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ROBERT FIELD: Last time, we
talked about photochemistry.
00:00:25.930 --> 00:00:29.710
And the crucial thing
in photochemistry
00:00:29.710 --> 00:00:35.320
is that the density
of vibrational states
00:00:35.320 --> 00:00:38.930
increases extremely rapidly--
00:00:38.930 --> 00:00:43.610
extremely, extremely--
and especially
00:00:43.610 --> 00:00:46.140
for larger molecules.
00:00:46.140 --> 00:00:50.910
And this enables you to
understand intermolecular
00:00:50.910 --> 00:00:54.180
vibrational redistribution,
intersystem crossing,
00:00:54.180 --> 00:00:55.700
internal conversion.
00:00:55.700 --> 00:00:57.960
The matrix elements
are different in each
00:00:57.960 --> 00:00:59.440
of these cases.
00:00:59.440 --> 00:01:04.860
But what happens when the
vibrational density of states
00:01:04.860 --> 00:01:12.610
gets large is that you
have a special state, which
00:01:12.610 --> 00:01:14.830
is a localized
state or something
00:01:14.830 --> 00:01:19.440
that you care about that
you think you understand--
00:01:19.440 --> 00:01:24.170
a bright state-- which is
different from the masses.
00:01:24.170 --> 00:01:29.390
And because of the density
of states, that bright state
00:01:29.390 --> 00:01:32.480
or that special
state, gets diluted
00:01:32.480 --> 00:01:40.470
into an extremely dense manifold
of uninteresting states.
00:01:40.470 --> 00:01:43.460
And as a result,
the system forgets
00:01:43.460 --> 00:01:47.550
whatever the bright state
wanted it to tell you.
00:01:47.550 --> 00:01:53.690
And so you get faster
decay of fluorescence,
00:01:53.690 --> 00:01:56.810
not because the
population goes away,
00:01:56.810 --> 00:02:01.180
but because the unique
thing-- the bright state,
00:02:01.180 --> 00:02:04.820
the localized state-- the
thing that you can prepare
00:02:04.820 --> 00:02:08.030
by excitation from the
ground state, which is always
00:02:08.030 --> 00:02:13.370
localized, its
character is dissipated.
00:02:13.370 --> 00:02:20.640
And so you get what's
called statistical behavior.
00:02:20.640 --> 00:02:25.270
Now I have a very strong feeling
about statistical behavior,
00:02:25.270 --> 00:02:29.430
and that it's mostly a fraud.
00:02:29.430 --> 00:02:32.820
But the current limit
for statistical behavior
00:02:32.820 --> 00:02:38.790
is when the density of
states is on the order of 100
00:02:38.790 --> 00:02:40.920
per wave number.
00:02:40.920 --> 00:02:45.660
That's the usual threshold
for statistical behavior.
00:02:45.660 --> 00:02:49.650
As we know more, as we develop
better experimental techniques,
00:02:49.650 --> 00:02:53.610
we're going to push that curtain
of statistical behavior back.
00:02:53.610 --> 00:02:58.020
And we'll understand more
about the short-time dynamics
00:02:58.020 --> 00:03:04.500
and whatever we want to do
about manipulating systems.
00:03:04.500 --> 00:03:10.860
So statistical just
means we don't know,
00:03:10.860 --> 00:03:13.360
but it's not what we expected.
00:03:13.360 --> 00:03:16.160
And it's usually boring.
00:03:16.160 --> 00:03:18.420
OK.
00:03:18.420 --> 00:03:22.110
Today I'm going to talk
about the discrete variable
00:03:22.110 --> 00:03:26.760
representation, which is really
weird and wonderful thing,
00:03:26.760 --> 00:03:30.270
which is entirely inappropriate
for a course at this level.
00:03:33.560 --> 00:03:37.240
But I think you'll like it.
00:03:37.240 --> 00:03:40.360
So in order to talk about
the discrete variable
00:03:40.360 --> 00:03:45.620
representation, I'll introduce
you to delta functions.
00:03:45.620 --> 00:03:48.730
And you sort of know
what delta functions are.
00:03:48.730 --> 00:03:53.440
And then I'm going to say,
well for any one-dimensional
00:03:53.440 --> 00:03:55.310
problem--
00:03:55.310 --> 00:03:58.210
and remember, you can have
a many-dimensional problem
00:03:58.210 --> 00:04:01.570
treated as a whole bunch of
one-dimensional problems.
00:04:01.570 --> 00:04:05.510
And so it really is
a general problem.
00:04:05.510 --> 00:04:09.840
But if you have a
one-dimensional potential,
00:04:09.840 --> 00:04:14.730
you can obtain the energy
levels and wave functions
00:04:14.730 --> 00:04:17.850
resulting from that
one-dimensional potential.
00:04:17.850 --> 00:04:20.459
Regardless of how
horrible it is,
00:04:20.459 --> 00:04:22.770
without using perturbation
theory, which is
00:04:22.770 --> 00:04:25.110
should make you
feel pretty good.
00:04:25.110 --> 00:04:28.260
Because perturbation
theory is labor intensive
00:04:28.260 --> 00:04:30.826
before you use the computer.
00:04:30.826 --> 00:04:35.450
The computer helps you,
but this is labor free
00:04:35.450 --> 00:04:37.482
because the computer
does everything.
00:04:41.180 --> 00:04:43.490
At the beginning of
the course, I said,
00:04:43.490 --> 00:04:47.940
you cannot experimentally
measure a wave function.
00:04:47.940 --> 00:04:50.005
And that's true.
00:04:50.005 --> 00:04:57.330
but if you can deal
with any potential
00:04:57.330 --> 00:05:06.390
and have a set of experimental
observations, energy levels,
00:05:06.390 --> 00:05:11.250
which you fit to an
effective Hamiltonian,
00:05:11.250 --> 00:05:15.480
you can generate the
wave functions associated
00:05:15.480 --> 00:05:18.150
with the effective
Hamiltonian directly
00:05:18.150 --> 00:05:20.620
from experimental observations.
00:05:20.620 --> 00:05:23.400
So it's not a
direct observation,
00:05:23.400 --> 00:05:25.770
but you get wave functions.
00:05:25.770 --> 00:05:31.010
And you can have whatever
complexity you want.
00:05:31.010 --> 00:05:32.430
OK, so let's begin.
00:05:36.380 --> 00:05:37.350
Delta functions.
00:05:45.570 --> 00:05:48.140
So this is not the
Kronecker delta.
00:05:48.140 --> 00:05:51.800
This is the delta
function that is
00:05:51.800 --> 00:05:54.440
a useful computational trick.
00:05:54.440 --> 00:05:55.610
And it's more than that.
00:06:04.100 --> 00:06:05.775
We write something like this.
00:06:15.770 --> 00:06:18.480
OK, this is a delta function.
00:06:18.480 --> 00:06:28.460
And it says that this thing is
non-zero when x is equal to xi.
00:06:28.460 --> 00:06:30.170
And it's really big.
00:06:30.170 --> 00:06:33.000
And it's 0 everywhere else.
00:06:33.000 --> 00:06:42.380
And as a result, you can say
that we get xi delta x xi.
00:06:42.380 --> 00:06:46.040
So that's an
eigenvalue equation.
00:06:46.040 --> 00:06:51.960
So we have the operator, x,
operating on this function.
00:06:51.960 --> 00:06:54.270
And this function has
the magical properties
00:06:54.270 --> 00:06:58.190
that it returns an
eigenvalue and the function.
00:07:02.410 --> 00:07:06.770
So this is more than
a mathematical trick.
00:07:06.770 --> 00:07:13.510
It's an entry into a
form of quantum mechanics
00:07:13.510 --> 00:07:16.300
that is truly wonderful.
00:07:16.300 --> 00:07:17.890
So this is part one.
00:07:17.890 --> 00:07:20.800
Part two will be DVR--
00:07:20.800 --> 00:07:24.830
discrete variable
representation.
00:07:24.830 --> 00:07:26.780
And the issue varies.
00:07:26.780 --> 00:07:32.117
Suppose we have a matrix
representation of an operator.
00:07:35.940 --> 00:07:41.380
Well, suppose we wanted some
function of that operator.
00:07:41.380 --> 00:07:43.915
How do we generate that?
00:07:43.915 --> 00:07:45.290
And there are lots
of cases where
00:07:45.290 --> 00:07:46.890
you care about such a thing.
00:07:46.890 --> 00:07:50.690
For example, you
might want to know
00:07:50.690 --> 00:08:00.010
about something like this--
e to the ih t over h bar.
00:08:02.780 --> 00:08:07.370
This would tell you something
about how a system propagates
00:08:07.370 --> 00:08:09.710
under the influence
of a Hamiltonian,
00:08:09.710 --> 00:08:12.600
which is not diagonal.
00:08:12.600 --> 00:08:15.560
This is kind of important.
00:08:15.560 --> 00:08:20.600
Almost all of NMR is based
on this sort of thing.
00:08:20.600 --> 00:08:22.430
Also-- well, we'll get to it.
00:08:25.780 --> 00:08:39.339
So extend DVR to
include rotation.
00:08:41.929 --> 00:08:43.669
And then the last part will be--
00:08:48.640 --> 00:08:55.470
can be determined for even
the most horrible situations,
00:08:55.470 --> 00:08:59.730
like a potential that
does something like this,
00:08:59.730 --> 00:09:12.270
or something like that.
00:09:12.270 --> 00:09:15.700
Now we don't want
to have a continuum.
00:09:15.700 --> 00:09:18.300
So I put a wall here.
00:09:18.300 --> 00:09:23.610
But this is a summarization
from an unstable isomer
00:09:23.610 --> 00:09:25.350
to a stable isomer.
00:09:25.350 --> 00:09:27.690
This is multiple minima.
00:09:27.690 --> 00:09:29.400
Anything you want,
you would never
00:09:29.400 --> 00:09:32.175
want to perturbation
theory on it.
00:09:32.175 --> 00:09:38.770
And you can solve these problems
automatically and wonderfully.
00:09:38.770 --> 00:09:43.320
OK, so let's just play
with delta functions.
00:09:43.320 --> 00:09:47.190
And a very good section
on delta functions
00:09:47.190 --> 00:09:56.040
is in Cohen-Tannoudji, and
it's pages 1468 to 1472,
00:09:56.040 --> 00:09:58.410
right at the end of the book.
00:09:58.410 --> 00:09:59.960
But it's not because
it's so hard,
00:09:59.960 --> 00:10:04.190
it's just because they
decided to put it there.
00:10:04.190 --> 00:10:07.700
And so we have notation--
00:10:07.700 --> 00:10:10.430
x xi.
00:10:10.430 --> 00:10:16.410
It's the same thing
as x minus xi.
00:10:16.410 --> 00:10:24.920
So basically, if you see a
variable at a specified value,
00:10:24.920 --> 00:10:27.560
it's equivalent to
this, and this thing
00:10:27.560 --> 00:10:32.980
is 0 everywhere except
when x is equal to xi.
00:10:32.980 --> 00:10:39.110
So the thing in parentheses
is the critical thing.
00:10:39.110 --> 00:10:41.480
And it has the
property that if we
00:10:41.480 --> 00:10:45.500
do an integral from minus
infinity to infinity,
00:10:45.500 --> 00:10:47.523
some function of x--
00:10:47.523 --> 00:10:54.980
dx-- delta x xi dx.
00:10:54.980 --> 00:10:57.680
We get f of xi.
00:10:57.680 --> 00:10:58.886
Isn't that wonderful?
00:11:02.290 --> 00:11:07.910
I mean, it's a very
lovely mathematical trick.
00:11:07.910 --> 00:11:08.910
But it's more than that.
00:11:13.860 --> 00:11:19.090
It says this thing is big
when x is equal to xi.
00:11:19.090 --> 00:11:22.570
It's 0 everywhere else.
00:11:22.570 --> 00:11:24.760
And it's normalized
to 1 in the sense
00:11:24.760 --> 00:11:28.210
that, well you get
back the function
00:11:28.210 --> 00:11:31.300
that you started with,
but at a particular value.
00:11:31.300 --> 00:11:37.510
So it's infinite,
but it's normalized.
00:11:37.510 --> 00:11:40.530
It should bother you,
but it's fantastic
00:11:40.530 --> 00:11:42.790
and you can deal with this.
00:11:49.880 --> 00:11:52.094
It's an eigenvalue equation.
00:11:52.094 --> 00:11:53.510
I mean, you can
say, all right, we
00:11:53.510 --> 00:12:01.710
have delta x xi xi delta x xi.
00:12:07.040 --> 00:12:11.420
And we can have delta functions
in position, in momentum,
00:12:11.420 --> 00:12:14.590
in anything you want.
00:12:14.590 --> 00:12:15.820
And they're useful.
00:12:22.850 --> 00:12:31.300
So suppose we have some
function, psi of x,
00:12:31.300 --> 00:12:34.930
and we want to
find out something
00:12:34.930 --> 00:12:37.600
about how it's composed.
00:12:37.600 --> 00:12:51.000
And so we say, well, we have
c of xi j delta x minus xi dx.
00:12:51.000 --> 00:12:57.000
This is the standard method
for expanding a function.
00:12:57.000 --> 00:13:00.240
OK, what we want is these
expansion coefficients.
00:13:00.240 --> 00:13:03.180
And you get them in
the standard way.
00:13:03.180 --> 00:13:06.520
And that is by--
00:13:06.520 --> 00:13:10.030
normally, when you
have a function,
00:13:10.030 --> 00:13:15.120
you write it ck j the function--
00:13:17.670 --> 00:13:18.170
I'm sorry.
00:13:24.620 --> 00:13:28.160
So here we have an
expansion of members
00:13:28.160 --> 00:13:30.610
of a complete set of functions.
00:13:30.610 --> 00:13:34.940
And in order to get the
expansion coefficients,
00:13:34.940 --> 00:13:44.800
you do the standard trick of
integrating phi j star, psi j
00:13:44.800 --> 00:13:46.490
dx.
00:13:46.490 --> 00:13:48.810
OK, so this is familiar.
00:13:48.810 --> 00:13:50.470
This is not.
00:13:50.470 --> 00:13:53.720
But it's the same business.
00:13:53.720 --> 00:13:59.250
In fact, we've been using
a notation incorrectly--
00:13:59.250 --> 00:14:01.550
the Dirac notation.
00:14:01.550 --> 00:14:05.030
We normally think that
if we have something
00:14:05.030 --> 00:14:14.197
like k and psi k of x,
they're just different ways
00:14:14.197 --> 00:14:15.280
of writing the same thing.
00:14:21.030 --> 00:14:26.130
But the equivalent to
the Schrodinger picture
00:14:26.130 --> 00:14:31.730
is really x psi.
00:14:31.730 --> 00:14:34.170
This is a vector,
not a function.
00:14:34.170 --> 00:14:38.790
This is a set of vectors.
00:14:38.790 --> 00:14:43.850
And this is how you relate
vectors to functions.
00:14:43.850 --> 00:14:46.620
We just have suppressed that.
00:14:46.620 --> 00:14:49.620
Because it's easier to
think that the wave function
00:14:49.620 --> 00:14:53.601
is equivalent to
the symbol here.
00:14:53.601 --> 00:14:55.840
It's not.
00:14:55.840 --> 00:14:57.760
And if you're going to
be doing derivations
00:14:57.760 --> 00:15:00.450
where you flip back and forth
between representations,
00:15:00.450 --> 00:15:03.835
you better remember this,
otherwise it won't make sense.
00:15:09.290 --> 00:15:12.170
This is familiar
stuff, you just didn't
00:15:12.170 --> 00:15:15.230
realize you understood it.
00:15:15.230 --> 00:15:19.460
OK, and so now, often,
you want to have
00:15:19.460 --> 00:15:23.914
a mathematical representation
of a delta function.
00:15:23.914 --> 00:15:25.455
And so this thing
has to be localized
00:15:25.455 --> 00:15:28.124
and it has to be normalized.
00:15:28.124 --> 00:15:30.040
And there are a whole
bunch of representations
00:15:30.040 --> 00:15:31.660
that work really well.
00:15:31.660 --> 00:15:35.490
One of them is we
take the limit,
00:15:35.490 --> 00:15:43.710
epsilon goes to 0 from
the positive side,
00:15:43.710 --> 00:15:50.330
and 1 over 2 epsilon e to
the minus x over epsilon.
00:15:53.090 --> 00:16:05.340
Well this guy, as
epsilon goes to 0,
00:16:05.340 --> 00:16:07.860
this becomes minus infinity.
00:16:07.860 --> 00:16:15.520
And it's 0 everywhere
except for x is equal to 0.
00:16:15.520 --> 00:16:20.800
And so if we wanted to
put x minus xi, then fine.
00:16:20.800 --> 00:16:24.820
Then it would be 0 everywhere
except when x is equal to xi.
00:16:24.820 --> 00:16:25.680
So that's one.
00:16:25.680 --> 00:16:27.970
That's a simple one.
00:16:27.970 --> 00:16:34.120
Another one is limit
as epsilon goes
00:16:34.120 --> 00:16:42.070
to 0 from the positive side of
1 over pi times epsilon over x
00:16:42.070 --> 00:16:46.020
squared plus epsilon squared.
00:16:46.020 --> 00:16:53.610
This also has the properties
of being an infinite spike at x
00:16:53.610 --> 00:16:55.950
equals 0.
00:16:55.950 --> 00:16:58.807
And it's also Lorentzian.
00:16:58.807 --> 00:16:59.890
And it has the full width.
00:16:59.890 --> 00:17:02.340
It'd have maximum-- or a
half width it would have--
00:17:02.340 --> 00:17:06.417
a full width at half
maximum of epsilon.
00:17:06.417 --> 00:17:08.250
If you make the width
as narrow as you want,
00:17:08.250 --> 00:17:10.410
well that's what this is.
00:17:10.410 --> 00:17:12.510
OK, there are a whole
bunch of representations.
00:17:12.510 --> 00:17:14.640
I'm going to stop giving
them because there's
00:17:14.640 --> 00:17:17.040
a lot of stuff I
want to say, and you
00:17:17.040 --> 00:17:18.270
can see them in the notes.
00:17:22.460 --> 00:17:29.070
OK, so what if we have
something like xi,
00:17:29.070 --> 00:17:34.210
that just is the same thing
as saying x minus xi is 0.
00:17:40.500 --> 00:17:45.300
You can produce a delta
function localized at any point
00:17:45.300 --> 00:17:48.110
by using this trick.
00:17:48.110 --> 00:17:49.830
OK, there are some
other tricks that you
00:17:49.830 --> 00:17:55.440
can do with delta functions,
which is a little surprising.
00:17:55.440 --> 00:18:01.420
And that's really
for the future.
00:18:01.420 --> 00:18:08.424
Delta of minus x is
equal to delta of x.
00:18:08.424 --> 00:18:09.340
It's an even function.
00:18:12.610 --> 00:18:14.950
Derivative of a delta
function is an odd function.
00:18:19.140 --> 00:18:24.270
Delta-- a constant times x
is 1 over the absolute value
00:18:24.270 --> 00:18:26.550
of that constant, delta x.
00:18:29.070 --> 00:18:33.450
A little surprising, but
if you have x, which--
00:18:38.710 --> 00:18:39.710
I better not say that.
00:18:39.710 --> 00:18:45.970
OK, we have now another
fantastic thing-- g of x.
00:18:45.970 --> 00:18:49.780
So what is the delta
function of a function?
00:18:52.530 --> 00:19:07.670
Well it's a sum over j of the
g ex and x of j times delta
00:19:07.670 --> 00:19:08.776
x minus xj.
00:19:18.270 --> 00:19:22.670
This is something
where you're harvesting
00:19:22.670 --> 00:19:27.190
the zeros of this function.
00:19:27.190 --> 00:19:33.200
OK, when this is 0,
we get a big thing.
00:19:36.350 --> 00:19:40.130
So the delta function
of our function
00:19:40.130 --> 00:19:45.640
is a sum over the derivative--
00:19:45.640 --> 00:19:49.260
the zeros of g of x.
00:19:49.260 --> 00:19:50.990
And at times, it's kind of neat.
00:19:54.140 --> 00:19:58.210
OK, there's also stuff
in Cohen-Tannoudji
00:19:58.210 --> 00:20:01.870
on Fourier transforms
of the delta functions.
00:20:01.870 --> 00:20:04.530
I'm not going to
talk about that.
00:20:04.530 --> 00:20:08.340
But one of the things is, if
you have a delta function and x,
00:20:08.340 --> 00:20:10.340
and you take the Fourier
transformer with it,
00:20:10.340 --> 00:20:13.040
you get a delta
function of p, momentum.
00:20:15.620 --> 00:20:16.730
Kind of useful.
00:20:16.730 --> 00:20:21.040
It enables you to do a transform
between an x representation
00:20:21.040 --> 00:20:26.500
and a position representation
and a momentum representation.
00:20:26.500 --> 00:20:28.340
It's kind of useful.
00:20:28.340 --> 00:20:30.320
OK, now we're going to
get to the good stuff.
00:20:35.710 --> 00:20:40.900
So for every system
that has a potential,
00:20:40.900 --> 00:20:45.010
and where the
potential has minima,
00:20:45.010 --> 00:20:46.950
what is the minimum potential?
00:20:46.950 --> 00:20:50.370
What is the condition for
a minimum of the potential?
00:20:53.970 --> 00:20:54.590
Yes?
00:20:54.590 --> 00:20:57.600
AUDIENCE: The first derivative
has to be 0 with respect
00:20:57.600 --> 00:20:58.277
to coordinates.
00:20:58.277 --> 00:20:59.110
ROBERT FIELD: Right.
00:20:59.110 --> 00:21:03.450
And so, any time
you have a minimum,
00:21:03.450 --> 00:21:07.190
the first term in the potential
is the quadratic term.
00:21:07.190 --> 00:21:10.320
And that's the same thing
as a harmonic oscillator.
00:21:10.320 --> 00:21:12.645
The rest is just excess baggage.
00:21:15.280 --> 00:21:17.440
I mean, that's what we do
for perturbation theory.
00:21:17.440 --> 00:21:21.070
We say, OK, we're going to
represent some arbitrary
00:21:21.070 --> 00:21:25.250
potential as a
harmonic oscillator.
00:21:25.250 --> 00:21:28.390
And all of the bad stuff,
all of the higher powers
00:21:28.390 --> 00:21:31.720
of the coordinate get treated
by perturbation theory.
00:21:34.410 --> 00:21:35.530
And you know how to do it.
00:21:35.530 --> 00:21:37.980
And you're not
excited about doing it
00:21:37.980 --> 00:21:41.100
because it's kind of
algebraically horrible.
00:21:43.780 --> 00:21:46.410
And nobody is going
to check your algebra.
00:21:46.410 --> 00:21:51.100
And almost guaranteed, you're
going to make a mistake.
00:21:51.100 --> 00:21:55.300
(Exam 2!)
00:21:55.300 --> 00:21:58.420
So it would be nice
to be able to deal
00:21:58.420 --> 00:22:00.730
with arbitrary potentials--
00:22:00.730 --> 00:22:04.690
potentials that might have
multiple minima, or might
00:22:04.690 --> 00:22:09.890
have all sorts of strange
stuff without doing
00:22:09.890 --> 00:22:10.820
perturbation theory.
00:22:15.450 --> 00:22:17.000
And that's what DVR does.
00:22:21.320 --> 00:22:27.400
So for example, we
want to know how
00:22:27.400 --> 00:22:41.630
to derive a matrix
representation of a matrix.
00:22:41.630 --> 00:22:46.360
So often, we have operators
like the overlap integral
00:22:46.360 --> 00:22:48.280
or the Hamiltonian.
00:22:48.280 --> 00:22:51.880
So we have the
operator, the S matrix,
00:22:51.880 --> 00:22:53.890
or the Hamiltonian matrix.
00:22:53.890 --> 00:22:57.460
And often, we want
to have something
00:22:57.460 --> 00:23:01.300
that is a matrix
representation of a matrix.
00:23:04.300 --> 00:23:10.150
For example, if we're dealing
with a problem in quantum
00:23:10.150 --> 00:23:17.470
chemistry, where our basis
set is not orthonormal,
00:23:17.470 --> 00:23:22.120
there is a trick
using the S matrix
00:23:22.120 --> 00:23:26.520
to orthonormalize everything.
00:23:26.520 --> 00:23:29.970
And that's useful because
then your secular equation
00:23:29.970 --> 00:23:32.550
is the standard secular
equation, which computers just
00:23:32.550 --> 00:23:33.512
love.
00:23:33.512 --> 00:23:35.220
And so you just have
to tell the computer
00:23:35.220 --> 00:23:39.060
to do something special before,
and it orthonormalizes stuff.
00:23:39.060 --> 00:23:41.920
And you can do this
in matrix language.
00:23:41.920 --> 00:23:44.340
And if you're interested
in time evolution,
00:23:44.340 --> 00:23:50.960
you often want to have e to
the minus I h t over h bar.
00:23:50.960 --> 00:23:52.770
And that's horrible.
00:23:52.770 --> 00:23:58.600
But so we'd like to know how to
obtain a matrix representation
00:23:58.600 --> 00:23:59.965
of a function of a matrix.
00:24:02.870 --> 00:24:09.070
Well, suppose we
have some matrix,
00:24:09.070 --> 00:24:18.510
and we can transform
it to be a1, a n, 0, 0.
00:24:18.510 --> 00:24:20.070
We diagnose it.
00:24:26.880 --> 00:24:33.440
And if a is real and
symmetric, or Hermitian.
00:24:33.440 --> 00:24:35.960
We know that the
transformation that
00:24:35.960 --> 00:24:43.040
diagonalized it has the
property that t dagger
00:24:43.040 --> 00:24:46.550
is equal to the inverse.
00:24:46.550 --> 00:24:50.360
And we also know that
if we diagonalize
00:24:50.360 --> 00:24:56.830
a matrix the eigenvectors
that-- they say,
00:24:56.830 --> 00:25:01.630
if you want the first
eigenvector, the thing that
00:25:01.630 --> 00:25:04.300
belongs to this
eigenvalue, we want
00:25:04.300 --> 00:25:07.830
the first column of t dagger.
00:25:07.830 --> 00:25:11.370
And if you want to use
perturbation theory instead
00:25:11.370 --> 00:25:16.080
of the computer to calculate
t dagger, you can do that,
00:25:16.080 --> 00:25:18.180
and you have a
good approximation.
00:25:18.180 --> 00:25:23.250
And you can write the vector,
the linear combination
00:25:23.250 --> 00:25:27.180
of basis functions, that
corresponds to t dagger
00:25:27.180 --> 00:25:28.440
as fast as you can write.
00:25:31.344 --> 00:25:32.760
And the computer
can do it faster.
00:25:38.150 --> 00:25:42.380
So if the matrix is Hermitian,
then all of these guys
00:25:42.380 --> 00:25:43.130
are real.
00:25:43.130 --> 00:25:45.350
And it's just real
and symmetric.
00:25:45.350 --> 00:25:48.140
Well then these guys
are still numbers
00:25:48.140 --> 00:25:49.520
that you could
generate, but they
00:25:49.520 --> 00:25:51.710
might be imaginary or complex.
00:25:59.390 --> 00:26:02.550
Suppose we want some
function of a matrix.
00:26:02.550 --> 00:26:06.050
So this is a matrix, this
has to be a matrix too.
00:26:06.050 --> 00:26:09.590
It has to be the same dimension
as the original matrix.
00:26:12.390 --> 00:26:14.575
So you can do this.
00:26:23.790 --> 00:26:25.980
Well we can call that f twiddle.
00:26:30.530 --> 00:26:33.150
But we don't want f twiddle.
00:26:33.150 --> 00:26:44.000
And so if we do this, we have
the matrix representation
00:26:44.000 --> 00:26:46.570
of the function.
00:26:46.570 --> 00:26:50.540
So this is something that could
be proven in linear algebra,
00:26:50.540 --> 00:26:52.790
but not in 5.61.
00:26:52.790 --> 00:26:56.090
You can do power
series expansions,
00:26:56.090 --> 00:26:57.920
and you can show term
by term that this
00:26:57.920 --> 00:27:02.930
is true for small
matrices, but it's true.
00:27:02.930 --> 00:27:07.750
So if your computer
can diagonalize this,
00:27:07.750 --> 00:27:11.860
then what happens is
that we have this--
00:27:16.380 --> 00:27:26.400
we can write that
f twiddle is the--
00:27:44.300 --> 00:27:47.830
So this is the crucial thing.
00:27:47.830 --> 00:27:50.950
We've diagonalized
this, so we have
00:27:50.950 --> 00:27:55.060
a bunch of eigenvalues of a.
00:27:55.060 --> 00:27:59.720
So the f twiddle
is just the values
00:27:59.720 --> 00:28:04.910
of the function at each
of the eigenvalues.
00:28:10.270 --> 00:28:11.420
And we have zeros here.
00:28:14.590 --> 00:28:18.460
And now we don't like
this because this isn't
00:28:18.460 --> 00:28:20.920
the matrix representation of f.
00:28:20.920 --> 00:28:26.440
It's f in a different
representation.
00:28:26.440 --> 00:28:29.730
So we have to go back to
the original representation.
00:28:29.730 --> 00:28:31.320
And so that's another
transformation.
00:28:31.320 --> 00:28:33.600
But it uses the same matrix.
00:28:36.190 --> 00:28:41.800
So if you did the
work to diagonalize a,
00:28:41.800 --> 00:28:46.135
well then you can go back
and undiagonalize this
00:28:46.135 --> 00:28:49.120
f twiddle to make the
representation of f.
00:28:52.410 --> 00:28:55.720
And so the only work involved
is asking the computer
00:28:55.720 --> 00:29:01.450
to find t dagger
for the a matrix.
00:29:01.450 --> 00:29:05.650
And then you get the true
matrix representation
00:29:05.650 --> 00:29:07.810
of this function of a.
00:29:10.910 --> 00:29:11.600
Is it useful?
00:29:11.600 --> 00:29:12.100
You bet.
00:29:24.010 --> 00:29:27.795
Now suppose a is
infinite dimension.
00:29:32.920 --> 00:29:36.130
And we know that this
is a very common case.
00:29:36.130 --> 00:29:38.240
Because, even for the
harmonic oscillator,
00:29:38.240 --> 00:29:43.000
we have an infinite
number of basis functions.
00:29:43.000 --> 00:29:46.520
But what about using
the delta function?
00:29:46.520 --> 00:29:48.390
We also have an
infinite number of them.
00:29:52.430 --> 00:29:54.520
So what do we do?
00:29:54.520 --> 00:29:56.590
We can't diagonalize
an infinite matrix.
00:30:00.440 --> 00:30:04.160
So what we do is we truncate it.
00:30:09.820 --> 00:30:12.820
Now the computer is quite
happy to deal with matrices
00:30:12.820 --> 00:30:16.000
of dimension 1,000.
00:30:16.000 --> 00:30:19.060
Your computer can
diagonalize a 1,000
00:30:19.060 --> 00:30:25.900
by 1,000 matrix in a few
minutes, maybe a few seconds
00:30:25.900 --> 00:30:30.250
depending on how up
to date this thing is.
00:30:30.250 --> 00:30:39.280
And so what we do is we say, oh,
well let's just take a 1,000.
00:30:39.280 --> 00:30:42.100
So here is an infinite matrix.
00:30:42.100 --> 00:30:45.870
And here is a 1,000
by 1,000 block.
00:30:45.870 --> 00:30:48.980
That's a million elements.
00:30:48.980 --> 00:30:50.810
The computer doesn't care.
00:30:50.810 --> 00:30:53.210
And we're just going to
throw away everything else.
00:30:57.350 --> 00:30:59.100
We don't care.
00:30:59.100 --> 00:31:00.930
Now this is an approximation.
00:31:00.930 --> 00:31:03.440
Now you can truncate
in clever ways,
00:31:03.440 --> 00:31:06.500
or just tell the computer to
throw away everything above
00:31:06.500 --> 00:31:09.530
the 1,000th basis function.
00:31:09.530 --> 00:31:10.940
That's very convenient.
00:31:10.940 --> 00:31:13.780
The computer doesn't care.
00:31:13.780 --> 00:31:17.160
Now there are
transformations that say,
00:31:17.160 --> 00:31:20.760
well you can fold in the effects
of the remote basis states,
00:31:20.760 --> 00:31:25.050
and do an augmented
representation.
00:31:25.050 --> 00:31:28.020
But usually, you just
throw everything away.
00:31:28.020 --> 00:31:32.700
And now you look at this
1,000 by 1,000 matrix.
00:31:32.700 --> 00:31:36.510
And you get the eigenvalues.
00:31:36.510 --> 00:31:46.300
And so this might be
the matrix of x, or q
00:31:46.300 --> 00:31:48.760
if we're talking in
the usual notation.
00:31:48.760 --> 00:31:52.193
This is the displacement
from equilibrium.
00:31:56.140 --> 00:31:57.990
Well, it seems a little--
00:32:01.590 --> 00:32:04.721
so we would like to
find this matrix.
00:32:04.721 --> 00:32:05.845
Well, we know what that is.
00:32:10.670 --> 00:32:14.220
Because we know the relationship
between x and a plus
00:32:14.220 --> 00:32:17.320
a dagger are friends.
00:32:17.320 --> 00:32:22.360
And so we have a matrix, which
is zeros along the diagonal,
00:32:22.360 --> 00:32:26.230
and numbers here and here,
and zeros everywhere else.
00:32:26.230 --> 00:32:28.330
It doesn't take much
to program a computer
00:32:28.330 --> 00:32:33.520
to fill in as many one-off
the diagonal, especially
00:32:33.520 --> 00:32:37.090
because they're square
roots of integers.
00:32:37.090 --> 00:32:38.920
So that's just a
few lines of code,
00:32:38.920 --> 00:32:43.560
and you have the matrix
representation of x.
00:32:43.560 --> 00:32:45.570
Now that is infinite.
00:32:45.570 --> 00:32:48.180
And you're going to
say, well, I don't care.
00:32:48.180 --> 00:32:52.280
I'm going to just
keep the first 1,000.
00:32:52.280 --> 00:32:56.690
We know that we
can always write--
00:32:56.690 --> 00:33:03.150
so we have v of x, and
this is a matrix now.
00:33:03.150 --> 00:33:05.300
And it's an infinite matrix.
00:33:05.300 --> 00:33:12.870
But we say, oh, we just
want v of x to the 1000th,
00:33:12.870 --> 00:33:15.830
and we'll get v of 1,000.
00:33:15.830 --> 00:33:18.496
We have a 1,000
by 1,000 v matrix.
00:33:18.496 --> 00:33:19.620
And we know how to do this.
00:33:19.620 --> 00:33:21.690
We diagonalize that.
00:33:21.690 --> 00:33:29.670
Then we write at each
eigenvalue of x, what v of x
00:33:29.670 --> 00:33:31.070
is at that eigenvalue.
00:33:33.820 --> 00:33:39.480
And so now we have a v
matrix, which is diagonal,
00:33:39.480 --> 00:33:44.040
but in the wrong
representation for us.
00:33:44.040 --> 00:33:46.740
And then we transform back
to the harmonic oscillator
00:33:46.740 --> 00:33:49.180
representation.
00:33:49.180 --> 00:33:51.250
And so everything is fine.
00:33:51.250 --> 00:33:53.830
We've done this.
00:33:53.830 --> 00:33:55.930
And we don't know how
good it's going to be.
00:33:58.700 --> 00:34:02.990
But what we do is
we do this problem.
00:34:02.990 --> 00:34:05.020
So we have the
Hamiltonian, which
00:34:05.020 --> 00:34:06.910
is equal to the
kinetic energy, I'm
00:34:06.910 --> 00:34:11.630
going to call it
k, plus v. We know
00:34:11.630 --> 00:34:16.880
how to generate the
representation of v
00:34:16.880 --> 00:34:21.900
in the harmonic oscillator
basis by writing v
00:34:21.900 --> 00:34:27.570
of x, diagonlizing x, and
then undiagonalizing--
00:34:27.570 --> 00:34:33.300
or then writing v at each
of the eigenvalues of x,
00:34:33.300 --> 00:34:37.980
and then going back to the
dramatic oscillator basis.
00:34:37.980 --> 00:34:41.679
We know k in the harmonic
oscillator basis.
00:34:41.679 --> 00:34:43.830
It's just tri-diagonal,
and it has
00:34:43.830 --> 00:34:49.500
matrix elements delta v
equals 0 plus minus two.
00:34:49.500 --> 00:34:53.550
So now we have a
matrix representation
00:34:53.550 --> 00:34:58.320
of k, which is simple,
add a v which is--
00:35:00.950 --> 00:35:03.380
computer makes it simple.
00:35:03.380 --> 00:35:06.890
Add any v you want, there it is.
00:35:06.890 --> 00:35:09.140
So that's a matrix,
the Hamiltonian.
00:35:09.140 --> 00:35:10.910
You solve the
Schrodinger equation
00:35:10.910 --> 00:35:12.290
by diagonalizing this matrix.
00:35:15.880 --> 00:35:26.260
And so you have this h matrix,
and it's for the 1,000-member x
00:35:26.260 --> 00:35:30.023
matrix, and you get a
bunch of eigenvalues.
00:35:34.280 --> 00:35:36.950
And so then you do it again.
00:35:36.950 --> 00:35:42.640
And maybe use a 900 by 900,
or maybe you use a 1,100--
00:35:42.640 --> 00:35:45.780
you do it again.
00:35:45.780 --> 00:35:50.340
And then you look
at the eigenenergies
00:35:50.340 --> 00:35:55.710
of the Hamiltonian,
e1, say, up to e100.
00:35:55.710 --> 00:35:57.540
Now if you did a
1,000 by 1,000, you
00:35:57.540 --> 00:35:59.790
have reasonable expectation
that the first 100
00:35:59.790 --> 00:36:02.830
eigenvalues will be right.
00:36:02.830 --> 00:36:07.030
And so you compare the results
you get for the 1,000 by 1,000
00:36:07.030 --> 00:36:10.660
to the 900 by 900,
or 1,100 by 1,100.
00:36:10.660 --> 00:36:15.070
And you see how accurate
your representation
00:36:15.070 --> 00:36:17.067
is for the first 100.
00:36:17.067 --> 00:36:19.150
Normally, you don't even
care about the first 100.
00:36:19.150 --> 00:36:20.483
You might care about 10 of them.
00:36:23.200 --> 00:36:27.725
So the computer is happy to
deal with 1,000 by 1,000s.
00:36:27.725 --> 00:36:29.100
There's no least
squares fitting,
00:36:29.100 --> 00:36:30.180
so you only do it once.
00:36:33.050 --> 00:36:37.090
And all of a sudden,
you've got the eigenvalues,
00:36:37.090 --> 00:36:40.380
and you've demonstrated
how accurate they are.
00:36:40.380 --> 00:36:44.610
And so depending on what
precision you want--
00:36:44.610 --> 00:36:48.720
you can trust this up to the
100th, or maybe the 73rd,
00:36:48.720 --> 00:36:52.470
or whatever, to a part in
a million, or whatever.
00:36:52.470 --> 00:36:54.900
And so you know how
it's going to work.
00:36:54.900 --> 00:36:57.000
And you have a check
for convergence.
00:36:59.750 --> 00:37:01.520
So it doesn't matter.
00:37:01.520 --> 00:37:03.700
So the only thing
that you want to do
00:37:03.700 --> 00:37:16.620
is you want to choose a
basis set where we have--
00:37:16.620 --> 00:37:18.900
x is the displacement
from equilibrium.
00:37:25.650 --> 00:37:29.360
OK, so this is the
equilibrium value.
00:37:29.360 --> 00:37:33.410
This is the definition
of the displacement.
00:37:33.410 --> 00:37:36.920
And so you want to
be able to choose
00:37:36.920 --> 00:37:41.450
your basis set, which is
centered at the equilibrium
00:37:41.450 --> 00:37:43.040
value.
00:37:43.040 --> 00:37:45.260
You could do it somewhere
else, it would be stupid.
00:37:45.260 --> 00:37:47.290
It wouldn't converge so well.
00:37:47.290 --> 00:37:52.400
And you want to use the
harmonic oscillator, k over u.
00:37:58.360 --> 00:38:01.910
You have a couple of choices
before you start telling
00:38:01.910 --> 00:38:04.060
the computer to go to work.
00:38:04.060 --> 00:38:07.320
And you tell it, well
I think the best basis
00:38:07.320 --> 00:38:10.690
that will be what works
at the equilibrium--
00:38:10.690 --> 00:38:13.730
the lowest minimum
of the potential,
00:38:13.730 --> 00:38:18.410
and matches the curvature there.
00:38:18.410 --> 00:38:20.360
You don't have to do that.
00:38:20.360 --> 00:38:23.470
But it would be a
good idea to ask
00:38:23.470 --> 00:38:28.760
it to do a problem that's likely
to be a good representation.
00:38:28.760 --> 00:38:32.120
And all this you've done.
00:38:32.120 --> 00:38:33.200
You get the energy level.
00:38:33.200 --> 00:38:34.760
So what you end up getting--
00:38:38.980 --> 00:38:44.230
So you produced
your Hamiltonian.
00:38:44.230 --> 00:38:50.770
And since it's not an infinite
dimension Hamiltonian,
00:38:50.770 --> 00:38:54.030
we can call it an
effective Hamiltonian.
00:38:54.030 --> 00:38:56.370
It contains everything
that is going
00:38:56.370 --> 00:38:58.350
to generate the eigenvalues.
00:38:58.350 --> 00:39:04.620
And we get from that a
set of energy levels--
00:39:04.620 --> 00:39:10.050
e vi-- and a set of functions.
00:39:12.920 --> 00:39:17.100
So these guys are the
true energy levels.
00:39:17.100 --> 00:39:18.950
And these are the
linear combinations
00:39:18.950 --> 00:39:21.020
of harmonic oscillator
functions that
00:39:21.020 --> 00:39:23.640
correspond to each of them.
00:39:23.640 --> 00:39:24.684
Who gives that to you?
00:39:27.350 --> 00:39:31.020
And so then you
say, well, I want
00:39:31.020 --> 00:39:35.760
to represent the Hamiltonian
by a traditional thing.
00:39:35.760 --> 00:39:43.660
Like, I want to say that we have
omega v plus 1/2 minus omega x
00:39:43.660 --> 00:39:48.090
equals 1/2 squared, et cetera.
00:39:48.090 --> 00:39:52.830
Now we do a least squares
fit of molecular constants
00:39:52.830 --> 00:39:55.900
to the energy levels
of the Hamiltonian.
00:40:00.900 --> 00:40:04.230
And there's lots of other
things we could do, but--
00:40:04.230 --> 00:40:08.730
so we say, well in the spectrum,
we would observe these things,
00:40:08.730 --> 00:40:14.220
but we're representing them as
a power series in v plus 1/2.
00:40:14.220 --> 00:40:21.810
Or maybe in-- where we would
have not just the energy--
00:40:21.810 --> 00:40:23.670
the vibrational quantum number--
00:40:23.670 --> 00:40:27.110
but the rotational constant.
00:40:27.110 --> 00:40:41.530
We could say the potential is
v of 0 plus b x j j plus 1.
00:40:41.530 --> 00:40:46.410
Well, that means we could extend
this to allow the molecule
00:40:46.410 --> 00:40:47.580
to rotate.
00:40:47.580 --> 00:40:52.430
And we just need to evaluate
the rotational constant
00:40:52.430 --> 00:40:57.800
as a function of x, and just
add that to what we have here.
00:41:02.100 --> 00:41:10.130
Another thing, you have t
dagger and t for our problem.
00:41:10.130 --> 00:41:13.820
And you have, say,
the 1,000 by 1,000,
00:41:13.820 --> 00:41:18.660
and maybe the 900 by
900 representations.
00:41:18.660 --> 00:41:21.370
You keep them.
00:41:21.370 --> 00:41:25.000
Because any problem you would
have, you could use these for.
00:41:27.620 --> 00:41:29.660
So you still have to
do a diagonalization
00:41:29.660 --> 00:41:33.250
of the Hamiltonian,
but the other
00:41:33.250 --> 00:41:36.330
stuff you don't
have to do anymore.
00:41:36.330 --> 00:41:45.750
Now maybe it's too bothersome to
store a 1,000 by 1,000 t dagger
00:41:45.750 --> 00:41:46.360
matrix.
00:41:46.360 --> 00:41:48.752
It's a million elements.
00:41:48.752 --> 00:41:50.960
Maybe you don't, and you
can just calculate it again.
00:41:50.960 --> 00:41:54.660
It takes 20 minutes
or maybe less.
00:41:54.660 --> 00:41:58.630
And so this is a pretty good.
00:41:58.630 --> 00:42:02.020
OK I've skipped a lot of
stuff in the notes, because I
00:42:02.020 --> 00:42:04.090
wanted to get to the end.
00:42:04.090 --> 00:42:09.610
But the end is really
just a correction
00:42:09.610 --> 00:42:13.045
of what I said was impossible
at the beginning of the course.
00:42:18.624 --> 00:42:20.040
We have in the
Schrodinger picture
00:42:20.040 --> 00:42:24.060
H psi is equal to E psi, right?
00:42:24.060 --> 00:42:27.320
That's the Schrodinger equation.
00:42:27.320 --> 00:42:31.750
So this wave function
is the essential thing
00:42:31.750 --> 00:42:34.460
in quantum mechanics.
00:42:34.460 --> 00:42:38.840
And I also told you,
you can't observe this.
00:42:38.840 --> 00:42:41.500
So it's a very
strange theory where
00:42:41.500 --> 00:42:45.490
the central quality
in the theory
00:42:45.490 --> 00:42:50.450
is experimentally inaccessible.
00:42:50.450 --> 00:42:51.590
But the theory works.
00:42:51.590 --> 00:42:55.500
The theory gives you
everything you need.
00:42:55.500 --> 00:43:00.000
It enables you to find
the eigenfunctions,
00:43:00.000 --> 00:43:02.460
if you have the
exact Hamiltonian.
00:43:02.460 --> 00:43:06.660
Or it says, well we can
take a model problem
00:43:06.660 --> 00:43:12.240
and we can find the
eigenvalues and wave functions
00:43:12.240 --> 00:43:14.250
for the model problem.
00:43:18.810 --> 00:43:21.180
We can generate an
effective Hamiltonian,
00:43:21.180 --> 00:43:23.970
which is expressed in terms
of molecular constants.
00:43:35.170 --> 00:43:42.490
We can then determine
the potential.
00:43:42.490 --> 00:43:45.820
And we can also determine psi.
00:43:45.820 --> 00:43:46.330
All of them.
00:43:50.630 --> 00:43:57.250
So what I told you was
true, but only a little bit
00:43:57.250 --> 00:44:02.200
of a lie in the sense that
you can get as accurate
00:44:02.200 --> 00:44:06.220
as you want a representation
of the wave function,
00:44:06.220 --> 00:44:08.150
if you want it.
00:44:08.150 --> 00:44:12.690
And DVR gives it to
you without any effort.
00:44:12.690 --> 00:44:18.480
And so it doesn't matter
how terrible the potential
00:44:18.480 --> 00:44:26.010
is, as long as it is more
or less well behaved.
00:44:26.010 --> 00:44:28.290
I mean, if you had
a potential, which--
00:44:34.300 --> 00:44:39.070
Even if I had a v potential--
the discontinuity here--
00:44:39.070 --> 00:44:45.900
you would still get a
reasonable result from DVR.
00:44:45.900 --> 00:44:47.620
What it doesn't
like is something
00:44:47.620 --> 00:44:53.060
like that, because then you
have a continuum over here.
00:44:53.060 --> 00:44:59.440
And the continuum uses up your
basis functions pretty fast.
00:44:59.440 --> 00:45:01.650
I mean, yeah, it
will work for this.
00:45:01.650 --> 00:45:08.250
But you don't quite know
how it's going to work,
00:45:08.250 --> 00:45:10.680
and you have to do very
careful convergence tests,
00:45:10.680 --> 00:45:12.150
because this might be good.
00:45:12.150 --> 00:45:17.910
But up here, it's going to
use a lot of basis functions.
00:45:17.910 --> 00:45:20.480
So for the first
time in a long time,
00:45:20.480 --> 00:45:24.410
I'm finishing on time
or even a little early.
00:45:24.410 --> 00:45:28.730
But DVR is really a
powerful computational tool
00:45:28.730 --> 00:45:36.950
that, if you are doing any kind
of theoretical calculation,
00:45:36.950 --> 00:45:41.660
you may very well much want
to use something like this
00:45:41.660 --> 00:45:45.710
rather than an infinite
set of basis functions
00:45:45.710 --> 00:45:49.260
and perturbation theory.
00:45:49.260 --> 00:45:53.550
It's something where you
leave almost all of the work
00:45:53.550 --> 00:45:54.390
to the computer.
00:45:54.390 --> 00:45:57.480
You don't have to do
much besides say, well
00:45:57.480 --> 00:46:00.750
what is the equilibrium value?
00:46:00.750 --> 00:46:03.030
And what vibrational
frequency have
00:46:03.030 --> 00:46:06.120
I got to use for my basis set?
00:46:06.120 --> 00:46:10.320
And if you choose
something that's
00:46:10.320 --> 00:46:13.110
appropriate for the curvature
at the absolute minimum
00:46:13.110 --> 00:46:17.190
of the potential, you're
likely to be doing very well.
00:46:17.190 --> 00:46:22.720
Now other choices might mean
you don't get the first 100,
00:46:22.720 --> 00:46:23.940
you only get the first 50.
00:46:23.940 --> 00:46:27.520
But you might only care
about the first 10.
00:46:27.520 --> 00:46:29.590
Or you could say, I'm
going to choose something
00:46:29.590 --> 00:46:34.640
which is a compromise
between two minima,
00:46:34.640 --> 00:46:36.960
and maybe I'll do better.
00:46:36.960 --> 00:46:40.390
But it doesn't cost you
anything in the computer.
00:46:40.390 --> 00:46:43.650
Your computer is mostly
sitting idly on your desk,
00:46:43.650 --> 00:46:46.740
and you could have it
doing these calculations.
00:46:46.740 --> 00:46:54.540
And there is no problem
where you can't use DVR.
00:46:54.540 --> 00:46:57.460
Because if you have a
function of two variables,
00:46:57.460 --> 00:47:00.810
you do a two-variable DVR.
00:47:00.810 --> 00:47:02.970
It gets a little
bit more complicated
00:47:02.970 --> 00:47:05.747
because, if you
have two DVRs, now
00:47:05.747 --> 00:47:07.080
you're talking about a million--
00:47:10.150 --> 00:47:14.070
1,000 by 1,000-- two of them,
and couplings between them.
00:47:14.070 --> 00:47:17.600
And so maybe you have
to be a little bit more
00:47:17.600 --> 00:47:21.680
thoughtful about how
you employ this trick.
00:47:21.680 --> 00:47:23.810
But it's a very powerful trick.
00:47:23.810 --> 00:47:27.300
And there are other
powerful tricks
00:47:27.300 --> 00:47:31.260
that you can use in conjunction
with quantum chemistry that
00:47:31.260 --> 00:47:35.320
enable you to deal
with things like--
00:47:35.320 --> 00:47:41.900
I've chosen a basis set,
which is not orthonormal.
00:47:41.900 --> 00:47:47.130
And my computer only
knows how to diagonalize
00:47:47.130 --> 00:47:51.860
an ordinary Hamiltonian
without subtracting
00:47:51.860 --> 00:47:54.170
an overlap matrix from it.
00:47:54.170 --> 00:47:58.790
And so if I transform to
diagonalize the overlap matrix,
00:47:58.790 --> 00:48:02.220
well then I can fix the problem.
00:48:02.220 --> 00:48:06.080
And so there is a way of
using this kind of theory
00:48:06.080 --> 00:48:11.630
to fix the problem,
which is based
00:48:11.630 --> 00:48:15.650
on choosing a convenient way of
solving the problem, as opposed
00:48:15.650 --> 00:48:21.280
to the most rigorous
way of doing it.
00:48:21.280 --> 00:48:26.120
OK, so I'm hoping
that I will have
00:48:26.120 --> 00:48:28.310
a sensible lecture on
the two-level problem
00:48:28.310 --> 00:48:29.420
for Wednesday.
00:48:29.420 --> 00:48:32.000
I've been struggling with
this for a long time.
00:48:32.000 --> 00:48:33.840
And maybe I can do it.
00:48:33.840 --> 00:48:37.680
If not, I'll review
the whole course.
00:48:37.680 --> 00:48:40.709
OK, see you on Wednesday.