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PROFESSOR: All right.
00:00:22.890 --> 00:00:27.800
So, this homework
that is due on Friday
00:00:27.800 --> 00:00:31.000
contains some questions on
the harmonic oscillator.
00:00:31.000 --> 00:00:34.630
And the harmonic oscillator
is awfully important.
00:00:34.630 --> 00:00:36.240
I gave you notes on that.
00:00:36.240 --> 00:00:39.930
And I want to use about
half of the lecture,
00:00:39.930 --> 00:00:42.440
perhaps a little
less, to go over
00:00:42.440 --> 00:00:45.470
some of those points
in the notes concerning
00:00:45.470 --> 00:00:47.490
the harmonic oscillator.
00:00:47.490 --> 00:00:52.150
After that, we're going
to begin, essentially,
00:00:52.150 --> 00:00:55.140
our study of dynamics.
00:00:55.140 --> 00:00:58.560
And we will give the revision,
today, of the Schrodinger
00:00:58.560 --> 00:01:00.990
equation.
00:01:00.990 --> 00:01:06.410
It's the way Dirac, in his
textbook on quantum mechanics,
00:01:06.410 --> 00:01:08.510
presents the
Schrodinger equation.
00:01:08.510 --> 00:01:12.000
I think it's actually,
extremely insightful.
00:01:12.000 --> 00:01:13.890
It's probably not
the way you should
00:01:13.890 --> 00:01:17.180
see it the first
time in your life.
00:01:17.180 --> 00:01:19.290
But it's a good way
to think about it.
00:01:19.290 --> 00:01:23.280
And it will give
you a nice feeling
00:01:23.280 --> 00:01:25.540
that this Schrodinger
equation is something
00:01:25.540 --> 00:01:28.190
so fundamental and
so basic that it
00:01:28.190 --> 00:01:32.563
would be very hard to
change or do anything to it
00:01:32.563 --> 00:01:34.600
and tinker with it.
00:01:34.600 --> 00:01:40.280
It's a rather complete theory
and quite beautiful [? idea. ?]
00:01:40.280 --> 00:01:42.370
So we begin with the
harmonic oscillator.
00:01:53.240 --> 00:01:56.150
And this will be a bit quick.
00:01:56.150 --> 00:01:59.190
I won't go over every detail.
00:01:59.190 --> 00:02:00.190
You have the notes.
00:02:00.190 --> 00:02:03.640
I think that's pretty
much all you need to know.
00:02:03.640 --> 00:02:07.880
So we'll leave it at that.
00:02:07.880 --> 00:02:11.740
So the harmonic oscillator
is a quantum system.
00:02:11.740 --> 00:02:14.740
And as quantum
systems go, they're
00:02:14.740 --> 00:02:17.480
inspired by classical systems.
00:02:17.480 --> 00:02:21.030
And the classical system
is very famous here.
00:02:21.030 --> 00:02:24.460
It's the system in
which, for example, you
00:02:24.460 --> 00:02:26.130
have a mass and a spring.
00:02:26.130 --> 00:02:29.950
And it does an oscillation for
which the energy is written
00:02:29.950 --> 00:02:37.730
as p squared over 2m plus 1/2
m, omega squared, x squared.
00:02:37.730 --> 00:02:42.220
And m omega squared is
sometimes called k squared,
00:02:42.220 --> 00:02:43.200
the spring constant.
00:02:46.350 --> 00:02:50.010
And you are supposed to do
quantum mechanics with this.
00:02:50.010 --> 00:02:55.460
So nobody can tell you this is
what the harmonic oscillators
00:02:55.460 --> 00:02:56.580
in quantum mechanics.
00:02:56.580 --> 00:02:58.240
You have to define it.
00:02:58.240 --> 00:03:00.920
But since there's
only one logical way
00:03:00.920 --> 00:03:03.790
to define the quantum
system, everybody
00:03:03.790 --> 00:03:08.130
agrees on what the harmonic
oscillator quantum system is.
00:03:08.130 --> 00:03:10.770
Basically, you use
the inspiration
00:03:10.770 --> 00:03:13.430
of the classical system
and declare, well,
00:03:13.430 --> 00:03:16.320
energy will be the
Hamiltonian operator.
00:03:16.320 --> 00:03:20.350
p will be the momentum operator.
00:03:20.350 --> 00:03:26.410
And x will be the
position operator.
00:03:26.410 --> 00:03:29.270
And given that
these are operators,
00:03:29.270 --> 00:03:34.980
will have a basic commutation
relation between x and p
00:03:34.980 --> 00:03:37.190
being equal to i h-bar.
00:03:37.190 --> 00:03:38.690
And that's it.
00:03:38.690 --> 00:03:40.790
This is your quantum system.
00:03:48.840 --> 00:03:52.180
Hamiltonian is--
the set of operators
00:03:52.180 --> 00:03:56.720
that are relevant for this are
the x the p, and the energy
00:03:56.720 --> 00:04:01.450
operator that will
control the dynamics.
00:04:01.450 --> 00:04:06.990
You know also you should specify
a vector space, the vector
00:04:06.990 --> 00:04:08.490
space where this acts.
00:04:08.490 --> 00:04:13.420
And this will be complex
functions on the real line.
00:04:13.420 --> 00:04:21.430
So this will act
in wave functions
00:04:21.430 --> 00:04:26.715
that define the vector space,
sometimes called Hilbert space.
00:04:31.136 --> 00:04:37.790
It will be the set of integrable
functions on the real line,
00:04:37.790 --> 00:04:46.675
so complex functions
on the real line.
00:04:50.640 --> 00:04:54.140
These are your wave functions,
a set of states of the theory.
00:04:54.140 --> 00:04:57.640
All these complex functions
on the real line work.
00:04:57.640 --> 00:05:00.080
I won't try to be more precise.
00:05:00.080 --> 00:05:02.300
You could say they're
square integrable.
00:05:02.300 --> 00:05:04.233
That for sure is necessary.
00:05:08.070 --> 00:05:10.210
And we'll leave it at that.
00:05:10.210 --> 00:05:12.720
Now you have to
solve this problem.
00:05:12.720 --> 00:05:17.730
And in 804, we discussed this by
using the differential equation
00:05:17.730 --> 00:05:22.070
and then through the creation
annihilation operators.
00:05:22.070 --> 00:05:24.440
And we're going
do it, today, just
00:05:24.440 --> 00:05:26.880
through creation and
annihilation operators.
00:05:26.880 --> 00:05:30.940
But we want to
emphasize something
00:05:30.940 --> 00:05:35.460
about this Hamiltonian and
something very general, which
00:05:35.460 --> 00:05:38.500
is that you can right
the Hamiltonian as say
00:05:38.500 --> 00:05:44.580
1/2m, omega squared, x squared.
00:05:44.580 --> 00:05:50.530
And then you have plus
p squared, m squared,
00:05:50.530 --> 00:05:51.525
omega squared.
00:05:55.060 --> 00:05:58.760
And a great solution
to the problem
00:05:58.760 --> 00:06:03.910
of solving the Hamiltonian--
and it's the best you could ever
00:06:03.910 --> 00:06:08.040
hope-- is what is
called the factorization
00:06:08.040 --> 00:06:11.720
of the Hamiltonian, in
which you would manage
00:06:11.720 --> 00:06:17.001
to write this Hamiltonian as
some operator times the dagger
00:06:17.001 --> 00:06:17.500
operator.
00:06:21.410 --> 00:06:24.190
So this is the ideal situation.
00:06:24.190 --> 00:06:27.240
It's just wonderful,
as you will see,
00:06:27.240 --> 00:06:28.975
if you can manage to do that.
00:06:33.130 --> 00:06:36.460
If you could manage to
do this factorization,
00:06:36.460 --> 00:06:39.400
you would know immediately
what is the ground state
00:06:39.400 --> 00:06:42.930
energy, how low can
it go, something
00:06:42.930 --> 00:06:44.310
about the Hamiltonian.
00:06:44.310 --> 00:06:47.060
You're way on your way
of solving the problem.
00:06:47.060 --> 00:06:48.920
If you could just factorize it.
00:06:48.920 --> 00:06:49.969
Yes?
00:06:49.969 --> 00:06:52.548
AUDIENCE: [INAUDIBLE] if
you could just factorize it
00:06:52.548 --> 00:06:57.540
in terms of v and v
instead of v dagger and v?
00:06:57.540 --> 00:07:01.561
PROFESSOR: You want to factorize
in which way instead of that?
00:07:01.561 --> 00:07:03.686
AUDIENCE: Would it be
helpful, if it were possible,
00:07:03.686 --> 00:07:07.530
to factor it in terms of v
times v instead of v dagger?
00:07:07.530 --> 00:07:09.730
PROFESSOR: No, no, I
want, really, v dagger.
00:07:09.730 --> 00:07:14.010
I don't want v v. That
that's not so good.
00:07:14.010 --> 00:07:17.490
I want that this factorization
has a v dagger there.
00:07:17.490 --> 00:07:21.350
It will make things
much, much better.
00:07:21.350 --> 00:07:23.860
So how can you achieve that?
00:07:23.860 --> 00:07:26.750
Well, it almost looks possible.
00:07:26.750 --> 00:07:31.860
If you have something like this,
like a squared plus b squared,
00:07:31.860 --> 00:07:36.640
you write it as a minus
ib times a plus ib.
00:07:41.150 --> 00:07:43.180
And that works out.
00:07:43.180 --> 00:07:49.390
So you try here, 1/2
m, omega squared,
00:07:49.390 --> 00:07:59.053
x minus ip over m omega,
x plus ip over m omega.
00:08:02.320 --> 00:08:07.870
And beware that's
not quite right.
00:08:07.870 --> 00:08:13.080
Because here, you have
cross terms that cancel.
00:08:13.080 --> 00:08:17.540
You have aib b and minus iba.
00:08:17.540 --> 00:08:20.380
And they would only
cancel if a and b commute.
00:08:20.380 --> 00:08:22.220
And here they don't commute.
00:08:22.220 --> 00:08:24.940
So it's almost perfect.
00:08:24.940 --> 00:08:29.710
But if you expand this out,
you get the x squared for sure.
00:08:29.710 --> 00:08:30.750
You get this term.
00:08:30.750 --> 00:08:36.006
But then you get an extra term
coming from the cross terms.
00:08:36.006 --> 00:08:37.750
And please calculate it.
00:08:37.750 --> 00:08:40.400
Happily, it's just a number,
because the commutator
00:08:40.400 --> 00:08:42.919
of x and b is just a number.
00:08:42.919 --> 00:08:48.790
So the answer for
this thing is that you
00:08:48.790 --> 00:08:52.980
get, here, x
squared plus this is
00:08:52.980 --> 00:09:00.425
equal to this, plus h-bar over m
omega, times the unit operator.
00:09:05.060 --> 00:09:15.760
So here is what you
could call v dagger.
00:09:15.760 --> 00:09:21.080
And this is what we'd call v.
00:09:21.080 --> 00:09:24.530
So what is your Hamiltonian?
00:09:24.530 --> 00:09:31.400
Your Hamiltonian has become
1/2 m, omega squared, v dagger
00:09:31.400 --> 00:09:37.460
v, plus, if you multiply out,
H omega times the identity.
00:09:41.300 --> 00:09:44.250
So we basically succeeded.
00:09:44.250 --> 00:09:48.350
And it's as good as what we
could hope or want, actually.
00:09:52.130 --> 00:09:56.670
I multiply this
out, so h-bar omega
00:09:56.670 --> 00:09:59.160
was the only thing
that was left.
00:09:59.160 --> 00:10:00.610
And there's your Hamiltonian.
00:10:00.610 --> 00:10:07.490
Now, in order to see
what this tells you,
00:10:07.490 --> 00:10:10.670
just sandwich it
between any two states.
00:10:13.220 --> 00:10:21.700
Well, this is 1/2 m, omega
squared, psi, v dagger, v,
00:10:21.700 --> 00:10:26.920
psi, plus 1/2 half h, omega.
00:10:26.920 --> 00:10:29.750
And assume it's a
normalized state,
00:10:29.750 --> 00:10:32.560
so it just gives you that.
00:10:32.560 --> 00:10:41.100
So this thing is the
norm of the state, v psi.
00:10:41.100 --> 00:10:45.080
You'd think it's
dagger and it's this.
00:10:45.080 --> 00:10:48.370
So this is the norm
squared of v psi.
00:10:52.920 --> 00:10:55.340
And therefore that's positive.
00:10:55.340 --> 00:11:01.540
So H, between any
normalized state,
00:11:01.540 --> 00:11:05.920
is greater than or equal
to 1/2 h-bar omega.
00:11:08.810 --> 00:11:16.760
In particular, if psi
is an energy eigenstate,
00:11:16.760 --> 00:11:21.420
so that H psi is equal to E psi.
00:11:24.580 --> 00:11:28.440
If psi is an energy
eigenstate, then you have this.
00:11:28.440 --> 00:11:33.060
And back here, you
get that the energy
00:11:33.060 --> 00:11:38.070
must be greater than or
equal to 1/2 h omega,
00:11:38.070 --> 00:11:42.980
because H and psi gives
you an E. The E goes out.
00:11:42.980 --> 00:11:46.530
And you're left with
psi, psi, which is 1.
00:11:46.530 --> 00:11:48.685
So you already know
that the energy
00:11:48.685 --> 00:11:53.140
is at least greater than
or equal to 1/2 h omega.
00:11:53.140 --> 00:11:56.650
So this factorization
has been very powerful.
00:11:56.650 --> 00:11:58.830
It has taught you
something extremely
00:11:58.830 --> 00:12:02.790
nontrivial about the
spectrum of the Hamiltonian.
00:12:02.790 --> 00:12:06.210
All energy eigenstates
must be greater than
00:12:06.210 --> 00:12:09.940
or equal to 1/2 h omega.
00:12:09.940 --> 00:12:11.860
In fact, this is
so good that people
00:12:11.860 --> 00:12:14.520
try to do this for
almost any problem.
00:12:14.520 --> 00:12:17.585
Any Hamiltonian, probably
the first thing you can try
00:12:17.585 --> 00:12:23.150
is to establish a
factorization of this kind.
00:12:23.150 --> 00:12:28.140
For the hydrogen atom, that
factorization is also possible.
00:12:28.140 --> 00:12:32.690
There will be some
homework sometime later on.
00:12:32.690 --> 00:12:38.420
It's less well known and
doesn't lead to useful creation
00:12:38.420 --> 00:12:39.940
and annihilation operators.
00:12:39.940 --> 00:12:42.580
But you can get the ground
state energy in a proof
00:12:42.580 --> 00:12:46.730
that you kind of go below
that energy very quickly.
00:12:46.730 --> 00:12:52.780
So a few things are done
now to clean up this system.
00:12:52.780 --> 00:12:59.380
And basically, here I have the
definition of v and v dagger.
00:12:59.380 --> 00:13:09.470
Then you define a to be square
root of m omega over 2 h-bar,
00:13:09.470 --> 00:13:20.420
v. And a dagger must be m
omega over 2 h-bar v dagger.
00:13:20.420 --> 00:13:25.540
And I have not written for
you the commutator of v and v
00:13:25.540 --> 00:13:26.040
dagger.
00:13:26.040 --> 00:13:30.880
We might as well do the
commutator of a and a dagger.
00:13:30.880 --> 00:13:34.600
And that commutator turns
out to be extremely simple.
00:13:34.600 --> 00:13:39.135
a with a dagger is
just equal to 1.
00:13:41.890 --> 00:13:45.360
Now things that are useful,
relations that are useful
00:13:45.360 --> 00:13:49.300
is-- just write
what v is in here
00:13:49.300 --> 00:13:52.770
so that you have a
formula for a and a dagger
00:13:52.770 --> 00:13:54.820
in terms of x and p.
00:13:54.820 --> 00:13:59.360
So I will not bother writing it.
00:13:59.360 --> 00:14:01.210
But it's here already.
00:14:01.210 --> 00:14:03.300
Maybe I'll do the first one.
00:14:03.300 --> 00:14:08.080
m omega over 2 h-bar.
00:14:08.080 --> 00:14:16.330
v is here would be x,
plus ip over m omega.
00:14:16.330 --> 00:14:19.460
And you can write
the other one there.
00:14:19.460 --> 00:14:22.730
So you have an expression
for a and a dagger
00:14:22.730 --> 00:14:25.040
in terms of x and p.
00:14:25.040 --> 00:14:27.390
And that can be
inverted as well.
00:14:27.390 --> 00:14:29.620
And it's pretty useful.
00:14:29.620 --> 00:14:32.200
And it's an example
of formulas that you
00:14:32.200 --> 00:14:33.860
don't need to know by heart.
00:14:33.860 --> 00:14:35.900
And they would be in
any formula sheet.
00:14:41.130 --> 00:14:48.670
And the units and
all those constants
00:14:48.670 --> 00:14:50.405
make it hard to remember.
00:14:54.460 --> 00:14:55.570
But here they are.
00:14:58.330 --> 00:15:08.250
So you should know that x is a
plus a dagger up to a constant.
00:15:08.250 --> 00:15:11.750
And p is a dagger minus a.
00:15:11.750 --> 00:15:15.610
Now p is Hermitian, that's
why there is an i here.
00:15:15.610 --> 00:15:18.115
So that this, this
anti-Hermitian,
00:15:18.115 --> 00:15:20.990
the i becomes a
Hermitian operator.
00:15:20.990 --> 00:15:28.310
x is manifestly Hermitian,
because a plus a dagger is.
00:15:28.310 --> 00:15:31.980
Finally, you want to
write the Hamiltonian.
00:15:31.980 --> 00:15:40.500
And the Hamiltonian is given
by the following formula.
00:15:40.500 --> 00:15:44.380
You know you just have to put
the v and v dagger, what they
00:15:44.380 --> 00:15:47.970
are in terms of the creation,
annihilation operators.
00:15:47.970 --> 00:15:51.630
So v dagger, you
substitute a dagger.
00:15:51.630 --> 00:15:55.290
v, you go back here
and just calculate it.
00:15:58.150 --> 00:16:01.740
And these calculations
really should be done.
00:16:01.740 --> 00:16:07.080
It's something that is
good practice and make sure
00:16:07.080 --> 00:16:11.520
you don't make silly mistakes.
00:16:11.520 --> 00:16:17.100
So this operator is so important
it has been given a name.
00:16:17.100 --> 00:16:28.810
It's called the
number operator, N.
00:16:28.810 --> 00:16:37.010
And its eigenvalues are numbers,
0, 1, 2, 3, all these things.
00:16:37.010 --> 00:16:39.950
And the good thing about
it is that, once you
00:16:39.950 --> 00:16:46.430
are with a's and a
daggers, all this m omega,
00:16:46.430 --> 00:16:48.480
h-bar are all gone.
00:16:48.480 --> 00:16:51.910
This is all that
is happening here.
00:16:51.910 --> 00:16:55.510
The basic energy is h-bar omega.
00:16:55.510 --> 00:17:00.410
Ground state energies, what
we'll see is 1/2 h-bar omega.
00:17:00.410 --> 00:17:02.300
And this is the number operator.
00:17:02.300 --> 00:17:08.430
So this is written as h-bar
omega, number operator--
00:17:08.430 --> 00:17:15.300
probably with a hat-- like that.
00:17:15.300 --> 00:17:26.520
So when you're talking
about eigenvalues,
00:17:26.520 --> 00:17:30.320
as we will talk soon, or
states for which these thing's
00:17:30.320 --> 00:17:33.430
are numbers, saying that
you have a state that
00:17:33.430 --> 00:17:35.550
is an eigenstate
of the Hamiltonian
00:17:35.550 --> 00:17:38.730
is exactly the same
thing as saying
00:17:38.730 --> 00:17:42.360
that it's an eigenstate
of the number operator.
00:17:42.360 --> 00:17:45.790
Because that's the only thing
that is an operator here.
00:17:45.790 --> 00:17:47.440
There's this plus this number.
00:17:47.440 --> 00:17:49.750
So this number
causes no problem.
00:17:49.750 --> 00:17:54.750
Any state multiplied by a number
is proportional to itself.
00:17:54.750 --> 00:17:58.160
But it's not true that every
state multiplied by a dagger a
00:17:58.160 --> 00:18:00.320
is proportional to itself.
00:18:00.320 --> 00:18:03.580
So being an
eigenstate of N means
00:18:03.580 --> 00:18:06.660
that acting on a state,
N, gives you a number.
00:18:06.660 --> 00:18:09.430
But then H is just
N times the number.
00:18:09.430 --> 00:18:12.450
So H is also an eigenstate.
00:18:12.450 --> 00:18:15.470
So eigenstates of N
or eigenstates of H
00:18:15.470 --> 00:18:19.970
are exactly the same thing.
00:18:19.970 --> 00:18:23.020
Now there's a couple
more properties
00:18:23.020 --> 00:18:29.060
that maybe need to be mentioned.
00:18:29.060 --> 00:18:32.010
So I wanted to talk in
terms of eigenvalues.
00:18:32.010 --> 00:18:35.330
I would just simply write
the energy eigenvalue
00:18:35.330 --> 00:18:41.130
is therefore equal h-bar
omega, the number eigenvalue--
00:18:41.130 --> 00:18:46.420
so the operator is
with a hat-- plus 1/2.
00:18:46.420 --> 00:18:51.130
So in terms of
eigenvalues, you have that.
00:18:51.130 --> 00:18:57.140
From here, the energy is
greater than 1/2 h omega.
00:18:57.140 --> 00:19:07.130
So the number must be greater
or equal than 0 on any state.
00:19:07.130 --> 00:19:12.080
And that's also clear from the
definition of this operator.
00:19:12.080 --> 00:19:16.770
On any state, the expectation
value of this operator
00:19:16.770 --> 00:19:19.200
has to be positive.
00:19:19.200 --> 00:19:23.070
And therefore, you have this.
00:19:23.070 --> 00:19:29.520
So two more properties
that are crucial here
00:19:29.520 --> 00:19:34.320
are that the
Hamiltonian commuted
00:19:34.320 --> 00:19:40.530
with a is equal
to minus h omega a
00:19:40.530 --> 00:19:46.100
and that the Hamiltonian
committed with a dagger
00:19:46.100 --> 00:19:52.350
is plus h omega a dagger.
00:19:52.350 --> 00:19:59.160
Now there is a
reasonably precise way
00:19:59.160 --> 00:20:01.250
of going through
the whole spectrum
00:20:01.250 --> 00:20:05.700
of the harmonic oscillator
without solving differential
00:20:05.700 --> 00:20:10.560
equations, almost to
any degree, and trying
00:20:10.560 --> 00:20:14.680
to be just very
logical about it.
00:20:14.680 --> 00:20:18.810
It's possible to deduce the
properties of the spectrum.
00:20:18.810 --> 00:20:24.040
So I will do that right now.
00:20:24.040 --> 00:20:26.630
And we begin with the
following statement.
00:20:26.630 --> 00:20:33.850
We assume there is
some energy eigenstate.
00:20:33.850 --> 00:20:47.046
So assume there is a state E
such that the Hamiltonian--
00:20:47.046 --> 00:20:48.670
for some reason in
the notes apparently
00:20:48.670 --> 00:20:53.190
I put hats on the Hamiltonian,
so I'll start putting hats
00:20:53.190 --> 00:21:03.890
here-- so that the states
are labeled by the energy.
00:21:03.890 --> 00:21:08.280
And this begins a tiny bit of
confusion about the notation.
00:21:08.280 --> 00:21:11.500
Many times you want to label
the states by the energy.
00:21:11.500 --> 00:21:17.340
We'll end up labeling them
with the number operator.
00:21:17.340 --> 00:21:19.300
And then, I said,
it will turn out,
00:21:19.300 --> 00:21:23.640
when the number operator is
0, we'll put a 0 in here.
00:21:23.640 --> 00:21:25.385
And that doesn't mean 0 energy.
00:21:25.385 --> 00:21:31.040
It means energy equal
1/2 h-bar omega.
00:21:31.040 --> 00:21:37.470
So if you assume there
is an energy eigenstate,
00:21:37.470 --> 00:21:40.190
that's the first step
in the construction.
00:21:40.190 --> 00:21:42.400
You assume there is one.
00:21:42.400 --> 00:21:44.920
And what does that mean?
00:21:44.920 --> 00:21:46.980
It means that this
is a good state.
00:21:46.980 --> 00:21:49.680
So it may be normalized.
00:21:49.680 --> 00:21:51.760
It may not be normalized.
00:21:51.760 --> 00:21:55.105
In any case, it
should be positive.
00:21:57.680 --> 00:22:01.860
I put first the equal, but
I shouldn't put the equal.
00:22:01.860 --> 00:22:05.840
Because we know in a
complex vector space,
00:22:05.840 --> 00:22:08.630
if a state has 0 norm, it's 0.
00:22:08.630 --> 00:22:10.930
And I want to say
that there's really
00:22:10.930 --> 00:22:16.470
some state that is non-0,
that has this energy.
00:22:16.470 --> 00:22:19.650
If the state would be 0, this
would become a triviality.
00:22:19.650 --> 00:22:21.710
So this state is good.
00:22:21.710 --> 00:22:27.410
It's all good.
00:22:27.410 --> 00:22:34.430
Now with this state, you can
define, now, two other states,
00:22:34.430 --> 00:22:38.620
acting with the creation,
annihilation operators.
00:22:38.620 --> 00:22:40.700
I didn't mention that name.
00:22:40.700 --> 00:22:44.910
But a dagger is going to be
called the creation operator.
00:22:44.910 --> 00:22:48.770
And this is the destruction
or annihilation operator.
00:22:48.770 --> 00:23:00.190
And we built two states, E
plus is a dagger acting on E.
00:23:00.190 --> 00:23:07.730
And E minus is a
acting on E. Now
00:23:07.730 --> 00:23:10.310
you could fairly
ask a this moment
00:23:10.310 --> 00:23:14.240
and say, well, how do you
know these states are good?
00:23:14.240 --> 00:23:16.010
How do you know they even exist?
00:23:16.010 --> 00:23:18.230
How do you know that
if you act with this,
00:23:18.230 --> 00:23:20.810
don't you get an
inconsistent state?
00:23:20.810 --> 00:23:23.540
How do you know
this makes sense?
00:23:23.540 --> 00:23:25.560
And these are perfectly
good questions.
00:23:25.560 --> 00:23:29.590
And in fact, this is exactly
what you have to understand.
00:23:29.590 --> 00:23:34.040
This procedure can
give some funny things.
00:23:34.040 --> 00:23:37.700
And we want to
discuss algebraically
00:23:37.700 --> 00:23:42.670
why some things are safe and
why some things may not quite
00:23:42.670 --> 00:23:44.410
be safe.
00:23:44.410 --> 00:23:49.090
And adding an a dagger,
we will see it's safe.
00:23:49.090 --> 00:23:54.920
While adding a's to the
state could be fairly unsafe.
00:23:54.920 --> 00:23:59.040
So what can be bad
about the state?
00:23:59.040 --> 00:24:04.750
It could be a 0 state, or it
could be an inconsistent state.
00:24:04.750 --> 00:24:07.650
And what this an
inconsistent state?
00:24:07.650 --> 00:24:13.030
Well, all our states are
represented by wave functions.
00:24:13.030 --> 00:24:16.070
And they should be normalizable.
00:24:16.070 --> 00:24:20.910
And therefore they have
norms that are positive,
00:24:20.910 --> 00:24:22.990
norms squared that are positive.
00:24:22.990 --> 00:24:26.050
Well you may find,
here, that you
00:24:26.050 --> 00:24:29.280
have states that have
norms that are negative,
00:24:29.280 --> 00:24:30.940
norm squareds that are negative.
00:24:30.940 --> 00:24:34.580
So this thing that
should be positive,
00:24:34.580 --> 00:24:37.420
algebraically you may
show that actually you
00:24:37.420 --> 00:24:39.830
can get into trouble.
00:24:39.830 --> 00:24:42.730
And trouble, of course,
is very interesting.
00:24:42.730 --> 00:24:51.020
So I want to skip this
calculation and state something
00:24:51.020 --> 00:24:54.350
that you probably checked
in 804, several times,
00:24:54.350 --> 00:24:57.380
that this state has
more energy than E
00:24:57.380 --> 00:25:03.930
and, in fact, has as much
energy as E plus h-bar omega.
00:25:03.930 --> 00:25:07.090
Because a dagger, the
creation operator,
00:25:07.090 --> 00:25:10.050
adds energy, h-bar omega.
00:25:10.050 --> 00:25:13.395
And this subtracts
energy, h-bar omega.
00:25:18.010 --> 00:25:21.890
This state has an
energy, E plus,
00:25:21.890 --> 00:25:26.410
which is equal to
E plus h-bar omega.
00:25:26.410 --> 00:25:31.080
And E minus is equal
to E minus h-bar omega.
00:25:31.080 --> 00:25:33.650
Now how do you check that?
00:25:33.650 --> 00:25:37.980
You're supposed to act
with a Hamiltonian on this,
00:25:37.980 --> 00:25:42.000
use the commutation relation
that we wrote up there,
00:25:42.000 --> 00:25:44.720
and prove that those are
the energy eigenvalues.
00:25:47.930 --> 00:25:54.350
So at this moment, you
can do the following.
00:25:54.350 --> 00:25:59.460
So these states have energies,
they have number operators,
00:25:59.460 --> 00:26:01.170
they have number eigenvalues.
00:26:01.170 --> 00:26:08.070
So we can test, if
these states are good,
00:26:08.070 --> 00:26:10.630
by computing their norms.
00:26:10.630 --> 00:26:17.300
So let's compute the
norm, a dagger on E,
00:26:17.300 --> 00:26:21.050
a dagger on E for the first one.
00:26:21.050 --> 00:26:29.780
And we'll compute a E, a E.
We'll do this computation.
00:26:29.780 --> 00:26:33.640
We just want to
see what this is.
00:26:33.640 --> 00:26:37.630
Now remember how you do this.
00:26:37.630 --> 00:26:40.550
An operator acting
here goes with a dagger
00:26:40.550 --> 00:26:41.960
into the other side.
00:26:41.960 --> 00:26:54.550
So this is equal to
E a, a dagger, E.
00:26:54.550 --> 00:26:59.700
Now a, a dagger is
not quite perfect.
00:26:59.700 --> 00:27:03.040
It differs from the
one that we know
00:27:03.040 --> 00:27:07.350
is an eigenvalue for this state,
which is the number operator.
00:27:07.350 --> 00:27:12.460
So what is a, a
dagger in terms of N?
00:27:12.460 --> 00:27:15.180
Well, a, a dagger--
it's something
00:27:15.180 --> 00:27:18.280
you will use many,
many times-- is
00:27:18.280 --> 00:27:25.140
equal to a commutator with
a dagger plus a dagger a.
00:27:25.140 --> 00:27:32.870
So that's 1 plus
the number operator.
00:27:32.870 --> 00:27:39.340
So this thing is E
1 plus the number
00:27:39.340 --> 00:27:44.510
operator acting on the state E.
00:27:44.510 --> 00:27:49.380
Well, the 1 is clear what it is.
00:27:49.380 --> 00:27:51.510
And the number operate is clear.
00:27:51.510 --> 00:27:55.330
If this has some
energy E, well, I
00:27:55.330 --> 00:27:58.380
can now what is the eigenvalue
of the number operator
00:27:58.380 --> 00:28:03.650
because the energy on
the number eigenvalues
00:28:03.650 --> 00:28:05.520
are related that way.
00:28:05.520 --> 00:28:10.740
So I will simply call
it the number of E
00:28:10.740 --> 00:28:12.100
and leave it at that.
00:28:12.100 --> 00:28:12.820
Times EE.
00:28:19.230 --> 00:28:24.080
So in here, the
computation is easier
00:28:24.080 --> 00:28:29.480
because it's just E a dagger
a E. That's the number,
00:28:29.480 --> 00:28:33.150
so that's just NE times EE.
00:28:39.578 --> 00:28:44.380
OK, so these are
the key equations
00:28:44.380 --> 00:28:46.290
we're going to be
using to understand
00:28:46.290 --> 00:28:49.700
the spectrum quickly.
00:28:49.700 --> 00:28:58.910
And let me say a couple
of things about them.
00:28:58.910 --> 00:29:03.500
So I'll repeat what we
have there, a dagger
00:29:03.500 --> 00:29:15.160
E a dagger E is equal
to 1 plus NE EE.
00:29:15.160 --> 00:29:23.870
On the other hand, 888
aE aE is equal to NE EE.
00:29:27.340 --> 00:29:31.550
OK, so here it goes.
00:29:31.550 --> 00:29:35.020
Here is the main thing that
you have to think about.
00:29:35.020 --> 00:29:39.800
Suppose this state
was good, which
00:29:39.800 --> 00:29:46.500
means this state has
a good norm here.
00:29:46.500 --> 00:29:49.240
And moreover, we've
already learned
00:29:49.240 --> 00:29:52.180
that the energy is
greater than some value.
00:29:52.180 --> 00:29:55.620
So the number
operator of this state
00:29:55.620 --> 00:30:00.350
could be 0-- could
take eigenvalue 0.
00:30:00.350 --> 00:30:07.040
But it could be bigger
than 0, so that's all good.
00:30:07.040 --> 00:30:18.240
Now, at this stage, we have
that-- for example, this state,
00:30:18.240 --> 00:30:23.290
a dagger E has number
one higher than this one,
00:30:23.290 --> 00:30:29.810
than the state E because it has
an extra factor of the a dagger
00:30:29.810 --> 00:30:33.100
which adds an energy of h omega.
00:30:33.100 --> 00:30:36.050
Which means that it
adds number of 1,
00:30:36.050 --> 00:30:39.950
So if this state
has some number,
00:30:39.950 --> 00:30:43.160
this state has a
number which is bigger.
00:30:43.160 --> 00:30:47.150
So suppose you keep adding.
00:30:47.150 --> 00:30:49.170
Now, look at the
norm of this state.
00:30:49.170 --> 00:30:52.760
The norm of this state is pretty
good because this is positive
00:30:52.760 --> 00:30:54.380
and this is positive.
00:30:54.380 --> 00:30:58.970
If you keep adding
a daggers here,
00:30:58.970 --> 00:31:05.520
you always have that this state,
the state with two a daggers,
00:31:05.520 --> 00:31:08.030
you could use that
to find its norm.
00:31:08.030 --> 00:31:10.575
You could use this formula,
put in the states with one
00:31:10.575 --> 00:31:12.140
a dagger here.
00:31:12.140 --> 00:31:15.980
But the states with one a
dagger already has a good norm.
00:31:15.980 --> 00:31:20.440
So this state with two a daggers
would have also good norm.
00:31:20.440 --> 00:31:24.610
So you can go on step by
step using this equation
00:31:24.610 --> 00:31:28.530
to show that as long as
you keep adding a daggers,
00:31:28.530 --> 00:31:32.820
all these states will
have positive norms.
00:31:32.820 --> 00:31:37.260
And they have positive norms
because their number eigenvalue
00:31:37.260 --> 00:31:39.320
is bigger and bigger.
00:31:39.320 --> 00:31:41.250
And therefore,
the recursion says
00:31:41.250 --> 00:31:43.210
that when you add
one a dagger, you
00:31:43.210 --> 00:31:47.540
don't change the sign of this
norm because this is positive
00:31:47.540 --> 00:31:50.900
and this is positive,
and this keeps happening.
00:31:50.900 --> 00:31:53.390
On the other hand,
this is an equation
00:31:53.390 --> 00:31:55.940
that's a lot more dangerous.
00:31:55.940 --> 00:32:02.370
Because this says that in this
equation, a lowers the number.
00:32:02.370 --> 00:32:09.870
So if this has some number,
NE, this has NE minus 1.
00:32:09.870 --> 00:32:13.050
And if you added
another a here, you
00:32:13.050 --> 00:32:15.960
would use this
equation again and try
00:32:15.960 --> 00:32:19.980
to find, what is the norm
of things with two a's here?
00:32:19.980 --> 00:32:23.320
And put in the one
with one a here
00:32:23.320 --> 00:32:26.230
and the number of that state.
00:32:26.230 --> 00:32:32.080
But eventually, the number can
turn into a negative number.
00:32:32.080 --> 00:32:36.850
And as soon as the number turns
negative, you run into trouble.
00:32:36.850 --> 00:32:40.350
So this is the equation
that is problematic
00:32:40.350 --> 00:32:43.610
and the equation that
you need to understand.
00:32:43.610 --> 00:32:47.810
So let me do it in two stages.
00:32:47.810 --> 00:32:51.030
Here are the numbers.
00:32:51.030 --> 00:32:57.390
And here is 5 4, 3, 2, 1, 0.
00:32:57.390 --> 00:33:04.010
Possibly minus 1, minus
2, and all these numbers.
00:33:04.010 --> 00:33:11.335
Now, suppose you start with
a number that is an integer.
00:33:14.320 --> 00:33:16.160
Well, you go with this equation.
00:33:16.160 --> 00:33:18.090
This has number 4.
00:33:18.090 --> 00:33:19.780
Well, you put an a.
00:33:19.780 --> 00:33:24.310
Now it's a state with
number 3, but its norm
00:33:24.310 --> 00:33:26.050
is given 4 times that.
00:33:26.050 --> 00:33:27.690
So it's good.
00:33:27.690 --> 00:33:32.930
Now you go down another 1, you
have a state with number 3,
00:33:32.930 --> 00:33:36.310
with number 2, with
number 1, with number 0.
00:33:36.310 --> 00:33:39.320
And then if you keep lowering,
you will get minus 1,
00:33:39.320 --> 00:33:41.430
which is not so good.
00:33:41.430 --> 00:33:42.810
We'll see what happens.
00:33:42.810 --> 00:33:46.360
Well, here you go
on and you start
00:33:46.360 --> 00:33:49.050
producing the states--
the state with number 4,
00:33:49.050 --> 00:33:53.220
state with number 3, state with
number 2, state with number 1.
00:33:53.220 --> 00:33:59.710
And state here, let's call
it has an energy E prime.
00:33:59.710 --> 00:34:05.540
And it has number equal 0.
00:34:05.540 --> 00:34:10.130
Number of E prime equals 0.
00:34:10.130 --> 00:34:12.360
So you look at this
equation and it
00:34:12.360 --> 00:34:24.169
says aE prime times aE prime is
equal N E prime times E prime E
00:34:24.169 --> 00:34:24.669
prime.
00:34:30.010 --> 00:34:33.980
Well, you obtain this
state at E prime,
00:34:33.980 --> 00:34:36.929
and it was a good
state because it
00:34:36.929 --> 00:34:39.969
came from a state
that was good before.
00:34:39.969 --> 00:34:42.250
And therefore, when
you did the last step,
00:34:42.250 --> 00:34:46.850
you had the state at 1
here, with n equals to 1,
00:34:46.850 --> 00:34:49.230
and then that was the
norm of this state.
00:34:49.230 --> 00:34:54.639
So this E E prime is a
fine number positive.
00:34:54.639 --> 00:34:57.285
But the number E prime is 0.
00:35:01.060 --> 00:35:11.005
So this equation says that aE
prime aE prime is equal to 0.
00:35:11.005 --> 00:35:14.612
And if that's equal
to 0, the state
00:35:14.612 --> 00:35:22.010
aE prime must be equal to 0.
00:35:22.010 --> 00:35:27.190
And 0 doesn't mean the
vacuum state or anything.
00:35:27.190 --> 00:35:28.660
It's just not there.
00:35:28.660 --> 00:35:30.150
There's no such state.
00:35:30.150 --> 00:35:32.210
You can't create it.
00:35:32.210 --> 00:35:37.650
You see, aE prime would be a
state here with number minus 1.
00:35:37.650 --> 00:35:42.730
And everything suggests to
us that that's not possible.
00:35:42.730 --> 00:35:44.550
It's an inconsistent state.
00:35:44.550 --> 00:35:47.750
The number must be less than 1.
00:35:47.750 --> 00:35:53.310
And we avoided the inconsistency
because this procedure
00:35:53.310 --> 00:35:56.660
said that as you go ahead
and do these things,
00:35:56.660 --> 00:36:03.370
you eventually run into this
state E prime at 0 number.
00:36:03.370 --> 00:36:08.150
But then, you get that
the next state is 0
00:36:08.150 --> 00:36:09.560
and there's no inconsistency.
00:36:12.710 --> 00:36:15.920
Now, that's one possibility.
00:36:15.920 --> 00:36:19.910
The other possibility
that could happen
00:36:19.910 --> 00:36:31.710
is that there are
energy eigenstates that
00:36:31.710 --> 00:36:38.430
have numbers which are not--
well, I'll put it here.
00:36:38.430 --> 00:36:40.230
That are not integer.
00:36:40.230 --> 00:36:47.910
So maybe you have a state
here with some number E
00:36:47.910 --> 00:36:49.710
which is not an integer.
00:36:49.710 --> 00:36:54.320
It doesn't belong
to the integers.
00:36:54.320 --> 00:36:56.350
OK, so what happens now?
00:36:59.120 --> 00:37:03.300
Well, this number is positive.
00:37:03.300 --> 00:37:07.470
So you can lower it and you can
put another state with number
00:37:07.470 --> 00:37:08.530
1 less.
00:37:08.530 --> 00:37:13.300
Also, not integer
and it has good norm.
00:37:13.300 --> 00:37:16.830
And this thing has
number 2.5, say.
00:37:16.830 --> 00:37:19.620
Well, if I use the
equation again,
00:37:19.620 --> 00:37:23.390
I put the 2.5 state
with its number 2.5
00:37:23.390 --> 00:37:26.516
and now I get the
state with number 1.5
00:37:26.516 --> 00:37:29.990
and it still has positive norm.
00:37:29.990 --> 00:37:35.300
Do it again, you find
the state with 0.5 number
00:37:35.300 --> 00:37:39.340
and still positive norm.
00:37:39.340 --> 00:37:41.940
And looking at this,
you start with a state
00:37:41.940 --> 00:37:45.060
with 0.5, with 0.5 here.
00:37:45.060 --> 00:37:50.202
And oops, you get a
state that minus 0.5.
00:37:50.202 --> 00:37:56.110
And it seems to be
good, positive norm.
00:37:56.110 --> 00:37:59.640
But then, if this
is possible, you
00:37:59.640 --> 00:38:03.760
could also build another
state acting with another a.
00:38:03.760 --> 00:38:09.410
And this state is now very bad
because the N for this state
00:38:09.410 --> 00:38:11.200
was minus 1/2.
00:38:11.200 --> 00:38:13.370
And therefore, if
you put that state,
00:38:13.370 --> 00:38:15.985
that state at the minus
1/2, you get the norm
00:38:15.985 --> 00:38:21.020
of the next one
that has one less.
00:38:21.020 --> 00:38:23.010
And this state now
is inconsistent.
00:38:30.640 --> 00:38:33.940
So you run into a difficulty.
00:38:33.940 --> 00:38:37.730
So what are the ways in
which this difficulty
00:38:37.730 --> 00:38:40.240
could be avoided?
00:38:40.240 --> 00:38:43.260
What are the escape hatches?
00:38:43.260 --> 00:38:46.900
There are two possibilities.
00:38:46.900 --> 00:38:52.170
Well, the simplest one would
be that the assumption is bad.
00:38:52.170 --> 00:38:56.090
There's no state with
fractional number
00:38:56.090 --> 00:38:59.550
because it leads to
inconsistent states.
00:38:59.550 --> 00:39:02.580
You can build them and
they should be good,
00:39:02.580 --> 00:39:05.850
but they're bad.
00:39:05.850 --> 00:39:07.950
The other possibility
is that just
00:39:07.950 --> 00:39:11.050
like this one sort
of terminated,
00:39:11.050 --> 00:39:15.160
and when you hit 0--
boom, the state became 0.
00:39:15.160 --> 00:39:18.990
Maybe this one with
a fractional one,
00:39:18.990 --> 00:39:24.260
before you run into trouble
you hit a 0 and the state
00:39:24.260 --> 00:39:26.240
becomes 0.
00:39:26.240 --> 00:39:29.760
So basically, what you
really need to know now
00:39:29.760 --> 00:39:38.160
on the algebraic method cannot
tell you is how many states are
00:39:38.160 --> 00:39:41.040
killed by a.
00:39:41.040 --> 00:39:45.450
If maybe the state of
1/2 is also killed by a,
00:39:45.450 --> 00:39:49.710
then we would have trouble.
00:39:49.710 --> 00:39:54.524
Now, as we will see now,
that's a simple problem.
00:39:54.524 --> 00:39:55.940
And it's the only
place where it's
00:39:55.940 --> 00:39:58.340
interesting to
solve some equation.
00:39:58.340 --> 00:40:00.760
So the equation that
we want to solve
00:40:00.760 --> 00:40:05.105
is the equation a on
some state is equal to 0.
00:40:09.360 --> 00:40:14.070
Now, that equation already says
that this possibility is not
00:40:14.070 --> 00:40:15.520
going to happen.
00:40:15.520 --> 00:40:16.020
Why?
00:40:16.020 --> 00:40:20.100
Because from this equation, you
can put an a dagger on this.
00:40:25.340 --> 00:40:31.400
And therefore, you get
that NE is equal to 0.
00:40:31.400 --> 00:40:35.320
This is the number operator,
so the eigenvalue of the number
00:40:35.320 --> 00:40:37.660
operator, we call it NE.
00:40:37.660 --> 00:40:43.470
So in order to be killed by a,
you have to have NE equals 0.
00:40:43.470 --> 00:40:48.470
So in the fractional case,
no state will be killed
00:40:48.470 --> 00:40:51.260
and you would arrive
to an inconsistency.
00:40:51.260 --> 00:40:54.585
So the only possibility is that
there's no fractional states.
00:40:57.500 --> 00:41:03.610
So it's still interesting to
figure out this differential
00:41:03.610 --> 00:41:05.850
equation, what it gives you.
00:41:05.850 --> 00:41:08.780
And why do we call it a
differential equation?
00:41:08.780 --> 00:41:13.260
Because a is this
operator over there.
00:41:13.260 --> 00:41:15.030
It has x and ip.
00:41:15.030 --> 00:41:22.840
So the equation
is x a E equals 0,
00:41:22.840 --> 00:41:29.390
which is square root of
m omega over 2 h bar x
00:41:29.390 --> 00:41:36.635
x plus ip over m
omega on E equals 0.
00:41:39.980 --> 00:41:45.050
And you've translated
these kind of things.
00:41:45.050 --> 00:41:49.990
The first term is an x
multiplying the wave function.
00:41:49.990 --> 00:41:55.290
We can call it psi E of x.
00:41:55.290 --> 00:41:58.200
The next term, the
coefficient in front
00:41:58.200 --> 00:42:01.020
is something you don't
have to worry, of course.
00:42:01.020 --> 00:42:02.800
It's just multiplying
everything,
00:42:02.800 --> 00:42:04.990
so it's just irrelevant.
00:42:04.990 --> 00:42:08.620
So have i over m omega.
00:42:08.620 --> 00:42:18.250
And p, as you remember, is h bar
over i d dx of psi E of x zero.
00:42:20.990 --> 00:42:26.600
So it's so simple
differential equation,
00:42:26.600 --> 00:42:39.380
x plus h bar over m omega d dx
on psi E of x is equal to 0.
00:42:39.380 --> 00:42:42.070
Just one solution
up to a constant
00:42:42.070 --> 00:42:46.160
is the Gaussian that
you know represents
00:42:46.160 --> 00:42:47.810
a simple harmonic oscillator.
00:42:56.460 --> 00:43:00.500
So that's pretty
much the end of it.
00:43:00.500 --> 00:43:05.310
This ground state wave
function is a number
00:43:05.310 --> 00:43:11.720
times the exponential of minus
m omega over 2 h bar x squared.
00:43:15.840 --> 00:43:21.954
And that's that.
00:43:21.954 --> 00:43:24.070
This is called the ground state.
00:43:24.070 --> 00:43:27.790
It has N equals 0
represented as a state.
00:43:31.130 --> 00:43:36.720
We say this number
is N equals 0.
00:43:36.720 --> 00:43:41.870
So this state is the thing
that represents this psi
00:43:41.870 --> 00:43:50.290
E. In other words, psi
E of x is x with 0.
00:43:50.290 --> 00:43:54.680
And that 0 is a
little confusing.
00:43:54.680 --> 00:43:57.000
Some people think
it's the 0 vector.
00:43:57.000 --> 00:43:59.580
That's not good.
00:43:59.580 --> 00:44:01.700
This is not the 0 vector.
00:44:01.700 --> 00:44:04.290
The 0 vector is not a state.
00:44:04.290 --> 00:44:06.430
It's not in the Hilbert space.
00:44:06.430 --> 00:44:08.410
This is the ground state.
00:44:08.410 --> 00:44:12.070
Then, the worst confusion is
to think it's the 0 vector.
00:44:12.070 --> 00:44:16.540
The next confusion is
to think it's 0 energy.
00:44:16.540 --> 00:44:20.350
That's not 0 energy,
it's number equals 0.
00:44:20.350 --> 00:44:25.050
The energy is, therefore,
1/2 h bar omega.
00:44:28.510 --> 00:44:32.030
And now, given
our discussion, we
00:44:32.030 --> 00:44:35.880
can start building states
with more oscillators.
00:44:35.880 --> 00:44:40.050
So we build a state
with number equal 1,
00:44:40.050 --> 00:44:44.970
which is constructed by
an a dagger on the vacuum.
00:44:44.970 --> 00:44:48.530
This has energy 1
h bar omega more.
00:44:48.530 --> 00:44:51.630
It has number equal to 1.
00:44:51.630 --> 00:44:53.550
And that's sometimes
useful to just
00:44:53.550 --> 00:45:00.150
make sure you understand why
N on a dagger on the vacuum
00:45:00.150 --> 00:45:06.680
is a dagger a a
dagger on the vacuum.
00:45:06.680 --> 00:45:10.140
Now, a kills the
vacuum, so this can
00:45:10.140 --> 00:45:14.190
be replaced by the
commutator, which is 1.
00:45:14.190 --> 00:45:16.890
And therefore, you're left
with a dagger on the vacuum.
00:45:20.300 --> 00:45:23.410
And that means that
the eigenvalue of n hat
00:45:23.410 --> 00:45:26.300
is 1 for this state.
00:45:26.300 --> 00:45:32.010
Moreover, this state is
where normalized 1 with 1
00:45:32.010 --> 00:45:35.270
actually gives you
a good normalization
00:45:35.270 --> 00:45:37.120
if 0 is well-normalized.
00:45:37.120 --> 00:45:42.940
So we'll take 0 with 0
to be 1, the number 1.
00:45:42.940 --> 00:45:48.660
And that requires fixing
that N0 over here.
00:45:48.660 --> 00:45:52.470
Now, these are things
that you've mostly seen,
00:45:52.470 --> 00:45:57.920
so I don't want to say
much more about them.
00:45:57.920 --> 00:46:02.040
I'd rather go through
the Schrodinger thing
00:46:02.040 --> 00:46:03.180
that we have later.
00:46:03.180 --> 00:46:11.610
So let me conclude by just
listing the general states,
00:46:11.610 --> 00:46:15.220
and then leaving for you to read
what is left there in the notes
00:46:15.220 --> 00:46:18.960
so that you can just get an
appreciation of how you use it.
00:46:18.960 --> 00:46:21.800
And with the practice
problems, you'll be done.
00:46:21.800 --> 00:46:25.000
So here it is.
00:46:25.000 --> 00:46:26.430
Here is the answer.
00:46:26.430 --> 00:46:32.160
The n state is given by 1 over
square root of n factorial
00:46:32.160 --> 00:46:35.960
a dagger to the n
acting on the vacuum.
00:46:38.790 --> 00:46:47.400
And these n states are such
that m with n is delta mn.
00:46:47.400 --> 00:46:51.360
So here we're using
all kinds of things.
00:46:51.360 --> 00:46:55.270
First, you should check
this is well normalized,
00:46:55.270 --> 00:46:57.980
or read it and do
the calculations.
00:46:57.980 --> 00:47:02.350
And these are, in
fact, orthogonal
00:47:02.350 --> 00:47:06.980
unless they have the same
number of creation operators
00:47:06.980 --> 00:47:08.790
are the same number.
00:47:08.790 --> 00:47:11.300
Now, that had to be expected.
00:47:11.300 --> 00:47:16.145
These are eigenstates
of a Hermitian operator.
00:47:16.145 --> 00:47:19.330
The N operator is Hermitian.
00:47:19.330 --> 00:47:22.010
Eigenstates of a
Hermitian operator
00:47:22.010 --> 00:47:25.320
with different
eigenvalues are always
00:47:25.320 --> 00:47:27.950
orthogonal to each other.
00:47:27.950 --> 00:47:30.330
If you have eigenstates
of a Hermitian operator
00:47:30.330 --> 00:47:33.850
with the same eigenvalue,
if you have a degeneracy,
00:47:33.850 --> 00:47:38.240
you can always arrange them
to make them orthogonal.
00:47:38.240 --> 00:47:42.280
But if the eigenvalues are
different, they are orthogonal.
00:47:42.280 --> 00:47:47.400
And there's no degeneracies
in this spectrum whatsoever.
00:47:47.400 --> 00:47:52.230
You will, in fact, argue that
because there's no degeneracy
00:47:52.230 --> 00:47:56.050
in the ground state, there
cannot be degeneracy anywhere
00:47:56.050 --> 00:47:58.210
else.
00:47:58.210 --> 00:48:00.740
So this result,
this orthonormality
00:48:00.740 --> 00:48:04.880
is really a consequence of
all the theorems we've proven.
00:48:04.880 --> 00:48:08.240
And you could check it
by doing the algebra
00:48:08.240 --> 00:48:10.840
and you would start
moving a and a daggers.
00:48:10.840 --> 00:48:13.570
And you would be left
with either some a's or
00:48:13.570 --> 00:48:14.610
some a daggers.
00:48:14.610 --> 00:48:16.850
If you're left
with some a's, they
00:48:16.850 --> 00:48:18.330
would kill the
thing on the right.
00:48:18.330 --> 00:48:19.800
If you're left with
some a daggers,
00:48:19.800 --> 00:48:22.350
it would kill the
thing on the left.
00:48:22.350 --> 00:48:23.840
So this can be proven.
00:48:23.840 --> 00:48:27.760
But this is just a consequence
that these are eigenstates
00:48:27.760 --> 00:48:34.620
of the Hermitian operator n
that have different eigenvalues.
00:48:34.620 --> 00:48:38.130
And therefore, you've
succeeded in constructing
00:48:38.130 --> 00:48:42.090
a full decomposition
of the state
00:48:42.090 --> 00:48:45.940
space of the
harmonic oscillator.
00:48:45.940 --> 00:48:49.680
We spoke about
the Hilbert space.
00:48:49.680 --> 00:48:52.000
Are now very
precisely, see we can
00:48:52.000 --> 00:48:59.690
say this is u0 plus
u1 plus u2 where
00:48:59.690 --> 00:49:13.030
uk is the states of the form
alpha k, where N on k-- maybe
00:49:13.030 --> 00:49:15.760
I should put n here.
00:49:15.760 --> 00:49:17.515
It looks nicer.
00:49:17.515 --> 00:49:18.878
n.
00:49:18.878 --> 00:49:25.000
Where N n equal n n.
00:49:25.000 --> 00:49:28.140
So every
one-dimensional subspace
00:49:28.140 --> 00:49:31.480
is spanned by that
state of number n.
00:49:31.480 --> 00:49:34.590
So you have the states of
number 0, states of number 1,
00:49:34.590 --> 00:49:36.220
states of number 2.
00:49:36.220 --> 00:49:39.050
These are all
orthogonal subspaces.
00:49:39.050 --> 00:49:40.910
They add up to form everything.
00:49:44.170 --> 00:49:46.640
It's a nice description.
00:49:46.640 --> 00:49:48.930
So the general
state in this system
00:49:48.930 --> 00:49:52.490
is a complex number times
the state with number 0
00:49:52.490 --> 00:49:56.240
plus the complex number states
of number 1, complex number,
00:49:56.240 --> 00:49:57.590
and that.
00:49:57.590 --> 00:50:03.350
Things couldn't have
been easier in a sense.
00:50:03.350 --> 00:50:07.350
The other thing that you
already know from 804
00:50:07.350 --> 00:50:11.060
is that if you try to compute
expectation values, most
00:50:11.060 --> 00:50:17.110
of the times you want to
use a's and a daggers.
00:50:17.110 --> 00:50:20.800
So the typical thing
that one wants to compete
00:50:20.800 --> 00:50:29.440
is on the state n, what is the
uncertainty in x on the state
00:50:29.440 --> 00:50:31.610
n?
00:50:31.610 --> 00:50:33.310
How much is it?
00:50:33.310 --> 00:50:36.260
What is the
uncertainty of momentum
00:50:36.260 --> 00:50:40.815
on the energy
eigenstate of number n?
00:50:43.800 --> 00:50:47.100
These are relatively
straightforward calculations.
00:50:47.100 --> 00:50:51.650
If you have to do the integrals,
each one-- by the time you
00:50:51.650 --> 00:50:56.200
organize all your constants--
half an hour, maybe 20 minutes.
00:50:56.200 --> 00:50:58.790
If you do it with
a and a daggers,
00:50:58.790 --> 00:51:01.430
this computation
should be five minutes,
00:51:01.430 --> 00:51:03.890
or something like that.
00:51:03.890 --> 00:51:06.460
We'll see that
done on the notes.
00:51:06.460 --> 00:51:09.780
You can also do them yourselves.
00:51:09.780 --> 00:51:13.000
You probably have
played with them a bit.
00:51:13.000 --> 00:51:19.860
So this was a brief review and
discussion of them spectrum.
00:51:19.860 --> 00:51:21.770
It was a little detailed.
00:51:21.770 --> 00:51:25.710
We had to argue things
carefully to make
00:51:25.710 --> 00:51:28.820
sure we don't assume things.
00:51:28.820 --> 00:51:31.110
And this is the
way we'll do also
00:51:31.110 --> 00:51:34.750
with angular momentum
in a few weeks from now.
00:51:34.750 --> 00:51:38.510
But now I want to leave that,
so I'm going to take questions.
00:51:38.510 --> 00:51:46.640
If there are any questions
on this logic, please ask.
00:51:46.640 --> 00:51:47.270
Yes.
00:51:47.270 --> 00:51:50.574
AUDIENCE: [INAUDIBLE] for
how you got a dagger, a,
00:51:50.574 --> 00:51:53.880
a dagger, 0, 2 dagger, 0?
00:51:53.880 --> 00:51:55.640
PROFESSOR: Yes,
that calculation.
00:51:55.640 --> 00:52:00.220
So let me do at the step
that I did in words.
00:52:00.220 --> 00:52:03.570
So at this place--
so the question was,
00:52:03.570 --> 00:52:07.560
how did I do this computation?
00:52:07.560 --> 00:52:10.060
Here I just copied what N is.
00:52:10.060 --> 00:52:11.910
So I just copied that.
00:52:11.910 --> 00:52:16.476
Then, the next step was to
say, since a kills this,
00:52:16.476 --> 00:52:26.280
this is equal to a dagger times
a a dagger minus a dagger a.
00:52:26.280 --> 00:52:28.140
Because a kills it.
00:52:28.140 --> 00:52:31.160
And I can add this, it
doesn't cost me anything.
00:52:31.160 --> 00:52:33.810
Now, I added something
that is convenient,
00:52:33.810 --> 00:52:37.430
so that this is a
dagger commutator of a
00:52:37.430 --> 00:52:41.270
with a dagger on 0.
00:52:41.270 --> 00:52:44.062
This is 1, so you get that.
00:52:47.290 --> 00:52:49.000
It's a little more
interesting when
00:52:49.000 --> 00:52:51.780
you have, for
example, the state 2,
00:52:51.780 --> 00:52:57.730
which is 1 over square root
of 2 a dagger a dagger on 0.
00:52:57.730 --> 00:53:01.330
I advise you to try to
calculate n on that.
00:53:01.330 --> 00:53:04.240
And in general,
convince yourselves
00:53:04.240 --> 00:53:07.750
that n is a number
operator, which means
00:53:07.750 --> 00:53:11.170
counts the number of a daggers.
00:53:11.170 --> 00:53:18.956
You'll have to use that
property if you have N with AB.
00:53:18.956 --> 00:53:25.610
It's N with A B and then A N
with B. The derivative property
00:53:25.610 --> 00:53:30.460
of the bracket has to
be used all the time.
00:53:30.460 --> 00:53:36.010
So Schrodinger dynamics,
let's spend the last 20
00:53:36.010 --> 00:53:37.780
minutes of our lecture on this.
00:53:46.100 --> 00:53:54.360
So basically, it's
a postulate of how
00:53:54.360 --> 00:53:57.040
evolution occurs in
quantum mechanics.
00:53:57.040 --> 00:53:59.935
So we'll state it as follows.
00:54:03.950 --> 00:54:06.140
What is time in
quantum mechanics?
00:54:06.140 --> 00:54:09.080
Well, you have a state space.
00:54:09.080 --> 00:54:12.090
And you see the
state space, you've
00:54:12.090 --> 00:54:14.040
seen it in the
harmonic oscillator
00:54:14.040 --> 00:54:15.680
is this sum of vectors.
00:54:15.680 --> 00:54:19.570
And these vectors were wave
functions, if you wish.
00:54:19.570 --> 00:54:22.090
There's no time anywhere there.
00:54:22.090 --> 00:54:24.820
There's no time on
this vector space.
00:54:24.820 --> 00:54:28.300
This vector space is an
abstract vector space
00:54:28.300 --> 00:54:37.150
of functions or states, but time
comes because you have clocks.
00:54:37.150 --> 00:54:39.300
And then you can ask,
where is my state?
00:54:39.300 --> 00:54:41.840
And that's that vector
on that state space.
00:54:41.840 --> 00:54:43.960
And you ask the
question a littler later
00:54:43.960 --> 00:54:45.270
and the state has moved.
00:54:45.270 --> 00:54:46.940
It's another vector.
00:54:46.940 --> 00:54:53.000
So these are vectors and
the vectors change in time.
00:54:53.000 --> 00:54:56.850
And that's all the dynamics
is in quantum mechanics.
00:54:56.850 --> 00:55:00.010
The time is sort of
auxiliary to all this.
00:55:02.570 --> 00:55:04.580
So we must have a
picture of that.
00:55:04.580 --> 00:55:13.610
And the way we do this is to
imagine that we have a vector
00:55:13.610 --> 00:55:16.870
space H. And here is a vector.
00:55:16.870 --> 00:55:20.960
And that H is for Hilbert space.
00:55:20.960 --> 00:55:24.490
We used to call it in our
math part of the course
00:55:24.490 --> 00:55:27.920
V, the complex vector space.
00:55:27.920 --> 00:55:32.280
And this state is the
state of the system.
00:55:32.280 --> 00:55:35.170
And we sometimes
put the time here
00:55:35.170 --> 00:55:36.985
to indicate that's what it is.
00:55:36.985 --> 00:55:39.400
At time t0, that's it.
00:55:39.400 --> 00:55:46.440
Well, at time t, some arbitrary
later time, it could be here.
00:55:46.440 --> 00:55:48.180
And the state moves.
00:55:48.180 --> 00:55:50.310
But one thing is clear.
00:55:50.310 --> 00:55:54.255
If it's a state of a
system, if we normalize it,
00:55:54.255 --> 00:55:57.780
it should be of unit length.
00:55:57.780 --> 00:56:03.970
And we can think of a sphere
in which this unit sphere is
00:56:03.970 --> 00:56:10.170
the set of all the tips of the
vectors that have unit norm.
00:56:10.170 --> 00:56:15.330
And this vector will move here
in time, trace a trajectory,
00:56:15.330 --> 00:56:18.340
and reach this one.
00:56:18.340 --> 00:56:22.870
And it should do it preserving
the length of the vector.
00:56:22.870 --> 00:56:25.735
And in fact, if you don't
use a normalized vector,
00:56:25.735 --> 00:56:28.320
it has a norm of 3.
00:56:28.320 --> 00:56:30.770
Well, it should preserve
that 3 because you'd
00:56:30.770 --> 00:56:34.060
normalize the state
once and forever.
00:56:34.060 --> 00:56:38.790
So we proved in our
math part of the subject
00:56:38.790 --> 00:56:43.020
that an operator
that always preserves
00:56:43.020 --> 00:56:46.620
the length of all vectors
is a unitary operator.
00:56:46.620 --> 00:56:50.060
So this is the fundamental
thing that we want.
00:56:50.060 --> 00:56:59.010
And the idea of quantum
mechanics is that psi at time t
00:56:59.010 --> 00:57:04.420
is obtained by the action
of a unitary operator
00:57:04.420 --> 00:57:07.130
from the state psi at time t0.
00:57:12.020 --> 00:57:17.090
And this is for all t and t0.
00:57:17.090 --> 00:57:18.583
And this being unitary.
00:57:22.760 --> 00:57:27.520
Now, I want to make
sure this is clear.
00:57:27.520 --> 00:57:32.570
It can be misinterpreted,
this equation.
00:57:32.570 --> 00:57:39.660
Here, psi at t0 is
an arbitrary state.
00:57:39.660 --> 00:57:42.765
If you had another
state, psi prime of t0,
00:57:42.765 --> 00:57:45.150
it would also evolve
with this formula.
00:57:45.150 --> 00:57:48.060
And this U is the same.
00:57:48.060 --> 00:57:51.120
So the postulate of
unitary time evolution
00:57:51.120 --> 00:57:56.350
is that there is this
magical U operator that
00:57:56.350 --> 00:58:00.230
can evolve any state.
00:58:00.230 --> 00:58:01.990
Any state that you
give me at time
00:58:01.990 --> 00:58:05.170
equal 0, any possible
state in the Hilbert space,
00:58:05.170 --> 00:58:07.580
you plug it in here.
00:58:07.580 --> 00:58:11.150
And by acting with
this unitary operator,
00:58:11.150 --> 00:58:13.230
you get the state
at the later time.
00:58:16.360 --> 00:58:22.550
Now, you've slipped an
extraordinary amount of physics
00:58:22.550 --> 00:58:24.570
into that statement.
00:58:24.570 --> 00:58:28.430
If you've bought it, you've
bought the Schrodinger equation
00:58:28.430 --> 00:58:29.440
already.
00:58:29.440 --> 00:58:31.920
That is going to
come out by just
00:58:31.920 --> 00:58:35.290
doing a little
calculation from this.
00:58:35.290 --> 00:58:40.070
So the Schrodinger equation
is really fundamentally,
00:58:40.070 --> 00:58:42.865
at the end of the
day, the statement
00:58:42.865 --> 00:58:45.520
that this unitary
time evolution, which
00:58:45.520 --> 00:58:48.510
is to mean there's a
unitary operator that
00:58:48.510 --> 00:58:51.700
evolves any physical state.
00:58:51.700 --> 00:58:55.200
So let's try to discuss this.
00:58:55.200 --> 00:58:57.540
Are there any questions?
00:58:57.540 --> 00:58:58.362
Yes.
00:58:58.362 --> 00:59:00.028
AUDIENCE: So you
mentioned at first that
00:59:00.028 --> 00:59:01.652
in the current
formulation [INAUDIBLE]?
00:59:04.590 --> 00:59:05.916
PROFESSOR: A little louder.
00:59:05.916 --> 00:59:08.005
We do what in our
current formulation?
00:59:08.005 --> 00:59:11.370
AUDIENCE: So if you don't
include time [INAUDIBLE].
00:59:11.370 --> 00:59:12.370
PROFESSOR: That's right.
00:59:12.370 --> 00:59:13.825
There's no start of
the vector space.
00:59:13.825 --> 00:59:14.491
AUDIENCE: Right.
00:59:14.491 --> 00:59:17.390
So is it possible to consider
a vector space with time?
00:59:20.970 --> 00:59:21.930
PROFESSOR: Unclear.
00:59:21.930 --> 00:59:25.250
I don't think so.
00:59:25.250 --> 00:59:28.720
It's just nowhere there.
00:59:28.720 --> 00:59:34.070
What would it mean, even, to
add time to the vector space?
00:59:34.070 --> 00:59:36.480
I think you would
have a hard time even
00:59:36.480 --> 00:59:37.915
imagining what it means.
00:59:41.500 --> 00:59:43.920
Now, people try to
change quantum mechanics
00:59:43.920 --> 00:59:46.510
in all kinds of ways.
00:59:46.510 --> 00:59:49.305
Nobody has succeeded in
changing quantum mechanics.
00:59:52.210 --> 00:59:55.770
That should not be a
deterrent for you to try,
00:59:55.770 --> 00:59:58.690
but should give you
a little caution
00:59:58.690 --> 01:00:03.300
that is not likely to be easy.
01:00:03.300 --> 01:00:07.770
So we'll not try to do that.
01:00:07.770 --> 01:00:11.960
Now, let me follow on this
and see what it gives us.
01:00:16.860 --> 01:00:20.120
Well, a few things.
01:00:20.120 --> 01:00:22.790
This operator is unique.
01:00:22.790 --> 01:00:24.650
If it exists, it's unique.
01:00:24.650 --> 01:00:28.220
If there's another operator that
evolves states the same way,
01:00:28.220 --> 01:00:30.520
it must be the same as that one.
01:00:30.520 --> 01:00:33.420
Easy to prove.
01:00:33.420 --> 01:00:36.410
Two operators that attack
the same way on every state
01:00:36.410 --> 01:00:39.060
are the same, so that's it.
01:00:39.060 --> 01:00:48.360
Unitary, what does it mean
that u t, t0 dagger times u t,
01:00:48.360 --> 01:00:54.390
t0 is equal to 1?
01:00:54.390 --> 01:00:58.660
Now, here these parentheses
are a little cumbersome.
01:00:58.660 --> 01:01:01.320
This is very clear,
you take this operator
01:01:01.320 --> 01:01:02.280
and you dagger it.
01:01:02.280 --> 01:01:05.980
But it's cumbersome, so
we write it like this.
01:01:12.740 --> 01:01:15.300
This means the dagger
of the whole operator.
01:01:15.300 --> 01:01:18.550
So this is just the same thing.
01:01:21.310 --> 01:01:24.065
OK, what else?
01:01:27.620 --> 01:01:36.180
u of t0, t0, it's
the unit operator.
01:01:36.180 --> 01:01:40.990
If the times are the same,
you get the unit operator
01:01:40.990 --> 01:01:48.160
for all t0 because you're
getting psi of t0 here
01:01:48.160 --> 01:01:49.210
and psi of t0 here.
01:01:49.210 --> 01:01:52.570
And the only operator that
leaves all states the same
01:01:52.570 --> 01:01:53.960
is the unit operator.
01:01:53.960 --> 01:01:59.760
So this unitary operator must
become the unit operator,
01:01:59.760 --> 01:02:05.100
in fact, for the two
arguments being equal.
01:02:05.100 --> 01:02:07.050
Composition.
01:02:07.050 --> 01:02:15.550
If you have psi t2, that can be
obtained as U of t2, t1 times
01:02:15.550 --> 01:02:19.350
the psi of t1.
01:02:19.350 --> 01:02:30.310
And it can be obtained as u of
t2, t1, u of t1, t0, psi of t0.
01:02:34.110 --> 01:02:36.310
So what do we learn from here?
01:02:36.310 --> 01:02:43.890
That this state itself is u of
t2, t0 on the original state.
01:02:43.890 --> 01:02:55.390
So u of t2, t0 is u of
t2, t1 times u of t1, t0.
01:02:59.500 --> 01:03:06.120
It's like time composition is
like matrix multiplication.
01:03:06.120 --> 01:03:10.020
You go from t0 to t1,
then from t1 to t2.
01:03:10.020 --> 01:03:14.520
It's like the second
index of this matrix.
01:03:14.520 --> 01:03:16.270
In the first index
of this matrix,
01:03:16.270 --> 01:03:21.280
you are multiplying them
and you get this thing.
01:03:21.280 --> 01:03:23.220
So that's composition.
01:03:23.220 --> 01:03:26.055
And then, you have
inverses as well.
01:03:33.640 --> 01:03:37.340
And here are the inverses.
01:03:37.340 --> 01:03:45.270
In that equation, you
take t2 equal to t0.
01:03:45.270 --> 01:03:47.635
So the left-hand side becomes 1.
01:03:50.400 --> 01:04:02.280
And t1 equal to t, so you get
u of t0, t be times u of t,
01:04:02.280 --> 01:04:07.610
t0 is equal to 1,
which makes sense.
01:04:07.610 --> 01:04:10.770
You propagate from t0 to t.
01:04:10.770 --> 01:04:14.210
And then from t to
t0, you get nothing.
01:04:14.210 --> 01:04:21.570
Or if it's to say that the
inverse of an operator--
01:04:21.570 --> 01:04:24.300
the inverse of this
operator is this one.
01:04:24.300 --> 01:04:29.190
So to take the inverse of a
u, you flip the arguments.
01:04:29.190 --> 01:04:39.170
So I'll write it like that,
the inverse minus 1 of t, t0.
01:04:39.170 --> 01:04:40.700
You just flip the arguments.
01:04:40.700 --> 01:04:42.120
It's u of t0, t.
01:04:45.040 --> 01:04:48.160
And since the
operator is Hermitian,
01:04:48.160 --> 01:04:51.380
the dagger is equal
to the inverse.
01:04:51.380 --> 01:04:56.660
So the inverse of an operator
is equal to the dagger.
01:04:56.660 --> 01:05:00.780
so t, t0 as well.
01:05:00.780 --> 01:05:03.930
So this one we got here.
01:05:03.930 --> 01:05:09.564
And Hermiticity says that the
dagger is equal to the inverse.
01:05:09.564 --> 01:05:12.120
Inverse and dagger are the same.
01:05:12.120 --> 01:05:16.090
So basically, you can
delete the word "inverse"
01:05:16.090 --> 01:05:18.170
by flipping the order
of the arguments.
01:05:18.170 --> 01:05:20.620
And since dagger is
the same as inverse,
01:05:20.620 --> 01:05:22.890
you can delete the
dagger by flipping
01:05:22.890 --> 01:05:24.383
the order of the arguments.
01:05:28.170 --> 01:05:32.065
All right, so let's try to
find the Schrodinger equation.
01:05:41.210 --> 01:05:44.185
So how c we c the
Schrodinger equation?
01:05:47.322 --> 01:05:51.940
Well, we try obtaining
the differential equation
01:05:51.940 --> 01:05:54.375
using that time
evolution over there.
01:05:57.120 --> 01:05:59.500
So the time evolution
is over there.
01:05:59.500 --> 01:06:05.860
Let's try to find
what is d dt of psi t.
01:06:10.240 --> 01:06:15.230
So d dt of psi of
t is just the d dt
01:06:15.230 --> 01:06:21.070
of this operator u
of t, t0 psi of t0.
01:06:24.170 --> 01:06:29.060
And I should only
differentiate that operate.
01:06:29.060 --> 01:06:33.610
Now, I want an
equation for psi of t.
01:06:33.610 --> 01:06:35.855
So I have here psi of t0.
01:06:35.855 --> 01:06:44.810
So I can write this
as du of t, t0 dt.
01:06:44.810 --> 01:06:48.990
And now put a psi at t.
01:06:48.990 --> 01:06:52.670
And then, I could
put a u from t to t0.
01:07:01.920 --> 01:07:10.640
Now, this u of t and t0 just
brings it back to time t0.
01:07:10.640 --> 01:07:15.880
And this is all good now, I have
this complicated operator here.
01:07:15.880 --> 01:07:18.410
But there's nothing too
complicated about it.
01:07:18.410 --> 01:07:21.560
Especially if I
reverse the order here,
01:07:21.560 --> 01:07:31.110
I'll have du dt of t, t0
and u dagger of t, t0.
01:07:31.110 --> 01:07:36.000
And I reverse the order there
in order that this operator
01:07:36.000 --> 01:07:38.980
is the same as that, the one
that is being [INAUDIBLE] that
01:07:38.980 --> 01:07:44.140
has the same order of
arguments, t and t0.
01:07:44.140 --> 01:07:47.510
So I've got something now.
01:07:47.510 --> 01:07:53.770
And I'll call this
lambda of t and t0.
01:07:56.530 --> 01:07:58.150
So what have I learned?
01:07:58.150 --> 01:08:07.860
That d dt of psi and t is equal
to lambda of t, t0 psi of t.
01:08:12.650 --> 01:08:13.770
Questions?
01:08:13.770 --> 01:08:16.950
I don't want to loose
you in their derivation.
01:08:16.950 --> 01:08:18.760
Look at it.
01:08:18.760 --> 01:08:23.250
Anything-- you got lost,
notation, anything.
01:08:23.250 --> 01:08:26.090
It's a good time to ask.
01:08:26.090 --> 01:08:26.734
Yes.
01:08:26.734 --> 01:08:29.275
AUDIENCE: Just to make sure when
you differentiated the state
01:08:29.275 --> 01:08:32.470
by t, the reason that you don't
put that in the derivative
01:08:32.470 --> 01:08:35.110
because it doesn't have a
time [INAUDIBLE] necessarily,
01:08:35.110 --> 01:08:38.240
or because-- oh, because
you're using the value at t0.
01:08:38.240 --> 01:08:39.180
PROFESSOR: Right.
01:08:39.180 --> 01:08:42.434
Here I looked at that
equation and the only part
01:08:42.434 --> 01:08:45.130
that has anything
to do with time t
01:08:45.130 --> 01:08:46.729
is the operator, not the state.
01:08:50.790 --> 01:08:52.555
Any other comments or questions?
01:08:56.029 --> 01:08:58.620
OK, so what have we learned?
01:08:58.620 --> 01:09:04.550
We want to know some important
things about this operator
01:09:04.550 --> 01:09:08.319
lambda because
somehow, it's almost
01:09:08.319 --> 01:09:10.210
looking like a
Schrodinger equation.
01:09:10.210 --> 01:09:12.660
So we want to see a
couple of things about it.
01:09:16.330 --> 01:09:20.060
So the first thing
that I will show to you
01:09:20.060 --> 01:09:26.680
is that lambda is, in
fact, anti-Hermitian.
01:09:31.510 --> 01:09:33.430
Here is lambda.
01:09:33.430 --> 01:09:36.790
I could figure out,
what is lambda dagger?
01:09:36.790 --> 01:09:40.500
Well, lambda dagger is you
take the dagger of this.
01:09:40.500 --> 01:09:43.609
You have to think when you
take the dagger of this thing.
01:09:43.609 --> 01:09:47.490
It looks a little worrisome,
but this is an operator.
01:09:47.490 --> 01:09:50.450
This is another operator,
which is a time derivative.
01:09:50.450 --> 01:09:54.810
So you take the dagger by
doing the reverse operators
01:09:54.810 --> 01:09:55.620
and daggers.
01:09:55.620 --> 01:10:01.370
So the first factor
is clearly u of t, t0.
01:10:01.370 --> 01:10:04.880
And then the dagger of this.
01:10:04.880 --> 01:10:09.830
Now, dagger doesn't interfere
at all with time derivatives.
01:10:09.830 --> 01:10:12.800
Think of the time derivative--
operator at one time,
01:10:12.800 --> 01:10:15.050
operator at another
slightly different time.
01:10:15.050 --> 01:10:16.590
Subtract it.
01:10:16.590 --> 01:10:19.210
You take the dagger
and the dagger
01:10:19.210 --> 01:10:20.730
goes through the derivative.
01:10:20.730 --> 01:10:28.430
So this is d u dagger t, t0 dt.
01:10:28.430 --> 01:10:32.220
So I wrote here what
lambda dagger is.
01:10:32.220 --> 01:10:34.840
You have here what lambda is.
01:10:34.840 --> 01:10:39.370
And the claim is that one
is minus the other one.
01:10:39.370 --> 01:10:41.150
It doesn't look
obvious because it's
01:10:41.150 --> 01:10:42.400
supposed to be anti-Hermitian.
01:10:45.810 --> 01:10:50.580
But you can show it is true by
doing the following-- u of t,
01:10:50.580 --> 01:11:00.470
t0 u dagger of t, t0
is a unitary operator.
01:11:00.470 --> 01:11:02.850
So this is 1.
01:11:02.850 --> 01:11:05.760
And now you differentiate
with respect to t.
01:11:10.480 --> 01:11:12.600
If you differentiate
with respect to t,
01:11:12.600 --> 01:11:24.390
you get du dt of t, t0 u dagger
of t, t0 plus u of t, t0 du
01:11:24.390 --> 01:11:32.840
dagger of t, t0 equals 0 because
the right-hand side is 1.
01:11:32.840 --> 01:11:38.000
And this term is lambda.
01:11:38.000 --> 01:11:43.332
And the second term
is lambda dagger.
01:11:43.332 --> 01:11:49.150
And they add up to 0, so
lambda dagger is minus lambda.
01:11:49.150 --> 01:11:52.920
Lambda is, therefore,
anti-Hermitian as claimed.
01:12:03.800 --> 01:12:04.990
Now, look.
01:12:04.990 --> 01:12:09.470
This is starting to
look pretty good.
01:12:09.470 --> 01:12:12.360
This lambda depends on t and t0.
01:12:12.360 --> 01:12:14.663
That's a little nasty though.
01:12:14.663 --> 01:12:15.162
Why?
01:12:17.980 --> 01:12:19.570
Here is t.
01:12:19.570 --> 01:12:24.270
What is t0 doing here?
01:12:24.270 --> 01:12:27.400
It better not be there.
01:12:27.400 --> 01:12:30.580
So what I want to show to
you is that even though this
01:12:30.580 --> 01:12:35.740
looks like it has a t0
in there, there's no t0.
01:12:35.740 --> 01:12:41.200
So we want to show this operator
is actually independent of t0.
01:12:41.200 --> 01:12:49.570
So I will show that if
you have lambda of t, t0,
01:12:49.570 --> 01:12:56.590
it's actually equal to
lambda of t, t1 for any t1.
01:12:59.270 --> 01:13:00.250
We'll show that.
01:13:03.240 --> 01:13:05.202
Sorry.
01:13:05.202 --> 01:13:09.820
[LAUGHTER]
01:13:09.820 --> 01:13:15.290
PROFESSOR: So this will
show that you could take t1
01:13:15.290 --> 01:13:18.110
to be t0 plus epsilon.
01:13:18.110 --> 01:13:20.440
And take the limit
and say the derivative
01:13:20.440 --> 01:13:22.900
of this with respect of t0 is 0.
01:13:22.900 --> 01:13:25.580
Or take this to
mean that it's just
01:13:25.580 --> 01:13:30.900
absolutely independent of t0
and t0 is really not there.
01:13:30.900 --> 01:13:33.960
So if you take t1 equal
t dot plus epsilon,
01:13:33.960 --> 01:13:36.295
you could just
conclude from these
01:13:36.295 --> 01:13:41.230
that this lambda with
respect to t0 is 0.
01:13:41.230 --> 01:13:42.670
No dependence on t0.
01:13:42.670 --> 01:13:43.980
So how do we do that?
01:13:43.980 --> 01:13:46.370
Let's go a little quick.
01:13:46.370 --> 01:13:54.170
This is du t, t0 dt
times u dagger of t, t0.
01:13:56.960 --> 01:14:00.770
Complete set of states
said add something.
01:14:00.770 --> 01:14:03.800
We want to put the t1 here.
01:14:03.800 --> 01:14:08.090
So let's add something
that will help us do that.
01:14:08.090 --> 01:14:14.680
So let's add t, t0 and
put here a u of t0,
01:14:14.680 --> 01:14:18.160
t1 and a u dagger of t0, t1.
01:14:22.600 --> 01:14:27.710
This thing is 1, and I've put
the u dagger of t, t0 here.
01:14:30.260 --> 01:14:36.910
OK, look at this.
01:14:36.910 --> 01:14:44.855
T0 and t1 here and t
dot t1 there like that.
01:14:49.100 --> 01:14:53.545
So actually, we'll do
it the following way.
01:14:56.240 --> 01:15:03.345
Think of this whole
thing, this d dt
01:15:03.345 --> 01:15:06.150
is acting just on this factor.
01:15:06.150 --> 01:15:09.070
But since it's time,
it might as well
01:15:09.070 --> 01:15:13.680
be acting on all of this factor
because this has no time.
01:15:13.680 --> 01:15:22.640
So this is d dt on
u t, t0 u t0, t1.
01:15:28.690 --> 01:15:37.260
And this thing is u of t1m t0.
01:15:37.260 --> 01:15:39.840
The dagger can be
compensated by this.
01:15:39.840 --> 01:15:47.330
And this dagger is u of t0, t.
01:15:47.330 --> 01:15:49.470
This at a t and that's a comma.
01:15:53.325 --> 01:15:54.510
t0, t.
01:15:54.510 --> 01:15:56.490
Yes.
01:15:56.490 --> 01:16:00.391
OK, so should I go there?
01:16:00.391 --> 01:16:00.890
Yes.
01:16:04.190 --> 01:16:05.890
We're almost there.
01:16:05.890 --> 01:16:11.120
You see that the first
derivative is already
01:16:11.120 --> 01:16:20.210
d dt of u of t, t1.
01:16:20.210 --> 01:16:23.370
And the second operator
by compensation
01:16:23.370 --> 01:16:37.490
is u of t1, t, which is the
same as u dagger of t, t1.
01:16:37.490 --> 01:16:43.970
And then, du of t, t1 u dagger
of t, t1 is lambda of t, t1.
01:16:46.540 --> 01:16:50.680
So it's a little
sneaky, the proof,
01:16:50.680 --> 01:16:53.590
but it's totally rigorous.
01:16:53.590 --> 01:16:55.420
And I don't think
there's any step
01:16:55.420 --> 01:16:56.830
you should be worried there.
01:16:56.830 --> 01:17:00.420
They're all very
logical and reasonable.
01:17:00.420 --> 01:17:03.100
So we have two things.
01:17:03.100 --> 01:17:06.150
First of all, that
this quantity,
01:17:06.150 --> 01:17:10.170
even though it looks
like it depends on t0,
01:17:10.170 --> 01:17:14.790
we finally realized that
it does not depend on t0.
01:17:14.790 --> 01:17:19.405
So I will rewrite this
equation as lambda of t.
01:17:26.020 --> 01:17:31.620
And lambda of t
is anti-Hermitian,
01:17:31.620 --> 01:17:39.400
so we will multiply by an
i to make it Hermitian.
01:17:39.400 --> 01:17:45.490
And in fact, lambda has
units of 1 over time.
01:17:45.490 --> 01:17:49.890
Unitary operators have no units.
01:17:49.890 --> 01:17:54.060
They're like numbers,
like 1 or e to the i phi,
01:17:54.060 --> 01:17:55.820
or something like
that-- have no units.
01:17:55.820 --> 01:17:58.880
So this has units
of 1 over time.
01:17:58.880 --> 01:18:07.110
So if I take i h
bar lambda of t,
01:18:07.110 --> 01:18:13.410
this goes from lambda being
anti-Hermitian-- this operator
01:18:13.410 --> 01:18:15.520
is now Hermitian.
01:18:15.520 --> 01:18:19.190
This goes from lambda
having units of 1 over time
01:18:19.190 --> 01:18:23.680
to this thing having
units of energy.
01:18:23.680 --> 01:18:38.185
So this is a Hermitian
operator with units of energy.
01:18:41.260 --> 01:18:45.390
Well, I guess not much
more needs to be said.
01:18:45.390 --> 01:18:49.360
If that's a Hermitian
operator with units of energy,
01:18:49.360 --> 01:18:55.450
we will give it a name
called H, or Hamiltonian.
01:18:55.450 --> 01:19:00.040
i h bar lambda of t.
01:19:00.040 --> 01:19:07.510
Take this equation and multiply
by i h bar to get i h bar
01:19:07.510 --> 01:19:13.990
d dt of psi is equal
to this i h bar
01:19:13.990 --> 01:19:19.581
lambda, which is
h of t psi of t.
01:19:22.880 --> 01:19:25.680
Schrodinger equation.
01:19:25.680 --> 01:19:28.820
So we really got it.
01:19:28.820 --> 01:19:31.220
That's the Schrodinger equation.
01:19:31.220 --> 01:19:33.440
That's the question
that must be satisfied
01:19:33.440 --> 01:19:38.020
by any system governed by
unitary time evolution.
01:19:38.020 --> 01:19:41.080
There's not more information
in the Schrodinger equation
01:19:41.080 --> 01:19:44.580
than unitary time evolution.
01:19:44.580 --> 01:19:49.290
But it allows you to
turn the problem around.
01:19:49.290 --> 01:19:53.970
You see, when you went to
invent a quantum system,
01:19:53.970 --> 01:19:58.580
you don't quite know how
to find this operator u.
01:19:58.580 --> 01:20:03.210
If you knew u, you know
how to evolve anything.
01:20:03.210 --> 01:20:06.550
And you don't have
any more questions.
01:20:06.550 --> 01:20:10.820
All your questions in life
have been answered by that.
01:20:10.820 --> 01:20:12.880
You know how to find the future.
01:20:12.880 --> 01:20:14.900
You can invest in
the stock market.
01:20:14.900 --> 01:20:16.380
You can do anything now.
01:20:19.660 --> 01:20:24.860
Anyway, but the unitary operator
then gives you the Hamiltonian.
01:20:24.860 --> 01:20:28.420
So if somebody tells you,
here's my unitary operator.
01:20:28.420 --> 01:20:31.150
And they ask you, what
is the Hamiltonian?
01:20:31.150 --> 01:20:36.167
You go here and calculate
I h bar lambda, where
01:20:36.167 --> 01:20:37.250
lambda is this derivative.
01:20:37.250 --> 01:20:38.375
And that's the Hamiltonian.
01:20:41.360 --> 01:20:45.160
And we conversely,
if you are lucky--
01:20:45.160 --> 01:20:47.560
and that's what we're
going to do next time.
01:20:47.560 --> 01:20:50.390
If you have a
Hamiltonian, you try
01:20:50.390 --> 01:20:52.870
to find the unitary
time evolution.
01:20:52.870 --> 01:20:55.260
That's all you want to know.
01:20:55.260 --> 01:20:59.640
But that's a harder problem
because you have a differential
01:20:59.640 --> 01:21:00.920
equation.
01:21:00.920 --> 01:21:06.490
You have h, which is here
, and you are to find u.
01:21:06.490 --> 01:21:11.025
So it's a first-order matrix
differential equation.
01:21:11.025 --> 01:21:13.860
So it's not a simple problem.
01:21:13.860 --> 01:21:16.020
But why do we like Hamiltonians?
01:21:16.020 --> 01:21:18.710
Because Hamiltonians
have to do with energy.
01:21:18.710 --> 01:21:22.460
And we can get inspired
and write quantum systems
01:21:22.460 --> 01:21:26.120
because we know the energy
functional of systems.
01:21:26.120 --> 01:21:29.760
So we invent a Hamiltonian
and typically try
01:21:29.760 --> 01:21:33.520
to find the unitary
time operator.
01:21:33.520 --> 01:21:35.970
But logically
speaking, there's not
01:21:35.970 --> 01:21:39.590
more and no less in the
Schrodinger equation
01:21:39.590 --> 01:21:42.970
than the postulate of
unitary time evolution.
01:21:42.970 --> 01:21:46.400
All right, we'll
see you next week.
01:21:46.400 --> 01:21:46.900
In fact--
01:21:46.900 --> 01:21:47.800
[APPLAUSE]
01:21:47.800 --> 01:21:49.650
Thank you.