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ROBERT FIELD: Last lecture, I
did a fairly standard treatment
00:00:26.990 --> 00:00:29.750
of the harmonic
oscillator, which is not
00:00:29.750 --> 00:00:31.430
supposed to make you
excited, but just
00:00:31.430 --> 00:00:33.320
to see that you can do this.
00:00:33.320 --> 00:00:38.000
And the path to the
solution was to define
00:00:38.000 --> 00:00:41.630
dimensionless
position coordinate,
00:00:41.630 --> 00:00:45.680
and then you get a dimensionless
Schrodinger equation.
00:00:45.680 --> 00:00:51.210
And the solution of
that involves two steps.
00:00:51.210 --> 00:00:57.950
One is to insist that the
solutions to the Schrodinger
00:00:57.950 --> 00:01:02.150
equation have an
exponentially damped form.
00:01:02.150 --> 00:01:03.710
And then the
Schrodinger equation
00:01:03.710 --> 00:01:07.441
is transformed into a new
equation called the Hermite
00:01:07.441 --> 00:01:07.940
equation.
00:01:10.499 --> 00:01:13.040
You shouldn't get all excited
about that-- the mathematicians
00:01:13.040 --> 00:01:14.650
take care of it.
00:01:14.650 --> 00:01:20.420
And so the solutions to this
Hermite differential equation
00:01:20.420 --> 00:01:25.170
gives you a set of orthogonal
and normalized wave functions.
00:01:25.170 --> 00:01:29.090
They give you the energy levels
expressed as a quantum number
00:01:29.090 --> 00:01:31.940
plus 1/2, times constant.
00:01:31.940 --> 00:01:37.920
The energy levels with
the quantum number v
00:01:37.920 --> 00:01:42.320
are even or odd in v, and
they're even or odd in psi.
00:01:42.320 --> 00:01:46.600
Even v corresponds to even psi.
00:01:46.600 --> 00:01:50.000
V is the number
of internal nodes,
00:01:50.000 --> 00:01:53.570
and there are all sorts of
things you can do to say,
00:01:53.570 --> 00:01:57.080
well, I expect if I know
how to do certain things
00:01:57.080 --> 00:01:58.772
in classical
mechanics, they're are
00:01:58.772 --> 00:02:00.980
going to come out pretty
much the same way in quantum
00:02:00.980 --> 00:02:02.490
mechanics.
00:02:02.490 --> 00:02:04.940
So that's the
standard structure,
00:02:04.940 --> 00:02:10.130
but more importantly, I want
you to have in your head
00:02:10.130 --> 00:02:14.570
the pictures of the wave
functions, the idea that there
00:02:14.570 --> 00:02:19.476
is a zero point energy, and
that there's a reason for that--
00:02:19.476 --> 00:02:22.520
that the wave functions
have tails extending out
00:02:22.520 --> 00:02:26.550
into the non-classical, or the
classically forbidden region.
00:02:26.550 --> 00:02:30.380
And this turns out to be
the beginning of tunneling.
00:02:30.380 --> 00:02:36.680
You'll be looking at
tunneling more specifically.
00:02:36.680 --> 00:02:40.650
I also want you to know how
the spacing of nodes is,
00:02:40.650 --> 00:02:45.950
and that involves generalization
of the bright idea
00:02:45.950 --> 00:02:47.960
that the wavelength is h over p.
00:02:47.960 --> 00:02:51.260
But if the potential
is not constant,
00:02:51.260 --> 00:02:53.720
then p is a function of x.
00:02:53.720 --> 00:02:58.020
This is not the quantum
mechanical operator,
00:02:58.020 --> 00:03:00.950
this is a function
which provides you
00:03:00.950 --> 00:03:03.770
with a lot of intuition.
00:03:03.770 --> 00:03:08.110
And then if you know where
the node spacings are, and you
00:03:08.110 --> 00:03:12.260
know the shape of
the envelope, you
00:03:12.260 --> 00:03:14.150
have basically
everything you need
00:03:14.150 --> 00:03:18.470
to have a classical
sense of what's going on.
00:03:18.470 --> 00:03:23.090
And then-- I guess it's
supposed to be hidden--
00:03:23.090 --> 00:03:27.410
I did a little bit of
semiclassical theory,
00:03:27.410 --> 00:03:30.650
and I showed that if you
integrate from the left turning
00:03:30.650 --> 00:03:32.690
point to the right
turning point at a given
00:03:32.690 --> 00:03:37.230
energy of this
momentum function,
00:03:37.230 --> 00:03:42.010
you get h over 2 times
the number of nodes.
00:03:42.010 --> 00:03:44.380
And this is the
semiclassical quantization--
00:03:44.380 --> 00:03:47.380
it's incredibly
important, and it's
00:03:47.380 --> 00:03:51.070
useful either as an exact
or approximate result
00:03:51.070 --> 00:03:53.710
for all one-dimensional
problems.
00:03:53.710 --> 00:03:57.160
And so it tells
you how to begin.
00:03:57.160 --> 00:03:59.500
Now, before I start
talking about what
00:03:59.500 --> 00:04:05.800
we're going to do today, I want
to stress where we're going.
00:04:05.800 --> 00:04:09.525
So we're going to be looking at
some exactly solved problems.
00:04:24.300 --> 00:04:29.140
And so we have a particle in a
box, the harmonic oscillator,
00:04:29.140 --> 00:04:31.980
the hydrogen atoms--
you have to count them--
00:04:31.980 --> 00:04:32.910
and the rigid rotor.
00:04:35.980 --> 00:04:40.660
Now, all of these problems
have an infinite number
00:04:40.660 --> 00:04:46.180
eigenfunctions, an infinite
number of energy levels,
00:04:46.180 --> 00:04:52.020
and that's intimidating,
but it's true.
00:04:52.020 --> 00:04:55.370
Now, these infinite
number of functions
00:04:55.370 --> 00:04:59.980
are explicit functions
of the quantum number.
00:04:59.980 --> 00:05:04.160
And so we have an
infinite number,
00:05:04.160 --> 00:05:07.610
but in order to
describe systems,
00:05:07.610 --> 00:05:10.320
we're going to be
calculating integrals.
00:05:10.320 --> 00:05:12.480
We're going to be calculating
a lot of integrals
00:05:12.480 --> 00:05:15.910
between these infinite
number of functions.
00:05:15.910 --> 00:05:19.320
So we have an infinity
squared of integrals.
00:05:22.120 --> 00:05:25.300
Well, that shouldn't
scare you because what
00:05:25.300 --> 00:05:28.960
I'm going to show you is that
all of the integrals that we
00:05:28.960 --> 00:05:33.430
are going to encounter
are explicit functions
00:05:33.430 --> 00:05:35.690
of the quantum
numbers, and they have
00:05:35.690 --> 00:05:38.090
relatively selection rules.
00:05:38.090 --> 00:05:42.100
In other words, which
integrals are non-zero
00:05:42.100 --> 00:05:43.780
based on the
difference in quantum
00:05:43.780 --> 00:05:48.290
numbers between the left-hand
side and the right-hand side?
00:05:48.290 --> 00:05:53.320
So we're collecting
these things in order
00:05:53.320 --> 00:05:56.320
to calculate a whole
bunch of stuff.
00:05:56.320 --> 00:05:59.410
Now, I told you that
this is a course for use
00:05:59.410 --> 00:06:03.440
rather than
philosophy or history.
00:06:03.440 --> 00:06:07.210
And so if you encounter any
quantum mechanical problem,
00:06:07.210 --> 00:06:11.020
you'd like to be
able to describe
00:06:11.020 --> 00:06:13.180
what you could observe with it.
00:06:13.180 --> 00:06:17.860
And so if you're armed with
the infinite number of energy
00:06:17.860 --> 00:06:23.350
levels and eigen
solutions for our problem,
00:06:23.350 --> 00:06:26.990
you can calculate any property.
00:06:26.990 --> 00:06:30.350
So you define some property
you're interested in--
00:06:30.350 --> 00:06:32.870
there is a quantum
mechanical operator that
00:06:32.870 --> 00:06:35.290
corresponds to that property.
00:06:35.290 --> 00:06:40.010
And in order to be able
to describe observations
00:06:40.010 --> 00:06:43.010
of that property,
you need to calculate
00:06:43.010 --> 00:06:47.280
integrals of that operator.
00:06:47.280 --> 00:06:49.060
Well, la dee dah.
00:06:49.060 --> 00:06:51.860
That should be
intimidating, but it's not
00:06:51.860 --> 00:06:54.830
because almost all
of these integrals
00:06:54.830 --> 00:06:59.850
can be expressed as a
simple constant times
00:06:59.850 --> 00:07:02.340
a function of the
quantum numbers
00:07:02.340 --> 00:07:04.680
or the difference
of quantum numbers,
00:07:04.680 --> 00:07:08.360
and that's a fantastic thing.
00:07:08.360 --> 00:07:15.290
So we have any operator--
00:07:15.290 --> 00:07:20.600
suppose the Hamiltonian is
an exactly solved problem
00:07:20.600 --> 00:07:24.530
plus something else,
which we'll call h1.
00:07:24.530 --> 00:07:27.860
And this is a
complexity in the--
00:07:27.860 --> 00:07:30.330
or it's the reality
in the problem.
00:07:30.330 --> 00:07:35.450
And in order to deal
with this, again, you're
00:07:35.450 --> 00:07:40.410
going to need to calculate
integrals of this operator.
00:07:40.410 --> 00:07:43.520
And the last thing that's
really going to be exciting
00:07:43.520 --> 00:07:48.287
is once we look at the time
dependent Schrodinger equation,
00:07:48.287 --> 00:07:49.620
we're going to get wave packets.
00:07:55.720 --> 00:08:01.020
And these are functions
of position and time,
00:08:01.020 --> 00:08:04.350
and these wave packets are
classical-like, localized
00:08:04.350 --> 00:08:08.610
objects that move following
the Newton's equations
00:08:08.610 --> 00:08:11.640
of motion with the center
of the wave packet.
00:08:11.640 --> 00:08:13.110
And again, there
are a whole bunch
00:08:13.110 --> 00:08:15.210
of integrals you're
going to need in order
00:08:15.210 --> 00:08:17.050
to do these things.
00:08:17.050 --> 00:08:21.090
And so right now, we're
starting with the best problem
00:08:21.090 --> 00:08:24.300
for these integrals, because
a harmonic oscillator
00:08:24.300 --> 00:08:26.880
has some special properties.
00:08:26.880 --> 00:08:31.170
And the lecture notes
are incredibly tedious,
00:08:31.170 --> 00:08:32.520
and they're mostly proofs.
00:08:32.520 --> 00:08:37.289
And I'm going to try to go
fast over the tedious stuff,
00:08:37.289 --> 00:08:42.390
and give you the
important ideas,
00:08:42.390 --> 00:08:44.970
but since there is
some important logic,
00:08:44.970 --> 00:08:46.680
you should really
look at these notes.
00:08:49.590 --> 00:08:53.070
So what we're going
to be doing today
00:08:53.070 --> 00:08:59.900
is we start with the
coordinate momentum operators,
00:08:59.900 --> 00:09:07.560
we're going to get
these operators
00:09:07.560 --> 00:09:11.400
in dimensionless
form, and then we're
00:09:11.400 --> 00:09:14.165
going to get these
a and a-dagger guys.
00:09:18.460 --> 00:09:20.880
So this step is
reminiscent of what
00:09:20.880 --> 00:09:23.920
I did at the beginning
of the previous lecture,
00:09:23.920 --> 00:09:28.380
and then this is magic
because this magic enables
00:09:28.380 --> 00:09:32.610
you to just look at
integrals and say,
00:09:32.610 --> 00:09:35.760
I know that integral is zero,
or I know that area is not zero.
00:09:35.760 --> 00:09:37.800
And with a little
bit more effort--
00:09:37.800 --> 00:09:40.260
maybe something that you'd
put on the back of a postage
00:09:40.260 --> 00:09:41.250
stamp--
00:09:41.250 --> 00:09:45.150
you can evaluate that integral,
not by knowing integral tables,
00:09:45.150 --> 00:09:47.920
but by knowing the properties
of these simple little a
00:09:47.920 --> 00:09:49.080
and a-dagger.
00:09:49.080 --> 00:09:51.420
And that's a fantastic thing.
00:09:51.420 --> 00:09:53.790
And it's so fantastic
that this is
00:09:53.790 --> 00:09:59.200
one of the reasons why almost
all problems in quantum
00:09:59.200 --> 00:10:03.310
mechanics start with a harmonic
oscillator approximation,
00:10:03.310 --> 00:10:07.030
because there is so much you
can do with this a and a-dagger
00:10:07.030 --> 00:10:08.770
formalism.
00:10:08.770 --> 00:10:11.530
Now, at the beginning
I also told you
00:10:11.530 --> 00:10:16.030
that in quantum mechanics,
the important thing that
00:10:16.030 --> 00:10:18.981
contains everything we're
allowed to know about a system
00:10:18.981 --> 00:10:19.855
is the wave function.
00:10:22.850 --> 00:10:27.310
But I also told you we can
never measure the wave function.
00:10:27.310 --> 00:10:31.760
We can never experimentally
determine it,
00:10:31.760 --> 00:10:35.690
and so we need to be able
to calculate what this wave
00:10:35.690 --> 00:10:40.580
function does as far
as what we can observe,
00:10:40.580 --> 00:10:43.670
and these a's and a-daggers
are really important in being
00:10:43.670 --> 00:10:44.360
able to do that.
00:10:48.090 --> 00:10:51.300
So I'm going to
start with covering
00:10:51.300 --> 00:10:56.040
what I did in the notes, but I'm
going to jump to final results
00:10:56.040 --> 00:10:57.300
at some point--
00:10:57.300 --> 00:10:59.250
governed by the clock.
00:10:59.250 --> 00:11:03.870
And so the first thing
we're going to do is these.
00:11:03.870 --> 00:11:10.920
And so what we do is we
define the relationship
00:11:10.920 --> 00:11:15.720
between the ordinary
position coordinate.
00:11:15.720 --> 00:11:25.050
And this little twiddle
means it's dimensionless,
00:11:25.050 --> 00:11:29.260
and so we can write
the inverse of that.
00:11:29.260 --> 00:11:32.120
And that's the one we
are going to want to--
00:11:32.120 --> 00:11:33.730
well, actually, we go both ways.
00:11:42.230 --> 00:11:45.500
And we do the same
thing for the momentum--
00:11:45.500 --> 00:11:52.880
p is equal to h-bar
mu omega square
00:11:52.880 --> 00:12:00.030
root, p twiddle and the inverse
which I don't need to write.
00:12:00.030 --> 00:12:04.190
And finally, we get
the Hamiltonian,
00:12:04.190 --> 00:12:10.760
which is p squared over
2 mu plus 1/2 kx squared.
00:12:14.080 --> 00:12:17.390
And we'll put that
into these new units.
00:12:17.390 --> 00:12:27.630
So we have h-bar mu omega
over 2 mu, p twiddle squared
00:12:27.630 --> 00:12:29.430
plus k over 2.
00:12:33.160 --> 00:12:38.130
This is all very tedious,
but the payoff is very soon.
00:12:40.980 --> 00:12:50.640
k over 2 times hr mu
omega, x twiddle squared.
00:12:55.200 --> 00:12:56.900
Oh, isn't that interesting?
00:12:56.900 --> 00:13:05.630
We can combine-- we can
absorb a k over mu in omega,
00:13:05.630 --> 00:13:08.360
and so we get, actually,
a big simplification.
00:13:08.360 --> 00:13:14.140
We get h bar omega
over 2 times p twiddle
00:13:14.140 --> 00:13:17.545
squared plus x twiddle squared.
00:13:20.110 --> 00:13:21.505
Well, that looks simpler.
00:13:26.770 --> 00:13:30.760
And so the next thing we do is--
it looks like a simple problem
00:13:30.760 --> 00:13:35.069
from algebra, let's factor this.
00:13:35.069 --> 00:13:36.610
Now, it's a little
tricky because you
00:13:36.610 --> 00:13:40.630
know you can factor
something in real terms
00:13:40.630 --> 00:13:42.500
if this is a minus sign.
00:13:42.500 --> 00:13:45.400
But we are allowed to talk
about complex quantities,
00:13:45.400 --> 00:13:48.400
so we can factor that.
00:13:48.400 --> 00:14:00.030
And so this term, p twiddle
squared plus x twiddle squared
00:14:00.030 --> 00:14:16.280
is equal to ip twiddle, plus x
twiddle times minus ip twiddle,
00:14:16.280 --> 00:14:17.610
plus x.
00:14:22.390 --> 00:14:27.990
And you can work that out-- that
ip times minus ip is p squared,
00:14:27.990 --> 00:14:30.810
and x times x is x squared.
00:14:30.810 --> 00:14:34.800
And then we have these
cross-terms, ip times
00:14:34.800 --> 00:14:37.410
x and x times minus ip.
00:14:37.410 --> 00:14:41.102
Whoops, they don't commute.
00:14:41.102 --> 00:14:42.530
If this were algebra--
00:14:42.530 --> 00:14:45.640
well, they would go
away, but they don't.
00:14:45.640 --> 00:14:52.010
And so what you
end up getting is
00:14:52.010 --> 00:15:06.160
p twiddle squared plus x twiddle
squared, plus i times p--
00:15:15.360 --> 00:15:17.580
I'm going to stop
writing the twiddles.
00:15:17.580 --> 00:15:18.870
So we have this.
00:15:24.820 --> 00:15:28.330
I want to make sure that I
haven't sabotaged myself--
00:15:28.330 --> 00:15:31.550
that's going to be--
yeah, that's right.
00:15:31.550 --> 00:15:36.941
So we have something
here that isn't zero.
00:15:36.941 --> 00:15:40.340
And it looks like i times
the commutator of p twiddle
00:15:40.340 --> 00:15:42.746
with x twiddle.
00:15:42.746 --> 00:15:44.550
But we can work
that out because we
00:15:44.550 --> 00:15:51.440
know the commutator of p
ordinary with x ordinary.
00:15:51.440 --> 00:15:54.190
And so I did that.
00:15:54.190 --> 00:16:07.690
And so we have this commutator,
p twiddle, x twiddle.
00:16:07.690 --> 00:16:10.495
After some algebra,
we get plus 1.
00:16:15.187 --> 00:16:17.950
A number-- pure number--
00:16:17.950 --> 00:16:18.940
no?
00:16:18.940 --> 00:16:23.270
I want you to check my algebra.
00:16:23.270 --> 00:16:28.000
So you just substitute
in what this
00:16:28.000 --> 00:16:31.960
is in terms of ordinary
p and the ordinary x.
00:16:31.960 --> 00:16:36.640
Use a commutator for
ordinary xp, which is ih-bar,
00:16:36.640 --> 00:16:40.160
and magically, you get plus 1.
00:16:40.160 --> 00:16:50.585
So this very strange and
boring derivation says, OK--
00:16:55.900 --> 00:17:02.830
well, let's now give
these two things names.
00:17:02.830 --> 00:17:09.069
This guy, we're going to call
as the square root of 2 times a,
00:17:09.069 --> 00:17:11.585
and this one is going
to be the square root
00:17:11.585 --> 00:17:13.869
of 2 times a-dagger.
00:17:13.869 --> 00:17:31.890
So H is going to be
h-bar omega over 2,
00:17:31.890 --> 00:17:40.280
times square root
of 2a-hat times
00:17:40.280 --> 00:17:47.496
the square root of
2a-dagger-hat minus 1.
00:17:47.496 --> 00:17:49.080
Remember, when we
factored it, we
00:17:49.080 --> 00:17:51.150
got this extra term which was 1.
00:17:51.150 --> 00:17:54.240
And in order to make it correct,
we have to subtract it away.
00:17:58.940 --> 00:18:12.130
And so this becomes h-bar
omega, a-dagger-hat minus 1/2.
00:18:12.130 --> 00:18:13.750
Well, isn't that nice?
00:18:13.750 --> 00:18:17.620
Now, we have the Hamiltonian
expressed as a constant, which
00:18:17.620 --> 00:18:20.500
we know is important because
it's related to the energy
00:18:20.500 --> 00:18:23.940
levels, and times these
two little things,
00:18:23.940 --> 00:18:29.040
which turn out to be
the gift from God.
00:18:29.040 --> 00:18:31.380
It's an incredible
thing, what these do.
00:18:35.740 --> 00:18:38.710
So we have gone
through some algebra,
00:18:38.710 --> 00:18:43.450
and we know the relationship
between a, and the x and p
00:18:43.450 --> 00:18:46.660
twiddles, and
similarly for a-dagger.
00:18:46.660 --> 00:18:49.340
And we can go in
the other direction,
00:18:49.340 --> 00:18:51.610
and we know the
commutator, and now we're
00:18:51.610 --> 00:18:53.940
going to start doing
some really great stuff.
00:18:59.860 --> 00:19:02.700
Well, one thing we're going to
want to know about is a-hat.
00:19:02.700 --> 00:19:09.640
a-hat, a-dagger,
that commutator-hat.
00:19:09.640 --> 00:19:12.340
And that turns out to be--
00:19:15.820 --> 00:19:20.960
well, I already derived it--
it turns out to be plus 1.
00:19:20.960 --> 00:19:23.910
And as a result, we can say
things like this-- a a-dagger--
00:19:56.290 --> 00:20:03.040
So using this trick, we can show
we can always replace something
00:20:03.040 --> 00:20:10.780
like a, a-dagger by a-dagger
a plus this commutator, which
00:20:10.780 --> 00:20:11.280
is 1.
00:20:15.580 --> 00:20:19.070
And so we have this
really neat way
00:20:19.070 --> 00:20:25.330
of reversing the order
of the a's and a-daggers.
00:20:25.330 --> 00:20:28.300
So with this, we're
going to soon discover
00:20:28.300 --> 00:20:33.980
the a operating on the
eigenfunction gives square root
00:20:33.980 --> 00:20:37.845
of v times psi v minus 1.
00:20:37.845 --> 00:20:42.430
And a-dagger operating
on this wave function
00:20:42.430 --> 00:20:48.550
gives v plus 1 square
root of psi v plus 1.
00:20:48.550 --> 00:20:55.010
Which is the reason these
things are valuable,
00:20:55.010 --> 00:21:01.980
because if you have
any eigenfunction,
00:21:01.980 --> 00:21:04.780
you can get all the others.
00:21:04.780 --> 00:21:07.350
So suppose you have the
lowest eigenfunction--
00:21:07.350 --> 00:21:10.310
you apply a-dagger
on it as many times
00:21:10.310 --> 00:21:12.126
as you need to get
to, say vth function.
00:21:15.070 --> 00:21:17.790
So you don't actually--
00:21:17.790 --> 00:21:20.640
you're not going to be
evaluating integrals, you're
00:21:20.640 --> 00:21:24.060
going to be counting
a's and a-daggers,
00:21:24.060 --> 00:21:26.410
and permuting them around,
and getting 1's, and stuff
00:21:26.410 --> 00:21:26.910
like that.
00:21:26.910 --> 00:21:27.760
Yes?
00:21:27.760 --> 00:21:29.551
AUDIENCE: In this line
with the commutator,
00:21:29.551 --> 00:21:30.904
you didn't move the dagger.
00:21:30.904 --> 00:21:32.070
ROBERT FIELD: I didn't what?
00:21:32.070 --> 00:21:34.785
AUDIENCE: For a, a times the
square root of a a-dagger plus
00:21:34.785 --> 00:21:38.945
a-dagger a, it
should be a a-dagger.
00:21:38.945 --> 00:21:42.470
And on the right-hand side,
you need to move the dagger.
00:21:47.870 --> 00:21:52.110
ROBERT FIELD: OK, so this
is to switch the order,
00:21:52.110 --> 00:21:53.610
and I've done that.
00:21:53.610 --> 00:21:55.035
And that then is--
00:22:00.150 --> 00:22:03.030
no, I think-- wait a second.
00:22:03.030 --> 00:22:04.400
So we have a a-dagger--
00:22:04.400 --> 00:22:10.070
so that's a a-dagger minus
a-dagger a, and that's--
00:22:10.070 --> 00:22:12.020
oh, yeah.
00:22:12.020 --> 00:22:13.940
Thank you.
00:22:13.940 --> 00:22:17.990
It's very, very easy to get
lost, and once you're lost,
00:22:17.990 --> 00:22:19.910
you can never be
found because you've
00:22:19.910 --> 00:22:23.300
made a mistake that's
built into your logic,
00:22:23.300 --> 00:22:25.330
and you're never
going to see it.
00:22:25.330 --> 00:22:27.870
You see it took me a couple
of minutes to even accept--
00:22:27.870 --> 00:22:31.640
that the insight from my TA
who's sitting there calmly
00:22:31.640 --> 00:22:34.300
thinking, and I'm trying to
do several things in addition
00:22:34.300 --> 00:22:35.810
to the thinking.
00:22:35.810 --> 00:22:39.730
So we can do things like this.
00:22:39.730 --> 00:22:42.080
Suppose we have psi--
00:22:42.080 --> 00:22:45.530
I can't use this notation yet.
00:22:45.530 --> 00:22:50.660
So suppose we have
psi-star v, and we
00:22:50.660 --> 00:22:58.990
have a-dagger, a-dagger,
a-dagger, psi, v prime, dx.
00:23:04.490 --> 00:23:06.140
These are raising
operators, so this
00:23:06.140 --> 00:23:11.120
is going to take v
prime to v prime plus 3.
00:23:11.120 --> 00:23:15.440
That's the only
integral that's not 0.
00:23:15.440 --> 00:23:29.554
And we get v prime plus 1, v
prime plus 2, v prime plus 3,
00:23:29.554 --> 00:23:31.760
square what?
00:23:31.760 --> 00:23:33.280
It has the constants.
00:23:36.800 --> 00:23:38.960
And this would be
v prime plus 3.
00:23:42.760 --> 00:23:45.430
So instead of
evaluating an integral,
00:23:45.430 --> 00:23:47.730
looking at what the
x's and p's are,
00:23:47.730 --> 00:23:49.840
we just have a
little game we play.
00:23:55.700 --> 00:24:05.530
So now, we have to prove
some of the things I've said.
00:24:05.530 --> 00:24:15.890
So we have h, and we're going
to operate on a-dagger psi v.
00:24:15.890 --> 00:24:20.430
So what does the Hamiltonian
do to this thing?
00:24:20.430 --> 00:24:22.250
So what we're
going to want to do
00:24:22.250 --> 00:24:28.880
is to show that this
thing is an eigenvalue--
00:24:28.880 --> 00:24:32.060
eigenfunction of v
plus 1, and that's
00:24:32.060 --> 00:24:33.180
what we are going to get.
00:24:33.180 --> 00:24:35.640
So let's just go through this.
00:24:35.640 --> 00:24:47.560
So we have h-bar omega, a-dagger
a plus 1/2, times psi v.
00:24:47.560 --> 00:24:49.935
So what I did is--
00:24:53.566 --> 00:24:59.830
where did I-- yeah, I showed
that the Hamiltonian--
00:24:59.830 --> 00:25:01.230
or did I not do that yet?
00:25:06.100 --> 00:25:07.500
Oh, yeah-- I did it right here.
00:25:07.500 --> 00:25:13.770
The Hamiltonian is h-bar
omega, a a-dagger minus 1/2,
00:25:13.770 --> 00:25:21.030
or if we reverse these, it's
equal to h-bar omega, a-dagger,
00:25:21.030 --> 00:25:25.410
a plus 1/2.
00:25:25.410 --> 00:25:35.000
So we can use either one,
so I'm using that one--
00:25:44.900 --> 00:25:48.140
except I wanted
an a-dagger here,
00:25:48.140 --> 00:25:52.520
because we want to show what
the Hamiltonian does to this.
00:25:52.520 --> 00:26:00.180
Now, we can pull in a-dagger
out to the right, because this--
00:26:06.942 --> 00:26:10.130
if it's 1/2 times a-dagger,
well, that doesn't matter.
00:26:10.130 --> 00:26:12.410
This a-dagger, a,
a-dagger-- well,
00:26:12.410 --> 00:26:14.840
we can pull this a-dagger out.
00:26:14.840 --> 00:26:25.854
So we have h-bar omega, a-dagger
is equal to a a-dagger plus
00:26:25.854 --> 00:26:26.354
1/2--
00:26:30.242 --> 00:26:31.214
so iv.
00:26:35.590 --> 00:26:38.740
Now, we use our magic
commutator trick
00:26:38.740 --> 00:26:52.152
to replace this by
a-dagger a plus 1.
00:26:59.580 --> 00:27:05.050
So now, we have h-bar
omega, a-dagger,
00:27:05.050 --> 00:27:12.760
and we have a-dagger a
plus three halves is iv.
00:27:22.650 --> 00:27:34.640
Well, this is Ev
plus h-bar omega.
00:27:34.640 --> 00:27:39.260
We've increased the
number that started here.
00:27:39.260 --> 00:27:42.800
Here is Ev-- that was Ev plus 1.
00:27:46.130 --> 00:27:48.740
And so now, we have
no operators in here,
00:27:48.740 --> 00:27:51.910
and we can stick the
a-dagger back here.
00:27:51.910 --> 00:28:10.080
And so we have h-bar
omega, Ev plus 1, a-dagger
00:28:10.080 --> 00:28:11.840
Well, what do we have here?
00:28:11.840 --> 00:28:15.800
We have an operator,
we have this function,
00:28:15.800 --> 00:28:20.650
we have some constant
times the same function.
00:28:20.650 --> 00:28:25.600
So what we've shown
is that this thing
00:28:25.600 --> 00:28:28.650
is an eigenfunction
of the Hamiltonian
00:28:28.650 --> 00:28:33.720
that belongs to the
eigenvalue Ev plus 1.
00:28:33.720 --> 00:28:36.690
We've increased the energy by 1.
00:28:42.390 --> 00:28:43.260
So what we have--
00:29:01.420 --> 00:29:05.590
so we can show that
we apply a-dagger
00:29:05.590 --> 00:29:12.700
to any function-- we
increase its energy,
00:29:12.700 --> 00:29:15.820
and we can do this forever.
00:29:15.820 --> 00:29:21.820
We could also do a similar
thing if we apply a to psi v. We
00:29:21.820 --> 00:29:27.550
can go down, but at some
point, we run out of steam
00:29:27.550 --> 00:29:33.740
because we've gone to the lowest
energy, and if we go lower,
00:29:33.740 --> 00:29:35.290
we get 0.
00:29:35.290 --> 00:29:43.810
So a operating on
psi min gives 0.
00:29:43.810 --> 00:29:50.050
So we have this stack of energy
levels and wave functions,
00:29:50.050 --> 00:29:53.290
and we have the same stack
being repeated as we go down,
00:29:53.290 --> 00:29:55.240
but this one has an end.
00:30:08.880 --> 00:30:17.000
We bring back what a is,
and so a psi min is 0--
00:30:17.000 --> 00:30:18.390
that's the equation.
00:30:18.390 --> 00:30:32.060
We bring in what a is, and it's
ip twiddle x plus x twiddle.
00:30:39.980 --> 00:30:42.040
So we do some algebra,
and what we end up
00:30:42.040 --> 00:30:46.060
with is a differential
equation, psi min.
00:30:46.060 --> 00:30:52.630
dxx twiddle is equal to--
00:30:52.630 --> 00:30:54.400
again, a little
bit more algebra--
00:30:54.400 --> 00:31:00.790
minus mu omega over
h-bar times psi min.
00:31:04.320 --> 00:31:09.880
So what function gives--
00:31:09.880 --> 00:31:13.360
there's an x in here too.
00:31:13.360 --> 00:31:16.940
So what function
has a derivative,
00:31:16.940 --> 00:31:19.690
which is the function
you had started
00:31:19.690 --> 00:31:22.330
with, times the variable,
times a constant?
00:31:27.370 --> 00:31:31.660
And so the answer to
that is that psi min
00:31:31.660 --> 00:31:33.680
has to have the form--
00:31:33.680 --> 00:31:39.700
some normalization factor
times e to the minus m omega--
00:31:39.700 --> 00:31:40.760
or mu omega-- sorry--
00:31:43.600 --> 00:31:49.680
over 2 h-bar x squared--
00:31:49.680 --> 00:31:50.420
a Gaussian.
00:31:52.950 --> 00:31:54.990
Well, it had to be
a Gaussian, right?
00:31:54.990 --> 00:31:57.990
We know when we did
the algebra that we're
00:31:57.990 --> 00:32:01.890
going to get some
function times a Gaussian.
00:32:01.890 --> 00:32:05.910
But for the lowest function,
the Hermite polynomial is 1,
00:32:05.910 --> 00:32:08.550
and all there is
is the Gaussian.
00:32:08.550 --> 00:32:15.430
And so we found the lowest
level, and we can normalize it.
00:32:21.880 --> 00:32:23.320
So let's start over here.
00:32:29.640 --> 00:32:35.030
So what we have
found is psi min of x
00:32:35.030 --> 00:32:47.260
is equal to mu omega over
pi h-bar to the 1/4 power, e
00:32:47.260 --> 00:32:55.797
to the minus mu omega
over 2 h-bar squared.
00:32:55.797 --> 00:32:56.630
Well, that's useful.
00:32:56.630 --> 00:33:00.050
We knew that, but this
time we got it out
00:33:00.050 --> 00:33:01.530
of a completely different path.
00:33:05.900 --> 00:33:16.120
And now, we can
get all higher v by
00:33:16.120 --> 00:33:21.140
a-dagger, a-dagger, et cetera.
00:33:21.140 --> 00:33:26.170
So remember, we
don't care anything
00:33:26.170 --> 00:33:29.890
about what the
function is, we just
00:33:29.890 --> 00:33:34.100
know that we can bring it in
and get rid of it at will,
00:33:34.100 --> 00:33:37.880
because what we want is the
values of integrals involving
00:33:37.880 --> 00:33:41.280
that function and some operator.
00:33:41.280 --> 00:33:44.280
So yeah, we can have
all of those functions,
00:33:44.280 --> 00:33:46.820
and this is a way of generating
all of the functions.
00:33:46.820 --> 00:33:56.880
And so if we wanted psi v, we
would do a-dagger to the vth
00:33:56.880 --> 00:34:04.850
power divided by v-dagger--
00:34:04.850 --> 00:34:12.300
v-- what do you call this
with an exclamation point?
00:34:12.300 --> 00:34:13.419
Factorial-- ha!
00:34:18.909 --> 00:34:24.719
So we apply this operator that
raises us to whatever level
00:34:24.719 --> 00:34:29.080
we want starting from this
Gaussian at the bottom,
00:34:29.080 --> 00:34:32.710
and we have this normalization
factor which cancels out
00:34:32.710 --> 00:34:35.800
the fact the stuff that
you get by applying av.
00:34:47.120 --> 00:34:49.300
Now, there is some
more logic in my notes,
00:34:49.300 --> 00:34:51.340
and I don't want to
do that, but what
00:34:51.340 --> 00:34:57.080
we'd like to be able to show
is that a-dagger on psi v
00:34:57.080 --> 00:35:04.850
gives some constant, and
that this constant has
00:35:04.850 --> 00:35:07.460
some value-- we're going
to evaluate what it is.
00:35:07.460 --> 00:35:16.280
And similarly, a psi v gives dv,
and some constant v minus that.
00:35:16.280 --> 00:35:20.030
We can derive those
things, and I'm not going
00:35:20.030 --> 00:35:23.120
to waste time deriving them--
00:35:23.120 --> 00:35:25.150
I'm going to just
give you the values.
00:35:25.150 --> 00:35:32.105
But we already know that cv
is square of v plus 1/2--
00:35:32.105 --> 00:35:34.355
e plus 1 and dv--
00:35:38.170 --> 00:35:40.210
and you can see the
derivation in my notes.
00:35:40.210 --> 00:35:43.950
I don't think going through
them is going to be instructive.
00:35:43.950 --> 00:35:49.650
And that's just going
to be v and 1/2.
00:35:49.650 --> 00:35:55.780
So now, we have something
that's wonderful,
00:35:55.780 --> 00:36:01.270
because everything you need
to know about getting numbers
00:36:01.270 --> 00:36:05.890
concerning harmonic oscillator
is obtained from these five
00:36:05.890 --> 00:36:07.300
equations.
00:36:07.300 --> 00:36:18.810
a-dagger on psi v is v plus
1 square root psi v plus 1.
00:36:18.810 --> 00:36:24.260
a on psi v is v square
root psi, v minus 1.
00:36:24.260 --> 00:36:28.450
I've said this before, but
these are the most useful things
00:36:28.450 --> 00:36:29.650
you'll ever encounter.
00:36:29.650 --> 00:36:32.150
We have this thing called
the number operator,
00:36:32.150 --> 00:36:35.890
and that number operator
is a-dagger-hat,
00:36:35.890 --> 00:36:43.140
and the number operator
operating on psi v
00:36:43.140 --> 00:36:49.410
gives v psi v. And so that's
a kind of benign operator
00:36:49.410 --> 00:36:56.310
that can suck up all sorts
of factors of a-dagger a,
00:36:56.310 --> 00:37:00.720
because it just
gives a useful thing.
00:37:00.720 --> 00:37:07.350
And then we have a,
a-dagger, and this is 1.
00:37:07.350 --> 00:37:08.940
Well, you sort of
know it's going
00:37:08.940 --> 00:37:13.980
to be 1, because a a-dagger
gives an increase-- it gives v
00:37:13.980 --> 00:37:22.260
plus 1, and a gives v minus 1.
00:37:22.260 --> 00:37:26.735
So it's plus 1, not minus 1--
you know that it's hardwired.
00:37:31.264 --> 00:37:31.930
Well, I did it--
00:37:31.930 --> 00:37:36.400
I got to the point where it
starts to get interesting.
00:37:36.400 --> 00:37:43.120
So we're going to be using
this notation, a-dagger and a,
00:37:43.120 --> 00:37:45.930
for all sorts of stuff.
00:37:45.930 --> 00:37:57.070
And one sort of thing is
transition intensities
00:37:57.070 --> 00:37:58.250
and selection rules.
00:38:03.350 --> 00:38:05.255
So you have a
harmonic oscillator.
00:38:05.255 --> 00:38:09.160
A harmonic oscillator is,
say a diatomic molecule
00:38:09.160 --> 00:38:11.410
which is heteronuclear.
00:38:11.410 --> 00:38:15.250
And so as the
molecule vibrates, you
00:38:15.250 --> 00:38:18.792
have a dipole moment
which is oscillating.
00:38:21.690 --> 00:38:25.420
And so any oscillating
electric field
00:38:25.420 --> 00:38:27.940
will grab a hold of
that dipole moment
00:38:27.940 --> 00:38:31.900
and stretch or
compress it, especially
00:38:31.900 --> 00:38:37.390
if that field is in
resonance with h-bar omega.
00:38:37.390 --> 00:38:41.675
And I've got some beautiful
animation showing this,
00:38:41.675 --> 00:38:44.050
but we can't do that until we
have time dependent quantum
00:38:44.050 --> 00:38:45.260
mechanics.
00:38:45.260 --> 00:38:48.550
So we have a time
dependent radiation
00:38:48.550 --> 00:38:50.770
field, which is
going to interact
00:38:50.770 --> 00:38:56.170
with the dipole associated
with the vibrating molecule,
00:38:56.170 --> 00:38:59.060
and it's going to
cause transitions.
00:38:59.060 --> 00:39:06.760
And so we can write the
quantum mechanical operator
00:39:06.760 --> 00:39:08.860
that causes the transitions--
00:39:08.860 --> 00:39:14.380
this is the electric
dipole moment operator--
00:39:14.380 --> 00:39:17.410
as a function of coordinate.
00:39:17.410 --> 00:39:21.670
And we can do a power
series expansion of this,
00:39:21.670 --> 00:39:22.270
and we have--
00:39:36.580 --> 00:39:40.240
so we have mu 0--
the constant term--
00:39:40.240 --> 00:39:42.130
the first derivative
of the dipole
00:39:42.130 --> 00:39:45.100
with respect to x, and the
second derivative of the dipole
00:39:45.100 --> 00:39:46.930
with respect to x.
00:39:46.930 --> 00:39:50.840
And we have the x cofactor
and the x squared cofactor.
00:39:50.840 --> 00:39:55.000
And so this guy doesn't have
any x on it-- it's a constant.
00:39:55.000 --> 00:39:57.730
The only integrals involving--
00:40:09.880 --> 00:40:18.070
the only integrals are delta v,
v prime following the selection
00:40:18.070 --> 00:40:19.190
rule delta v v prime.
00:40:19.190 --> 00:40:24.440
So these integrals are 0 unless
v and v prime are the same.
00:40:26.980 --> 00:40:30.220
And that says, well,
an isolating field
00:40:30.220 --> 00:40:33.580
isn't going to do
anything key to it,
00:40:33.580 --> 00:40:36.160
it's just going to leave it
in the same vibration level.
00:40:36.160 --> 00:40:39.880
But it might have an
electric Stark effect,
00:40:39.880 --> 00:40:41.320
but that's something else.
00:40:41.320 --> 00:40:45.940
So this term does nothing as
far as vibration is concerned.
00:40:45.940 --> 00:40:50.395
This guy, which is
a plus a-dagger,
00:40:50.395 --> 00:40:54.544
has a selection rule, delta
v of plus and minus 1,
00:40:54.544 --> 00:40:58.255
and this guy has a
selection rule, delta v
00:40:58.255 --> 00:41:00.220
of plus and minus 2 and 0.
00:41:03.600 --> 00:41:06.270
So if we're interested
in the intensities
00:41:06.270 --> 00:41:08.910
of vibrational
transitions, it says,
00:41:08.910 --> 00:41:12.420
well, this is the important
term and it causes transitions,
00:41:12.420 --> 00:41:16.080
changing the vibrational
quantum number by one, which
00:41:16.080 --> 00:41:18.540
is called the fundamental.
00:41:18.540 --> 00:41:22.530
This gives rise to overtones.
00:41:22.530 --> 00:41:27.330
So all of a sudden, we're
in real problem land, where
00:41:27.330 --> 00:41:31.590
if we're looking at vibrational
transitions in a molecule,
00:41:31.590 --> 00:41:34.740
that this enables us to
calculate what's important,
00:41:34.740 --> 00:41:37.560
or to say these are the
intensities I measure,
00:41:37.560 --> 00:41:41.850
and these are the first
and second derivative
00:41:41.850 --> 00:41:45.330
of the dipole moment operator
as a function of internuclear
00:41:45.330 --> 00:41:46.160
distance.
00:41:46.160 --> 00:41:46.860
Isn't that neat?
00:41:53.370 --> 00:41:59.460
I've gone so fast, I'm more or
less at the end of my notes,
00:41:59.460 --> 00:42:01.120
but I can blather
on for a while.
00:42:03.790 --> 00:42:09.420
So suppose you have
some integral involving
00:42:09.420 --> 00:42:13.680
an operator and a
vibrational wave function.
00:42:13.680 --> 00:42:21.810
So we have psi v star, some
operator, psi v prime dx.
00:42:25.300 --> 00:42:28.630
And we'd like to know how
to focus our energies.
00:42:28.630 --> 00:42:30.070
We're very busy people--
00:42:30.070 --> 00:42:31.590
we don't want any
value integrals
00:42:31.590 --> 00:42:36.280
that come out to be 0,
we'd like to just know.
00:42:36.280 --> 00:42:45.100
So if this operator is some
function of x or function of p,
00:42:45.100 --> 00:42:48.550
we'd have a power series
expansion of the operator,
00:42:48.550 --> 00:42:51.580
and we then know what
the selection rules are.
00:42:56.460 --> 00:42:58.600
So usually, you
look at the operator
00:42:58.600 --> 00:43:05.040
and you find that it's a linear
quadratic cubic function of x--
00:43:05.040 --> 00:43:07.580
the leading term
is usually linear.
00:43:07.580 --> 00:43:10.920
Bang-- you have a
delta v of plus 1--
00:43:10.920 --> 00:43:12.900
selection rule.
00:43:12.900 --> 00:43:16.770
Or if someone has bothered
to actually convert
00:43:16.770 --> 00:43:18.250
the operator to some form--
00:43:20.855 --> 00:43:23.666
oh, it's the operator--
00:43:29.130 --> 00:43:34.772
this might have some
form, a-dagger cubed times
00:43:34.772 --> 00:43:35.355
some constant.
00:43:40.100 --> 00:43:43.940
So if the operator looks
like a-dagger cubed,
00:43:43.940 --> 00:43:49.740
we know that the selection
rule is v to v plus 3,
00:43:49.740 --> 00:43:54.550
and we know that the matrix
of the integral is v plus 1,
00:43:54.550 --> 00:43:59.550
times v plus 2, times v
plus 3, square root of that,
00:43:59.550 --> 00:44:01.870
times the constant.
00:44:01.870 --> 00:44:04.740
So there is a huge
number of problems
00:44:04.740 --> 00:44:08.860
that, instead of being
pages and pages of algebra,
00:44:08.860 --> 00:44:14.520
are just reduced something that
you can tell by inspection.
00:44:14.520 --> 00:44:19.520
So one of the tricks
is we have an operator
00:44:19.520 --> 00:44:21.290
like x squared or x cubed--
00:44:24.320 --> 00:44:27.560
what we want to do is
write this in terms
00:44:27.560 --> 00:44:36.720
of a squared, a-dagger
squared, and maybe
00:44:36.720 --> 00:44:40.920
some combination of a-dagger a.
00:44:40.920 --> 00:44:44.340
So we want to take the a-dagger
a's with the a a-daggers
00:44:44.340 --> 00:44:47.550
and combine them using
the commutation rule.
00:44:47.550 --> 00:44:51.720
And then we have expressed this
in this maximally simple form,
00:44:51.720 --> 00:44:54.600
and then you just
apply a squared,
00:44:54.600 --> 00:44:59.050
apply a-dagger squared,
and you apply this,
00:44:59.050 --> 00:45:02.800
then you've got the
value of the integral.
00:45:02.800 --> 00:45:04.930
So if you're a
busy person and you
00:45:04.930 --> 00:45:08.640
want to actually
calculate stuff,
00:45:08.640 --> 00:45:12.960
you want to know how
to reduce operators--
00:45:12.960 --> 00:45:17.010
usually expressed as some
power of the coordinate
00:45:17.010 --> 00:45:18.240
or the momentum--
00:45:18.240 --> 00:45:23.730
into a sum of terms
involving these organized
00:45:23.730 --> 00:45:25.500
products of a's and a-daggers.
00:45:28.150 --> 00:45:32.980
And you're going to be
absolutely shocked at how
00:45:32.980 --> 00:45:35.140
perturbation theory--
00:45:35.140 --> 00:45:41.910
which leads to basically all of
the formulas and spectroscopy--
00:45:41.910 --> 00:45:46.230
it's an ugly theory, but it
reduces everything to things
00:45:46.230 --> 00:45:50.280
that you can just write down
at the speed of your pen
00:45:50.280 --> 00:45:54.040
or pencil, and that's
a fantastic thing.
00:45:54.040 --> 00:45:58.140
So you can't do this with
the particle in a box,
00:45:58.140 --> 00:46:00.480
you can't do this
with a hydrogen atom,
00:46:00.480 --> 00:46:02.040
you can't do this
with the rigid--
00:46:02.040 --> 00:46:04.230
well, you can do some of
this with a rigid rotor.
00:46:04.230 --> 00:46:07.920
But the harmonic
oscillator is so ubiquitous
00:46:07.920 --> 00:46:09.930
because every
one-dimensional problem
00:46:09.930 --> 00:46:12.647
is harmonic at the bottom.
00:46:12.647 --> 00:46:15.230
And so you can use it and then
you can put in the corrections.
00:46:15.230 --> 00:46:21.530
But also because you want
to describe dynamics,
00:46:21.530 --> 00:46:24.830
you almost always use
the harmonic oscillator,
00:46:24.830 --> 00:46:27.440
because not only do
you know the integrals,
00:46:27.440 --> 00:46:29.540
but you know there's only a few.
00:46:29.540 --> 00:46:31.640
Normally, you're
going to be summing
00:46:31.640 --> 00:46:36.290
over an infinite number
of quantum numbers,
00:46:36.290 --> 00:46:39.950
and that takes time, and
it takes judgment to say,
00:46:39.950 --> 00:46:44.230
well, only certain of
these are important.
00:46:44.230 --> 00:46:49.240
But for the harmonic
oscillator, the sums are finite.
00:46:49.240 --> 00:46:50.710
All of these things
are wonderful,
00:46:50.710 --> 00:46:54.880
and that's why whenever
you look at a theory,
00:46:54.880 --> 00:46:58.390
you're going to discover
that hidden in there
00:46:58.390 --> 00:47:01.550
is the harmonic
oscillator approximation,
00:47:01.550 --> 00:47:07.290
because everything is
doable in no effort.
00:47:07.290 --> 00:47:10.880
And sometimes when you
look at a paper like that,
00:47:10.880 --> 00:47:13.280
it doesn't show you
the intermediate steps,
00:47:13.280 --> 00:47:17.480
because everybody knows what
a harmonic oscillator does.
00:47:17.480 --> 00:47:18.950
And there's also
a lot of insight
00:47:18.950 --> 00:47:23.730
because something like this--
00:47:23.730 --> 00:47:27.440
this is an odd symmetry term,
this is an even symmetry term,
00:47:27.440 --> 00:47:30.370
and there are all sorts of
things that have to do with,
00:47:30.370 --> 00:47:33.050
are you're conserving
symmetry or changing symmetry?
00:47:33.050 --> 00:47:38.660
And sometimes, the issue is how
does the molecule spontaneously
00:47:38.660 --> 00:47:40.610
change symmetry
by doing something
00:47:40.610 --> 00:47:43.370
interacting with a
field, or interacting
00:47:43.370 --> 00:47:47.280
with some feature of
the potential surface.
00:47:47.280 --> 00:47:51.560
So this is a place where
it's labor saving and insight
00:47:51.560 --> 00:47:55.520
generating, and
it's really amazing.
00:47:55.520 --> 00:47:59.110
So maybe I've bored
you with this,
00:47:59.110 --> 00:48:03.070
but this is the beginning
of almost every theory
00:48:03.070 --> 00:48:07.670
that you encounter just because
of the simplicity of the a's
00:48:07.670 --> 00:48:10.210
and a-daggers.
00:48:10.210 --> 00:48:11.470
OK, I'm done.
00:48:11.470 --> 00:48:13.054
Thank you.