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ROBERT FIELD: So as I said at
the beginning of the course,
00:00:25.550 --> 00:00:30.010
this is quantum mechanics for
use, not admiration, and not
00:00:30.010 --> 00:00:31.780
historical.
00:00:31.780 --> 00:00:34.480
You're going to leave
this course knowing
00:00:34.480 --> 00:00:40.840
how to solve a very large number
of quantum mechanical problems.
00:00:40.840 --> 00:00:42.880
Or if not to solve
it, to get insight
00:00:42.880 --> 00:00:45.120
into how you would solve it.
00:00:45.120 --> 00:00:53.600
And so I presented a couple
of exactly solved problems,
00:00:53.600 --> 00:00:56.940
and that's not because they're
historically important.
00:00:56.940 --> 00:00:59.190
It's because you're
going to use them,
00:00:59.190 --> 00:01:03.030
and you're going to embed the
results of those exactly solved
00:01:03.030 --> 00:01:09.750
problems in an approximate
approach to almost any problem.
00:01:09.750 --> 00:01:12.180
And the vast
majority of problems
00:01:12.180 --> 00:01:16.340
that you would face use
the harmonic oscillator
00:01:16.340 --> 00:01:19.700
as the core of
your understanding
00:01:19.700 --> 00:01:24.260
because almost all
potentials have a minimum,
00:01:24.260 --> 00:01:25.880
and so that means
the first derivative
00:01:25.880 --> 00:01:27.530
is zero at the minimum.
00:01:27.530 --> 00:01:30.710
So you don't care about
the first derivative.
00:01:30.710 --> 00:01:32.330
You care about where
the minimum is.
00:01:32.330 --> 00:01:37.970
And the second derivative
is the dominant thing,
00:01:37.970 --> 00:01:40.770
and that's the harmonic aspect.
00:01:40.770 --> 00:01:43.780
And so using a harmonic
oscillator basis set,
00:01:43.780 --> 00:01:48.750
you're going to be able
to attack every problem.
00:01:48.750 --> 00:01:51.330
And another thing about
the harmonic oscillator
00:01:51.330 --> 00:01:54.345
is that it uses these As and
A daggers, which are magic.
00:01:57.490 --> 00:02:02.110
And what happens is
you forget, totally,
00:02:02.110 --> 00:02:04.120
about the wave functions.
00:02:04.120 --> 00:02:06.580
The wave functions
are there if you
00:02:06.580 --> 00:02:09.039
want to calculate some kind
of probability amplitude
00:02:09.039 --> 00:02:12.040
distribution, but you
never look at them
00:02:12.040 --> 00:02:16.790
when you're solving the problem,
and that's a fantastic thing.
00:02:16.790 --> 00:02:19.950
Now, there is one more
exactly solved problem
00:02:19.950 --> 00:02:22.580
that I want to talk about
today, which is not usually
00:02:22.580 --> 00:02:26.120
included in the list of
exactly solved problems
00:02:26.120 --> 00:02:28.520
because it's kind of special.
00:02:28.520 --> 00:02:30.850
The two-level problem.
00:02:30.850 --> 00:02:32.730
The two-level problem
is exactly solved
00:02:32.730 --> 00:02:34.830
because there are
only two levels,
00:02:34.830 --> 00:02:37.877
and the solution of
that problem involves
00:02:37.877 --> 00:02:38.835
the quadratic equation.
00:02:42.780 --> 00:02:45.330
So there is an exact solution.
00:02:45.330 --> 00:02:50.610
And this is used a lot in
introducing new techniques
00:02:50.610 --> 00:02:53.070
in quantum mechanics,
and so you're
00:02:53.070 --> 00:02:56.760
going to see the two-level
problem again and again
00:02:56.760 --> 00:02:59.670
as opening the door
to being able to deal
00:02:59.670 --> 00:03:01.980
with much more
complicated problems,
00:03:01.980 --> 00:03:04.320
and I'm going to try
to refer to that a lot.
00:03:04.320 --> 00:03:10.230
OK, so we're about to go
from the Schrodinger picture
00:03:10.230 --> 00:03:13.290
to matrix mechanics,
and I'd like
00:03:13.290 --> 00:03:16.746
to have some comments
about what are the elements
00:03:16.746 --> 00:03:17.870
of the Schrodinger picture.
00:03:20.670 --> 00:03:22.410
And there's no
wrong answer here,
00:03:22.410 --> 00:03:24.165
but I'm looking
for certain things.
00:03:26.920 --> 00:03:28.320
Anybody want to tell me?
00:03:28.320 --> 00:03:29.840
Yes?
00:03:29.840 --> 00:03:33.530
AUDIENCE: Is it based off wave
equation or the Schrodinger
00:03:33.530 --> 00:03:36.589
equation, which is very
similar to the wave equation?
00:03:36.589 --> 00:03:37.380
ROBERT FIELD: Yeah.
00:03:41.320 --> 00:03:44.300
So we have a
differential equation,
00:03:44.300 --> 00:03:48.151
and the solutions
are wave functions.
00:03:48.151 --> 00:03:48.650
More?
00:03:57.488 --> 00:03:58.480
AUDIENCE: [INAUDIBLE]
00:03:58.480 --> 00:04:02.170
ROBERT FIELD: Well, mathematics
is challenging to some people.
00:04:02.170 --> 00:04:03.920
Some people really love it.
00:04:03.920 --> 00:04:07.030
But yes, it's much
more mathematical
00:04:07.030 --> 00:04:11.410
because you're faced with
solving differential equations,
00:04:11.410 --> 00:04:14.650
coupled differential
equations, and you're
00:04:14.650 --> 00:04:18.160
challenged with calculating
a lot of intervals,
00:04:18.160 --> 00:04:21.070
and sometimes the intervals
are not over simple functions.
00:04:21.070 --> 00:04:21.940
They're complicated.
00:04:27.110 --> 00:04:30.800
But the main thing is you have
this thing which you could
00:04:30.800 --> 00:04:35.230
never observe, but is somehow
the core of everything
00:04:35.230 --> 00:04:38.410
that you can know,
and that really--
00:04:38.410 --> 00:04:41.301
as I get older, it
bothers me more and more.
00:04:41.301 --> 00:04:43.050
It's not that I'm going
to invent some way
00:04:43.050 --> 00:04:45.720
to do quantum mechanics totally
without a wave function.
00:04:49.410 --> 00:04:51.850
The most important
thing, I think,
00:04:51.850 --> 00:04:55.050
is that when you work in
the Schrodinger picture,
00:04:55.050 --> 00:05:03.970
you get one wave function
and one eigenvalue at a time.
00:05:03.970 --> 00:05:07.770
And so yes, you can
solve these problems,
00:05:07.770 --> 00:05:09.830
but no, you don't get insight.
00:05:12.340 --> 00:05:15.040
You don't see the
overall structure.
00:05:15.040 --> 00:05:17.260
You just see well, if
you did this experiment,
00:05:17.260 --> 00:05:19.480
this is what you would observe.
00:05:19.480 --> 00:05:21.730
That's perfectly wonderful.
00:05:21.730 --> 00:05:25.420
And now, some anticipation.
00:05:25.420 --> 00:05:28.210
What's so special about
the Heisenberg picture?
00:05:28.210 --> 00:05:30.130
Or what is the
Heisenberg picture?
00:05:30.130 --> 00:05:33.430
Now, I haven't lectured on it
so if you read my lecture notes,
00:05:33.430 --> 00:05:34.930
you know the answers
to all of this,
00:05:34.930 --> 00:05:39.570
but can we speculate
about what goes here?
00:05:39.570 --> 00:05:40.810
Yes?
00:05:40.810 --> 00:05:43.406
AUDIENCE: [INAUDIBLE]
00:05:55.760 --> 00:05:59.190
ROBERT FIELD: OK,
so every operator
00:05:59.190 --> 00:06:06.150
can be represented by a matrix,
and the elements of that matrix
00:06:06.150 --> 00:06:10.290
are calculated
usually automatically.
00:06:10.290 --> 00:06:15.930
Often, they're an infinite
number of basis states,
00:06:15.930 --> 00:06:18.130
so you sort of refer to
the Schrodinger picture,
00:06:18.130 --> 00:06:23.030
but anyway, these matrices
are arrays of numbers,
00:06:23.030 --> 00:06:27.450
and they can be infinite
arrays of numbers,
00:06:27.450 --> 00:06:30.660
but you don't write an
infinite array of numbers down.
00:06:30.660 --> 00:06:34.470
You write a few numbers and you
recognize what the pattern is,
00:06:34.470 --> 00:06:37.410
and usually, you have some
function of the quantum
00:06:37.410 --> 00:06:43.620
numbers, and that generates all
the elements in this matrix.
00:06:43.620 --> 00:06:50.160
And the solutions are going
to be certain eigenvectors,
00:06:50.160 --> 00:06:53.330
and instead of differential
equations, it's linear algebra.
00:06:57.350 --> 00:07:01.490
Now, for differential equations,
you'll learn a lot of tricks.
00:07:01.490 --> 00:07:04.710
For integrals, you
learn a lot of tricks.
00:07:04.710 --> 00:07:07.230
For linear algebra,
there aren't any tricks.
00:07:07.230 --> 00:07:08.190
It's all right there.
00:07:10.740 --> 00:07:16.150
And so it's a much
more transparent--
00:07:16.150 --> 00:07:19.270
now, it might be numerically
demanding because you're
00:07:19.270 --> 00:07:22.480
dealing with very large
arrays of numbers,
00:07:22.480 --> 00:07:24.400
and you're trying to
get something from it,
00:07:24.400 --> 00:07:28.270
but the linear
algebra is simple.
00:07:28.270 --> 00:07:34.310
OK, and we have these things
called matrix elements.
00:07:34.310 --> 00:07:38.880
I can use this word now
because the numbers in a matrix
00:07:38.880 --> 00:07:40.110
are called matrix elements.
00:07:40.110 --> 00:07:43.800
They're integrals,
but they're integrals
00:07:43.800 --> 00:07:45.360
that are, more or
less, given to you.
00:07:49.537 --> 00:07:50.995
And then we have
infinite matrices.
00:07:57.070 --> 00:08:01.450
And in the Schrodinger picture,
we might have infinite basis
00:08:01.450 --> 00:08:05.200
sets, but we don't
really think too much
00:08:05.200 --> 00:08:07.960
about the problem of infinities
because we're looking
00:08:07.960 --> 00:08:09.580
at things one at a time.
00:08:09.580 --> 00:08:14.280
Here, the operators are
implicitly infinite,
00:08:14.280 --> 00:08:17.900
and we have to do
something about that
00:08:17.900 --> 00:08:22.370
because no computer can find
the eigenvalues and eigenvectors
00:08:22.370 --> 00:08:24.470
of an infinite matrix.
00:08:24.470 --> 00:08:26.780
So somehow, you
have to truncate it,
00:08:26.780 --> 00:08:29.180
and you have to use
some approximations,
00:08:29.180 --> 00:08:31.820
and the framework for that
is perturbation theory.
00:08:37.690 --> 00:08:39.350
Now, I love perturbation theory.
00:08:39.350 --> 00:08:41.750
I've made my career out
of perturbation theory
00:08:41.750 --> 00:08:45.120
and you're going to see a
lot of it, but not today.
00:08:51.230 --> 00:08:54.170
The next exam is going to have
a lot of perturbation theory.
00:08:57.480 --> 00:09:01.080
OK, so let's review
where we've been.
00:09:01.080 --> 00:09:13.220
With a two-level problem,
we have two energy levels,
00:09:13.220 --> 00:09:19.580
and there is some
interaction between them,
00:09:19.580 --> 00:09:26.190
and we get E plus, psi plus,
and E minus, psi minus.
00:09:35.280 --> 00:09:43.620
So we have these
integrals, H2,2, H1,2,
00:09:43.620 --> 00:09:46.590
and because it's a
Hermitian matrix,
00:09:46.590 --> 00:09:48.570
it's equal to H2,1 star.
00:09:51.390 --> 00:09:55.790
That's the definition
of a Hermitian matrix,
00:09:55.790 --> 00:10:01.520
and that means if the
H1,2 element is imaginary,
00:10:01.520 --> 00:10:04.750
the H2,1 matrix
element is imaginary,
00:10:04.750 --> 00:10:08.369
but with the opposite sign.
00:10:08.369 --> 00:10:09.910
Now, almost all of
the problems we're
00:10:09.910 --> 00:10:17.230
going to face are expressed in
terms of real matrix elements,
00:10:17.230 --> 00:10:22.090
and so all you're really doing
to go from the 1,2 to the 2,1
00:10:22.090 --> 00:10:24.940
is just reversing the
order of the index.
00:10:24.940 --> 00:10:28.120
Nothing else is happening,
and that has also
00:10:28.120 --> 00:10:30.940
some great simplicity
in the solutions
00:10:30.940 --> 00:10:35.140
we use to solve the
two-level problems.
00:10:35.140 --> 00:10:40.160
OK, the solutions to
the two-level problem
00:10:40.160 --> 00:10:44.113
are based on some
simplifications.
00:10:47.220 --> 00:10:50.430
So what we do is we make
the two by two matrix
00:10:50.430 --> 00:10:53.610
symmetric by subtracting
out the average energy.
00:10:56.180 --> 00:11:01.850
We define then, two numbers,
which are H1,1 minus H2,2 over
00:11:01.850 --> 00:11:03.670
two and this.
00:11:03.670 --> 00:11:11.230
So we have E bar and delta,
and we have the interaction,
00:11:11.230 --> 00:11:22.160
which is H1,2, and we call it
v. And we have to be a little
00:11:22.160 --> 00:11:28.190
careful because v could
be complex or imaginary,
00:11:28.190 --> 00:11:34.340
but for now, we're just going
to treat it always as real.
00:11:34.340 --> 00:11:40.140
OK, so for this
two-level problem,
00:11:40.140 --> 00:11:50.880
the energy levels are given by E
bar plus or minus delta squared
00:11:50.880 --> 00:11:55.810
plus v-squared square root.
00:11:55.810 --> 00:12:00.280
Or E bar plus or
minus the square root
00:12:00.280 --> 00:12:03.840
of x, where that's x.
00:12:06.820 --> 00:12:12.280
And the eigenfunctions
are expressed in terms
00:12:12.280 --> 00:12:13.960
of these coefficients--
00:12:13.960 --> 00:12:21.730
one plus or minus is equal
to 1/2 one plus or minus
00:12:21.730 --> 00:12:27.790
delta over x to the
1/2 square root.
00:12:27.790 --> 00:12:30.250
Square root is
outside the bracket.
00:12:30.250 --> 00:12:38.940
And c2 plus or minus
looks almost the same,
00:12:38.940 --> 00:12:47.240
except we have 1 minus or plus
delta over square root of x.
00:12:50.480 --> 00:12:52.460
So in this reduced
picture, there's
00:12:52.460 --> 00:12:55.400
not too much to remember.
00:12:55.400 --> 00:13:05.220
The difference between the
eigenfunctions for the higher
00:13:05.220 --> 00:13:09.780
energy and the lower
energy eigenstates
00:13:09.780 --> 00:13:12.120
differ only by these signs.
00:13:12.120 --> 00:13:17.330
OK, now, this is a lot.
00:13:17.330 --> 00:13:20.610
I derived it in
lecture last time,
00:13:20.610 --> 00:13:23.540
and the algebra is
horrible, and I'm not
00:13:23.540 --> 00:13:26.880
good at presenting algebra,
but it's all in the notes.
00:13:26.880 --> 00:13:32.510
But if you take these formulas
and you try them on for size,
00:13:32.510 --> 00:13:36.140
you will be able to
verify that these
00:13:36.140 --> 00:13:42.770
give normalized and orthogonal
eigenfunctions, which
00:13:42.770 --> 00:13:45.700
I recommend, mostly.
00:13:45.700 --> 00:13:50.360
OK, we take this expression
for the eigenvalues,
00:13:50.360 --> 00:13:53.660
and let's just take one
of the eigenfunctions
00:13:53.660 --> 00:13:58.850
and find out whether it belongs
to the correct eigenvalue.
00:14:02.120 --> 00:14:06.850
If it doesn't, you've
made an algebraic mistake.
00:14:06.850 --> 00:14:08.120
It's a very useful thing.
00:14:08.120 --> 00:14:13.030
OK, so now we're going
to take the results
00:14:13.030 --> 00:14:16.450
for this two-level problem
solved algebraically,
00:14:16.450 --> 00:14:21.610
and at the core of that was
the quadratic energy formula.
00:14:21.610 --> 00:14:26.650
The quadratic formula
for the equation
00:14:26.650 --> 00:14:29.448
that determines the energy.
00:14:29.448 --> 00:14:32.420
The quadratic formula
is always applicable.
00:14:32.420 --> 00:14:33.020
It's exact.
00:14:35.720 --> 00:14:39.000
So what we end up getting
is analytical expressions.
00:14:39.000 --> 00:14:44.000
It doesn't matter what the value
of the two critical quantities,
00:14:44.000 --> 00:14:46.570
delta and x, are.
00:14:46.570 --> 00:14:47.720
You have solutions.
00:14:54.770 --> 00:14:57.280
So now we're going to
take all this stuff
00:14:57.280 --> 00:14:59.590
and we're going to start over.
00:14:59.590 --> 00:15:03.520
We're going to rearrange it so
that we have a different way
00:15:03.520 --> 00:15:04.660
of approaching the problem.
00:15:08.370 --> 00:15:12.790
So we're going to start
talking about matrices
00:15:12.790 --> 00:15:14.410
rather than operators.
00:15:14.410 --> 00:15:22.380
And so I represent a
matrix by a boldface--
00:15:22.380 --> 00:15:28.050
this symbol is how you ask
a computer to make it--
00:15:28.050 --> 00:15:32.760
yes, it's how I represent
a matrix, instead of a hat,
00:15:32.760 --> 00:15:35.590
which is the way you
represent an operator.
00:15:35.590 --> 00:15:38.790
But now we're going to
say every operator is
00:15:38.790 --> 00:15:44.080
represented by a matrix rather
than a differential operator.
00:15:44.080 --> 00:15:54.580
And so this Hamiltonian is E
bar, zero, E bar plus delta, v,
00:15:54.580 --> 00:16:00.240
v star, minus delta.
00:16:00.240 --> 00:16:03.000
And v is usually real.
00:16:03.000 --> 00:16:05.480
And another way
we can say this is
00:16:05.480 --> 00:16:14.350
it's E bar times this fancy 1
with an under bar plus H bar.
00:16:14.350 --> 00:16:20.530
So this is a symmetric
Hamiltonian matrix.
00:16:20.530 --> 00:16:23.450
This is the unit matrix.
00:16:23.450 --> 00:16:31.190
And now, since we're going to
be doing matrix multiplications,
00:16:31.190 --> 00:16:33.620
let me just give
you some mnemonics.
00:16:33.620 --> 00:16:39.380
So if we have a square matrix
multiplying a square matrix,
00:16:39.380 --> 00:16:51.919
what we do is we multiply
this row by that column,
00:16:51.919 --> 00:16:54.335
and we get one number, and you
fill out the square matrix.
00:16:57.830 --> 00:17:00.680
And with a little practice,
this will be permanently
00:17:00.680 --> 00:17:03.340
ingrained in your head.
00:17:03.340 --> 00:17:08.900
We also can have a matrix
multiplying a vector.
00:17:08.900 --> 00:17:16.890
And so a matrix multiplying
a vector gives a vector,
00:17:16.890 --> 00:17:19.690
and this product
gives a number here.
00:17:24.690 --> 00:17:27.660
And you've probably all
seen these sorts of things
00:17:27.660 --> 00:17:31.250
or could grasp
them very quickly,
00:17:31.250 --> 00:17:34.240
but it's useful just
to not get confused.
00:17:34.240 --> 00:17:40.920
We can also do
something like this,
00:17:40.920 --> 00:17:46.640
and again, we use the usual
vector and we get a vector.
00:17:46.640 --> 00:17:48.195
I'm sorry, we get a column.
00:17:53.480 --> 00:18:00.560
And this is a difficult
symbol to make on a computer,
00:18:00.560 --> 00:18:05.560
but you get this first
element here like that.
00:18:05.560 --> 00:18:12.280
And of course, you can
do this times that,
00:18:12.280 --> 00:18:14.800
and you get a number.
00:18:14.800 --> 00:18:17.120
And you can also do it
the other way around.
00:18:17.120 --> 00:18:18.670
You can do this times that--
00:18:21.990 --> 00:18:23.730
I'm sorry, don't do that.
00:18:28.224 --> 00:18:29.390
And you get a square matrix.
00:18:33.150 --> 00:18:35.720
So those are the things
that you have to practice,
00:18:35.720 --> 00:18:40.790
and it becomes second
nature very quickly,
00:18:40.790 --> 00:18:45.410
and it's a lot easier than
doing differential equations,
00:18:45.410 --> 00:18:48.421
or matrices, or integrals.
00:18:51.010 --> 00:18:56.380
OK, now, we use
this superscript.
00:18:59.570 --> 00:19:01.810
This means transpose.
00:19:01.810 --> 00:19:05.530
This means complex
conjugate and transpose.
00:19:09.866 --> 00:19:15.260
The theory deals
with this, but when
00:19:15.260 --> 00:19:19.970
the Hamiltonian or the matrix
you're interested in is real,
00:19:19.970 --> 00:19:23.150
the transformation
that diagonalizes it
00:19:23.150 --> 00:19:24.600
is always just given.
00:19:24.600 --> 00:19:26.660
You need this to transpose.
00:19:26.660 --> 00:19:28.940
So these two
symbols look similar
00:19:28.940 --> 00:19:31.670
and you won't have
any trouble with that.
00:19:31.670 --> 00:19:37.920
And now, we have
this kind of symbol.
00:19:37.920 --> 00:19:41.900
So this is a normalization.
00:19:41.900 --> 00:19:44.480
That should be
equal to one, and it
00:19:44.480 --> 00:19:48.350
is because one times one is
one plus zero times zero.
00:19:52.280 --> 00:19:57.600
And we can also look at
this, and that's zero
00:19:57.600 --> 00:20:03.150
because zero times one is zero
and one times zero is zero.
00:20:03.150 --> 00:20:05.807
So this is normalization,
this is orthogonality.
00:20:09.150 --> 00:20:14.260
OK, we're playing with
numbers, and we don't really
00:20:14.260 --> 00:20:18.580
look at the size, even
though the numbers all
00:20:18.580 --> 00:20:23.290
are obtained by doing an
integral over the wave
00:20:23.290 --> 00:20:28.520
functions and the operator,
but that's something
00:20:28.520 --> 00:20:31.949
that you sort of do in first
grade of quantum mechanics,
00:20:31.949 --> 00:20:33.740
and you forget how you
did it, and you just
00:20:33.740 --> 00:20:35.870
know that they're there
to be played with.
00:20:38.770 --> 00:20:46.660
OK, now, a unitary
operator is one
00:20:46.660 --> 00:20:52.220
where the conjugate transpose--
00:20:52.220 --> 00:20:59.280
the unitary matrix-- is equal
to the inverse, which means
00:20:59.280 --> 00:21:06.790
t minus one times t equals one.
00:21:06.790 --> 00:21:10.780
So it's nice to be able to
derive the inverse of a matrix.
00:21:10.780 --> 00:21:13.370
And for certain
kinds of matrices,
00:21:13.370 --> 00:21:15.580
this is really easy
because all you do
00:21:15.580 --> 00:21:19.080
is tip along the main diagonal.
00:21:19.080 --> 00:21:22.800
There are other matrices where
you have to do a lot of work,
00:21:22.800 --> 00:21:25.150
but whenever you're
dealing with matrices,
00:21:25.150 --> 00:21:26.670
you're not doing the work.
00:21:26.670 --> 00:21:28.470
The computer is doing the work.
00:21:28.470 --> 00:21:32.280
You teach the computer how
to do matrix operations,
00:21:32.280 --> 00:21:36.580
and even if it's a hard one, the
computer says OK, here it is.
00:21:40.140 --> 00:21:54.380
OK, so if you have a
real symmetric matrix,
00:21:54.380 --> 00:22:03.100
then you can say OK,
T transpose matrix T
00:22:03.100 --> 00:22:12.240
gives a1, a n, zero, zero.
00:22:12.240 --> 00:22:17.990
So you can diagonalize
a real symmetric matrix
00:22:17.990 --> 00:22:20.630
by this kind of
a transformation.
00:22:20.630 --> 00:22:23.250
That's called an
orthogonal transformation.
00:22:23.250 --> 00:22:26.930
And if it's not real,
then you use the conjugate
00:22:26.930 --> 00:22:31.250
transpose, use the
dagger, and you still
00:22:31.250 --> 00:22:34.980
get the diagonalization.
00:22:34.980 --> 00:22:37.320
Now, in most of the
books that you'll ever
00:22:37.320 --> 00:22:40.110
look at about unitary
transformations,
00:22:40.110 --> 00:22:42.060
they actually are
giving you what's called
00:22:42.060 --> 00:22:45.550
the orthogonal
transformation, and it's
00:22:45.550 --> 00:22:52.040
what works for a real matrix,
and I'm going to do that too.
00:22:52.040 --> 00:22:55.400
So when we have
something like this,
00:22:55.400 --> 00:23:00.440
we say that this transformation
diagonalizes A, or H,
00:23:00.440 --> 00:23:01.810
or whatever.
00:23:01.810 --> 00:23:05.716
And the word "diagonalizes"
is really important
00:23:05.716 --> 00:23:07.340
because that's what
you want because it
00:23:07.340 --> 00:23:09.620
presents all the eigenvalues.
00:23:09.620 --> 00:23:11.900
Remember, one of the things
about quantum mechanics
00:23:11.900 --> 00:23:14.300
is you have an operator.
00:23:14.300 --> 00:23:16.330
You're going to
observe something
00:23:16.330 --> 00:23:18.010
connected with an operator.
00:23:18.010 --> 00:23:21.860
The only things you can
get are the eigenvalues.
00:23:21.860 --> 00:23:24.154
So here they are, all of them.
00:23:24.154 --> 00:23:25.070
That's kind of useful.
00:23:28.090 --> 00:23:37.030
OK, so in the
Heisenberg picture,
00:23:37.030 --> 00:23:41.020
the key equation
is the Hamiltonian
00:23:41.020 --> 00:23:44.260
as a matrix, some vector--
00:23:47.920 --> 00:23:52.240
OK, this is the analog of
the Schrodinger equation,
00:23:52.240 --> 00:23:55.180
but it's the
Heisenberg equation.
00:23:55.180 --> 00:23:58.159
And mostly, it's just
notation, and you
00:23:58.159 --> 00:23:59.200
have to get used to that.
00:24:07.360 --> 00:24:13.060
We want to find the vectors
that are eigenvectors
00:24:13.060 --> 00:24:16.672
of this Hamiltonian
with eigenvalue E.
00:24:16.672 --> 00:24:18.130
It's just like this
the Schrodinger
00:24:18.130 --> 00:24:22.270
equation, except it's now
looking for eigenvectors rather
00:24:22.270 --> 00:24:26.290
than eigenfunctions
and eigenenergies.
00:24:26.290 --> 00:24:29.180
Now, in order to
solve this problem,
00:24:29.180 --> 00:24:33.820
we exploit this kind
of transformation,
00:24:33.820 --> 00:24:45.074
and we insert T T dagger between
H and c, and that's just one.
00:24:45.074 --> 00:24:47.240
So we don't even have to
worry about the other side.
00:24:50.860 --> 00:24:52.510
We're just playing
with matrices,
00:24:52.510 --> 00:24:57.610
but they look like functions or
variables, and everything is--
00:24:57.610 --> 00:25:01.030
it's really neat how
beautiful linear algebra
00:25:01.030 --> 00:25:05.830
is because you are now dealing
with an infinite number
00:25:05.830 --> 00:25:07.300
of equations at once.
00:25:07.300 --> 00:25:10.150
You're dealing
with these objects
00:25:10.150 --> 00:25:13.450
and you're using your
insight from algebra
00:25:13.450 --> 00:25:18.940
as much as possible in order
to figure out what's going on.
00:25:18.940 --> 00:25:21.070
Really beautiful.
00:25:21.070 --> 00:25:25.640
OK, and so now we must multiply
this equation on the left by T
00:25:25.640 --> 00:25:26.140
dagger.
00:25:36.060 --> 00:25:39.490
OK, I'm dropping
the under-bars now.
00:25:39.490 --> 00:25:45.310
OK, so now we say OK,
here we have H twiddle,
00:25:45.310 --> 00:25:50.830
and here we have c twiddle,
and here we have c twiddle.
00:25:50.830 --> 00:25:59.350
So this is now a different
eigenvector equation,
00:25:59.350 --> 00:26:04.450
but we insist that
this guy, H twiddle,
00:26:04.450 --> 00:26:11.921
looks like E1, E2, En, 0,0.
00:26:11.921 --> 00:26:16.000
A diagonal matrix, where
all the eigenvalues
00:26:16.000 --> 00:26:19.310
are along the diagonal.
00:26:19.310 --> 00:26:25.820
And so this is what we
want, and lo and behold,
00:26:25.820 --> 00:26:29.060
this is what we need
in order to say,
00:26:29.060 --> 00:26:35.980
well, yeah, this thing has to be
the eigenvector of this for one
00:26:35.980 --> 00:26:40.360
of the eigenvalues because
this is an eigenvalue equation
00:26:40.360 --> 00:26:42.500
or an eigenvector equation.
00:26:42.500 --> 00:26:45.760
So if we can diagonalize
the Hamiltonian,
00:26:45.760 --> 00:26:49.090
the transformation that
diagonalizes the Hamiltonian
00:26:49.090 --> 00:26:55.630
gives you the linear
combination of basis vectors
00:26:55.630 --> 00:27:00.070
that is the
eigenvector, and we'll
00:27:00.070 --> 00:27:02.830
talk about this some more.
00:27:02.830 --> 00:27:06.370
So for the two-level
problem, we want
00:27:06.370 --> 00:27:13.510
to find E plus, 0, 0, E minus.
00:27:13.510 --> 00:27:21.770
And usually, E plus is the
higher energy eigenvalue
00:27:21.770 --> 00:27:23.480
than E minus.
00:27:23.480 --> 00:27:29.450
Always, when you do this
stuff, you get eigenvalues
00:27:29.450 --> 00:27:32.390
and you get eigenvectors.
00:27:32.390 --> 00:27:35.210
And frequently, when you
do the algebra as opposed
00:27:35.210 --> 00:27:36.980
to the computer
during the algebra,
00:27:36.980 --> 00:27:39.590
you don't know that a
particular eigenvector
00:27:39.590 --> 00:27:43.160
belongs to which eigenvalues.
00:27:43.160 --> 00:27:45.220
So it's useful to have
a couple of things
00:27:45.220 --> 00:27:47.830
that you normally insist on.
00:27:47.830 --> 00:27:51.280
And so I like to
label these things
00:27:51.280 --> 00:27:53.380
by plus and minus,
corresponding to which
00:27:53.380 --> 00:27:55.540
is higher energy
and which is lower.
00:27:55.540 --> 00:27:57.730
You could also say,
well, the plus is
00:27:57.730 --> 00:28:01.530
going to correspond to a plus
linear combination somewhere,
00:28:01.530 --> 00:28:03.960
but that's really dangerous.
00:28:09.080 --> 00:28:11.870
So now, let's just
play a little bit.
00:28:22.120 --> 00:28:30.220
So we have simplified
H magically so far
00:28:30.220 --> 00:28:32.890
to diagonal form.
00:28:32.890 --> 00:28:35.110
So H C twiddle--
00:28:39.020 --> 00:28:49.510
I'm sorry, yes H C twiddle
is going to be E c twiddle.
00:28:54.500 --> 00:29:10.790
So H c plus is going to be
E plus 0, 0, E minus, one,
00:29:10.790 --> 00:29:19.950
zero because that gives
us E plus times one
00:29:19.950 --> 00:29:21.840
and zero times one.
00:29:25.070 --> 00:29:26.440
So multiply these two things.
00:29:26.440 --> 00:29:29.740
We have a column vector, and
that's the same thing as E
00:29:29.740 --> 00:29:31.780
plus times one, zero.
00:29:39.060 --> 00:29:43.920
And we do the same
sort of thing to--
00:29:43.920 --> 00:29:46.770
instead of using c
plus, we use c minus.
00:29:46.770 --> 00:29:50.250
That's a zero, one, and that
will give us E minus times
00:29:50.250 --> 00:29:52.390
zero, one.
00:29:52.390 --> 00:29:55.840
This is all just
playing with notation
00:29:55.840 --> 00:29:58.190
and we're about to
start doing some work.
00:29:58.190 --> 00:30:12.350
OK, so T dagger times c is
supposedly equal to c plus.
00:30:12.350 --> 00:30:16.310
OK, and so well, we
can write this formula
00:30:16.310 --> 00:30:22.240
in a schematic way, and
so we have T dagger.
00:30:29.480 --> 00:30:32.540
Now, I always remember
this because there
00:30:32.540 --> 00:30:36.260
used to be something analogous
to Coke and Pepsi called Royal
00:30:36.260 --> 00:30:41.000
Crown Cola, and for Royal
Crown, that just reminds me
00:30:41.000 --> 00:30:43.910
that row first, column second.
00:30:43.910 --> 00:30:48.230
I don't believe that any of you
have ever heard of Royal Crown,
00:30:48.230 --> 00:30:52.321
but you could think of
some other mnemonic.
00:30:52.321 --> 00:30:55.100
Now, it's really important to
keep the rows and the columns
00:30:55.100 --> 00:30:57.710
straight.
00:30:57.710 --> 00:31:02.240
So we have 1, 1, and T dagger.
00:31:02.240 --> 00:31:03.800
Now, what goes here?
00:31:03.800 --> 00:31:08.280
This is in the first
row, second column.
00:31:08.280 --> 00:31:11.270
So what do I put here?
00:31:11.270 --> 00:31:12.690
Yeah.
00:31:12.690 --> 00:31:13.800
You could even say it.
00:31:16.530 --> 00:31:20.760
OK, and here we
have T dagger 2,1,
00:31:20.760 --> 00:31:29.940
and here we have T dagger 2,2.
00:31:29.940 --> 00:31:45.160
Now, if we multiply one,
zero, because that's
00:31:45.160 --> 00:32:02.450
what we're supposed to do here,
we'll get T 1,1 plus T 1,2
00:32:02.450 --> 00:32:11.850
times zero, and then T
2,1 plus T 2,2 times zero.
00:32:11.850 --> 00:32:19.870
OK, and that's simply
T 1,1 dagger times one,
00:32:19.870 --> 00:32:26.760
zero plus T 2,1 dagger
times zero, one.
00:32:26.760 --> 00:32:30.730
So this thing gives the linear
combination of the basis
00:32:30.730 --> 00:32:34.770
vectors that is equal to
a particular eigenvector.
00:32:46.930 --> 00:32:52.690
So that means if we can
find T, we can get T dagger,
00:32:52.690 --> 00:33:00.520
and we can get E plus and E
minus, and c plus and c minus.
00:33:00.520 --> 00:33:04.385
So we have completely solved the
problem if we know what T is.
00:33:07.290 --> 00:33:11.640
Well, with a
two-level problem, we
00:33:11.640 --> 00:33:14.970
know algebraically
that there is such a T,
00:33:14.970 --> 00:33:19.770
and that it's
analytically determined.
00:33:19.770 --> 00:33:22.770
There is another way of
approaching this, and that
00:33:22.770 --> 00:33:31.050
is to say the general
orthogonal transformation,
00:33:31.050 --> 00:33:33.570
which we will call a
unitary transformation,
00:33:33.570 --> 00:33:36.440
but it's missing a
little bit of stuff
00:33:36.440 --> 00:33:38.700
if it really wants
to be unitary.
00:33:38.700 --> 00:33:40.080
I'm going to call it--
00:33:40.080 --> 00:33:48.940
so T dagger is cos
theta, sin theta,
00:33:48.940 --> 00:33:52.060
minus sin theta, cos theta.
00:33:55.500 --> 00:33:58.530
So this is a matrix which is
determined by one thing, theta.
00:34:01.100 --> 00:34:04.750
We want to find what
theta needs to be in order
00:34:04.750 --> 00:34:06.400
to diagonalize the matrix.
00:34:09.440 --> 00:34:14.440
Now, since we know we're
talking about sines and cosines,
00:34:14.440 --> 00:34:18.010
and that there is one theta,
we abbreviate this to c, s,
00:34:18.010 --> 00:34:23.639
minus s, c because the
algebra is heinous.
00:34:23.639 --> 00:34:29.860
Not as bad as in the Schrodinger
picture, but it's terrible,
00:34:29.860 --> 00:34:34.070
and so you want to compress the
symbols as much as possible.
00:34:34.070 --> 00:34:42.620
OK, so we want T dagger HT
because that's H twiddle.
00:34:42.620 --> 00:34:49.219
That's the thing we want,
and we want T dagger HT.
00:34:49.219 --> 00:34:56.150
And now since we've
expressed the T in this form,
00:34:56.150 --> 00:34:58.760
we can multiply this out.
00:34:58.760 --> 00:35:09.190
And so we have c, c, minus
s, c, delta, v, v, delta.
00:35:09.190 --> 00:35:15.730
Delta, v, v, minus
delta, c, minus s, s, c.
00:35:15.730 --> 00:35:18.175
So we have three two by
two matrices to multiply.
00:35:20.680 --> 00:35:23.420
Now, that's not something
you do in your head.
00:35:23.420 --> 00:35:25.480
You could do two.
00:35:25.480 --> 00:35:27.430
So you multiply these
two, and then you
00:35:27.430 --> 00:35:32.810
multiply by that,
and the result--
00:35:32.810 --> 00:35:37.030
I would be here for
hours doing this,
00:35:37.030 --> 00:35:42.280
and you wouldn't learn anything
except that I'm a real klutz.
00:35:42.280 --> 00:35:44.710
I should write this
on its own board
00:35:44.710 --> 00:35:46.260
because it's really important.
00:36:00.910 --> 00:36:04.690
So that matrix
becomes a big matrix,
00:36:04.690 --> 00:36:15.640
c squared minus s squared
times delta plus 2sc times V,
00:36:15.640 --> 00:36:26.530
and c squared minus s squared
times V minus 2cs delta.
00:36:26.530 --> 00:36:31.650
And we have the same thing down
here, c squared minus s squared
00:36:31.650 --> 00:36:36.430
times V minus 2cs times delta.
00:36:36.430 --> 00:36:42.280
And the last one is minus
c squared minus s squared
00:36:42.280 --> 00:36:51.260
times delta minus
2cs times V. So this
00:36:51.260 --> 00:36:55.650
is what we get when we
take the general form
00:36:55.650 --> 00:36:59.010
for the unitary
transformation, and transform
00:36:59.010 --> 00:37:02.240
the Hamiltonian with it.
00:37:02.240 --> 00:37:04.530
And the first thing
we do is we say,
00:37:04.530 --> 00:37:09.480
well, we want this to be zero.
00:37:09.480 --> 00:37:11.230
If this is zero, then
this is zero, right?
00:37:14.980 --> 00:37:18.540
So this turns out to
be an equation that
00:37:18.540 --> 00:37:21.380
tells us what theta has to be.
00:37:21.380 --> 00:37:24.020
OK, and we also know
from trigonometry,
00:37:24.020 --> 00:37:28.480
c squared minus s
squared is what?
00:37:39.890 --> 00:37:46.010
I'm actually jumping ahead,
but it's just cosine two theta,
00:37:46.010 --> 00:37:54.010
and 2sc is sine two theta.
00:37:54.010 --> 00:37:57.570
So we're going to get a
simplification based on this,
00:37:57.570 --> 00:38:00.480
but now let's just say
we want this to be zero.
00:38:00.480 --> 00:38:05.550
So that means that c squared
minus s squared times V has
00:38:05.550 --> 00:38:07.980
to be equal to 2cs times delta.
00:38:14.050 --> 00:38:15.610
Which way am I going?
00:38:15.610 --> 00:38:20.770
2cs over c squared minus
s squared is V over delta.
00:38:27.060 --> 00:38:29.580
Well, that looks
pretty good, especially
00:38:29.580 --> 00:38:32.550
because this is cosine--
00:38:35.620 --> 00:38:42.930
this is sine two theta, and
this is cosine two theta, which
00:38:42.930 --> 00:38:47.220
is tan theta is equal to this.
00:38:50.060 --> 00:38:52.370
So now we have a
simple equation.
00:38:52.370 --> 00:38:56.750
We have the theta, and
we have the V and a D--
00:38:56.750 --> 00:38:58.500
a V over delta.
00:39:01.057 --> 00:39:01.890
I shouldn't do that.
00:39:07.030 --> 00:39:08.370
So now I can cover this again.
00:39:17.050 --> 00:39:20.500
So we can take this
equation and solve it,
00:39:20.500 --> 00:39:35.050
and we have that theta is equal
to 1/2 inverse tangent of V
00:39:35.050 --> 00:39:37.970
over delta.
00:39:37.970 --> 00:39:38.470
There it is.
00:39:38.470 --> 00:39:46.964
That's an analytic result. So
for any value of V over delta,
00:39:46.964 --> 00:39:47.880
we know what theta is.
00:39:51.750 --> 00:39:54.060
That's not an
iterative solution.
00:39:54.060 --> 00:39:58.590
That's complete analytical
result, and that's fantastic,
00:39:58.590 --> 00:40:04.680
and it says, just like with
the quadratic formula, which
00:40:04.680 --> 00:40:09.940
we used to look at the
original Hamiltonian
00:40:09.940 --> 00:40:12.670
at the eigenvalues
of the original--
00:40:12.670 --> 00:40:17.020
well, yeah, it says no matter
what, there is a solution,
00:40:17.020 --> 00:40:19.600
and you can express
this solution
00:40:19.600 --> 00:40:21.320
as some combination
of V and delta.
00:40:26.700 --> 00:40:32.440
OK, and so when you do that, you
get that the energy levels are
00:40:32.440 --> 00:40:44.400
E bar plus or minus delta times
cos two theta plus V times sin
00:40:44.400 --> 00:40:45.600
two theta.
00:40:49.240 --> 00:40:50.620
And there's no square root here.
00:40:50.620 --> 00:40:51.820
Why do you know that?
00:40:51.820 --> 00:40:53.350
Well, this is dimensionless.
00:40:53.350 --> 00:40:57.000
This is the units
of energy, and so
00:40:57.000 --> 00:40:58.704
square roots keep
coming popping up,
00:40:58.704 --> 00:41:00.120
but you don't want
to put one here
00:41:00.120 --> 00:41:01.286
because that would be wrong.
00:41:03.780 --> 00:41:05.820
Even if you didn't
do the derivation,
00:41:05.820 --> 00:41:07.320
if you saw a square
root here, you'd
00:41:07.320 --> 00:41:11.460
know somebody is just writing
down things from memory
00:41:11.460 --> 00:41:16.040
or copying badly, and
making corrections.
00:41:16.040 --> 00:41:24.810
And that leads to E plus minus
is equal to E bar plus or minus
00:41:24.810 --> 00:41:29.091
delta squared plus V squared,
and there is a square root
00:41:29.091 --> 00:41:30.270
there.
00:41:30.270 --> 00:41:34.350
This is what we derived
via the quadratic formula.
00:41:34.350 --> 00:41:38.130
We knew this, and it
came out to be the same.
00:41:38.130 --> 00:41:40.890
Well, it better have.
00:41:40.890 --> 00:41:43.950
And we can also
determine what T is.
00:41:46.890 --> 00:41:50.130
And I'm not going to write
it, it's in the notes.
00:41:50.130 --> 00:41:54.390
It's a lot of symbols,
but it's something--
00:41:54.390 --> 00:42:00.830
it's so compressed that
you can guess the form,
00:42:00.830 --> 00:42:04.600
and so you should look at that.
00:42:04.600 --> 00:42:09.850
We derived the eigenfunctions
of the Hamiltonian,
00:42:09.850 --> 00:42:14.140
and they are exactly the
same as what we get here.
00:42:14.140 --> 00:42:23.410
And remember that the column
of T dagger or T transpose
00:42:23.410 --> 00:42:32.980
is eigenvector, and
sometimes we want
00:42:32.980 --> 00:42:34.738
to know those eigenvectors.
00:42:42.230 --> 00:42:43.170
We're doing semi-OK.
00:43:02.080 --> 00:43:04.390
What happens if we go beyond
the two-level problem?
00:43:08.070 --> 00:43:14.100
Well, you know from algebra that
there is no general solution
00:43:14.100 --> 00:43:16.760
to a cubic equation.
00:43:16.760 --> 00:43:19.040
There are some limited
range over which
00:43:19.040 --> 00:43:23.240
there is an analytic
solution, but mostly, you
00:43:23.240 --> 00:43:26.960
don't use a cubic formula
to solve the cubic equation.
00:43:26.960 --> 00:43:30.930
You do some kind of iteration.
00:43:30.930 --> 00:43:36.180
So for the number of
levels greater than two,
00:43:36.180 --> 00:43:38.520
we know we're going to
have a problem because just
00:43:38.520 --> 00:43:42.330
approaching it by transformation
theory or linear algebra as
00:43:42.330 --> 00:43:48.130
opposed to the
Schrodinger picture--
00:43:48.130 --> 00:43:51.850
if you can't get a solution
in one picture which
00:43:51.850 --> 00:43:56.290
requires solving an
algebraic equation,
00:43:56.290 --> 00:43:59.410
you're not going to
get it by playing
00:43:59.410 --> 00:44:01.210
with these unitary matrices.
00:44:07.290 --> 00:44:14.430
So we're going to be
approaching a problem where
00:44:14.430 --> 00:44:20.550
we have to find the
eigenvalues and the elements
00:44:20.550 --> 00:44:23.700
of the transformation
matrix in some sort
00:44:23.700 --> 00:44:25.170
of computer-based way.
00:44:25.170 --> 00:44:26.700
We're not going to do it.
00:44:26.700 --> 00:44:29.850
It would be nuts, even
for a three by three.
00:44:29.850 --> 00:44:34.110
Although, I will give
you an exam problem which
00:44:34.110 --> 00:44:36.435
will be a three by
three, and you're
00:44:36.435 --> 00:44:38.310
going to use perturbation
theory to solve it.
00:44:41.030 --> 00:44:42.920
I haven't told you about
perturbation theory.
00:44:42.920 --> 00:44:48.230
That's going to be next week,
but we're leading up to it.
00:44:48.230 --> 00:44:54.470
OK, but now the point is we
have the machinery in place.
00:44:54.470 --> 00:45:08.300
We have exactly solved problems,
and we have the key parameters
00:45:08.300 --> 00:45:10.340
for exactly solved problems.
00:45:10.340 --> 00:45:11.360
So the structural--
00:45:18.530 --> 00:45:21.230
So for the harmonic
oscillator, we
00:45:21.230 --> 00:45:26.750
have the force constant
and the reduced mass.
00:45:26.750 --> 00:45:36.590
For the particle in the box, we
have the bottom of the box, v0,
00:45:36.590 --> 00:45:40.440
and we have the
width of the box.
00:45:40.440 --> 00:45:45.030
For the rigid rotor,
we're going to have
00:45:45.030 --> 00:45:52.780
the reduced mass and we're
going to have the internuclear
00:45:52.780 --> 00:45:53.280
distance.
00:45:55.810 --> 00:46:00.510
There's things that
determine all of the energy
00:46:00.510 --> 00:46:02.790
levels for exact
solved problems,
00:46:02.790 --> 00:46:04.740
and they are basically
the things that
00:46:04.740 --> 00:46:08.040
appear in the
Hamiltonian, and we call
00:46:08.040 --> 00:46:11.500
them structural parameters.
00:46:11.500 --> 00:46:12.820
And we have energy levels.
00:46:18.410 --> 00:46:22.020
And often, these are some
function of a quantum number.
00:46:25.170 --> 00:46:27.870
This is what we can observe.
00:46:27.870 --> 00:46:32.600
We observe the energy
levels, and we represent them
00:46:32.600 --> 00:46:36.530
by some formula,
and the coefficients
00:46:36.530 --> 00:46:42.307
of the quantum numbers
relate to these things
00:46:42.307 --> 00:46:43.140
that we really what.
00:46:46.390 --> 00:46:49.910
So when you're not dealing with
an exactly solved problem--
00:46:49.910 --> 00:46:52.030
like instead of having
a harmonic oscillator,
00:46:52.030 --> 00:46:55.030
you have a harmonic oscillator
with something at the bottom,
00:46:55.030 --> 00:46:58.570
or you have a particle in
a box with a slant bottom,
00:46:58.570 --> 00:47:02.200
or you have a rigid rotor
where it's not rigid,
00:47:02.200 --> 00:47:06.020
you have additional
terms in the Hamiltonian,
00:47:06.020 --> 00:47:08.540
and they are going to-- we
are going to use perturbation
00:47:08.540 --> 00:47:12.710
theory to relate the
numerical values of the things
00:47:12.710 --> 00:47:17.560
we want to know to the
things we can observe,
00:47:17.560 --> 00:47:23.860
and perturbation theory gives
us the formulas of the quantum
00:47:23.860 --> 00:47:30.220
numbers, and tells us the
explicit relationships
00:47:30.220 --> 00:47:35.380
of the coefficients of each
term in the quantum number
00:47:35.380 --> 00:47:38.640
expression to the
things we want.
00:47:38.640 --> 00:47:41.880
This is how we learn everything.
00:47:41.880 --> 00:47:46.500
When we do spectroscopy, we
measure these energy levels,
00:47:46.500 --> 00:47:52.290
and these energy levels
encode all of the structure
00:47:52.290 --> 00:47:53.340
and all of the dynamics.
00:47:56.950 --> 00:47:58.840
It's really neat.
00:47:58.840 --> 00:48:01.630
OK, now, the last thing I
want to-- do I have time?
00:48:01.630 --> 00:48:02.263
Yeah, maybe.
00:48:15.200 --> 00:48:19.150
Remember when we do time
dependent quantum mechanics
00:48:19.150 --> 00:48:21.627
with a time independent
Hamiltonian.
00:48:26.870 --> 00:48:32.090
We want to have psi of x and t.
00:48:35.450 --> 00:48:41.150
And usually, we're given
psi of x, t equals zero.
00:48:41.150 --> 00:48:46.040
We're given the initial
state, and that initial state,
00:48:46.040 --> 00:48:47.540
if this is an
interesting problem,
00:48:47.540 --> 00:48:51.050
is not an eigenstate
of the Hamiltonian.
00:48:51.050 --> 00:48:53.880
It's a linear combination
of eigenstates
00:48:53.880 --> 00:48:57.860
of that Hamiltonian,
and the kinds of flux
00:48:57.860 --> 00:49:01.070
we almost always use
to test our insight,
00:49:01.070 --> 00:49:04.010
or actually, because they're
feasible experimentally,
00:49:04.010 --> 00:49:09.440
is the initial state is one of
the eigenstates of an exactly
00:49:09.440 --> 00:49:11.500
solved problem.
00:49:11.500 --> 00:49:18.110
It's some special
combination of easy stuff.
00:49:18.110 --> 00:49:22.570
And we need to know
how the coefficient--
00:49:22.570 --> 00:49:29.892
this thing-- how that
is expressed as j
00:49:29.892 --> 00:49:40.280
equals one to n of c j psi j.
00:49:40.280 --> 00:49:46.790
Because if we can express
the t equals zero pluck
00:49:46.790 --> 00:49:49.220
as a linear combination
of the eigenstates,
00:49:49.220 --> 00:49:58.160
then it's just a matter
of mindless manipulation
00:49:58.160 --> 00:50:04.760
because we have c j, e to the
minus i e j t over h bar times
00:50:04.760 --> 00:50:05.780
j.
00:50:05.780 --> 00:50:08.190
Bang, it's done.
00:50:08.190 --> 00:50:13.280
So what we want to know is how
to go from a not-eigenstate
00:50:13.280 --> 00:50:15.530
to an eigenstate.
00:50:15.530 --> 00:50:17.540
And lo and behold,
that's just the inverse
00:50:17.540 --> 00:50:20.960
of the transformation.
00:50:20.960 --> 00:50:25.050
So what we want to know is
OK, since I don't have time
00:50:25.050 --> 00:50:28.210
to spell it out
for you exactly, we
00:50:28.210 --> 00:50:32.590
have t dagger, which
relates the zero order
00:50:32.590 --> 00:50:36.100
states to the eigenstates.
00:50:36.100 --> 00:50:39.940
We want to go in the
opposite direction.
00:50:39.940 --> 00:50:41.810
We want the inverse
transformation.
00:50:41.810 --> 00:50:47.890
So we want t, or we want to
take instead of the columns of t
00:50:47.890 --> 00:50:51.680
dagger, we take the rows.
00:50:51.680 --> 00:50:57.390
And so if you have a
machine or a brain--
00:50:57.390 --> 00:50:59.000
and I'm not doubting this!--
00:50:59.000 --> 00:51:03.770
that enables you to write down
all of the elements in the t
00:51:03.770 --> 00:51:08.990
dagger matrix, you are armed
to go both from zero order
00:51:08.990 --> 00:51:12.620
states to eigenstates,
and from plucks
00:51:12.620 --> 00:51:15.780
to time evolving wave packets.
00:51:15.780 --> 00:51:18.030
It's really
beautiful and simple.
00:51:18.030 --> 00:51:21.270
And normally, when
it's presented,
00:51:21.270 --> 00:51:24.390
these are presented as
separate project problems,
00:51:24.390 --> 00:51:27.960
and the whole point is you've
got a unified picture that
00:51:27.960 --> 00:51:32.310
enables you to go get whatever
you need in a simple way,
00:51:32.310 --> 00:51:36.240
as long as a computer
is able to diagonalize
00:51:36.240 --> 00:51:37.950
your critical matrices.
00:51:42.477 --> 00:51:47.800
Well, I don't have time to
talk about this in any detail,
00:51:47.800 --> 00:51:55.380
but if we look at the
eigenfunctions or eigenvectors
00:51:55.380 --> 00:52:00.550
for the two-level problem, and
we do power series expansions
00:52:00.550 --> 00:52:05.900
in theta, where
theta is V over d--
00:52:05.900 --> 00:52:09.620
theta is also called
the mixing angle--
00:52:09.620 --> 00:52:13.100
we discover that we
have some formulas which
00:52:13.100 --> 00:52:17.570
says that the energy levels--
00:52:17.570 --> 00:52:26.630
let's say the j-th energy
level is equal to E bar plus--
00:52:26.630 --> 00:52:30.490
now I'm doing this and I have
to somehow grab something
00:52:30.490 --> 00:52:31.080
from in here.
00:52:36.980 --> 00:52:45.360
We have a sum of k
not equal to j of V--
00:52:45.360 --> 00:52:57.890
of H.
00:52:57.890 --> 00:52:59.840
This is the formula
for the correction
00:52:59.840 --> 00:53:05.960
of the energy by a second
order perturbation theory,
00:53:05.960 --> 00:53:09.650
and we can also write the
formula for the corrected wave
00:53:09.650 --> 00:53:12.170
function.
00:53:12.170 --> 00:53:15.590
By doing a power
series expansion
00:53:15.590 --> 00:53:21.500
in terms of powers
of V/d or theta,
00:53:21.500 --> 00:53:26.540
we find what the structure
has to be for the solutions
00:53:26.540 --> 00:53:29.150
to the general problem
when n is not two.
00:53:31.860 --> 00:53:36.380
I'll develop the formal theory
for non-degenerate perturbation
00:53:36.380 --> 00:53:40.880
theory in Monday's lecture,
and that's really empowering.
00:53:40.880 --> 00:53:45.650
It's really ugly, but
it gives you the answers
00:53:45.650 --> 00:53:47.960
to essentially every
problem you will ever
00:53:47.960 --> 00:53:51.300
face in quantum mechanics.
00:53:51.300 --> 00:53:54.160
OK, have a nice weekend.