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PROFESSOR: So it's a
beautiful recording.
00:00:26.780 --> 00:00:30.180
OK, so to get
started, questions?
00:00:33.090 --> 00:00:34.040
From last time?
00:00:34.040 --> 00:00:35.530
Barton covered for me last time.
00:00:35.530 --> 00:00:36.610
I fled.
00:00:36.610 --> 00:00:38.350
I was out of town.
00:00:38.350 --> 00:00:40.654
I was at a math conference.
00:00:40.654 --> 00:00:41.570
It was pretty surreal.
00:00:41.570 --> 00:00:42.830
Questions, yes.
00:00:42.830 --> 00:00:44.590
AUDIENCE: [INAUDIBLE]
about the exam?
00:00:44.590 --> 00:00:45.000
PROFESSOR: About--?
00:00:45.000 --> 00:00:45.520
AUDIENCE: The exam.
00:00:45.520 --> 00:00:46.020
PROFESSOR: The exam.
00:00:46.020 --> 00:00:46.720
Yes, absolutely.
00:00:46.720 --> 00:00:52.340
So the exam is, as you all
know, on Thursday, a week hence.
00:00:52.340 --> 00:00:54.380
So on Tuesday we
will have a lecture.
00:00:54.380 --> 00:00:58.340
The material Tuesday will
not be covered on the exam.
00:00:58.340 --> 00:01:00.580
The exam will be a
review of everything
00:01:00.580 --> 00:01:03.010
through today's lecture,
including the problems that,
00:01:03.010 --> 00:01:04.486
which for some
technical reason I
00:01:04.486 --> 00:01:05.860
don't know why
didn't get posted.
00:01:05.860 --> 00:01:09.440
But it should be up
after lecture today.
00:01:09.440 --> 00:01:13.120
The exam will be a
combination of short questions
00:01:13.120 --> 00:01:15.012
and computations.
00:01:15.012 --> 00:01:17.345
It will not focus on an
enormous number of computations.
00:01:17.345 --> 00:01:19.030
It will focus more
on conceptual things.
00:01:19.030 --> 00:01:22.440
But there will be a few
calculations on the exam.
00:01:22.440 --> 00:01:25.230
And I will post some
practice problems
00:01:25.230 --> 00:01:27.092
over the next couple of days.
00:01:27.092 --> 00:01:29.050
AUDIENCE: Do we have a
problem set [INAUDIBLE]?
00:01:29.050 --> 00:01:31.230
PROFESSOR: You do have a
problem set due Tuesday.
00:01:31.230 --> 00:01:33.790
And that is part of your
preparation for the exam.
00:01:33.790 --> 00:01:36.560
Here's a basic strategy
for exams for this class.
00:01:36.560 --> 00:01:39.170
Anything that's on a
problem set is fair game.
00:01:39.170 --> 00:01:41.330
Anything that's not
covered on a problem set
00:01:41.330 --> 00:01:42.694
is not going to be fair game.
00:01:42.694 --> 00:01:44.360
If you haven't seen
a new problem on it,
00:01:44.360 --> 00:01:47.800
broadly construed, then you
won't-- I won't test you
00:01:47.800 --> 00:01:50.800
on a topic you haven't
done problems on before.
00:01:50.800 --> 00:01:53.180
But I will take
problems and ideas
00:01:53.180 --> 00:01:55.920
that you've studied before and
spin them slightly differently
00:01:55.920 --> 00:01:58.360
to make you think through
them in real time on the exam.
00:01:58.360 --> 00:01:58.860
OK?
00:02:01.210 --> 00:02:03.520
From my point of view,
the purpose of these exams
00:02:03.520 --> 00:02:04.682
is not to give you a grade.
00:02:04.682 --> 00:02:05.890
I don't care about the grade.
00:02:05.890 --> 00:02:08.830
The purpose of these
exams is to give you
00:02:08.830 --> 00:02:11.402
feedback on your understanding.
00:02:11.402 --> 00:02:13.860
It's very easy to slip through
quantum mechanics and think,
00:02:13.860 --> 00:02:15.345
oh yeah, I totally-- I got this.
00:02:15.345 --> 00:02:16.530
This is fine.
00:02:16.530 --> 00:02:20.890
But it's not always
an accurate read.
00:02:20.890 --> 00:02:22.662
So that's the point.
00:02:22.662 --> 00:02:25.010
Did that answer your question?
00:02:25.010 --> 00:02:26.060
Other questions?
00:02:26.060 --> 00:02:27.039
Exam or-- yeah.
00:02:27.039 --> 00:02:29.080
AUDIENCE: About the harmonic
oscillator actually.
00:02:29.080 --> 00:02:29.280
PROFESSOR: Excellent.
00:02:29.280 --> 00:02:31.474
AUDIENCE: So when
we solved it Tuesday
00:02:31.474 --> 00:02:34.470
using the series method,
so there are two solutions
00:02:34.470 --> 00:02:37.294
technically, the even solution
and the odd term solution.
00:02:37.294 --> 00:02:39.210
So did boundary conditions
force the other one
00:02:39.210 --> 00:02:42.560
to be completely zero, like
the coefficient in front of it?
00:02:42.560 --> 00:02:47.050
So there's like an A0 term which
determines all the other ones.
00:02:47.050 --> 00:02:50.104
But there's an A0 term and an
A1 term for the evens and odds.
00:02:50.104 --> 00:02:51.770
So did the other ones
just have to be 0?
00:02:51.770 --> 00:02:53.460
PROFESSOR: This is a
really good question.
00:02:53.460 --> 00:02:54.670
This is an excellent question.
00:02:54.670 --> 00:02:56.544
Let me ask the question
slightly differently.
00:02:56.544 --> 00:02:58.650
And tell me if this
is the same question.
00:02:58.650 --> 00:03:00.130
When we wrote down our
differential equation--
00:03:00.130 --> 00:03:01.963
so last time we did the
harmonic oscillator.
00:03:01.963 --> 00:03:04.840
And Barton did give you
the brute force strategy
00:03:04.840 --> 00:03:06.150
for the harmonic oscillator.
00:03:06.150 --> 00:03:07.775
We want to find the
energy eigenstates,
00:03:07.775 --> 00:03:10.780
because that's what we do to
solve the Schrodinger equation.
00:03:10.780 --> 00:03:13.000
And we turn that into a
differential equation.
00:03:13.000 --> 00:03:14.625
And we solve this
differential equation
00:03:14.625 --> 00:03:18.190
by doing an asymptotic analysis
and then a series expansion.
00:03:18.190 --> 00:03:21.590
Now, this is a second order
differential equation.
00:03:21.590 --> 00:03:22.980
Everyone agree with that?
00:03:22.980 --> 00:03:24.730
It's a second order
differential equation.
00:03:24.730 --> 00:03:27.420
However, in our
series expansion we
00:03:27.420 --> 00:03:33.010
ended up with one integration
constant, not two.
00:03:33.010 --> 00:03:34.494
How does that work?
00:03:34.494 --> 00:03:36.910
How can it be that there was
only one integration constant
00:03:36.910 --> 00:03:37.410
and not two?
00:03:37.410 --> 00:03:39.407
It's a second order
differential equation.
00:03:39.407 --> 00:03:40.240
Is this he question?
00:03:40.240 --> 00:03:40.540
AUDIENCE: Yeah.
00:03:40.540 --> 00:03:42.020
PROFESSOR: OK, and this
is an excellent question.
00:03:42.020 --> 00:03:44.830
Because it must be true,
that there are two solutions.
00:03:44.830 --> 00:03:46.930
It cannot be that there
is just one solution.
00:03:46.930 --> 00:03:48.750
It's a second order
differential equation.
00:03:48.750 --> 00:03:51.945
Their existence in uniqueness
theorems, which tell us there
00:03:51.945 --> 00:03:54.027
are two integration constants.
00:03:54.027 --> 00:03:56.110
So how can it possibly be
that there was only one?
00:03:56.110 --> 00:03:57.930
Well, we did something
rather subtle
00:03:57.930 --> 00:04:00.350
in that series expansion.
00:04:00.350 --> 00:04:02.950
For that series expansion
there was a critical moment,
00:04:02.950 --> 00:04:03.880
which I'm not going
to go through but you
00:04:03.880 --> 00:04:05.520
can come to my office
hours again, but just
00:04:05.520 --> 00:04:06.180
look through the notes.
00:04:06.180 --> 00:04:08.055
There's an important
moment in the notes when
00:04:08.055 --> 00:04:10.500
we say, aha, these terms matter.
00:04:10.500 --> 00:04:15.980
But what we did is we
suppressed a singular solution.
00:04:15.980 --> 00:04:18.500
There's a solution of
that differential equation
00:04:18.500 --> 00:04:20.640
which is not well-behaved,
which is not smooth,
00:04:20.640 --> 00:04:23.070
and in particular
which diverges.
00:04:23.070 --> 00:04:25.400
And we already did, from
the asymptotic analysis,
00:04:25.400 --> 00:04:26.930
we already fixed that the
asymptotic behavior was
00:04:26.930 --> 00:04:27.850
exponentially falling.
00:04:27.850 --> 00:04:30.620
But there's a second solution
which is exponentially growing.
00:04:30.620 --> 00:04:33.427
So what we did, remember
how we did this story?
00:04:33.427 --> 00:04:35.010
We took our wave
function and we said,
00:04:35.010 --> 00:04:36.130
OK, look, we're
going to pull off--
00:04:36.130 --> 00:04:37.838
we're going to first
asymptotic analysis.
00:04:37.838 --> 00:04:40.182
And asymptotic analysis
tells us that either we
00:04:40.182 --> 00:04:42.390
have exponentially growing
or exponentially shrinking
00:04:42.390 --> 00:04:43.070
solutions.
00:04:43.070 --> 00:04:45.410
Let's pick the exponentially
shrinking solutions.
00:04:45.410 --> 00:04:48.370
So phi e is equal
to e to the minus
00:04:48.370 --> 00:04:51.545
x over 2a squared
squared, times some--
00:04:51.545 --> 00:04:53.170
I don't remember what
Barton called it.
00:04:53.170 --> 00:04:55.300
I'll call it u of x.
00:04:55.300 --> 00:04:58.910
So we've extracted, because
we know that asymptotically it
00:04:58.910 --> 00:04:59.840
takes this form.
00:04:59.840 --> 00:05:01.060
Well, it could also
take the other form.
00:05:01.060 --> 00:05:02.250
It could be e to the
plus, which would
00:05:02.250 --> 00:05:03.417
be bad and not normalizable.
00:05:03.417 --> 00:05:05.249
We've extracted that,
and then we write down
00:05:05.249 --> 00:05:06.720
the differential equation for u.
00:05:06.720 --> 00:05:08.553
And then we solve that
differential equation
00:05:08.553 --> 00:05:10.570
by series analysis, yeah?
00:05:10.570 --> 00:05:17.151
However, if I have a secondary
differential equation for phi,
00:05:17.151 --> 00:05:19.150
this change of variables
doesn't change the fact
00:05:19.150 --> 00:05:20.840
that it's a secondary
differential equation for u,
00:05:20.840 --> 00:05:21.610
right?
00:05:21.610 --> 00:05:23.920
There's still two
solutions for u.
00:05:23.920 --> 00:05:25.984
One of those solutions
will be the solution
00:05:25.984 --> 00:05:27.900
of the equation that has
this asymptotic form.
00:05:27.900 --> 00:05:30.683
But the other solution
will be one that has an e
00:05:30.683 --> 00:05:33.320
to the plus x squared
over a squared
00:05:33.320 --> 00:05:35.070
so that it cancels off
this leading factor
00:05:35.070 --> 00:05:37.070
and gives me the exponentially
growing solution.
00:05:40.490 --> 00:05:42.650
Everyone cool with that?
00:05:42.650 --> 00:05:46.330
So in that series
analysis there's
00:05:46.330 --> 00:05:47.880
sort of a subtle
moment where you
00:05:47.880 --> 00:05:51.527
impose that you have
the convergent solution.
00:05:51.527 --> 00:05:53.860
So the answer of, why did we
get a first order relation,
00:05:53.860 --> 00:05:57.820
is that we very carefully,
although it may not
00:05:57.820 --> 00:06:00.740
have been totally obvious,
when doing this calculation
00:06:00.740 --> 00:06:03.360
one carefully chooses the
convergent solution that
00:06:03.360 --> 00:06:06.380
doesn't have this
function blowing up so
00:06:06.380 --> 00:06:08.957
as to overwhelm the envelope.
00:06:08.957 --> 00:06:10.040
That answer your question?
00:06:10.040 --> 00:06:10.530
AUDIENCE: Yep.
00:06:10.530 --> 00:06:11.260
PROFESSOR: Great.
00:06:11.260 --> 00:06:12.530
It's a very good question.
00:06:12.530 --> 00:06:13.500
This is an important
subtlety that
00:06:13.500 --> 00:06:16.190
comes up all over the place
when you do asymptotic analysis.
00:06:16.190 --> 00:06:17.210
I speak from my heart.
00:06:17.210 --> 00:06:18.420
It's an important
thing in the research
00:06:18.420 --> 00:06:19.795
that I'm doing
right now, getting
00:06:19.795 --> 00:06:22.490
these sorts of subtleties right.
00:06:22.490 --> 00:06:23.532
It can be very confusing.
00:06:23.532 --> 00:06:25.490
It's important to think
carefully through them.
00:06:25.490 --> 00:06:26.700
So it's a very good question.
00:06:26.700 --> 00:06:29.940
Other questions
before we move on?
00:06:29.940 --> 00:06:31.830
OK.
00:06:31.830 --> 00:06:34.386
So I'm going to erase
this, because it's not
00:06:34.386 --> 00:06:36.430
directly germane,
but it is great.
00:06:36.430 --> 00:06:40.690
OK, so one of the lessons
of this brute force analysis
00:06:40.690 --> 00:06:43.120
was that we constructed
the spectrum, i.e., the set
00:06:43.120 --> 00:06:48.190
of energy eigenvalues allowed
for the quantum harmonic
00:06:48.190 --> 00:06:51.100
oscillator, and we constructed
the wave functions.
00:06:51.100 --> 00:06:52.490
We constructed
the wave functions
00:06:52.490 --> 00:06:53.990
by solving the
differential equation
00:06:53.990 --> 00:06:57.180
through asymptotic analysis,
which give us the Gaussian
00:06:57.180 --> 00:06:59.050
envelope, and series
expansion, which
00:06:59.050 --> 00:07:01.490
give us the Hermite polynomials.
00:07:01.490 --> 00:07:04.059
And then there's some
normalization coefficient.
00:07:04.059 --> 00:07:06.100
And then we got the energy
eigenvalues by asking,
00:07:06.100 --> 00:07:09.290
when does this series
expansion converge?
00:07:09.290 --> 00:07:11.170
When does it, in fact
truncate, terminate,
00:07:11.170 --> 00:07:13.570
so that we can write
down an answer?
00:07:13.570 --> 00:07:15.700
And that was what gave
us these discrete values.
00:07:15.700 --> 00:07:17.950
But fine, we can see that
it would be discrete values.
00:07:17.950 --> 00:07:18.480
We're cool with that.
00:07:18.480 --> 00:07:21.104
In fact, Barton went through the
discussion of the node theorem
00:07:21.104 --> 00:07:26.570
and the lack of degeneracy
in one dimensional quantum
00:07:26.570 --> 00:07:27.590
mechanics.
00:07:27.590 --> 00:07:30.250
So it's reasonable that we
get a bunch of discrete energy
00:07:30.250 --> 00:07:32.800
eigenvalues, as we've talked
about now for two lectures.
00:07:32.800 --> 00:07:34.880
However, there's
a surprise here,
00:07:34.880 --> 00:07:36.680
which is that these
aren't just discrete,
00:07:36.680 --> 00:07:38.020
they're evenly spaced.
00:07:38.020 --> 00:07:41.040
We get a tower, starting with
the lowest possible energy
00:07:41.040 --> 00:07:44.170
corresponding to a--
sorry, E0-- starting
00:07:44.170 --> 00:07:48.190
with the lowest possible
energy, which is greater than 0,
00:07:48.190 --> 00:07:52.900
and a corresponding ground
state wave function.
00:07:52.900 --> 00:07:55.816
And then we have a whole bunch
of other states, phi 1, phi 2,
00:07:55.816 --> 00:08:00.009
phi 3, phi 4, labeled
by their energies
00:08:00.009 --> 00:08:01.550
where the energies
are evenly spaced.
00:08:04.890 --> 00:08:09.410
They needed to be discrete,
because these are bound states.
00:08:09.410 --> 00:08:11.700
But evenly spaced is a surprise.
00:08:11.700 --> 00:08:14.430
So why are they evenly spaced?
00:08:14.430 --> 00:08:18.904
Anyone, based on the
last lecture's analysis?
00:08:18.904 --> 00:08:20.320
Yeah, you don't
have a good answer
00:08:20.320 --> 00:08:21.540
to that from last
lecture's analysis.
00:08:21.540 --> 00:08:22.870
It's one of the
mysteries that comes out
00:08:22.870 --> 00:08:23.780
of the first analysis.
00:08:23.780 --> 00:08:25.363
When you take a
differential equation,
00:08:25.363 --> 00:08:28.440
you just beat the crap out of
it with a stick by solving it.
00:08:28.440 --> 00:08:31.500
With differential equations
strategies like this
00:08:31.500 --> 00:08:34.772
you don't necessarily get some
of the more subtle structure.
00:08:34.772 --> 00:08:36.230
One of the goals
of today's lecture
00:08:36.230 --> 00:08:39.909
is going to be to explain
why we get this structure.
00:08:39.909 --> 00:08:41.706
Why just from the
physics of the problem,
00:08:41.706 --> 00:08:43.122
the underlying
physics, should you
00:08:43.122 --> 00:08:46.300
know that the system is going to
have evenly spaced eigenvalues?
00:08:46.300 --> 00:08:47.600
What's the structure?
00:08:47.600 --> 00:08:50.121
And secondly, I want to
show you a way of repeating
00:08:50.121 --> 00:08:52.620
this calculation without doing
the brute force analysis that
00:08:52.620 --> 00:08:56.200
reveals some of that more fine
grain structure of the problem.
00:08:56.200 --> 00:08:57.820
And this is going
to turn out to be
00:08:57.820 --> 00:09:00.200
one of the canonical
moves in the analysis
00:09:00.200 --> 00:09:01.799
of quantum mechanical systems.
00:09:01.799 --> 00:09:03.840
So from quantum mechanics
to quantum field theory
00:09:03.840 --> 00:09:07.150
this is a basic
series of logic moves.
00:09:07.150 --> 00:09:10.120
What I'm going to do today
also has an independent life
00:09:10.120 --> 00:09:13.765
in mathematics, in algebra.
00:09:13.765 --> 00:09:15.140
And that will be
something you'll
00:09:15.140 --> 00:09:17.390
studying in more detail in
8.05, but I would encourage
00:09:17.390 --> 00:09:19.600
you to ask your recitation
instructors about it,
00:09:19.600 --> 00:09:21.386
or me in office hours.
00:09:25.520 --> 00:09:30.440
So our goal is to understand
that even spacing and also
00:09:30.440 --> 00:09:32.300
to re-derive these
results without the sort
00:09:32.300 --> 00:09:37.692
of brutal direct assault
methods we used last time.
00:09:37.692 --> 00:09:39.400
So what I'm going to
tell you about today
00:09:39.400 --> 00:09:40.950
is something called
the operator method.
00:09:40.950 --> 00:09:43.200
It usually goes under the
name of the operator method.
00:09:57.340 --> 00:10:01.780
To get us started let's go back
to look at the energy operator
00:10:01.780 --> 00:10:04.310
for the harmonic
oscillator, which
00:10:04.310 --> 00:10:08.310
is what, at the end of
the day, we want to solve.
00:10:08.310 --> 00:10:15.230
p squared over 2m plus m omega
squared upon 2, x squared.
00:10:15.230 --> 00:10:18.615
And this is the operator that we
want to-- whose eigenvalues we
00:10:18.615 --> 00:10:20.240
want to construct,
whose eigenfunctions
00:10:20.240 --> 00:10:21.600
we want to construct.
00:10:21.600 --> 00:10:25.720
Before we do anything else, we
should do dimensional analysis.
00:10:25.720 --> 00:10:27.470
First thing you when
you look at a problem
00:10:27.470 --> 00:10:28.600
is do some dimensional analysis.
00:10:28.600 --> 00:10:30.433
Identify the salient
scales and make things,
00:10:30.433 --> 00:10:32.210
to the degree possible,
dimensionless.
00:10:32.210 --> 00:10:33.470
Your life will be better.
00:10:33.470 --> 00:10:35.780
So what are the parameters
we have available to us?
00:10:35.780 --> 00:10:38.529
We have h bar, because
it's quantum mechanics.
00:10:38.529 --> 00:10:40.570
We have m, because we have
a particle of mass, m.
00:10:40.570 --> 00:10:42.850
We have omega,
because this potential
00:10:42.850 --> 00:10:44.740
has a characteristic
frequency of omega.
00:10:44.740 --> 00:10:46.940
What other parameters do
we have available to us?
00:10:46.940 --> 00:10:47.930
Well, we have c.
00:10:47.930 --> 00:10:49.230
That's available to us.
00:10:49.230 --> 00:10:51.360
But is it relevant?
00:10:51.360 --> 00:10:52.087
No.
00:10:52.087 --> 00:10:54.420
If you get an answer that
depends on the speed of light,
00:10:54.420 --> 00:10:56.510
you made some horrible mistake.
00:10:56.510 --> 00:10:57.260
So not there.
00:10:57.260 --> 00:11:00.230
What about the number
of students in 8.04?
00:11:00.230 --> 00:11:00.730
No.
00:11:00.730 --> 00:11:01.990
There are an infinite
number of parameters
00:11:01.990 --> 00:11:03.430
that don't matter
to this problem.
00:11:03.430 --> 00:11:05.270
What you want to know is, when
you do dimensional analysis,
00:11:05.270 --> 00:11:07.110
what parameters matter
for the problem.
00:11:07.110 --> 00:11:09.600
What parameters could possibly
appear during the answer?
00:11:09.600 --> 00:11:10.510
And that's it.
00:11:10.510 --> 00:11:14.120
There are no other
parameters in this problem.
00:11:14.120 --> 00:11:16.730
So that's a full set of
parameters available to us.
00:11:16.730 --> 00:11:19.470
This has dimensions of
momentum times length.
00:11:19.470 --> 00:11:22.850
This has dimensions of mass,
and this has dimensions of one
00:11:22.850 --> 00:11:23.840
upon the time.
00:11:23.840 --> 00:11:25.890
And so what
characteristic scales can
00:11:25.890 --> 00:11:29.050
we build using these
three parameters?
00:11:29.050 --> 00:11:33.310
Well, this is a
moment times a length.
00:11:33.310 --> 00:11:38.340
If we multiply by a mass,
that's momentum times mass
00:11:38.340 --> 00:11:41.020
times x, which is
almost momentum again.
00:11:41.020 --> 00:11:42.700
We need a velocity
and not a position,
00:11:42.700 --> 00:11:44.020
but we have 1 over time.
00:11:44.020 --> 00:11:48.570
So if we take h bar
times and omega,
00:11:48.570 --> 00:11:51.324
so that's px times
m over t, that
00:11:51.324 --> 00:11:52.573
has units of momentum squared.
00:11:59.330 --> 00:12:00.990
And similarly, this
is momentum which
00:12:00.990 --> 00:12:03.960
is x, which is length
mass over time.
00:12:03.960 --> 00:12:05.870
I can divide by mass
and divide by frequency
00:12:05.870 --> 00:12:11.200
or multiply by time,
so h bar upon m omega.
00:12:11.200 --> 00:12:15.411
And this is going to have
units of length squared.
00:12:15.411 --> 00:12:17.660
And with a little bit of
foresight from factors of two
00:12:17.660 --> 00:12:20.860
I'm going to use these to
define two link scales.
00:12:20.860 --> 00:12:26.390
x0 is equal to h
bar-- I want to be
00:12:26.390 --> 00:12:28.676
careful to get my coefficients.
00:12:28.676 --> 00:12:30.390
I always put the two
in the wrong place.
00:12:30.390 --> 00:12:33.100
So 2 h bar upon m omega.
00:12:33.100 --> 00:12:34.890
Square root.
00:12:34.890 --> 00:12:39.660
And I'm going to define p0 as
equal to square root of 2 h bar
00:12:39.660 --> 00:12:40.440
times m omega.
00:12:45.900 --> 00:12:46.860
So here's my claim.
00:12:46.860 --> 00:12:49.650
My claim is at the end of
the day the salient link
00:12:49.650 --> 00:12:51.250
scales for this
problem should be
00:12:51.250 --> 00:12:54.780
integers or dimensionless
numbers times this link scale.
00:12:54.780 --> 00:12:58.480
And salient momentum scales
should be this scale.
00:12:58.480 --> 00:12:59.930
Just from dimensional analysis.
00:12:59.930 --> 00:13:01.420
So if someone at
this point says,
00:13:01.420 --> 00:13:03.840
what do you think is
the typical scale, what
00:13:03.840 --> 00:13:06.670
is the typical size of
the ground state wave
00:13:06.670 --> 00:13:10.530
function, the typical link scale
over which the wave function is
00:13:10.530 --> 00:13:12.670
not 0?
00:13:12.670 --> 00:13:15.220
Well, that can't possibly
be the size of Manhattan.
00:13:15.220 --> 00:13:16.609
It's not the size of a proton.
00:13:16.609 --> 00:13:18.900
There's only one link scale
associated with the system.
00:13:18.900 --> 00:13:20.730
It should be of order x0.
00:13:20.730 --> 00:13:22.510
Always start with
dimensional analysis.
00:13:22.510 --> 00:13:23.180
Always.
00:13:23.180 --> 00:13:26.432
OK, so with that we can
rewrite this energy.
00:13:26.432 --> 00:13:29.050
Sorry, and there's
one last energy.
00:13:29.050 --> 00:13:32.780
We can write an energy,
the thing with energy,
00:13:32.780 --> 00:13:35.880
which is equal to h bar omega.
00:13:35.880 --> 00:13:43.390
And this times a frequency
gives us an energy.
00:13:43.390 --> 00:13:47.360
So we can rewrite
this energy operator
00:13:47.360 --> 00:13:52.784
as h bar omega times
p squared over p0.
00:13:52.784 --> 00:13:53.950
So this has units of energy.
00:13:53.950 --> 00:13:55.820
So everything here
must be dimensionless.
00:13:55.820 --> 00:13:58.340
And it turns out to be p
squared over p0 squared
00:13:58.340 --> 00:14:02.760
plus x operator squared
over x0 squared.
00:14:02.760 --> 00:14:05.515
So that's convenient.
00:14:05.515 --> 00:14:07.640
So this has nothing to do
with the operator method.
00:14:07.640 --> 00:14:08.889
This is just being reasonable.
00:14:15.190 --> 00:14:25.380
Quick thing to note, x0 times p0
is just-- the m omegas cancel,
00:14:25.380 --> 00:14:28.070
so we get root 2h
bar squared, 2 h bar.
00:14:31.410 --> 00:14:33.160
Little tricks like
that are useful to keep
00:14:33.160 --> 00:14:34.180
track of as you go.
00:14:36.760 --> 00:14:40.930
So we're interested in
this energy operator.
00:14:40.930 --> 00:14:42.030
And it has a nice form.
00:14:42.030 --> 00:14:43.140
It's a sum of squares.
00:14:43.140 --> 00:14:46.320
And we see the sum of squares,
a very tempting thing to do
00:14:46.320 --> 00:14:47.820
is to factor it.
00:14:47.820 --> 00:14:51.190
So for example, if I have
two classical numbers,
00:14:51.190 --> 00:14:55.030
c squared plus d squared,
the mathematician in me
00:14:55.030 --> 00:14:59.940
screams out to write c
minus id times c plus id.
00:15:02.640 --> 00:15:03.630
I have factored this.
00:15:03.630 --> 00:15:05.500
And that's usually a step
in the right direction.
00:15:05.500 --> 00:15:06.208
And is this true?
00:15:06.208 --> 00:15:08.910
Well yes, it's true. c
squared plus d squared
00:15:08.910 --> 00:15:10.312
and the cross terms cancel.
00:15:10.312 --> 00:15:11.020
OK, that's great.
00:15:11.020 --> 00:15:12.840
Four complex numbers,
or four C numbers.
00:15:15.819 --> 00:15:17.110
Now is this true for operators?
00:15:17.110 --> 00:15:17.920
Can I do this for operators?
00:15:17.920 --> 00:15:20.340
Here we have the energy
operator as a sum of squares.
00:15:22.910 --> 00:15:26.150
Well, let's try it.
00:15:26.150 --> 00:15:28.290
I'd like to write that
in terms of x and p.
00:15:28.290 --> 00:15:38.070
So what about writing the
quantity x minus ip over x0
00:15:38.070 --> 00:15:46.252
over p0, operator, times
x over x0 plus ip over p0.
00:15:49.647 --> 00:15:50.480
We can compute this.
00:15:50.480 --> 00:15:51.270
This is easy.
00:15:51.270 --> 00:15:56.700
So the first term gives us
the x squared over x0 squared.
00:15:56.700 --> 00:15:58.650
That last term gives
us-- the i's cancel,
00:15:58.650 --> 00:16:03.579
so we get p squared
over p0, squared.
00:16:03.579 --> 00:16:04.870
But then there are cross terms.
00:16:04.870 --> 00:16:09.950
We have an xp and a minus
px, with an overall i.
00:16:09.950 --> 00:16:17.320
So plus i times xp
over x0 times p0.
00:16:17.320 --> 00:16:19.350
x0 times p0,
however, is 2 h bar.
00:16:19.350 --> 00:16:20.570
So that's over 2 h bar.
00:16:23.300 --> 00:16:25.150
And then we have the
other term, minus px.
00:16:25.150 --> 00:16:25.650
Same thing.
00:16:25.650 --> 00:16:30.676
So I could write that as
a commutator, xp minus px.
00:16:30.676 --> 00:16:31.675
Everyone cool with what?
00:16:35.152 --> 00:16:36.860
Unfortunately this is
not what we wanted.
00:16:36.860 --> 00:16:38.740
We wanted just p
squared plus x squared.
00:16:38.740 --> 00:16:40.420
And what we got
instead was p squared
00:16:40.420 --> 00:16:42.200
plus x squared close
plus a commutator.
00:16:42.200 --> 00:16:43.616
Happily this
commutator is simple.
00:16:43.616 --> 00:16:45.054
What's the commutator
of x with p?
00:16:45.054 --> 00:16:45.970
AUDIENCE: [INAUDIBLE].
00:16:45.970 --> 00:16:47.011
PROFESSOR: Yeah, exactly.
00:16:47.011 --> 00:16:49.850
Commit this to memory.
00:16:49.850 --> 00:16:53.070
This is your friend.
00:16:53.070 --> 00:16:54.900
So this is just i
h bar, so this is
00:16:54.900 --> 00:16:58.780
equal to ditto plus
ditto plus i h bar.
00:17:01.390 --> 00:17:02.690
Somewhere I got a minus sign.
00:17:02.690 --> 00:17:06.255
Where did I get my minus
sign wrong? x with ip.
00:17:06.255 --> 00:17:06.900
Oh now, good.
00:17:06.900 --> 00:17:07.560
This is good.
00:17:07.560 --> 00:17:10.901
So x with p is i--
so we get an i h bar.
00:17:10.901 --> 00:17:12.359
No, I really did
screw up the sign.
00:17:12.359 --> 00:17:13.530
How did I screw up the sign?
00:17:16.160 --> 00:17:17.770
No I didn't.
00:17:17.770 --> 00:17:18.440
Wait.
00:17:18.440 --> 00:17:18.720
Oh!
00:17:18.720 --> 00:17:19.220
Of course.
00:17:19.220 --> 00:17:20.099
No, good.
00:17:20.099 --> 00:17:21.630
Sorry, sorry.
00:17:21.630 --> 00:17:24.030
Trust your calculation,
not your memory.
00:17:24.030 --> 00:17:25.407
So the calculation gave us this.
00:17:25.407 --> 00:17:26.490
So what does this give us?
00:17:26.490 --> 00:17:28.740
It gives us i h bar.
00:17:28.740 --> 00:17:29.360
So plus.
00:17:29.360 --> 00:17:31.675
But the i h bar times i is
going to give me a minus.
00:17:31.675 --> 00:17:33.050
And the h bar is
going to cancel,
00:17:33.050 --> 00:17:35.050
because I've got an h bar
from here and an h bar
00:17:35.050 --> 00:17:37.460
at the denominator minus 1/2.
00:17:37.460 --> 00:17:43.222
So this quantity is equal to the
quantity we wanted minus 1/2.
00:17:43.222 --> 00:17:44.680
And what is the
quantity we wanted,
00:17:44.680 --> 00:17:48.260
x0 squared plus p0 squared?
00:17:48.260 --> 00:17:50.902
This guy.
00:17:50.902 --> 00:17:52.360
So putting that
all together we can
00:17:52.360 --> 00:17:55.680
write that the energy
operator, which
00:17:55.680 --> 00:18:05.841
was equal to h bar omega
times the quantity we wanted,
00:18:05.841 --> 00:18:07.590
is equal to-- well,
the quantity we wanted
00:18:07.590 --> 00:18:10.290
is this quantity plus 1/2.
00:18:10.290 --> 00:18:14.290
h bar omega times--
I'll write this
00:18:14.290 --> 00:18:31.450
as x over x0 plus ip over p0,
x over x0 plus ip over p0, hat,
00:18:31.450 --> 00:18:33.700
hat, hat, plus 1/2.
00:18:37.860 --> 00:18:39.370
Everyone cool with that?
00:18:39.370 --> 00:18:40.230
So it almost worked.
00:18:40.230 --> 00:18:41.150
We can almost factor.
00:18:44.709 --> 00:18:46.500
So at this point it's
tempting to say, well
00:18:46.500 --> 00:18:48.800
that isn't really
much an improvement.
00:18:48.800 --> 00:18:50.590
You've just made it uglier.
00:18:50.590 --> 00:18:54.117
But consider the following.
00:18:54.117 --> 00:18:56.700
And just trust me on this one,
that this is not a stupid thing
00:18:56.700 --> 00:18:58.694
to do.
00:18:58.694 --> 00:19:00.360
That's a stupid symbol
to write, though.
00:19:00.360 --> 00:19:04.780
So let's define
an operator called
00:19:04.780 --> 00:19:13.740
a, which is equal to x
over x0 plus ip over p0,
00:19:13.740 --> 00:19:17.062
and an operator, which
I will call a dagger.
00:19:17.062 --> 00:19:19.550
"Is that a dagger
I see before me?"
00:19:19.550 --> 00:19:23.950
Sorry. x over x0
minus ip over p0.
00:19:27.420 --> 00:19:29.360
Hamlet quotes are harder.
00:19:29.360 --> 00:19:31.500
So this is a dagger.
00:19:31.500 --> 00:19:33.960
And we can now write
the energy operator
00:19:33.960 --> 00:19:37.930
for the harmonic oscillator
is equal to h bar omega times
00:19:37.930 --> 00:19:41.370
a dagger a plus 1/2.
00:19:45.740 --> 00:19:46.740
Everyone cool with that?
00:19:49.175 --> 00:19:50.550
Now, this should
look suggestive.
00:19:50.550 --> 00:19:52.966
You should say, aha, this looks
like h bar omega something
00:19:52.966 --> 00:19:53.620
plus 1/2.
00:19:53.620 --> 00:19:57.080
That sure looks familiar from
our brute force calculation.
00:19:57.080 --> 00:20:02.325
But, OK, that familiarity is
not an answer to the question.
00:20:02.325 --> 00:20:04.200
Meanwhile you should
say something like this.
00:20:04.200 --> 00:20:06.330
Look, this looks kind of
like the complex conjugate
00:20:06.330 --> 00:20:08.140
of this guy.
00:20:08.140 --> 00:20:10.859
Because there's an i and you
change the sign of the i.
00:20:10.859 --> 00:20:12.900
But what is the complex
conjugate of an operator?
00:20:12.900 --> 00:20:13.733
What does that mean?
00:20:15.885 --> 00:20:17.760
An operator is like take
a vector and rotate.
00:20:17.760 --> 00:20:19.920
What is the complex
conjugation of that?
00:20:19.920 --> 00:20:20.477
I don't know.
00:20:20.477 --> 00:20:21.560
So we have to define that.
00:20:24.740 --> 00:20:27.430
So I'm now going to start
with this quick math aside.
00:20:30.280 --> 00:20:32.249
And morally, this
is about what is
00:20:32.249 --> 00:20:33.790
the complex conjugate
of an operator.
00:20:33.790 --> 00:20:35.123
But before I move on, questions?
00:20:40.390 --> 00:20:42.630
OK.
00:20:42.630 --> 00:20:45.830
So here's a mathematical
series of a facts and claims.
00:20:48.630 --> 00:20:49.680
I claim the following.
00:20:49.680 --> 00:21:04.327
Given any linear operator we can
build-- there's a natural way
00:21:04.327 --> 00:21:06.410
to build without making
any additional assumptions
00:21:06.410 --> 00:21:07.660
or any additional ingredients.
00:21:07.660 --> 00:21:14.960
We can build another
operator, o dagger, hat, hat,
00:21:14.960 --> 00:21:16.860
in the following way.
00:21:16.860 --> 00:21:20.600
Consider the inner
product of f with g,
00:21:20.600 --> 00:21:22.280
or the bracket of f with g.
00:21:22.280 --> 00:21:27.130
So integral dx of f
complex conjugate g.
00:21:27.130 --> 00:21:29.830
Consider the function
we're taking here
00:21:29.830 --> 00:21:31.910
is actually the
operator we have on g.
00:21:35.102 --> 00:21:37.560
I'm going to define my-- so
this is a perfectly good thing.
00:21:37.560 --> 00:21:39.790
What this expression says
is, take your function g.
00:21:39.790 --> 00:21:41.744
Act on it with the operator o.
00:21:41.744 --> 00:21:43.160
Multiply by the
complex conjugate.
00:21:43.160 --> 00:21:43.860
Take the integral.
00:21:43.860 --> 00:21:45.401
This is what we
would have done if we
00:21:45.401 --> 00:21:50.430
had taken the inner product
of f with the function we get
00:21:50.430 --> 00:21:52.320
by taking o and acting
on the function g.
00:21:56.347 --> 00:21:57.180
So here's the thing.
00:21:57.180 --> 00:22:01.379
What we want-- just an
aside-- what we want to do
00:22:01.379 --> 00:22:02.420
is define a new operator.
00:22:02.420 --> 00:22:03.410
And here's how I'm
going to define it.
00:22:03.410 --> 00:22:04.932
We can define it by
choosing how it acts.
00:22:04.932 --> 00:22:06.790
I'm going to tell you
exactly how it acts,
00:22:06.790 --> 00:22:08.330
and then we'll
define the operator.
00:22:08.330 --> 00:22:12.837
So this operator, o with a
dagger, called the adjoint,
00:22:12.837 --> 00:22:14.170
is defined in the following way.
00:22:14.170 --> 00:22:17.030
This is whatever
operator you need, such
00:22:17.030 --> 00:22:21.035
that the integral
gives you-- such
00:22:21.035 --> 00:22:22.160
that the following is true.
00:22:22.160 --> 00:22:29.050
Integral dx o on f,
complex conjugate, g.
00:22:29.050 --> 00:22:31.960
So this is the
definition of this dagger
00:22:31.960 --> 00:22:35.530
action, the adjoint action.
00:22:35.530 --> 00:22:38.870
OK so o dagger is the adjoint.
00:22:41.670 --> 00:22:44.012
And sometimes it's called
the Hermetian adjoint.
00:22:44.012 --> 00:22:46.220
I'll occasionally say
Hermetian and occasionally not,
00:22:46.220 --> 00:22:49.580
with no particular order to it.
00:22:49.580 --> 00:22:51.970
So what does this mean?
00:22:51.970 --> 00:22:54.480
This means that
whatever o dagger is,
00:22:54.480 --> 00:22:57.760
it's that operator
that when acting on g
00:22:57.760 --> 00:22:59.440
and then taking the
inner product with f
00:22:59.440 --> 00:23:01.565
gives me same answer as
taking my original operator
00:23:01.565 --> 00:23:05.805
and acting on f and taking
the inner product with g.
00:23:05.805 --> 00:23:06.305
Cool?
00:23:09.210 --> 00:23:12.655
So we know how-- if we know
what our operator o is,
00:23:12.655 --> 00:23:15.280
the challenge now is going to be
to figure out what must this o
00:23:15.280 --> 00:23:18.220
dagger operator be such that
this expression is true.
00:23:18.220 --> 00:23:21.850
That's going to be my
definition of the adjoint.
00:23:21.850 --> 00:23:22.625
Cool?
00:23:22.625 --> 00:23:24.250
So I'm going to do
a bunch of examples.
00:23:24.250 --> 00:23:25.541
I'm going to walk through this.
00:23:30.100 --> 00:23:31.620
So the mathematical
definition is
00:23:31.620 --> 00:23:34.110
that an operator o
defined in this fashion
00:23:34.110 --> 00:23:39.920
is the Hermetian adjoint of o.
00:23:44.050 --> 00:23:47.330
So that's the
mathematical definition.
00:23:47.330 --> 00:23:51.784
Well, that's our version of
the mathematical definition.
00:23:51.784 --> 00:23:53.450
I just came back from
a math conference,
00:23:53.450 --> 00:23:55.866
so I'm particularly chastened
at the moment to be careful.
00:23:58.580 --> 00:24:00.660
So let's do some quick examples.
00:24:00.660 --> 00:24:02.780
Example one.
00:24:02.780 --> 00:24:05.010
Suppose c is a complex number.
00:24:09.554 --> 00:24:11.095
I claim a number is
also an operator.
00:24:11.095 --> 00:24:12.592
It acts by multiplication.
00:24:12.592 --> 00:24:14.800
The number 7 is an operator
because it takes a vector
00:24:14.800 --> 00:24:17.810
and it gives you 7
times that vector.
00:24:17.810 --> 00:24:21.980
So this number is a particularly
simple kind of operator.
00:24:21.980 --> 00:24:22.960
And what's the adjoint?
00:24:26.816 --> 00:24:27.440
We can do that.
00:24:27.440 --> 00:24:28.026
That's easy.
00:24:28.026 --> 00:24:30.400
So c adjoint is going to be
defined in the following way.
00:24:30.400 --> 00:24:35.740
It's integral dx f
star of c adjoint g
00:24:35.740 --> 00:24:43.430
is equal to the integral dx of c
on f complex conjugate times g.
00:24:43.430 --> 00:24:46.060
But what is this?
00:24:46.060 --> 00:24:47.399
Well, c is just a number.
00:24:47.399 --> 00:24:48.940
So when we take its
complex conjugate
00:24:48.940 --> 00:24:50.190
we can just pull it out.
00:24:50.190 --> 00:24:54.810
So this is equal to the
integral dx c complex conjugate,
00:24:54.810 --> 00:24:58.174
f star g.
00:24:58.174 --> 00:24:59.590
But I'm now going
to rewrite this,
00:24:59.590 --> 00:25:02.410
using the awesome power of
reordering multiplication,
00:25:02.410 --> 00:25:04.187
as c star.
00:25:04.187 --> 00:25:06.020
And I'm going to put
parentheses around this
00:25:06.020 --> 00:25:07.820
because it seems like fun.
00:25:07.820 --> 00:25:09.320
So now we have this
nice expression.
00:25:09.320 --> 00:25:12.540
The integral dx of
f c adjoint g is
00:25:12.540 --> 00:25:18.880
equal to the integral dx of f c
star g, c complex conjugate g.
00:25:18.880 --> 00:25:21.350
But notice that this
must be true for all f.
00:25:24.691 --> 00:25:25.440
It's true for all.
00:25:25.440 --> 00:25:27.064
Because I made no
assumption about what
00:25:27.064 --> 00:25:31.340
f and g are, true
for all f and g.
00:25:31.340 --> 00:25:34.269
And therefore the adjoint
of a complex number
00:25:34.269 --> 00:25:35.310
is its complex conjugate.
00:25:38.500 --> 00:25:41.470
And this is the basic
strategy for determining
00:25:41.470 --> 00:25:43.660
the adjoint of any operator.
00:25:43.660 --> 00:25:45.790
We're going to play
exactly this sort of game.
00:25:45.790 --> 00:25:47.460
We'll put the adjoint in here.
00:25:47.460 --> 00:25:49.220
We'll use the definition
of the adjoint.
00:25:49.220 --> 00:25:50.870
And then we'll do
whatever machinations
00:25:50.870 --> 00:25:54.280
are necessary to rewrite
this as some operator acting
00:25:54.280 --> 00:25:55.810
on the first factor.
00:25:55.810 --> 00:25:56.310
Cool?
00:25:59.050 --> 00:26:00.820
Questions?
00:26:00.820 --> 00:26:03.790
OK, let's do a more
interesting operator.
00:26:03.790 --> 00:26:05.330
By the way, to
check at home, and I
00:26:05.330 --> 00:26:07.830
think this might be
on your problem set--
00:26:07.830 --> 00:26:09.484
but I don't remember
if it's on or not.
00:26:09.484 --> 00:26:11.150
So if it's not, check
this for yourself.
00:26:11.150 --> 00:26:13.827
Check that the
adjoin of the adjoint
00:26:13.827 --> 00:26:15.160
is equal to the operator itself.
00:26:18.640 --> 00:26:20.380
It's an easy thing to check.
00:26:20.380 --> 00:26:21.650
So next example.
00:26:27.550 --> 00:26:32.550
What is the adjoint of
the operator derivative
00:26:32.550 --> 00:26:33.540
with respect to x?
00:26:35.421 --> 00:26:37.920
Consider the operator, which
is just derivative with respect
00:26:37.920 --> 00:26:38.750
to x.
00:26:38.750 --> 00:26:43.280
And I want to know what is
the adjoint of this beast.
00:26:43.280 --> 00:26:45.700
So how do we do this?
00:26:45.700 --> 00:26:46.820
Same logic as before.
00:26:46.820 --> 00:26:49.520
Whatever the operator is, it's
defined in the following way.
00:26:49.520 --> 00:26:59.020
Integral dx, f complex
conjugate on dx dagger on g.
00:26:59.020 --> 00:27:02.920
This is equal to the integral
dx of-- how we doing on time?
00:27:02.920 --> 00:27:12.961
Good-- integral dx of dx,
f complex conjugate on g.
00:27:12.961 --> 00:27:15.460
Now, what we want is we want
to turn this into an expression
00:27:15.460 --> 00:27:17.085
where the operator
is acting on g, just
00:27:17.085 --> 00:27:19.680
as our familiar operator ddx.
00:27:19.680 --> 00:27:23.370
So how do I get
the ddx over here?
00:27:26.160 --> 00:27:27.170
I need to do two things.
00:27:27.170 --> 00:27:28.380
First, what's the
complex conjugate
00:27:28.380 --> 00:27:30.796
of the derivative with respect
to x of a complex function?
00:27:35.210 --> 00:27:36.940
MIT has indigestion.
00:27:36.940 --> 00:27:39.460
So this is integral
dx, derivative
00:27:39.460 --> 00:27:42.200
with respect to x of f
complex conjugate, g.
00:27:44.870 --> 00:27:46.790
And now I want this operator.
00:27:46.790 --> 00:27:48.229
I want derivative acting on g.
00:27:48.229 --> 00:27:49.145
That's the definition.
00:27:57.630 --> 00:28:00.650
Because I want to know
what is this operator.
00:28:00.650 --> 00:28:04.300
And so I'm going to do
integration by parts.
00:28:04.300 --> 00:28:07.920
So this is equal to
the integral, dx.
00:28:07.920 --> 00:28:11.460
When I integrate by
parts I get an F complex
00:28:11.460 --> 00:28:16.610
conjugate, and then an overall
minus sign from the integration
00:28:16.610 --> 00:28:19.684
by parts minus f
complex conjugate dx g.
00:28:22.851 --> 00:28:24.350
Was I telling you
the truth earlier?
00:28:24.350 --> 00:28:25.183
Or did I lie to you?
00:28:30.870 --> 00:28:34.430
OK, keep thinking about that.
00:28:34.430 --> 00:28:37.670
And this is equal
to, well the integral
00:28:37.670 --> 00:28:42.420
of dx, f complex conjugate
if minus the derivative
00:28:42.420 --> 00:28:45.450
with respect to x acting on g.
00:28:45.450 --> 00:28:47.120
Everyone cool with that?
00:28:47.120 --> 00:28:50.110
But if you look at
these equalities,
00:28:50.110 --> 00:28:55.620
dx adjoint acting on g is the
same as minus dx acting on g.
00:28:55.620 --> 00:28:59.060
So this tells me that
the adjoint of dx
00:28:59.060 --> 00:29:00.973
is equal to minus dx.
00:29:05.403 --> 00:29:05.903
Yeah?
00:29:05.903 --> 00:29:07.880
AUDIENCE: Are you assuming
that your surface terms vanish?
00:29:07.880 --> 00:29:08.755
PROFESSOR: Thank you!
00:29:08.755 --> 00:29:09.360
I lied to you.
00:29:09.360 --> 00:29:12.445
So I assumed in this that
my surface terms vanished.
00:29:12.445 --> 00:29:13.570
I did a variation by parts.
00:29:13.570 --> 00:29:15.995
And that leaves me with
a total derivative.
00:29:15.995 --> 00:29:18.120
And that total derivative
gives me a boundary term.
00:29:18.120 --> 00:29:20.010
Remember how integration
by parts works.
00:29:20.010 --> 00:29:26.100
Integration by parts
says the integral of AdxB
00:29:26.100 --> 00:29:28.170
is equal to the
integral of-- well,
00:29:28.170 --> 00:29:31.530
AdxB can be written as
derivative with respect
00:29:31.530 --> 00:29:43.500
to x of AB minus B derivative
with respect to x of A.
00:29:43.500 --> 00:29:47.956
Because this is A prime B plus B
prime A. Here we have AB prime.
00:29:47.956 --> 00:29:49.830
So we just subtract off
the appropriate term.
00:29:49.830 --> 00:29:51.500
But this is a total derivative.
00:29:51.500 --> 00:29:53.740
So it only gives
us a boundary term.
00:29:53.740 --> 00:29:57.410
So this integral is equal
to-- can move the integral
00:29:57.410 --> 00:30:01.690
over here-- the integral
and the derivative,
00:30:01.690 --> 00:30:04.386
because an integral is
nothing but an antiderivative.
00:30:04.386 --> 00:30:06.010
The integral and the
derivative cancel,
00:30:06.010 --> 00:30:07.570
leaving us with
the boundary terms.
00:30:07.570 --> 00:30:09.320
And in this case, it's
from our boundaries
00:30:09.320 --> 00:30:13.880
which are minus infinity
plus infinity, minus infinity
00:30:13.880 --> 00:30:16.420
and plus infinity.
00:30:16.420 --> 00:30:18.480
Now, this tells us
something very important.
00:30:18.480 --> 00:30:19.990
And I'm not going to speak
about this in detail,
00:30:19.990 --> 00:30:21.360
but I encourage the
recitation instructors
00:30:21.360 --> 00:30:23.020
who might happen
to be here to think
00:30:23.020 --> 00:30:25.490
to mention this in recitation.
00:30:25.490 --> 00:30:29.170
And I encourage you
all to think about it.
00:30:29.170 --> 00:30:30.855
If I ask you, what
is the adjoint
00:30:30.855 --> 00:30:32.680
of the derivative
operator acting
00:30:32.680 --> 00:30:37.080
on the space of functions
which are normalizable,
00:30:37.080 --> 00:30:40.180
so that they vanish
at infinity, what
00:30:40.180 --> 00:30:43.490
is the adjoint of the
derivative operator acting
00:30:43.490 --> 00:30:46.060
on the space of functions which
is normalizable at infinity?
00:30:46.060 --> 00:30:47.143
We just derive the answer.
00:30:47.143 --> 00:30:51.300
Because we assume that
these surface terms vanish.
00:30:51.300 --> 00:30:55.110
Because our wave functions,
f and g, vanish at infinity.
00:30:55.110 --> 00:30:56.674
They're normalizable.
00:30:56.674 --> 00:30:59.090
However, if I had asked you a
slightly different question,
00:30:59.090 --> 00:31:01.410
if I had asked you,
what's the adjoint
00:31:01.410 --> 00:31:03.784
of the derivative
operator acting
00:31:03.784 --> 00:31:05.450
on a different set
of functions, the set
00:31:05.450 --> 00:31:06.900
of functions that
don't necessarily
00:31:06.900 --> 00:31:09.025
vanish at infinity, including
sinusoids that go off
00:31:09.025 --> 00:31:10.860
to infinity and don't vanish.
00:31:10.860 --> 00:31:13.400
Is this the correct answer?
00:31:13.400 --> 00:31:13.900
No.
00:31:13.900 --> 00:31:14.860
This would not be
the correct answer,
00:31:14.860 --> 00:31:16.630
because there are
boundary terms.
00:31:16.630 --> 00:31:20.292
So the point I'm making
here, first off, in physics
00:31:20.292 --> 00:31:22.750
we're always going to be talking
about normalizable beasts.
00:31:22.750 --> 00:31:24.380
At the end of the day, the
physical objects we care about
00:31:24.380 --> 00:31:25.230
are in a room.
00:31:25.230 --> 00:31:26.651
They're not off infinity.
00:31:26.651 --> 00:31:28.400
So everything is going
to be normalizable.
00:31:28.400 --> 00:31:30.610
That is just how
the world works.
00:31:30.610 --> 00:31:33.760
However, you've got to
be careful in making
00:31:33.760 --> 00:31:35.260
these sorts of
arguments and realize
00:31:35.260 --> 00:31:38.710
that when I ask you, what is
the adjoint of this operator,
00:31:38.710 --> 00:31:40.976
I need to tell you
something more precise.
00:31:40.976 --> 00:31:42.350
I need to say,
what's the adjoint
00:31:42.350 --> 00:31:45.640
of the derivative acting when
this operator's understood
00:31:45.640 --> 00:31:48.490
as acting on some
particular set of functions,
00:31:48.490 --> 00:31:50.225
acting on normalizable
functions?
00:31:58.390 --> 00:31:59.840
Good.
00:31:59.840 --> 00:32:02.920
So anyway, I'll leave that
aside as something to ponder.
00:32:06.430 --> 00:32:08.342
But with that
technical detail aside,
00:32:08.342 --> 00:32:10.550
as long as we're talking
about normalizable functions
00:32:10.550 --> 00:32:13.220
so these boundary terms from
the integration by parts cancel,
00:32:13.220 --> 00:32:15.010
the adjoint of the
derivative operator
00:32:15.010 --> 00:32:17.650
is minus the
derivative operator.
00:32:17.650 --> 00:32:20.300
Cool?
00:32:20.300 --> 00:32:23.760
OK, let's do another example.
00:32:23.760 --> 00:32:26.008
And where do I want to do this?
00:32:26.008 --> 00:32:27.490
I'll do it here.
00:32:27.490 --> 00:32:30.300
So another example.
00:32:30.300 --> 00:32:30.970
Actually, no.
00:32:34.300 --> 00:32:35.210
I will do it here.
00:32:41.910 --> 00:32:50.050
So we have another
example, which is three.
00:32:50.050 --> 00:32:52.650
What's the adjoint of
the position operator?
00:32:59.540 --> 00:33:00.480
OK, take two minutes.
00:33:00.480 --> 00:33:02.757
Do this on a piece of
paper in front of you.
00:33:02.757 --> 00:33:03.965
I'm not going to call on you.
00:33:03.965 --> 00:33:05.960
So you can raise you hand if
you-- OK, chat with the person
00:33:05.960 --> 00:33:06.610
next to you.
00:33:10.046 --> 00:33:11.420
I mean chat about
physics, right?
00:33:11.420 --> 00:33:13.380
Just not-- [LAUGHS].
00:33:13.380 --> 00:33:16.810
AUDIENCE: [CHATTING]
00:33:42.931 --> 00:33:44.930
PROFESSOR: OK, so how do
we go about doing this?
00:33:44.930 --> 00:33:49.750
We go about solving
this problem by using
00:33:49.750 --> 00:33:51.580
the definition of the adjoint.
00:33:51.580 --> 00:33:52.940
So what is x adjoint?
00:33:52.940 --> 00:33:54.690
It's that operator
such that the following
00:33:54.690 --> 00:34:01.210
is true, such that the integral
dx of f complex conjugate
00:34:01.210 --> 00:34:06.598
with x dagger acting on g
is equal to the integral dx
00:34:06.598 --> 00:34:08.889
of what I get by taking the
complex conjugate of taking
00:34:08.889 --> 00:34:12.850
x and acting on f and then
integrating this against g.
00:34:12.850 --> 00:34:15.179
But now we can use
the action of x
00:34:15.179 --> 00:34:19.510
and say that this is equal
to the integral dx of x f
00:34:19.510 --> 00:34:22.096
complex conjugate g.
00:34:22.096 --> 00:34:23.179
But here's the nice thing.
00:34:23.179 --> 00:34:24.730
What is the complex
conjugate of f
00:34:24.730 --> 00:34:26.890
times the complex function of f?
00:34:26.890 --> 00:34:28.719
x is real.
00:34:28.719 --> 00:34:29.610
Positions are real.
00:34:29.610 --> 00:34:32.989
So that's just x times the
complex conjugate of f.
00:34:32.989 --> 00:34:34.949
So that was
essential move there.
00:34:34.949 --> 00:34:37.790
And now we can
rewrite this as equal
00:34:37.790 --> 00:34:41.860
the integral of f
complex conjugate xg.
00:34:41.860 --> 00:34:44.359
And now, eyeballing
this, x dagger
00:34:44.359 --> 00:34:46.590
is that operator
which acts by acting
00:34:46.590 --> 00:34:49.630
by multiplying with little x.
00:34:49.630 --> 00:34:52.280
Therefore, the adjoint
of the operator x
00:34:52.280 --> 00:34:55.690
is equal to the same operator.
00:34:55.690 --> 00:34:59.330
x is equal to its own adjoint.
00:34:59.330 --> 00:35:00.450
OK?
00:35:00.450 --> 00:35:01.930
Cool?
00:35:01.930 --> 00:35:04.160
So we've just learned a
couple of really nice things.
00:35:04.160 --> 00:35:06.380
So the first is-- where
we I want to do this?
00:35:06.380 --> 00:35:07.690
Yeah, good.
00:35:07.690 --> 00:35:11.051
So we've learned a
couple of nice things.
00:35:11.051 --> 00:35:13.300
And I want to encode them
in the following definition.
00:35:15.830 --> 00:35:19.090
Definition-- an
operator, which I
00:35:19.090 --> 00:35:25.210
will call o, whose
adjoint is equal to o,
00:35:25.210 --> 00:35:28.470
so an operator whose
adjoint is equal to itself
00:35:28.470 --> 00:35:31.685
is called Hermetian.
00:35:36.500 --> 00:35:38.500
So an operator which is
equal to its own adjoint
00:35:38.500 --> 00:35:39.333
is called Hermetian.
00:35:42.760 --> 00:35:46.080
And so I want to note a couple
of nice examples of that.
00:35:46.080 --> 00:35:58.031
So note a number which
is Hermetian is what?
00:35:58.031 --> 00:35:58.530
Real.
00:36:01.110 --> 00:36:04.880
An operator-- we found
an operator which
00:36:04.880 --> 00:36:08.820
is equal to its own adjoint.
x dagger is equal to x.
00:36:08.820 --> 00:36:11.829
And what can you say about the
eigenvalues of this operator?
00:36:11.829 --> 00:36:12.370
They're real.
00:36:12.370 --> 00:36:13.920
We use that in the
proof, actually.
00:36:13.920 --> 00:36:15.200
So this is real.
00:36:15.200 --> 00:36:18.260
I will call an operator
real if it's Hermetian.
00:36:18.260 --> 00:36:21.190
And here's a
mathematical fact, which
00:36:21.190 --> 00:36:23.540
is that any operator
which is Hermetian
00:36:23.540 --> 00:36:25.640
has all real eigenvalues.
00:36:25.640 --> 00:36:27.820
So this is really-- I'll
state it as a theorem,
00:36:27.820 --> 00:36:29.620
but it's just a fact for us.
00:36:32.360 --> 00:36:38.045
o has all real eigenvalues.
00:36:48.338 --> 00:36:49.254
AUDIENCE: [INAUDIBLE].
00:36:49.254 --> 00:36:49.920
PROFESSOR: Yeah?
00:36:49.920 --> 00:36:52.777
AUDIENCE: Is it if
and only [INAUDIBLE]?
00:36:52.777 --> 00:36:53.360
PROFESSOR: No.
00:37:00.280 --> 00:37:01.410
Let's see.
00:37:01.410 --> 00:37:04.420
If you have all
real eigenvalues,
00:37:04.420 --> 00:37:06.190
it does not imply
that you're Hermetian.
00:37:06.190 --> 00:37:08.790
However, if you have
all real eigenvalues
00:37:08.790 --> 00:37:11.390
and you can be
diagonalized, it does imply.
00:37:11.390 --> 00:37:12.640
So let me give you an example.
00:37:12.640 --> 00:37:16.830
So consider the
following operator.
00:37:16.830 --> 00:37:18.330
We've done this
many times, rotation
00:37:18.330 --> 00:37:20.121
in real three-dimensional
space of a vector
00:37:20.121 --> 00:37:21.980
around the vertical axis.
00:37:21.980 --> 00:37:26.690
It has one eigenvector,
which is the vertical vector.
00:37:26.690 --> 00:37:30.859
And the eigenvalue
is 1, so it's real.
00:37:30.859 --> 00:37:32.650
But that's not enough
to make it Hermetian.
00:37:32.650 --> 00:37:33.920
Because there's another
fact that we haven't
00:37:33.920 --> 00:37:35.420
got to yet with
Hermetian operators,
00:37:35.420 --> 00:37:38.080
which is going to tell us
that a Hermetian operator has
00:37:38.080 --> 00:37:40.990
as many eigenvectors
as there are dimensions
00:37:40.990 --> 00:37:46.124
in the space, i.e., that the
eigenvectors form a basis.
00:37:46.124 --> 00:37:48.040
But there's only one
eigenvector for this guy,
00:37:48.040 --> 00:37:50.290
even though we're in a
three-dimensional vector space.
00:37:50.290 --> 00:37:52.520
So this operator, rotation
by an angle theta,
00:37:52.520 --> 00:37:55.540
is not Hermetian, even
though its only eigenvalue
00:37:55.540 --> 00:37:57.120
isn't in fact real.
00:37:57.120 --> 00:37:58.500
So it's not an only if.
00:37:58.500 --> 00:38:01.000
If you are Hermetian, your
eigenvalues are all real.
00:38:01.000 --> 00:38:03.560
And you'll prove this
on a problem set.
00:38:03.560 --> 00:38:04.060
Yeah?
00:38:04.060 --> 00:38:08.420
AUDIENCE: If you're Hermetian,
are your eigenfunctions normal?
00:38:10.565 --> 00:38:11.690
PROFESSOR: Not necessarily.
00:38:11.690 --> 00:38:13.400
But they can be made normal.
00:38:13.400 --> 00:38:15.970
We'll talk about this
in more detail later.
00:38:15.970 --> 00:38:16.470
OK.
00:38:19.730 --> 00:38:23.647
Let's do a quick
check, last example.
00:38:23.647 --> 00:38:25.980
And I'm not actually going
to go through this in detail,
00:38:25.980 --> 00:38:27.000
but what about p?
00:38:27.000 --> 00:38:28.750
What about the
momentum operator?
00:38:28.750 --> 00:38:31.800
First off, do you think
the momentum is real?
00:38:31.800 --> 00:38:33.480
It sure would be nice.
00:38:33.480 --> 00:38:37.070
Because its eigenvalues are the
observable values of momentum.
00:38:37.070 --> 00:38:39.070
And so its eigenvalues
should all be real.
00:38:39.070 --> 00:38:40.236
Does that make it Hermetian?
00:38:40.236 --> 00:38:41.910
Not necessarily,
but let's check.
00:38:41.910 --> 00:38:44.980
So what is the adjoint of p?
00:38:44.980 --> 00:38:47.140
Well, this actually
we can do very easily.
00:38:47.140 --> 00:38:49.390
And I'm not going to go
through an elaborate argument.
00:38:49.390 --> 00:38:50.931
I'm just going to
know the following.
00:38:50.931 --> 00:38:58.150
p is equal to h bar upon i ddx.
00:39:01.420 --> 00:39:02.420
And this is an operator.
00:39:02.420 --> 00:39:03.410
This is an operator.
00:39:03.410 --> 00:39:06.400
So what's the adjoint
of this operator?
00:39:06.400 --> 00:39:10.280
Well, this under an adjoint
gets a minus sign, right?
00:39:10.280 --> 00:39:11.860
It's itself up to a minus sign.
00:39:11.860 --> 00:39:14.414
So is the derivative Hermetian?
00:39:14.414 --> 00:39:16.080
No, it's in fact what
we anti-Hermetian.
00:39:16.080 --> 00:39:18.530
Its adjoint is minus itself.
00:39:18.530 --> 00:39:20.530
What about i?
00:39:20.530 --> 00:39:22.680
What's its adjoint?
00:39:22.680 --> 00:39:24.070
Minus i.
00:39:24.070 --> 00:39:24.602
Sweet.
00:39:24.602 --> 00:39:26.310
So this has an adjoint,
picks up a minus.
00:39:26.310 --> 00:39:28.460
This has an adjoint,
picks up a minus.
00:39:28.460 --> 00:39:29.800
The minuses cancel.
00:39:29.800 --> 00:39:30.890
p adjoint is p.
00:39:33.987 --> 00:39:35.070
So p is in fact Hermetian.
00:39:38.410 --> 00:39:40.150
And here's a stronger
physical fact.
00:39:40.150 --> 00:39:43.370
So now we've seen that each
of the operators we built
00:39:43.370 --> 00:39:45.930
is x and p, true
of the operators
00:39:45.930 --> 00:39:47.840
we've looked at so
far is Hermetian,
00:39:47.840 --> 00:39:49.810
those that correspond
to physical observables.
00:39:49.810 --> 00:39:50.990
Here's a physical fact.
00:39:57.310 --> 00:40:03.990
All the observables you
measure with sticks are real.
00:40:07.220 --> 00:40:09.740
And the corresponding
statement is
00:40:09.740 --> 00:40:21.190
that all operators
corresponding to observables,
00:40:21.190 --> 00:40:22.640
all operators must be Hermetian.
00:40:32.480 --> 00:40:34.749
To the postulate that
says, "Observables
00:40:34.749 --> 00:40:36.290
are represented by
operators," should
00:40:36.290 --> 00:40:37.930
be adjoined the
word "Hermetian."
00:40:37.930 --> 00:40:40.160
Observables are represented
in quantum mechanics
00:40:40.160 --> 00:40:42.787
by Hermetian operators,
which are operators
00:40:42.787 --> 00:40:44.370
that have a number
of nice properties,
00:40:44.370 --> 00:40:47.250
including they have
all real eigenvalues.
00:40:47.250 --> 00:40:47.750
Cool?
00:40:50.850 --> 00:40:53.250
OK.
00:40:53.250 --> 00:40:54.460
Questions?
00:40:54.460 --> 00:40:55.440
Yeah.
00:40:55.440 --> 00:40:57.739
AUDIENCE: If it has to be
Hermetian and not just have
00:40:57.739 --> 00:41:00.030
real eigenvalues, does that
mean the eigenvalues always
00:41:00.030 --> 00:41:01.650
need to form some kind of basis?
00:41:01.650 --> 00:41:04.539
PROFESSOR: Yeah, the
eigenvectors will.
00:41:04.539 --> 00:41:06.580
This is connected to the
fact we've already seen.
00:41:06.580 --> 00:41:07.850
If you take an
arbitrary wave function
00:41:07.850 --> 00:41:09.933
you can expand it in states
with definite momentum
00:41:09.933 --> 00:41:10.840
as a superposition.
00:41:10.840 --> 00:41:14.720
You can also expand it in a set
of states of definite energy
00:41:14.720 --> 00:41:16.210
or of definite position.
00:41:16.210 --> 00:41:18.510
Anytime you have a
Hermetian operator,
00:41:18.510 --> 00:41:24.880
its eigenvectors suffice
to expand any function.
00:41:24.880 --> 00:41:27.270
They provide a basis for
representing any function.
00:41:29.840 --> 00:41:31.750
So that's the end of
the mathematical side.
00:41:31.750 --> 00:41:36.350
Let's get back to
this physical point.
00:41:36.350 --> 00:41:39.800
So we've defined this operator
a and this other operator
00:41:39.800 --> 00:41:41.140
a dagger.
00:41:41.140 --> 00:41:42.800
And here's my question first.
00:41:42.800 --> 00:41:45.560
Is a Hermetian?
00:41:45.560 --> 00:41:46.060
No.
00:41:46.060 --> 00:41:47.000
That's Hermetian.
00:41:47.000 --> 00:41:47.708
That's Hermetian.
00:41:47.708 --> 00:41:48.620
But there's an i.
00:41:48.620 --> 00:41:50.080
That i will pick
up the minus sign
00:41:50.080 --> 00:41:52.490
when we do the
complex conjugation.
00:41:52.490 --> 00:41:54.510
Oh, look.
00:41:54.510 --> 00:41:57.180
Sure was fortuitous that
I called this a dagger,
00:41:57.180 --> 00:42:00.690
since this is equal to a dagger.
00:42:03.700 --> 00:42:05.175
So this is the adjoint of a.
00:42:05.175 --> 00:42:07.425
So this immediately tells
you something interesting. x
00:42:07.425 --> 00:42:08.440
and p are both observables.
00:42:08.440 --> 00:42:09.898
Does a correspond
to an observable?
00:42:13.750 --> 00:42:14.520
Is it Hermetian?
00:42:17.110 --> 00:42:21.280
Every intervals is associated
to a Hermetian operator.
00:42:21.280 --> 00:42:23.090
This is not Hermetian.
00:42:23.090 --> 00:42:27.210
So a does not represent
an observable operator.
00:42:32.200 --> 00:42:35.239
And I will post notes on
the web page, which give us
00:42:35.239 --> 00:42:36.780
a somewhat lengthy
discussion-- or it
00:42:36.780 --> 00:42:37.830
might be in one
of the solutions--
00:42:37.830 --> 00:42:39.371
a somewhat lengthy
discussion of what
00:42:39.371 --> 00:42:42.550
it means for a and a dagger
to not be observable.
00:42:42.550 --> 00:42:46.390
You'll get more
discussion of that there.
00:42:46.390 --> 00:42:49.155
Meanwhile, if a is not
observable, it's not Hermetian,
00:42:49.155 --> 00:42:50.405
does it have real eigenvalues?
00:42:53.030 --> 00:42:54.400
Well, here's an important thing.
00:42:54.400 --> 00:42:57.344
I said if you're Hermetian,
all the eigenvalues are real.
00:42:57.344 --> 00:42:59.260
If you're not Hermetian,
that doesn't tell you
00:42:59.260 --> 00:43:01.030
you can't have any
real eigenvalues.
00:43:01.030 --> 00:43:02.990
It just says that I
haven't guaranteed for you
00:43:02.990 --> 00:43:05.170
that all the
eigenvalues are real.
00:43:05.170 --> 00:43:07.337
So what we'll discover
towards the end of the course
00:43:07.337 --> 00:43:09.795
when we talk about something
called coherent states is that
00:43:09.795 --> 00:43:11.890
in fact, a does have a
nice set of eigenvectors.
00:43:11.890 --> 00:43:12.750
They're very nice.
00:43:12.750 --> 00:43:13.370
They're great.
00:43:13.370 --> 00:43:15.520
We use them for lasers.
00:43:15.520 --> 00:43:16.510
They're very useful.
00:43:16.510 --> 00:43:18.030
And they're called
coherent states.
00:43:18.030 --> 00:43:20.820
But their eigenvalues
are not in general real.
00:43:20.820 --> 00:43:23.320
They're generically
complex numbers.
00:43:23.320 --> 00:43:25.320
Are they things you can measure?
00:43:25.320 --> 00:43:26.447
Not directly.
00:43:26.447 --> 00:43:28.530
They're related to things
you can measure, though,
00:43:28.530 --> 00:43:30.030
in some pretty nice ways.
00:43:34.560 --> 00:43:36.240
So why are we bothering
with these guys
00:43:36.240 --> 00:43:38.800
if they're not observable?
00:43:38.800 --> 00:43:39.891
Yeah.
00:43:39.891 --> 00:43:40.890
AUDIENCE: E [INAUDIBLE].
00:43:40.890 --> 00:43:41.806
PROFESSOR: Yeah, good.
00:43:41.806 --> 00:43:42.307
Excellent.
00:43:42.307 --> 00:43:43.097
That's really good.
00:43:43.097 --> 00:43:44.150
So two things about it.
00:43:44.150 --> 00:43:46.900
So one thing is this form
for the energy operator
00:43:46.900 --> 00:43:49.150
is particularly simple.
00:43:49.150 --> 00:43:49.850
We see the 1/2.
00:43:49.850 --> 00:43:51.770
This looks suggestive
from before.
00:43:51.770 --> 00:43:55.150
But it makes it obvious
that E is Hermetian.
00:43:55.150 --> 00:43:56.830
And that may not be
obvious to you guys.
00:43:56.830 --> 00:43:58.360
So let's just check.
00:43:58.360 --> 00:44:00.740
Here's something that you'll
show on the problem set.
00:44:00.740 --> 00:44:05.310
AB adjoint is equal to
B adjoint A adjoint.
00:44:05.310 --> 00:44:06.920
The order matters.
00:44:06.920 --> 00:44:08.070
These are operators.
00:44:08.070 --> 00:44:09.760
And so if we take the
adjective of this,
00:44:09.760 --> 00:44:10.968
what's this going to give us?
00:44:10.968 --> 00:44:12.270
Well we change the order.
00:44:12.270 --> 00:44:13.520
So it's going to be
a dagger, and then we
00:44:13.520 --> 00:44:15.300
take the dagger of
both of the a dagger.
00:44:15.300 --> 00:44:17.730
So this is self-adjoint,
or Hermetian.
00:44:17.730 --> 00:44:18.590
So that's good.
00:44:18.590 --> 00:44:19.760
Of course, we already
knew that, because we
00:44:19.760 --> 00:44:21.880
could have written it
in terms of x and p.
00:44:21.880 --> 00:44:23.450
But this is somehow simpler.
00:44:23.450 --> 00:44:25.990
And it in particular
emphasizes the form,
00:44:25.990 --> 00:44:28.565
or recapitulates the form
of the energy eigenvalues.
00:44:31.170 --> 00:44:33.270
Why else would we care
about a and a dagger?
00:44:36.721 --> 00:44:39.179
OK, now this is a good moment.
00:44:39.179 --> 00:44:40.220
Here's the second reason.
00:44:40.220 --> 00:44:41.928
So the first reason
you care is this sort
00:44:41.928 --> 00:44:43.590
of structural
similarity and the fact
00:44:43.590 --> 00:44:46.270
that it's nicely Hermetian
in a different way.
00:44:46.270 --> 00:44:48.060
Here's the key thing.
00:44:48.060 --> 00:44:50.480
Key.
00:44:50.480 --> 00:44:55.330
a and a dagger satisfy the
simplest commutation relation
00:44:55.330 --> 00:44:57.160
in the world.
00:44:57.160 --> 00:44:58.280
Well, the second simplest.
00:44:58.280 --> 00:45:00.405
The simplest is that it's
0 on the right-hand side.
00:45:00.405 --> 00:45:02.830
But the simplest not trivial
commutation relationship.
00:45:02.830 --> 00:45:09.324
a with a dagger is equal to--
so what is a dagger equal to?
00:45:09.324 --> 00:45:10.490
We just take the definition.
00:45:10.490 --> 00:45:11.240
Let's put this in.
00:45:11.240 --> 00:45:21.050
So this is x over x0
plus ip over p0, comma,
00:45:21.050 --> 00:45:27.140
x over x0 minus i, p over p0,
hat, hat, hat, hat, bracket,
00:45:27.140 --> 00:45:28.080
bracket.
00:45:28.080 --> 00:45:28.690
Good.
00:45:28.690 --> 00:45:32.030
So here there are
going to be four terms.
00:45:32.030 --> 00:45:33.230
There's x commutator x.
00:45:33.230 --> 00:45:36.040
What is that?
00:45:36.040 --> 00:45:39.551
What is the commutator of
an operator with itself?
00:45:39.551 --> 00:45:40.050
0.
00:45:40.050 --> 00:45:43.145
Because remember the
definition of the commutator A,
00:45:43.145 --> 00:45:47.410
B is AB minus BA.
00:45:47.410 --> 00:45:52.270
So A with A is equal
to AA minus AA.
00:45:52.270 --> 00:45:55.680
And you have no options there.
00:45:55.680 --> 00:45:56.480
That's 0.
00:45:56.480 --> 00:45:59.667
So x with x is 0. p with p is 0.
00:45:59.667 --> 00:46:01.750
So the only terms that
matter are the cross terms.
00:46:01.750 --> 00:46:04.500
We have an x with p.
00:46:04.500 --> 00:46:08.192
And notice that's going to be
times a minus i with p0 and x0.
00:46:08.192 --> 00:46:09.650
And then we have
another term which
00:46:09.650 --> 00:46:14.650
is p with x, which
is i, p0 over x0.
00:46:14.650 --> 00:46:16.650
So you change the order
and you change the sign.
00:46:16.650 --> 00:46:18.483
But if you change the
order of a commutator,
00:46:18.483 --> 00:46:19.500
you change the side.
00:46:19.500 --> 00:46:21.845
So we can put them
both in the same order.
00:46:21.845 --> 00:46:22.970
Let me just write this out.
00:46:22.970 --> 00:46:24.950
So this is i over x0 p0.
00:46:24.950 --> 00:46:27.590
So this guy, minus i over x0p0.
00:46:27.590 --> 00:46:31.990
But x0p0 is equal to 2h
bar, as we checked before.
00:46:31.990 --> 00:46:34.520
This was x with p.
00:46:34.520 --> 00:46:40.190
And then the second
term was plus i, again
00:46:40.190 --> 00:46:43.400
over x0p0, which is
2h bar, p with x.
00:46:46.810 --> 00:46:50.610
This x with p is equal to?
00:46:50.610 --> 00:46:51.330
i h bar.
00:46:51.330 --> 00:46:52.280
So the h bar cancels.
00:46:52.280 --> 00:46:56.800
The i gives me a plus 1.
00:46:56.800 --> 00:46:59.370
And p with x gives
me minus i h bar.
00:46:59.370 --> 00:47:04.422
So the h bar and the
minus i gives me plus 1.
00:47:04.422 --> 00:47:05.130
Well that's nice.
00:47:05.130 --> 00:47:07.370
This is equal to 1.
00:47:07.370 --> 00:47:16.875
So plus 1/2, therefore a
with a dagger is equal to 1.
00:47:20.800 --> 00:47:25.110
As advertised, that is
about as simple as it gets.
00:47:25.110 --> 00:47:26.610
Notice a couple of
other commutators
00:47:26.610 --> 00:47:29.230
that follow from this.
00:47:29.230 --> 00:47:35.637
a dagger with a is
equal to minus 1.
00:47:35.637 --> 00:47:36.720
We just changed the order.
00:47:36.720 --> 00:47:38.520
And that's just an
overall minus sign.
00:47:38.520 --> 00:47:40.430
And a with a is what?
00:47:40.430 --> 00:47:41.390
0.
00:47:41.390 --> 00:47:43.310
a dagger with a dagger?
00:47:43.310 --> 00:47:43.940
Good.
00:47:43.940 --> 00:47:46.060
OK.
00:47:46.060 --> 00:47:53.930
So we are now going to use
this commutation relation
00:47:53.930 --> 00:47:56.960
to totally crush the
problem into submission.
00:47:56.960 --> 00:48:00.700
It's going to be weeping before
us like the Romans in front
00:48:00.700 --> 00:48:01.770
of the Visigoths.
00:48:01.770 --> 00:48:03.570
It's going to be dramatic.
00:48:03.570 --> 00:48:06.900
OK, so let's check.
00:48:06.900 --> 00:48:10.080
So let's combine the two things.
00:48:10.080 --> 00:48:12.360
So we had the first thing
is that this form is simple.
00:48:12.360 --> 00:48:14.010
The second is that the
commutator is simple.
00:48:14.010 --> 00:48:15.930
Let's combine these
together and really milk
00:48:15.930 --> 00:48:16.880
the system for what it's got.
00:48:16.880 --> 00:48:18.786
And to do that, I need
two more commutators.
00:48:21.880 --> 00:48:24.124
And the lesson of this
series of machinations,
00:48:24.124 --> 00:48:25.540
it's very tempting
to look at this
00:48:25.540 --> 00:48:26.650
and be like, why
are you doing this?
00:48:26.650 --> 00:48:28.233
And the reason is,
I want to encourage
00:48:28.233 --> 00:48:32.270
you to see the power of
these commutation relations.
00:48:32.270 --> 00:48:34.950
They're telling you a tremendous
amount about the system.
00:48:34.950 --> 00:48:36.440
So we're going through and
doing some relatively simple
00:48:36.440 --> 00:48:36.950
calculations.
00:48:36.950 --> 00:48:38.324
We're just computing
commutators.
00:48:38.324 --> 00:48:39.740
We're following our nose.
00:48:39.740 --> 00:48:43.030
And we're going to
derive something awesome.
00:48:43.030 --> 00:48:46.310
So don't just bear with it.
00:48:46.310 --> 00:48:48.830
Learn from this, that
there's something very useful
00:48:48.830 --> 00:48:50.740
and powerful about
commutation relations.
00:48:50.740 --> 00:48:51.510
You'll see that at the end.
00:48:51.510 --> 00:48:53.426
But I want you to on to
the slight awkwardness
00:48:53.426 --> 00:48:56.520
right now, that it's not
totally obvious beforehand where
00:48:56.520 --> 00:48:58.700
this is going.
00:48:58.700 --> 00:49:00.786
So what is E with a?
00:49:00.786 --> 00:49:01.660
That's easy.
00:49:01.660 --> 00:49:06.210
It's the h bar omega
a dagger a plus 1/2.
00:49:06.210 --> 00:49:09.385
So the 1/2, what's 1/2
commutator with an operator?
00:49:12.120 --> 00:49:12.620
0.
00:49:12.620 --> 00:49:14.494
Because any number
commutes with an operator.
00:49:14.494 --> 00:49:16.380
1/2 operator is operator 1/2.
00:49:16.380 --> 00:49:18.270
It's just a constant.
00:49:18.270 --> 00:49:19.020
That term is gone.
00:49:19.020 --> 00:49:20.436
So the only thing
that's left over
00:49:20.436 --> 00:49:25.764
is h bar omega, a
dagger a with a.
00:49:25.764 --> 00:49:27.180
The h bar omega's
just a constant.
00:49:27.180 --> 00:49:29.721
It's going to pull out no matter
which term we're looking at.
00:49:29.721 --> 00:49:33.450
So I could just pull
that factor out.
00:49:33.450 --> 00:49:40.840
So this is equal to h
bar omega times a dagger
00:49:40.840 --> 00:49:46.420
a minus a a dagger a.
00:49:46.420 --> 00:49:49.360
But this is equal to
h bar omega-- well,
00:49:49.360 --> 00:49:51.420
that's a dagger a
a, a a dagger a.
00:49:51.420 --> 00:49:53.690
You can just pull out
the a on the right.
00:49:53.690 --> 00:49:58.230
a dagger a minus a a dagger a.
00:49:58.230 --> 00:50:00.130
That's equal to h bar omega.
00:50:00.130 --> 00:50:08.360
Well, a dagger with a is equal
to a dagger with a minus 1
00:50:08.360 --> 00:50:10.150
is equal to minus h bar omega.
00:50:10.150 --> 00:50:12.070
And we have this a leftover, a.
00:50:12.070 --> 00:50:17.960
So E with a is equal to minus a.
00:50:17.960 --> 00:50:20.500
Well, that's interesting.
00:50:20.500 --> 00:50:22.000
Now, the second
commutator-- I'm not
00:50:22.000 --> 00:50:25.092
going to do it--
E with a dagger is
00:50:25.092 --> 00:50:26.800
going to be equal to--
let's just eyeball
00:50:26.800 --> 00:50:27.758
what's going to happen.
00:50:27.758 --> 00:50:30.600
They can be a dagger.
00:50:30.600 --> 00:50:32.870
So we're going to
have a dagger a dagger
00:50:32.870 --> 00:50:34.289
minus a dagger a dagger a.
00:50:34.289 --> 00:50:36.080
So we're going to have
an a dagger in front
00:50:36.080 --> 00:50:38.661
and then a dagger.
00:50:38.661 --> 00:50:40.160
So all we're going
to get is a sign.
00:50:40.160 --> 00:50:43.394
And it's going to be a
dagger plus a dagger.
00:50:46.355 --> 00:50:47.980
I shouldn't written
that in the center.
00:50:52.420 --> 00:50:54.195
Everyone cool with that?
00:50:54.195 --> 00:50:54.695
Yeah.
00:50:54.695 --> 00:50:57.360
AUDIENCE: The h bar where?
00:50:57.360 --> 00:50:59.740
PROFESSOR: Oh shoot, thank you!
00:50:59.740 --> 00:51:00.530
h bar here.
00:51:00.530 --> 00:51:01.105
Thank you.
00:51:03.860 --> 00:51:07.760
We would have
misruled the galaxy.
00:51:07.760 --> 00:51:08.859
OK, good.
00:51:08.859 --> 00:51:09.525
Other questions?
00:51:12.410 --> 00:51:16.102
You don't notice-- you haven't
noticed yet, but we just won.
00:51:16.102 --> 00:51:17.560
We just totally
solved the problem.
00:51:17.560 --> 00:51:18.590
And here's why.
00:51:18.590 --> 00:51:24.050
Once you see this, any
time you see this, anytime
00:51:24.050 --> 00:51:27.325
you see this commutator,
an operator with an a
00:51:27.325 --> 00:51:31.050
is equal to plus a times some
constant, anytime you see this,
00:51:31.050 --> 00:51:32.460
cheer.
00:51:32.460 --> 00:51:33.970
And here's why.
00:51:33.970 --> 00:51:34.500
Yeah, right.
00:51:34.500 --> 00:51:35.000
Exactly.
00:51:35.000 --> 00:51:35.680
Now.
00:51:35.680 --> 00:51:37.920
Whoo!
00:51:37.920 --> 00:51:39.064
Here's why.
00:51:39.064 --> 00:51:40.230
Here's why you should cheer.
00:51:40.230 --> 00:51:42.430
Because you no longer have
to solve any problems.
00:51:42.430 --> 00:51:44.800
You no longer have to solve
any differential equations.
00:51:44.800 --> 00:51:46.600
You can simply write
down the problem.
00:51:46.600 --> 00:51:48.850
And let's see that you can
just write down the answer.
00:51:51.740 --> 00:51:54.410
Suppose that we already
happened to have access--
00:51:54.410 --> 00:51:57.710
here in my sleeve I have
access to an eigenfunction
00:51:57.710 --> 00:51:58.810
of the energy operator.
00:51:58.810 --> 00:52:05.780
E on phi E is equal to E phi
E. Suppose I have this guy.
00:52:05.780 --> 00:52:08.500
Cool?
00:52:08.500 --> 00:52:10.260
Check this out.
00:52:10.260 --> 00:52:15.650
Consider a new
state, psi, which is
00:52:15.650 --> 00:52:18.810
equal to a-- which do
I want to do first?
00:52:18.810 --> 00:52:20.740
Doesn't really matter,
but let's do a.
00:52:20.740 --> 00:52:26.910
Consider psi is
equal to a on phi E.
00:52:26.910 --> 00:52:28.625
What can you say
about this state?
00:52:28.625 --> 00:52:31.000
Well, it's the state you get
by taking this wave function
00:52:31.000 --> 00:52:32.300
and acting with a.
00:52:32.300 --> 00:52:33.550
Not terribly illuminating.
00:52:33.550 --> 00:52:39.670
However, E on psi
is equal to what?
00:52:39.670 --> 00:52:41.730
Maybe this has some nice
property under acting
00:52:41.730 --> 00:52:47.390
with E. This is equal
to E on a with pfi E.
00:52:47.390 --> 00:52:48.660
Now, this is tantalizing.
00:52:48.660 --> 00:52:50.700
Because at this point
it's very-- look, that E,
00:52:50.700 --> 00:52:52.840
it really wants to hit this phi.
00:52:52.840 --> 00:52:53.840
It just really wants to.
00:52:53.840 --> 00:52:55.256
There's an E it
wants to pull out.
00:52:55.256 --> 00:52:56.130
It'll be great.
00:52:56.130 --> 00:52:57.920
The problem is it's not there.
00:52:57.920 --> 00:52:59.030
There's an a in the way.
00:52:59.030 --> 00:53:01.880
And so at this point we add 0.
00:53:01.880 --> 00:53:03.800
And this is a very
powerful technique.
00:53:03.800 --> 00:53:17.110
This is equal to Ea
minus aE plus aE, phi E.
00:53:17.110 --> 00:53:18.770
But that has a nice expression.
00:53:18.770 --> 00:53:21.610
This is equal to Ea minus aE.
00:53:21.610 --> 00:53:25.960
That's the commutator
of E with a.
00:53:25.960 --> 00:53:27.110
Plus a.
00:53:27.110 --> 00:53:29.960
What's E acting on phi E?
00:53:29.960 --> 00:53:31.960
Actually, let me just
leave this as aE.
00:53:31.960 --> 00:53:34.932
So what have we done here
before we actually act?
00:53:34.932 --> 00:53:36.390
What we've done is
something called
00:53:36.390 --> 00:53:37.905
commuting an operator through.
00:53:37.905 --> 00:53:40.030
So what do I mean by
commuting an operator through?
00:53:40.030 --> 00:53:44.016
If we have an operator A and
an operator B and a state f,
00:53:44.016 --> 00:53:46.800
and I want A to act
on f, I can always
00:53:46.800 --> 00:53:51.070
write this as-- this is
equal to the commutator of A
00:53:51.070 --> 00:53:55.470
would be plus BA acting on f.
00:53:55.470 --> 00:53:58.266
So this lets me act A
on f directly without B.
00:53:58.266 --> 00:54:00.390
But I have to know what
the commutator of these two
00:54:00.390 --> 00:54:01.365
operators is.
00:54:01.365 --> 00:54:03.490
So if I know what the
commutator is, I can do this.
00:54:03.490 --> 00:54:04.962
I can simplify.
00:54:04.962 --> 00:54:07.170
When one does this, when
one takes AB and replaces it
00:54:07.170 --> 00:54:10.310
by the commutator of A with B,
plus BA, changing the order,
00:54:10.310 --> 00:54:14.470
the phrase that one uses is
I have commuted A through B.
00:54:14.470 --> 00:54:16.520
And commuting
operators through other
00:54:16.520 --> 00:54:19.940
is an extraordinarily useful
tool, useful technique.
00:54:19.940 --> 00:54:20.860
Now let's do y.
00:54:20.860 --> 00:54:25.230
So here what's the
commutator of E with a?
00:54:25.230 --> 00:54:26.850
We just did that.
00:54:26.850 --> 00:54:30.360
It's minus h bar omega a.
00:54:30.360 --> 00:54:33.230
And what's aE on phi E?
00:54:33.230 --> 00:54:35.740
What's E on phi E?
00:54:35.740 --> 00:54:37.820
E. Exactly.
00:54:37.820 --> 00:54:42.290
Plus Ea on phi.
00:54:46.240 --> 00:54:48.180
And now we're cooking with gas.
00:54:48.180 --> 00:54:54.750
Because this is equal to minus
h bar omega a plus Ea, hat.
00:54:54.750 --> 00:54:57.450
I'm going to pull out
this common factor of a.
00:54:57.450 --> 00:55:04.904
So if I pull out that common
factor of a, plus E, a phi E,
00:55:04.904 --> 00:55:06.570
and now I'm going to
just slightly write
00:55:06.570 --> 00:55:08.752
this instead of minus
h bar omega plus E,
00:55:08.752 --> 00:55:10.710
I'm going to write this
as E minus h bar omega.
00:55:10.710 --> 00:55:12.918
I'm just literally changing
the order of the algebra.
00:55:12.918 --> 00:55:15.340
E minus h bar omega.
00:55:15.340 --> 00:55:18.330
And what is aE?
00:55:18.330 --> 00:55:18.830
Psi.
00:55:18.830 --> 00:55:22.820
That was the original
state we started with, psi.
00:55:22.820 --> 00:55:24.230
Well, that's cool.
00:55:24.230 --> 00:55:27.460
If I have a state with
energy E and I act on it
00:55:27.460 --> 00:55:32.420
with the operator a, I
get a new state, psi,
00:55:32.420 --> 00:55:36.900
which is also an eigenstate
of the energy operator,
00:55:36.900 --> 00:55:39.940
but with a slightly
different energy eigenvalue.
00:55:39.940 --> 00:55:44.115
The eigenvalue is now
decreased by h bar omega.
00:55:44.115 --> 00:55:44.615
Cool?
00:55:50.090 --> 00:55:54.350
And that is what we wanted.
00:55:54.350 --> 00:55:58.560
Let's explore the
consequences of this.
00:55:58.560 --> 00:56:03.160
So if we have a state
with eigenvalue E,
00:56:03.160 --> 00:56:08.440
we have phi E such that E on
phi E is equal to E phi E.
00:56:08.440 --> 00:56:17.780
Then the state a
phi E has eigenvalue
00:56:17.780 --> 00:56:26.100
as energy, eigenvalue
E minus h bar omega.
00:56:30.380 --> 00:56:39.230
So I could call this phi
sub E minus h bar omega.
00:56:39.230 --> 00:56:41.940
It's an eigenfunction
of the energy operator,
00:56:41.940 --> 00:56:44.530
the eigenvalue, E
minus h bar omega.
00:56:44.530 --> 00:56:45.740
Agreed?
00:56:45.740 --> 00:56:50.250
Do I know that this is in
fact properly normalized?
00:56:50.250 --> 00:56:51.830
No, because 12
times it would also
00:56:51.830 --> 00:56:53.204
be a perfectly
good eigenfunction
00:56:53.204 --> 00:56:54.210
of the energy operator.
00:56:54.210 --> 00:56:57.880
So this is proportional to
the properly normalized guy,
00:56:57.880 --> 00:57:00.937
with some, at the moment,
unknown constant coefficient
00:57:00.937 --> 00:57:01.520
normalization.
00:57:01.520 --> 00:57:03.590
Everyone cool with that?
00:57:03.590 --> 00:57:05.820
So now let's think about
what this tells us.
00:57:05.820 --> 00:57:07.335
This tells us if
we have a state phi
00:57:07.335 --> 00:57:10.330
E, which I will denote
its energy by this level,
00:57:10.330 --> 00:57:13.460
then if I act on
it with a phi E I
00:57:13.460 --> 00:57:16.790
get another state where the
energy, instead of being E,
00:57:16.790 --> 00:57:19.270
is equal E minus h bar omega.
00:57:19.270 --> 00:57:22.660
So this distance in
energy is h bar omega.
00:57:22.660 --> 00:57:24.180
Cool?
00:57:24.180 --> 00:57:26.110
Let me do it again.
00:57:26.110 --> 00:57:29.070
We'll tack a on phi E. By
exactly the same argument, if I
00:57:29.070 --> 00:57:34.660
make psi as equal to a on
a phi E, a squared phi E,
00:57:34.660 --> 00:57:38.819
I get another state, again
separated by h bar omega, E
00:57:38.819 --> 00:57:39.610
minus 2h bar omega.
00:57:44.699 --> 00:57:45.740
Turtles all the way down.
00:57:50.911 --> 00:57:51.910
Everyone cool with that?
00:57:54.485 --> 00:57:56.235
Let's do a slightly
different calculation.
00:58:00.590 --> 00:58:05.052
But before we do that,
I want to give a a name.
00:58:05.052 --> 00:58:06.260
a does something really cool.
00:58:06.260 --> 00:58:08.820
When you take the state phi
E that has definite energy E,
00:58:08.820 --> 00:58:10.860
it's an energy eigenfunction,
and you act on it
00:58:10.860 --> 00:58:13.880
with a, what happens?
00:58:13.880 --> 00:58:16.180
It lowers the energy
by h bar omega.
00:58:16.180 --> 00:58:18.470
So I'm going to call a
the lowering operator.
00:58:21.270 --> 00:58:24.890
Because what it does is it
takes a state with phi E,
00:58:24.890 --> 00:58:26.530
with energy
eigenvalue E to state
00:58:26.530 --> 00:58:28.730
with energy E minus h bar omega.
00:58:28.730 --> 00:58:30.980
And I can just keep doing
this as many times as I like
00:58:30.980 --> 00:58:31.813
and I build a tower.
00:58:31.813 --> 00:58:32.600
Yes?
00:58:32.600 --> 00:58:34.330
AUDIENCE: [INAUDIBLE]
00:58:34.330 --> 00:58:35.580
PROFESSOR: Very good question.
00:58:35.580 --> 00:58:37.832
Hold on to that for a second.
00:58:37.832 --> 00:58:39.540
We'll come back to
that in just a second.
00:58:42.920 --> 00:58:46.500
So this seems to build
for me a ladder downwards.
00:58:46.500 --> 00:58:47.684
Everyone cool with that?
00:58:47.684 --> 00:58:49.850
But we could have done the
same thing with a dagger.
00:58:49.850 --> 00:58:51.141
And how does this story change?
00:58:51.141 --> 00:58:53.240
What happens if we take
a dagger instead of a?
00:58:53.240 --> 00:58:54.865
Well, let's go through
every step here.
00:58:54.865 --> 00:58:57.290
So this is going to
be E on a dagger.
00:58:57.290 --> 00:59:01.610
And now we have E a dagger,
a dagger, E, a dagger.
00:59:01.610 --> 00:59:02.910
What's E with a dagger?
00:59:08.780 --> 00:59:11.990
E with a dagger is equal to
same thing but with a plus.
00:59:14.810 --> 00:59:15.840
And again, psi.
00:59:15.840 --> 00:59:20.020
Same thing, because the
a dagger factors out.
00:59:20.020 --> 00:59:22.050
Yeah?
00:59:22.050 --> 00:59:28.600
So we go down by acting with a.
00:59:28.600 --> 00:59:30.500
We go up by acting
with a dagger.
00:59:33.760 --> 00:59:36.510
And again, the spacing
is h bar omega.
00:59:36.510 --> 00:59:38.520
And we go up by acting
with a dagger again.
00:59:46.560 --> 00:59:53.560
So a and a dagger are called the
raising and lowering operators.
00:59:58.770 --> 01:00:01.650
a dagger, the raising operator.
01:00:12.670 --> 01:00:17.563
a dagger phi E plus h bar omega.
01:00:24.660 --> 01:00:31.990
So what that lets us do is
build a tower of states,
01:00:31.990 --> 01:00:34.360
an infinite number of
states where, given a state,
01:00:34.360 --> 01:00:37.400
we can walk up this ladder
with the raising operator,
01:00:37.400 --> 01:00:41.450
and we can walk down it
by the lowering operator.
01:00:41.450 --> 01:00:43.780
So now I ask you
the question, why
01:00:43.780 --> 01:00:45.865
is this ladder evenly spaced?
01:00:49.630 --> 01:00:52.089
There's one equation on the
board that you can point to-- I
01:00:52.089 --> 01:00:54.088
guess two, technically--
there are two equations
01:00:54.088 --> 01:00:55.860
on the board that you
could point to that
01:00:55.860 --> 01:00:58.100
suffice to immediately
answer the question,
01:00:58.100 --> 01:01:03.944
why is the tower of energy
eigenstates evenly spaced.
01:01:03.944 --> 01:01:04.860
What is that equation?
01:01:07.005 --> 01:01:07.880
AUDIENCE: [INAUDIBLE]
01:01:07.880 --> 01:01:10.260
PROFESSOR: Yeah,
those commutators.
01:01:10.260 --> 01:01:11.969
These commutators
are all we needed.
01:01:11.969 --> 01:01:13.510
We didn't need to
know anything else.
01:01:13.510 --> 01:01:17.120
We didn't even need to know
what the potential was.
01:01:17.120 --> 01:01:19.200
If I just told you there's
an energy operator E
01:01:19.200 --> 01:01:21.200
and there's an operator
a that you can build out
01:01:21.200 --> 01:01:22.700
of the observables
of the system, such
01:01:22.700 --> 01:01:24.366
that you have this
commutation relation,
01:01:24.366 --> 01:01:26.610
what do you immediately know?
01:01:26.610 --> 01:01:29.360
You immediately know that
you get a tower of operators.
01:01:29.360 --> 01:01:31.630
Because you can act with
a and raise the energy
01:01:31.630 --> 01:01:33.494
by a finite amount,
which is the coefficient
01:01:33.494 --> 01:01:34.660
of that a in the commutator.
01:01:38.010 --> 01:01:40.350
This didn't have to be
the quantum mechanics
01:01:40.350 --> 01:01:42.220
of the harmonic
oscillator at this point.
01:01:42.220 --> 01:01:45.390
We just needed this commutator
relation, E with a, E
01:01:45.390 --> 01:01:47.539
with a dagger.
01:01:47.539 --> 01:01:49.080
And one of the
totally awesome things
01:01:49.080 --> 01:01:50.149
is how often it shows up.
01:01:50.149 --> 01:01:52.190
If you take a bunch of
electrons and you put them
01:01:52.190 --> 01:01:55.290
in a magnetic field,
bunch of electrons,
01:01:55.290 --> 01:01:57.767
very strong magnetic
field, what you discover
01:01:57.767 --> 01:01:59.350
is the quantum
mechanics of those guys
01:01:59.350 --> 01:02:01.083
has nothing to do with the
harmonic oscillator on the face
01:02:01.083 --> 01:02:04.160
if it's magnetic fields, Lorentz
force law, the whole thing.
01:02:04.160 --> 01:02:07.275
What you discover is
there's an operator, which
01:02:07.275 --> 01:02:09.455
isn't usually called a, but
it depends on which book
01:02:09.455 --> 01:02:11.440
you use-- it's n
or m or l-- there's
01:02:11.440 --> 01:02:13.180
an operator that
commutes with the energy
01:02:13.180 --> 01:02:16.180
operator in precisely this
fashion, which tells you
01:02:16.180 --> 01:02:20.070
that the energy eigenstates
live in a ladder.
01:02:20.070 --> 01:02:21.926
They're called Landau levels.
01:02:21.926 --> 01:02:23.300
This turns out to
be very useful.
01:02:23.300 --> 01:02:24.883
Any of you who are
doing a [INAUDIBLE]
01:02:24.883 --> 01:02:27.970
in the lab that has
graphene or any material,
01:02:27.970 --> 01:02:30.620
really, with a magnetic
field, then this matters.
01:02:30.620 --> 01:02:33.800
So this commutator
encodes an enormous amount
01:02:33.800 --> 01:02:36.420
of the structure of
the energy eigenvalues.
01:02:36.420 --> 01:02:38.460
And the trick for us
was showing that we
01:02:38.460 --> 01:02:40.210
could write the harmonic
oscillator energy
01:02:40.210 --> 01:02:43.200
operator in terms of operators
that commute in this fashion.
01:02:46.040 --> 01:02:49.550
So we're going to run into this
structure over and over again.
01:02:49.550 --> 01:02:51.070
This operator
commutes with this one
01:02:51.070 --> 01:02:53.120
to the same operator times
a constant that tells you
01:02:53.120 --> 01:02:53.770
have a ladder.
01:02:53.770 --> 01:02:55.020
We're going to run
into that over and over
01:02:55.020 --> 01:02:57.030
again when we talk about
Landau levels, if we get there.
01:02:57.030 --> 01:02:58.260
When we talk about
angular momentum
01:02:58.260 --> 01:02:59.390
we'll get the same thing.
01:02:59.390 --> 01:03:01.139
When we talk about the
harmonic oscillator
01:03:01.139 --> 01:03:02.700
we'll get the same thing.
01:03:02.700 --> 01:03:04.960
Sorry, the hydrogen system.
01:03:04.960 --> 01:03:07.680
We'll get the same thing.
01:03:07.680 --> 01:03:15.285
So second question, does this
ladder extend infinitely up?
01:03:15.285 --> 01:03:16.430
Yeah, why not?
01:03:16.430 --> 01:03:17.785
Can it extend infinitely down?
01:03:17.785 --> 01:03:18.410
AUDIENCE: Nope.
01:03:18.410 --> 01:03:20.492
PROFESSOR: Why?
01:03:20.492 --> 01:03:21.450
AUDIENCE: Ground state.
01:03:21.450 --> 01:03:22.850
PROFESSOR: Well, people
are saying ground state.
01:03:22.850 --> 01:03:24.370
Well, we know that from the
brute force calculation.
01:03:24.370 --> 01:03:26.036
But without the brute
force calculation,
01:03:26.036 --> 01:03:27.953
can this ladder extend
infinitely down?
01:03:27.953 --> 01:03:30.120
AUDIENCE: [INAUDIBLE]
you can't go [INAUDIBLE].
01:03:30.120 --> 01:03:30.480
PROFESSOR: Brilliant.
01:03:30.480 --> 01:03:30.980
OK, good.
01:03:30.980 --> 01:03:32.140
And as you'll prove
on the problems,
01:03:32.140 --> 01:03:34.306
that you can't make the
energy arbitrarily negative.
01:03:34.306 --> 01:03:36.990
But let me make that sharp.
01:03:36.990 --> 01:03:40.930
I don't want to appeal to
something we haven't proven.
01:03:40.930 --> 01:03:42.695
Let me show you that concretely.
01:03:48.910 --> 01:03:52.370
In some state, in any state,
the energy expectation value
01:03:52.370 --> 01:03:54.492
can be written as the
integral of phi complex
01:03:54.492 --> 01:03:56.200
conjugate-- we'll say
in this state phi--
01:03:56.200 --> 01:03:58.110
phi complex conjugate E phi.
01:04:01.570 --> 01:04:03.510
But I can write this
as the integral,
01:04:03.510 --> 01:04:07.460
and let's say dx, integral dx.
01:04:07.460 --> 01:04:10.160
Let's just put in what the
energy operator looks like.
01:04:10.160 --> 01:04:12.235
So psi tilda, we can
take the 4a transfer
01:04:12.235 --> 01:04:16.145
and write the psi tilda p,
p squared upon 2m-- whoops,
01:04:16.145 --> 01:04:19.939
dp-- for the kinetic energy
term, plus the integral--
01:04:19.939 --> 01:04:21.730
and now I'm using the
harmonic oscillator--
01:04:21.730 --> 01:04:29.640
plus the integral dx of psi of
x norm squared, norm squared,
01:04:29.640 --> 01:04:32.030
m omega squared upon 2x squared.
01:04:34.960 --> 01:04:36.409
Little bit of a
quick move there,
01:04:36.409 --> 01:04:38.200
doing the 4a transfer
for the momentum term
01:04:38.200 --> 01:04:39.610
and not doing the 4a
[INAUDIBLE] but it's OK.
01:04:39.610 --> 01:04:40.735
They're separate integrals.
01:04:40.735 --> 01:04:41.450
I can do this.
01:04:41.450 --> 01:04:43.870
And the crucial thing here
is, this is positive definite.
01:04:43.870 --> 01:04:45.850
This is positive definite,
positive definite,
01:04:45.850 --> 01:04:46.800
positive definite.
01:04:46.800 --> 01:04:48.410
All these terms are
strictly positive.
01:04:48.410 --> 01:04:50.076
This must be greater
than or equal to 0.
01:04:50.076 --> 01:04:52.240
It can never be negative.
01:04:52.240 --> 01:04:54.890
Yeah?
01:04:54.890 --> 01:04:56.670
So what that tells
us is there must
01:04:56.670 --> 01:05:02.950
be a minimum E. There
must be a minimum energy.
01:05:02.950 --> 01:05:04.360
And I will call it minimum E0.
01:05:07.026 --> 01:05:08.400
We can't lower
the tower forever.
01:05:15.030 --> 01:05:16.250
So how is this possible?
01:05:16.250 --> 01:05:18.412
How is it possible that,
look, on the one hand,
01:05:18.412 --> 01:05:20.870
if we want, if we have a state,
we can always build a lower
01:05:20.870 --> 01:05:24.450
energy state by acting
with lowering operator a.
01:05:24.450 --> 01:05:26.140
And yet this is telling
me that I can't.
01:05:26.140 --> 01:05:29.030
There must be a last one where
I can't lower it anymore.
01:05:29.030 --> 01:05:30.870
So what reaches out
of the chalkboard
01:05:30.870 --> 01:05:34.170
and stops me from
acting with a again?
01:05:34.170 --> 01:05:36.410
How can it possibly be
true that a always lowers
01:05:36.410 --> 01:05:38.670
the eigenfunction but
there's at least one
01:05:38.670 --> 01:05:41.580
that can't be
lowered any further.
01:05:41.580 --> 01:05:43.070
Normalizable's a good guess.
01:05:43.070 --> 01:05:43.860
Very good guess.
01:05:43.860 --> 01:05:45.840
Not the case.
01:05:45.840 --> 01:05:49.615
Because from this argument we
don't even use wave functions.
01:05:49.615 --> 01:05:50.490
AUDIENCE: [INAUDIBLE]
01:05:55.720 --> 01:05:56.930
PROFESSOR: That would be bad.
01:05:56.930 --> 01:05:57.670
Yes, exactly.
01:05:57.670 --> 01:05:59.570
So that would be
bad, but that's just
01:05:59.570 --> 01:06:01.502
saying that there's
an inconsistency here.
01:06:01.502 --> 01:06:03.210
So I'm going to come
back to your answer,
01:06:03.210 --> 01:06:04.250
a non-normalizable.
01:06:04.250 --> 01:06:06.620
It's correct, but
in a sneaky way.
01:06:06.620 --> 01:06:09.950
Here's the way it's sneaky.
01:06:09.950 --> 01:06:13.420
Consider a state
a on phi-- let's
01:06:13.420 --> 01:06:16.530
say this is the lowest state,
the lowest possible state.
01:06:16.530 --> 01:06:18.930
It must be true that the
resulting statement is not
01:06:18.930 --> 01:06:20.450
phi minus h bar omega.
01:06:20.450 --> 01:06:22.947
There can't be any such state.
01:06:22.947 --> 01:06:23.780
And how can that be?
01:06:23.780 --> 01:06:26.370
That can be true if it's 0.
01:06:26.370 --> 01:06:30.660
So if the lowering operator acts
on some state and gives me 0,
01:06:30.660 --> 01:06:33.700
well, OK, that's an eigenstate.
01:06:33.700 --> 01:06:35.070
But it's a stupid eigenstate.
01:06:35.070 --> 01:06:36.520
It's not normalizable.
01:06:36.520 --> 01:06:39.287
It can't be used to describe
any real physical object.
01:06:39.287 --> 01:06:40.120
Because where is it?
01:06:40.120 --> 01:06:40.660
Well, it's nowhere.
01:06:40.660 --> 01:06:42.659
The probability density,
you'd find it anywhere.
01:06:42.659 --> 01:06:45.400
It's nowhere, nothing, zero.
01:06:45.400 --> 01:06:48.540
So the way that this
tower terminates
01:06:48.540 --> 01:06:50.850
is by having a last
state, which we'll
01:06:50.850 --> 01:06:55.135
call phi 0, such that
lowering it gives me 0.
01:06:58.110 --> 01:07:00.127
Not the state called 0,
which I would call this,
01:07:00.127 --> 01:07:02.710
but actually the function called
0, which is not normalizable,
01:07:02.710 --> 01:07:05.390
which is not a good state.
01:07:05.390 --> 01:07:06.590
So there's a minimum E0.
01:07:06.590 --> 01:07:11.470
Associated with that is a
lowest energy eigenstate
01:07:11.470 --> 01:07:14.670
called the ground state.
01:07:14.670 --> 01:07:18.930
Now, can the energy
get arbitrarily large?
01:07:18.930 --> 01:07:19.430
Sure.
01:07:19.430 --> 01:07:20.660
That's a positive
definite thing,
01:07:20.660 --> 01:07:22.220
and this could get
as large as you like.
01:07:22.220 --> 01:07:23.910
There's no problem with the
energy eigenvalues getting
01:07:23.910 --> 01:07:24.500
arbitrarily large.
01:07:24.500 --> 01:07:26.541
We can just keep raising
and raising and raising.
01:07:26.541 --> 01:07:28.240
I mention that
because later on in
01:07:28.240 --> 01:07:30.090
the semester we
will find a system
01:07:30.090 --> 01:07:32.617
with exactly that commutation
relation, precisely
01:07:32.617 --> 01:07:34.200
that commutation
relation, where there
01:07:34.200 --> 01:07:38.044
will be a minimum and a maximum.
01:07:38.044 --> 01:07:39.960
So the communication
relation is a good start,
01:07:39.960 --> 01:07:40.860
but it doesn't
tell you anything.
01:07:40.860 --> 01:07:42.990
We have to add in some
physics like the energy
01:07:42.990 --> 01:07:46.100
operators bounded below for
the harmonic oscillator.
01:07:49.190 --> 01:07:50.572
Questions at this point?
01:07:50.572 --> 01:07:51.880
Yeah?
01:07:51.880 --> 01:07:54.700
AUDIENCE: So you basically
[INAUDIBLE] this ladder
01:07:54.700 --> 01:07:57.206
has to [INAUDIBLE] my
particular energy eigenstate
01:07:57.206 --> 01:07:58.747
and I can kind of
construct a ladder.
01:07:58.747 --> 01:08:00.246
How do I know that
I can't construct
01:08:00.246 --> 01:08:01.693
other, intersecting ladders?
01:08:01.693 --> 01:08:04.150
PROFESSOR: Yeah, that's
an excellent question.
01:08:04.150 --> 01:08:08.790
I remember vividly when I
saw this lecture in 143A,
01:08:08.790 --> 01:08:10.290
and that question plagued me.
01:08:10.290 --> 01:08:11.992
And foolishly I didn't ask it.
01:08:11.992 --> 01:08:12.950
So here's the question.
01:08:12.950 --> 01:08:15.075
The question is, look, you
found a bunch of states.
01:08:15.075 --> 01:08:16.622
How do you know
that's all of them?
01:08:16.622 --> 01:08:18.080
How do you know
that's all of them?
01:08:18.080 --> 01:08:19.140
So let's think through that.
01:08:19.140 --> 01:08:20.040
That's a very good question.
01:08:20.040 --> 01:08:21.831
I'm not going to worry
about normalization.
01:08:21.831 --> 01:08:25.054
There's a discussion of
normalization in the notes.
01:08:25.054 --> 01:08:26.470
How do we know
that's all of them?
01:08:29.344 --> 01:08:30.540
That's a little bit tricky.
01:08:30.540 --> 01:08:32.960
So let's think through it.
01:08:32.960 --> 01:08:35.350
Imagine it's not all of them.
01:08:35.350 --> 01:08:37.120
In particular, what
would that mean?
01:08:37.120 --> 01:08:39.359
In order for there to be
more states than the ones
01:08:39.359 --> 01:08:41.170
that we've written
down, there must
01:08:41.170 --> 01:08:43.653
be states that are
not on that tower.
01:08:43.653 --> 01:08:46.319
And how can we possi-- wow, this
thing is totally falling apart.
01:08:46.319 --> 01:08:47.160
How do we do that?
01:08:47.160 --> 01:08:49.479
How is that possible?
01:08:49.479 --> 01:08:52.040
There are two ways to do it.
01:08:52.040 --> 01:08:53.215
Here's my tower of states.
01:08:56.109 --> 01:08:59.529
I'll call this one phi 0.
01:08:59.529 --> 01:09:05.576
so I raise with a dagger
and I lower with a.
01:09:05.576 --> 01:09:07.450
So how could it be that
I missed some states?
01:09:07.450 --> 01:09:08.700
Well, there are ways to do it.
01:09:08.700 --> 01:09:11.910
One is there could be extra
states that are in between.
01:09:11.910 --> 01:09:13.910
So let's say that there's
one extra state that's
01:09:13.910 --> 01:09:15.779
in between these two.
01:09:15.779 --> 01:09:17.430
Just imagine that's true.
01:09:17.430 --> 01:09:21.540
If there is such a state,
by that commutation relation
01:09:21.540 --> 01:09:24.124
there must be another tower.
01:09:24.124 --> 01:09:26.540
So there must be this state,
and there must be this state,
01:09:26.540 --> 01:09:29.319
and there must be this state,
and there must be this state.
01:09:29.319 --> 01:09:30.840
Yeah?
01:09:30.840 --> 01:09:32.260
OK, so that's good so far.
01:09:32.260 --> 01:09:33.240
But what happens?
01:09:33.240 --> 01:09:38.000
Well, A on this guy gave me 0.
01:09:38.000 --> 01:09:41.649
And this is going to
be some phi tilde 0.
01:09:41.649 --> 01:09:44.590
Suppose that this tower ends.
01:09:44.590 --> 01:09:46.426
And now you have to
ask the question,
01:09:46.426 --> 01:09:47.800
can there be two
different states
01:09:47.800 --> 01:09:50.734
with two different
energies with a0?
01:09:50.734 --> 01:09:52.109
Can there be two
different states
01:09:52.109 --> 01:09:54.610
that are annihilated by a0?
01:09:54.610 --> 01:09:56.290
Well, let's check.
01:09:56.290 --> 01:09:59.130
What must be true of any
state annihilated by a0?
01:09:59.130 --> 01:10:02.420
Well, let's write the energy
operator acting on that state.
01:10:02.420 --> 01:10:04.040
What's the energy of that state?
01:10:04.040 --> 01:10:06.268
Energy on phi 0 is
equal to h bar omega.
01:10:06.268 --> 01:10:08.476
This is a very good question,
so let's go through it.
01:10:08.476 --> 01:10:10.920
So it's equal to h bar
omega times a dagger
01:10:10.920 --> 01:10:15.967
a plus 1/2 on phi 0.
01:10:15.967 --> 01:10:17.300
But what can you say about this?
01:10:17.300 --> 01:10:20.220
Well, a annihilates phi 0.
01:10:20.220 --> 01:10:21.814
It gives us 0.
01:10:21.814 --> 01:10:24.480
So in addition to a being called
the lowering operator it's also
01:10:24.480 --> 01:10:26.730
called the
annihilation operator,
01:10:26.730 --> 01:10:30.320
because, I don't know, we're
a brutal and warlike species.
01:10:30.320 --> 01:10:32.950
So this is equal to h bar
omega-- this term kills phi
01:10:32.950 --> 01:10:36.440
0-- again with the kills-- and
gives me a 1/2 half leftover.
01:10:36.440 --> 01:10:39.650
1/2 h bar omega phi 0.
01:10:39.650 --> 01:10:46.080
So the ground state, any state--
any state annihilated by a
01:10:46.080 --> 01:10:49.640
must have the same energy.
01:10:49.640 --> 01:10:51.633
The only way you can
be annihilated by a
01:10:51.633 --> 01:10:53.750
is if your energy is this.
01:10:53.750 --> 01:10:54.590
Cool?
01:10:54.590 --> 01:10:56.673
So what does that tell you
about the second ladder
01:10:56.673 --> 01:10:58.290
of hidden seats that we missed?
01:10:58.290 --> 01:10:59.642
It's got to be degenerate.
01:10:59.642 --> 01:11:01.100
It's got to have
the same energies.
01:11:08.330 --> 01:11:09.850
I drew that really
badly, didn't I?
01:11:09.850 --> 01:11:12.402
Those are evenly spaced.
01:11:12.402 --> 01:11:13.610
So it's got to be degenerate.
01:11:13.610 --> 01:11:16.450
However, Barton proved for you
the node theorem last time,
01:11:16.450 --> 01:11:16.950
right?
01:11:16.950 --> 01:11:20.554
He gave you my spread
argument for the node theorem?
01:11:20.554 --> 01:11:22.470
In particular, one of
the consequences of that
01:11:22.470 --> 01:11:25.125
is it in a system
with bound states,
01:11:25.125 --> 01:11:28.790
in a system with
potential that goes up,
01:11:28.790 --> 01:11:32.420
you can never have
degeneracies in one dimension.
01:11:32.420 --> 01:11:34.520
We're not going to prove
that carefully in here.
01:11:34.520 --> 01:11:35.945
But it's relatively
easy to prove.
01:11:35.945 --> 01:11:37.570
In fact, if you come
to my office hours
01:11:37.570 --> 01:11:37.935
I'll prove it for you.
01:11:37.935 --> 01:11:38.680
It takes three minutes.
01:11:38.680 --> 01:11:40.150
But I don't want to set
up the math right now.
01:11:40.150 --> 01:11:41.990
So how many people know
about the Wronskian?
01:11:44.870 --> 01:11:45.630
That's awesome.
01:11:45.630 --> 01:11:48.640
OK, so I leave it to you as an
exercise to use the Wronskian
01:11:48.640 --> 01:11:52.070
to show that there cannot be
degeneracies in one dimension,
01:11:52.070 --> 01:11:52.960
which is cool.
01:11:52.960 --> 01:11:55.340
Anyway, so the Wronskian for
the differential equation,
01:11:55.340 --> 01:11:58.100
which is the energy
eigenvalue equation.
01:11:58.100 --> 01:12:01.610
There can be no degeneracies
in one dimensional potentials
01:12:01.610 --> 01:12:03.297
with bound states.
01:12:03.297 --> 01:12:05.630
So what we've just shown is
that the only way that there
01:12:05.630 --> 01:12:07.046
can be extra states
that we missed
01:12:07.046 --> 01:12:10.230
is if there's a tower with
exactly identical energies all
01:12:10.230 --> 01:12:11.040
the way up.
01:12:11.040 --> 01:12:13.527
But if they have exactly
identical energies,
01:12:13.527 --> 01:12:14.860
that means there's a degenerate.
01:12:14.860 --> 01:12:18.350
But we can prove that there
can't be degeneracies in 1D.
01:12:18.350 --> 01:12:21.130
So can there be an extra
tower of states we missed?
01:12:21.130 --> 01:12:21.630
No.
01:12:21.630 --> 01:12:24.390
Can we have missed any states?
01:12:24.390 --> 01:12:25.180
No.
01:12:25.180 --> 01:12:26.850
Those are all the
states there are.
01:12:26.850 --> 01:12:29.210
And we've done it without
ever solving a differential
01:12:29.210 --> 01:12:32.297
equation, just by using
that commutation relation.
01:12:32.297 --> 01:12:34.130
Now at this point it's
very tempting to say,
01:12:34.130 --> 01:12:36.140
that was just sort of
magical mystery stuff.
01:12:36.140 --> 01:12:38.610
But what we really did
last time was very honest.
01:12:38.610 --> 01:12:39.540
We wrote down a
differential equation.
01:12:39.540 --> 01:12:40.510
We found the solution.
01:12:40.510 --> 01:12:42.330
And we got the wave functions.
01:12:42.330 --> 01:12:44.342
So, Professor Adams,
you just monkeyed around
01:12:44.342 --> 01:12:46.300
at the chalkboard with
commutators for a while,
01:12:46.300 --> 01:12:47.930
but what are the
damn wave functions?
01:12:47.930 --> 01:12:48.430
Right?
01:12:51.190 --> 01:12:52.770
We already have the answer.
01:12:52.770 --> 01:12:54.870
This is really quite nice.
01:12:54.870 --> 01:12:57.470
Last time we solved that
differential equation.
01:12:57.470 --> 01:12:59.386
And we had to solve that
differential equation
01:12:59.386 --> 01:13:02.470
many, many times,
different levels.
01:13:02.470 --> 01:13:04.550
But now we have a very
nice thing we can do.
01:13:04.550 --> 01:13:06.849
What's true of the ground state?
01:13:06.849 --> 01:13:08.390
Well, the ground
state is annihilated
01:13:08.390 --> 01:13:10.200
by the lowering operator.
01:13:10.200 --> 01:13:15.400
So that means that a acting
on phi 0 of x is equal to 0.
01:13:15.400 --> 01:13:19.170
But a has a nice expression,
which unfortunately I erased.
01:13:19.170 --> 01:13:19.920
Sorry about that.
01:13:19.920 --> 01:13:21.750
So a has a nice expression.
01:13:21.750 --> 01:13:27.670
a is equal to x over
x0 plus ip over p0.
01:13:27.670 --> 01:13:30.510
And so if you write
that out and multiply
01:13:30.510 --> 01:13:33.640
from appropriate
constants, this becomes
01:13:33.640 --> 01:13:35.140
the following
differential equation.
01:13:35.140 --> 01:13:36.589
The x is just multiplied by x.
01:13:36.589 --> 01:13:38.380
And the p is take a
derivative with respect
01:13:38.380 --> 01:13:40.837
to x, multiply by h bar upon i.
01:13:40.837 --> 01:13:42.420
And multiplying by
i over h bar to get
01:13:42.420 --> 01:13:52.612
that equation, this gives us dx
plus p over h bar x0-- sorry,
01:13:52.612 --> 01:13:53.570
that shouldn't be an i.
01:13:53.570 --> 01:13:57.710
That should be p0.
01:13:57.710 --> 01:14:00.550
x on phi 0 is equal to 0.
01:14:03.780 --> 01:14:05.710
And you solved
this last time when
01:14:05.710 --> 01:14:08.420
you did the asymptotic analysis.
01:14:08.420 --> 01:14:10.770
This is actually a
ridiculously easy equation.
01:14:10.770 --> 01:14:13.820
It's a first order
differential equation.
01:14:13.820 --> 01:14:15.385
There's one
integration constant.
01:14:15.385 --> 01:14:17.260
That's going to be the
overall normalization.
01:14:17.260 --> 01:14:19.180
And so the form is
completely fixed.
01:14:19.180 --> 01:14:20.596
First order
differential equation.
01:14:20.596 --> 01:14:22.260
So what's the
solution of this guy?
01:14:22.260 --> 01:14:23.030
It's a Gaussian.
01:14:23.030 --> 01:14:24.613
And what's the width
of that Gaussian?
01:14:24.613 --> 01:14:28.000
Well, look at p0 over h bar x0.
01:14:28.000 --> 01:14:31.660
We know that p0 times
x0 is twice h bar.
01:14:31.660 --> 01:14:33.960
So if I multiply by x amount
on the top and bottom,
01:14:33.960 --> 01:14:35.520
you get 2 h bar.
01:14:35.520 --> 01:14:37.060
The h bars cancel.
01:14:37.060 --> 01:14:40.904
So this gives me
two upon x0 squared.
01:14:40.904 --> 01:14:42.820
Remember I said it would
be useful to remember
01:14:42.820 --> 01:14:45.310
that p0 times x0 is 2h bar?
01:14:45.310 --> 01:14:46.870
It's useful.
01:14:46.870 --> 01:14:48.990
So it gives us this.
01:14:48.990 --> 01:14:51.180
And so the result is
that phi 0 is equal to,
01:14:51.180 --> 01:14:55.280
up to an overall normalization
coefficient, e to the minus
01:14:55.280 --> 01:14:58.980
x squared over x0 squared.
01:14:58.980 --> 01:15:01.081
Solid.
01:15:01.081 --> 01:15:01.580
So there.
01:15:01.580 --> 01:15:02.640
We've solved that
differential equation.
01:15:02.640 --> 01:15:04.780
Is the easiest, second
easiest differential equation.
01:15:04.780 --> 01:15:05.780
It's our first order
differential equation
01:15:05.780 --> 01:15:07.820
with a linear term
rather than a constant.
01:15:07.820 --> 01:15:08.720
We get a Gaussian.
01:15:08.720 --> 01:15:10.950
And now that we've
got this guy-- look,
01:15:10.950 --> 01:15:12.990
do you remember the
third Hermite polynomial?
01:15:12.990 --> 01:15:15.520
Because we know the
third excited state
01:15:15.520 --> 01:15:19.090
is given by h3
times this Gaussian.
01:15:19.090 --> 01:15:21.290
Do you remember it off
the top of your head?
01:15:21.290 --> 01:15:23.110
How do you solve what it is?
01:15:23.110 --> 01:15:24.940
How do we get phi 3?
01:15:24.940 --> 01:15:26.539
First off, how do we get phi 1?
01:15:26.539 --> 01:15:28.330
How do we get the next
state in the ladder?
01:15:28.330 --> 01:15:30.042
How do we get the wave function?
01:15:30.042 --> 01:15:30.750
Raising operator.
01:15:30.750 --> 01:15:32.240
But what is the
raising operator?
01:15:32.240 --> 01:15:35.120
Oh, it's the differential
operator I take with-- OK,
01:15:35.120 --> 01:15:38.230
but if I had a dagger, it's just
going to change the sign here.
01:15:38.230 --> 01:15:40.590
So how do I get phi 1?
01:15:40.590 --> 01:15:43.380
Phi 1 is equal to up
to some normalization.
01:15:43.380 --> 01:15:50.180
dx minus 2 over x0
squared, x0, phi 0.
01:15:52.365 --> 01:15:54.490
So now do I have to solve
the differential equation
01:15:54.490 --> 01:15:56.601
to get the higher states?
01:15:56.601 --> 01:15:57.100
No.
01:15:57.100 --> 01:16:00.680
I take derivatives and
multiply by constants.
01:16:00.680 --> 01:16:03.610
So to get the third Hermite
polynomial what do you do?
01:16:03.610 --> 01:16:06.740
You do this three times.
01:16:06.740 --> 01:16:08.640
This is actually an
extremely efficient way--
01:16:08.640 --> 01:16:10.100
it's related to something
called the generating function,
01:16:10.100 --> 01:16:11.400
and an extremely efficient
way to write down
01:16:11.400 --> 01:16:12.752
the Hermite polynomials.
01:16:12.752 --> 01:16:14.460
They're the things
that you get by acting
01:16:14.460 --> 01:16:18.920
on this with this operator
as many times as you want.
01:16:18.920 --> 01:16:21.880
That is a nice formal definition
of the Hermite polynomials.
01:16:21.880 --> 01:16:24.610
The upshot of all of
this is the following.
01:16:24.610 --> 01:16:32.160
The upshot of all this is that
we've derived that without ever
01:16:32.160 --> 01:16:34.860
solving the differential
equation the spectrum just from
01:16:34.860 --> 01:16:37.190
that commutation relation,
just from that commutation
01:16:37.190 --> 01:16:39.030
relation-- I cannot emphasize
this strongly enough--
01:16:39.030 --> 01:16:40.488
just from the
commutation relation,
01:16:40.488 --> 01:16:42.270
Ea is minus a
times the constant,
01:16:42.270 --> 01:16:44.610
and Ea dagger is a dagger
times the constant.
01:16:44.610 --> 01:16:48.675
We derive that the energy
eigenstates come in a tower.
01:16:48.675 --> 01:16:51.050
You can move along this tower
by raising with the raising
01:16:51.050 --> 01:16:52.966
operator, lowering with
the lowering operator.
01:16:52.966 --> 01:16:54.610
You can construct
the ground state
01:16:54.610 --> 01:16:57.999
by building that simple
wave function, which
01:16:57.999 --> 01:16:59.665
is annihilated by the
lowering operator.
01:16:59.665 --> 01:17:01.210
You can build all
the other states
01:17:01.210 --> 01:17:03.860
by raising them, which is
just taking derivatives
01:17:03.860 --> 01:17:07.540
instead of solving differential
equations, which is hard.
01:17:07.540 --> 01:17:10.340
And all of this came from
this commutation relation.
01:17:10.340 --> 01:17:13.030
And since we are going to see
this over and over again--
01:17:13.030 --> 01:17:15.787
and depending on how
far you take physics,
01:17:15.787 --> 01:17:16.870
you will see this in 8.05.
01:17:16.870 --> 01:17:17.780
You will see this in 8.06.
01:17:17.780 --> 01:17:19.639
You will see this in
quantum field theory.
01:17:19.639 --> 01:17:20.680
This shows up everywhere.
01:17:20.680 --> 01:17:22.340
It's absolutely
at the core of how
01:17:22.340 --> 01:17:25.130
we organize the
degrees of freedom.
01:17:25.130 --> 01:17:29.220
This structure is something you
should see and declare victory
01:17:29.220 --> 01:17:30.020
upon seeing.
01:17:30.020 --> 01:17:32.420
Should see this and immediately
say, I know the answer,
01:17:32.420 --> 01:17:34.520
and I can write it down.
01:17:34.520 --> 01:17:36.310
OK?
01:17:36.310 --> 01:17:39.604
In the next lecture we're
going to do a review which
01:17:39.604 --> 01:17:42.270
is going to introduce a slightly
more formal presentation of all
01:17:42.270 --> 01:17:43.214
these ideas.
01:17:43.214 --> 01:17:45.380
That's not going to be
material covered on the exam,
01:17:45.380 --> 01:17:47.338
but it's going to help
you with the exam, which
01:17:47.338 --> 01:17:48.390
will be on Thursday.
01:17:48.390 --> 01:17:50.210
See you Tuesday.