15:11.180
don't think about curvy
bottoms but I just
01:15:11.180 --> 01:15:12.730
think about things
as being constant.
01:15:12.730 --> 01:15:16.460
So, my idealization
is going to be
01:15:16.460 --> 01:15:22.030
that the well is
infinitely high and square.
01:15:22.030 --> 01:15:27.630
So out here the
potential is infinite,
01:15:27.630 --> 01:15:29.205
and in here the
potential is zero.
01:15:29.205 --> 01:15:32.630
U equals inside, between
zero and l for x.
01:15:36.270 --> 01:15:38.310
So that's my system
particle in a box.
01:15:38.310 --> 01:15:40.987
So, let's find the
energy Eigenfunctions.
01:15:40.987 --> 01:15:42.945
And again, it's the same
differential equations
01:15:42.945 --> 01:15:43.444
as before.
01:15:47.264 --> 01:15:49.430
So, first off, before we
even solve anything, what's
01:15:49.430 --> 01:15:51.820
the probability that I
find x less than zero,
01:15:51.820 --> 01:15:58.822
or find the particle
at x greater than l?
01:15:58.822 --> 01:15:59.735
AUDIENCE: Zero.
01:15:59.735 --> 01:16:01.360
PROFESSOR: Right,
because the potential
01:16:01.360 --> 01:16:02.770
is infinitely large out there.
01:16:02.770 --> 01:16:03.650
It's just not going to happen.
01:16:03.650 --> 01:16:04.810
If you found it there,
that would correspond
01:16:04.810 --> 01:16:06.590
to a particle of
infinite energy,
01:16:06.590 --> 01:16:08.180
and that's not going to happen.
01:16:08.180 --> 01:16:09.770
So, our this tells
us effectively
01:16:09.770 --> 01:16:12.530
the boundary
condition Psi of x is
01:16:12.530 --> 01:16:14.200
equal to zero outside the box.
01:16:16.984 --> 01:16:18.400
So all we have to
do is figure out
01:16:18.400 --> 01:16:22.077
what the wave function is inside
the box between zero and l.
01:16:22.077 --> 01:16:23.910
And meanwhile, what
must be true of the wave
01:16:23.910 --> 01:16:25.934
function at zero and at l?
01:16:25.934 --> 01:16:27.850
It's got to actually
vanish at the boundaries.
01:16:27.850 --> 01:16:30.830
So this gives us boundary
conditions outside the box
01:16:30.830 --> 01:16:35.914
and at the boundaries x
equals zero, x equals l.
01:16:40.570 --> 01:16:43.310
But, what's our differential
equation inside the box?
01:16:43.310 --> 01:16:46.470
Inside the box, well,
the potential is zero.
01:16:46.470 --> 01:16:48.410
So the equation is the
same as the equation
01:16:48.410 --> 01:16:49.950
for a free particle.
01:16:49.950 --> 01:16:51.540
It's just this guy.
01:16:51.540 --> 01:16:53.900
And we know what
the solutions are.
01:16:53.900 --> 01:16:56.150
So the solutions can be
written in the following form.
01:16:56.150 --> 01:17:01.780
Therefore inside the
wave function-- whoops.
01:17:01.780 --> 01:17:03.490
Let me write this as
Phi sub E-- Phi sub
01:17:03.490 --> 01:17:05.390
E is a superposition of two.
01:17:05.390 --> 01:17:07.220
And instead of writing
it as exponentials,
01:17:07.220 --> 01:17:10.800
I'm going to write it
as sines and cosines,
01:17:10.800 --> 01:17:13.190
because you can express
them in terms of each other.
01:17:13.190 --> 01:17:23.030
Alpha cosine of kx
plus Beta sine of kx,
01:17:23.030 --> 01:17:27.950
where again Alpha and Beta
are general complex numbers.
01:17:27.950 --> 01:17:33.210
But, we must satisfy the
boundary conditions imposed
01:17:33.210 --> 01:17:35.830
by our potential at x
equals zero and x equals l.
01:17:35.830 --> 01:17:38.989
So from x equals
zero, we find that Phi
01:17:38.989 --> 01:17:40.280
must vanish when x equals zero.
01:17:40.280 --> 01:17:42.238
When x equals zero, this
is automatically zero.
01:17:42.238 --> 01:17:43.520
Sine of zero is zero.
01:17:43.520 --> 01:17:45.820
Cosine of zero is one.
01:17:45.820 --> 01:17:49.630
So that tells us that
Alpha is equal to zero.
01:17:49.630 --> 01:17:53.490
Meanwhile, the condition that at
x equals l-- the wave function
01:17:53.490 --> 01:17:57.355
must also vanish-- tells us
that-- so this term is gone,
01:17:57.355 --> 01:17:59.700
set off with zero-- this
term, when x is equal to l,
01:17:59.700 --> 01:18:01.580
had better also be zero.
01:18:01.580 --> 01:18:03.630
We can solve that by
setting Beta equal to zero,
01:18:03.630 --> 01:18:05.690
but then our wave
function is just zero.
01:18:05.690 --> 01:18:07.790
And that's a really
stupid wave function.
01:18:07.790 --> 01:18:08.360
So we don't want to do that.
01:18:08.360 --> 01:18:09.776
We don't want to
set Beta to zero.
01:18:09.776 --> 01:18:12.840
Instead, what must we do?
01:18:12.840 --> 01:18:16.566
Well, we've got a sine,
and depending on what k is,
01:18:16.566 --> 01:18:18.440
it starts at zero and
it ends somewhere else.
01:18:18.440 --> 01:18:19.810
But we need it hit zero.
01:18:19.810 --> 01:18:21.960
So only for a very
special value of k will
01:18:21.960 --> 01:18:25.400
it actually hit zero
at the end of l.
01:18:25.400 --> 01:18:26.600
We need kl equals zero.
01:18:29.170 --> 01:18:31.560
Or really, kl is
a multiple of pi.
01:18:31.560 --> 01:18:33.980
Kl is equal-- and we
want it to not be zero,
01:18:33.980 --> 01:18:37.880
so I'll call it n plus
1, an integer, times pi.
01:18:37.880 --> 01:18:42.940
Or equivalently, k is equal
to sub n is equal to n plus 1,
01:18:42.940 --> 01:18:46.540
where n goes from zero to any
large positive integer, pi
01:18:46.540 --> 01:18:49.500
over l.
01:18:49.500 --> 01:18:51.210
So the energy
Eigenfunction's here.
01:18:54.711 --> 01:18:56.710
The energy Eigenfunction
is some normalization--
01:18:56.710 --> 01:19:05.940
whoops-- a sub n
sine of k and x.
01:19:14.470 --> 01:19:17.720
And where kn is equal to
this-- and as a consequence,
01:19:17.720 --> 01:19:21.820
E is equal to h bar squared
kn can squared E sub
01:19:21.820 --> 01:19:27.160
n is h bar squared kn squared
over 2m, which is equal to h
01:19:27.160 --> 01:19:32.860
bar squared-- just plugging
in-- pi squared n plus 1 squared
01:19:32.860 --> 01:19:34.260
over 2ml squared.
01:19:37.180 --> 01:19:39.860
And what we found is
something really interesting.
01:19:39.860 --> 01:19:42.810
What we found is, first
off, that the wave functions
01:19:42.810 --> 01:19:45.170
look like-- well,
the ground state,
01:19:45.170 --> 01:19:48.350
the lowest possible energy
there is n equals zero.
01:19:48.350 --> 01:19:51.510
For n equals zero, this is
just a single half a sine wave.
01:19:51.510 --> 01:19:52.260
It does this.
01:19:52.260 --> 01:19:54.200
This is the n equals zero state.
01:19:54.200 --> 01:19:56.330
And it has some energy,
which is E zero.
01:19:56.330 --> 01:19:58.820
And in particular, E zero
is not equal to zero.
01:19:58.820 --> 01:20:03.905
E zero is equal to h bar squared
pi squared over 2ml squared.
01:20:08.784 --> 01:20:10.450
It is impossible for
a particle in a box
01:20:10.450 --> 01:20:14.800
to have an energy lower
than some minimal value E
01:20:14.800 --> 01:20:17.260
naught, which is not zero.
01:20:17.260 --> 01:20:21.060
You cannot have less
energy than this.
01:20:21.060 --> 01:20:22.620
Everyone agree with that?
01:20:22.620 --> 01:20:24.990
There is no such Eigenstate
with energy less than this.
01:20:24.990 --> 01:20:26.510
Meanwhile, it's worse.
01:20:26.510 --> 01:20:28.540
The next energy is
when n is equal to 1,
01:20:28.540 --> 01:20:32.730
because if we decrease
the wavelength or increase
01:20:32.730 --> 01:20:35.010
k a little bit, we get
something that looks like this,
01:20:35.010 --> 01:20:36.810
and that doesn't satisfy
our boundary condition.
01:20:36.810 --> 01:20:38.630
In order to satisfy
our boundary condition,
01:20:38.630 --> 01:20:40.055
we're going to have to
eventually have it cross over
01:20:40.055 --> 01:20:41.640
and get to zero again.
01:20:41.640 --> 01:20:47.390
And if I could only draw--
I'll draw it up here--
01:20:47.390 --> 01:20:48.480
it looks like this.
01:20:48.480 --> 01:20:50.080
And this has an
energy E one, which
01:20:50.080 --> 01:20:51.750
you can get by
plugging one in here.
01:20:51.750 --> 01:20:55.160
And that differs by one,
two, four, a factor of four
01:20:55.160 --> 01:20:56.230
from this guy.
01:20:56.230 --> 01:20:59.110
E one is four E zero.
01:20:59.110 --> 01:21:00.330
And so on and so forth.
01:21:00.330 --> 01:21:03.180
The energies are gaped.
01:21:03.180 --> 01:21:04.770
They're spread away
from each other.
01:21:04.770 --> 01:21:06.872
The energies are discrete.
01:21:06.872 --> 01:21:09.080
And they get further and
further away from each other
01:21:09.080 --> 01:21:12.640
as we go to higher
and higher energies.
01:21:12.640 --> 01:21:15.180
So this is already
a peculiar fact,
01:21:15.180 --> 01:21:18.087
and we'll explore some of
its consequences later on.
01:21:18.087 --> 01:21:19.920
But here's that I want
to emphasize for you.
01:21:19.920 --> 01:21:23.630
Already in the first
most trivial example
01:21:23.630 --> 01:21:25.460
of solving a
Schrodinger equation,
01:21:25.460 --> 01:21:27.850
or actually even before
that, just finding the energy
01:21:27.850 --> 01:21:31.090
Eigenvalues and the energy of
Eigenfunctions of the simplest
01:21:31.090 --> 01:21:33.575
system you possibly could,
either a free particular,
01:21:33.575 --> 01:21:35.950
or a particle in
a box, a particle
01:21:35.950 --> 01:21:38.800
trapped inside a potential
well, what we discovered
01:21:38.800 --> 01:21:41.220
is that the energy
Eigenvalues, the allowed values
01:21:41.220 --> 01:21:46.520
of the energy, are discrete, and
that they're greater than zero.
01:21:46.520 --> 01:21:48.522
You can never have zero energy.
01:21:48.522 --> 01:21:49.980
And if that doesn't
sound familiar,
01:21:49.980 --> 01:21:51.340
let me remind you of something.
01:21:51.340 --> 01:21:55.380
The spectrum of light coming
off of a gas of hot hydrogen
01:21:55.380 --> 01:21:57.420
is discrete.
01:21:57.420 --> 01:21:59.870
And no one's ever
found a zero energy
01:21:59.870 --> 01:22:02.440
beam of light coming out of it.
01:22:02.440 --> 01:22:05.550
And we're going to make contact
with that experimental data.
01:22:05.550 --> 01:22:07.550
That's going to be part
of the job for the rest.
01:22:07.550 --> 01:22:08.300
See you next time.
01:22:11.270 --> 01:22:15.220
[APPLAUSE]