WEBVTT

00:00:00.870 --> 00:00:02.490
PROFESSOR: So,
adiabatic results.

00:00:02.490 --> 00:00:07.380
So last time we just
followed and tried

00:00:07.380 --> 00:00:10.560
to use an instantaneous
eigenstate

00:00:10.560 --> 00:00:13.890
to construct the solution
of the Schrodinger equation.

00:00:13.890 --> 00:00:16.140
Our result was that
we couldn't quite

00:00:16.140 --> 00:00:19.510
construct the solution of
the Schrodinger equation.

00:00:19.510 --> 00:00:24.210
What we wrote didn't exactly
solve the Schrodinger equation.

00:00:24.210 --> 00:00:28.320
But we claimed that it was
important and interesting.

00:00:28.320 --> 00:00:32.250
And therefore let me
remind you of what we said.

00:00:32.250 --> 00:00:39.990
So we showed that the
wave function, psi of t,

00:00:39.990 --> 00:00:48.720
could be written as a constant
here, e to the i theta of t, e

00:00:48.720 --> 00:00:57.170
to the i gamma of t
times this wave function.

00:00:57.170 --> 00:01:01.540
And that wave
function here was what

00:01:01.540 --> 00:01:06.460
we called an instantaneous
eigenstate of the Hamiltonian.

00:01:06.460 --> 00:01:19.036
So H of t acting on psi of t
was, at any instant of time,

00:01:19.036 --> 00:01:20.310
an eigenstate.

00:01:24.190 --> 00:01:27.750
Nevertheless, we said that
these eigenstates are not

00:01:27.750 --> 00:01:30.600
solutions of the Schrodinger
equation in general.

00:01:30.600 --> 00:01:37.500
They solve this funny time--

00:01:37.500 --> 00:01:41.880
inspired by the time-independent
Schrodinger equation,

00:01:41.880 --> 00:01:44.400
this instantaneous
eigenstate condition.

00:01:44.400 --> 00:01:48.030
But when you try to solve
the Schrodinger equation,

00:01:48.030 --> 00:01:50.950
this psi of t
would not solve it.

00:01:50.950 --> 00:01:53.490
So we tried to
add things, and we

00:01:53.490 --> 00:01:57.780
constructed what seemed
to be close to a solution.

00:01:57.780 --> 00:02:00.420
And there it is.

00:02:00.420 --> 00:02:03.850
What was theta of t?

00:02:03.850 --> 00:02:07.190
It was a dynamical
phase, it's called,

00:02:07.190 --> 00:02:14.560
and is defined by this integral.

00:02:14.560 --> 00:02:21.580
If E of t was a constant, it
would be a minus E times t

00:02:21.580 --> 00:02:23.420
over H bar.

00:02:23.420 --> 00:02:26.410
And this would be the
familiar phase with which

00:02:26.410 --> 00:02:29.620
you evolve energy eigenstates.

00:02:29.620 --> 00:02:34.260
The second term, this gamma of
t, was a little more intricate.

00:02:34.260 --> 00:02:36.980
We define an
intermediate thing called

00:02:36.980 --> 00:02:43.720
nu of t, which was i
times the inner product

00:02:43.720 --> 00:02:47.980
of the instantaneous
eigenstate with the derivative

00:02:47.980 --> 00:02:51.790
with respect to time of the
instantaneous eigenstate.

00:02:51.790 --> 00:02:56.920
We argued that this factor
over here was imaginary.

00:02:56.920 --> 00:03:01.600
Therefore we put an
i so that nu is real.

00:03:01.600 --> 00:03:05.680
And the gamma of t-- so
these were definitions.

00:03:05.680 --> 00:03:11.170
It's defined as 0 to t
dt prime nu of t prime.

00:03:17.230 --> 00:03:18.850
OK.

00:03:18.850 --> 00:03:25.620
So the claim was
that if you happen

00:03:25.620 --> 00:03:31.350
to have a wave function
that, at time equals 0,

00:03:31.350 --> 00:03:35.190
it's in one of those
instantaneous eigenstates,

00:03:35.190 --> 00:03:40.980
it would remain in such
instantaneous eigenstate up

00:03:40.980 --> 00:03:45.421
to phases that are calculable.

00:03:45.421 --> 00:03:45.920
OK.

00:03:45.920 --> 00:03:51.580
So let's make this a
little more explicit,

00:03:51.580 --> 00:03:54.740
in the sense of an
adiabatic theorem.

00:03:54.740 --> 00:04:02.280
So I'll also put here that
this is not an exact solution.

00:04:02.280 --> 00:04:05.920
So I would say the
wave function at time t

00:04:05.920 --> 00:04:08.470
is roughly given
by that quantity.

00:04:08.470 --> 00:04:12.310
It's not exactly, because
it's not an exact solution

00:04:12.310 --> 00:04:14.450
of the Schrodinger equation.

00:04:14.450 --> 00:04:18.310
So if we want to state the
adiabatic theorem a little more

00:04:18.310 --> 00:04:27.340
precisely, we consider a set
of instantaneous eigenstates.

00:04:27.340 --> 00:04:32.065
Instantaneous eigenstates.

00:04:35.420 --> 00:04:47.903
And that is H of t psi n of
t equal E n of t psi n of t.

00:04:52.340 --> 00:04:58.700
And here n will go from 1
up to possibly infinity.

00:04:58.700 --> 00:05:00.980
It just doesn't have to stop.

00:05:00.980 --> 00:05:04.340
It can go on forever.

00:05:04.340 --> 00:05:04.840
OK.

00:05:04.840 --> 00:05:07.170
Equipped with the
instantaneous eigenstates,

00:05:07.170 --> 00:05:10.500
you might decide
that you initially

00:05:10.500 --> 00:05:12.990
want to be in a state.

00:05:12.990 --> 00:05:24.090
So psi at t equals 0
is given by psi K at 0.

00:05:24.090 --> 00:05:27.040
That's your initial condition.

00:05:27.040 --> 00:05:29.140
Now, you look at
this and say, OK,

00:05:29.140 --> 00:05:32.340
when is the adiabatic
result that you're

00:05:32.340 --> 00:05:35.450
going to stay roughly in
that energy eigenstate going

00:05:35.450 --> 00:05:36.810
to be true?

00:05:36.810 --> 00:05:44.640
This is going to require
that, as you have the energy E

00:05:44.640 --> 00:05:50.240
K as a function of time--

00:05:50.240 --> 00:05:54.730
here is the energy E K
as a function of time.

00:05:54.730 --> 00:05:56.541
0.

00:05:56.541 --> 00:05:57.600
So I'll write it here.

00:06:00.110 --> 00:06:03.500
You should have that
this doesn't get mixed,

00:06:03.500 --> 00:06:07.070
or doesn't coincide,
with some other energy,

00:06:07.070 --> 00:06:10.430
like something like
that, the crossing,

00:06:10.430 --> 00:06:13.610
or that they touch
with another one.

00:06:13.610 --> 00:06:15.710
You should keep them separate.

00:06:15.710 --> 00:06:18.020
You know that the
general perturbation

00:06:18.020 --> 00:06:20.610
theory makes life complicated.

00:06:20.610 --> 00:06:25.080
And if you find another
state that gets very close,

00:06:25.080 --> 00:06:28.400
the adiabatic theorem or
the adiabatic approximation

00:06:28.400 --> 00:06:29.490
might go wrong.

00:06:29.490 --> 00:06:32.640
So we want other
energy eigenstates.

00:06:32.640 --> 00:06:40.790
Here is E K minus 1,
and here is E K plus 1.

00:06:40.790 --> 00:06:45.590
So you have all these
energy eigenvalues

00:06:45.590 --> 00:06:47.390
as a function of time.

00:06:47.390 --> 00:06:51.480
And you want that,
for example, E K--

00:06:51.480 --> 00:06:56.300
you order all your energies,
and you want that E K of t

00:06:56.300 --> 00:06:59.690
is definitely less
than E K plus 1

00:06:59.690 --> 00:07:04.860
of t, which is less than
or equal to the other ones.

00:07:04.860 --> 00:07:08.660
And this is definitely
greater than E K minus 1

00:07:08.660 --> 00:07:12.690
of t, which is greater or
equal than the other one.

00:07:12.690 --> 00:07:16.700
So the instantaneous
energy eigenstates

00:07:16.700 --> 00:07:20.870
that you are focused in
should be well separated

00:07:20.870 --> 00:07:21.950
from the other ones.

00:07:21.950 --> 00:07:24.530
Otherwise you could
run into difficulties.

00:07:24.530 --> 00:07:31.710
So that's definitely
necessary for this to hold.

00:07:31.710 --> 00:07:37.530
Then the adiabatic
theorem says that psi of t

00:07:37.530 --> 00:07:43.450
will be approximately
equal to e to the i theta

00:07:43.450 --> 00:07:52.780
K of t, e to the i gamma
K of t times psi K of t.

00:07:55.550 --> 00:07:58.240
So I'm keeping a little
difference of notation,

00:07:58.240 --> 00:08:00.200
trying to be careful.

00:08:00.200 --> 00:08:06.310
These wave functions, at all
times, I put them with a line

00:08:06.310 --> 00:08:08.020
below the psi.

00:08:08.020 --> 00:08:12.070
And these instantaneous energy
eigenstates, I don't put that.

00:08:12.070 --> 00:08:17.470
So a little difference that
should help you determine this.

00:08:17.470 --> 00:08:21.400
Now, what are these
theta K or gamma K?

00:08:21.400 --> 00:08:25.360
Are just the same quantities.

00:08:25.360 --> 00:08:28.240
I might as well write
them for completeness.

00:08:28.240 --> 00:08:31.600
Theta K of t is
what you had before.

00:08:31.600 --> 00:08:34.960
Minus 1 over H bar 0 to t.

00:08:34.960 --> 00:08:40.630
But this time E
K of the t prime.

00:08:40.630 --> 00:08:43.990
Then you have nu of t.

00:08:43.990 --> 00:08:54.610
Nu K is equal to i psi
K of t psi K dot of t.

00:08:54.610 --> 00:09:01.150
And finally the gamma K of
t is the integral from 0

00:09:01.150 --> 00:09:09.670
to t of nu of t prime dt
prime K. Everything with K,

00:09:09.670 --> 00:09:13.000
corresponding to
the Kth eigenstate.

00:09:13.000 --> 00:09:16.240
So this is a more
precise version

00:09:16.240 --> 00:09:25.120
already of the statement of
the adiabatic approximation.

00:09:25.120 --> 00:09:29.890
You want to make it
even more precise?

00:09:29.890 --> 00:09:35.680
We'll say something about
the error in this formula.

00:09:35.680 --> 00:09:40.630
In a sense, the hard work in
the adiabatic approximation

00:09:40.630 --> 00:09:44.530
is telling you how
much error there is.

00:09:44.530 --> 00:09:49.400
So we will not go through
proof of the error.

00:09:49.400 --> 00:09:51.910
We'll just motivate it.

00:09:51.910 --> 00:09:54.970
And perhaps this will be
discussed in recitation.

00:09:54.970 --> 00:09:59.200
It's an interesting
subject, and goes back

00:09:59.200 --> 00:10:03.430
to work of Born, Max
Born, the one that

00:10:03.430 --> 00:10:06.040
gave the probabilistic
interpretation of quantum

00:10:06.040 --> 00:10:10.210
mechanics, and Fock,
of Fock space states

00:10:10.210 --> 00:10:12.730
of the harmonic oscillator.

00:10:12.730 --> 00:10:14.890
They did the first
work doing that.

00:10:14.890 --> 00:10:19.740
Then a Japanese fellow,
Kato, in the middle '50s--

00:10:19.740 --> 00:10:23.530
so Born and Fock
was 1928, I believe.

00:10:23.530 --> 00:10:28.180
And in the '50s, Kato
improved the analysis

00:10:28.180 --> 00:10:29.970
of the adiabatic theorem.

00:10:29.970 --> 00:10:35.890
And this whole thing
has many applications

00:10:35.890 --> 00:10:41.560
to the theory of molecules
and spin states and systems.

00:10:41.560 --> 00:10:46.190
We will begin to see
some of this today.

00:10:46.190 --> 00:10:52.420
And had the big revival
with the discovery of Berry

00:10:52.420 --> 00:10:58.840
that there's quite a lot
of physics in this phase.

00:10:58.840 --> 00:11:01.330
This phase here--
this is a normal phase

00:11:01.330 --> 00:11:04.100
that you know for
energy eigenstates,

00:11:04.100 --> 00:11:05.560
slightly generalized.

00:11:05.560 --> 00:11:07.820
But this phase is
very different.

00:11:07.820 --> 00:11:12.200
And that's what Barry
elaborated and explained

00:11:12.200 --> 00:11:16.360
and showed that it could be
observable in some cases.

00:11:16.360 --> 00:11:19.420
And it was a pretty
nice discovery.

00:11:19.420 --> 00:11:24.640
So basically, the
adiabatic statement here

00:11:24.640 --> 00:11:29.790
says that you don't jump to
another instantaneous energy

00:11:29.790 --> 00:11:34.030
eigenstate, you remain in
that instantaneous energy

00:11:34.030 --> 00:11:34.810
eigenstate.

00:11:39.460 --> 00:11:40.530
OK.

00:11:40.530 --> 00:11:45.240
Another thing that is important
is that sometimes you say,

00:11:45.240 --> 00:11:47.880
well, these are phases.

00:11:47.880 --> 00:11:50.940
When you have states,
the phases are

00:11:50.940 --> 00:11:54.120
unobservable over all
phases of a state.

00:11:54.120 --> 00:11:56.650
You calculate the
state after a while.

00:11:56.650 --> 00:11:57.480
It has a phase.

00:11:57.480 --> 00:12:00.180
Well, that phase of
your physical state

00:12:00.180 --> 00:12:03.360
doesn't make any difference.

00:12:03.360 --> 00:12:04.500
Well, that's true.

00:12:04.500 --> 00:12:07.950
The overall phase of a state
doesn't make any difference.

00:12:07.950 --> 00:12:13.650
But if you have a system that
is in a superposition of two

00:12:13.650 --> 00:12:16.470
quantum states,
one corresponding

00:12:16.470 --> 00:12:19.980
to one instantaneous
energy eigenstate and one

00:12:19.980 --> 00:12:21.880
corresponding to another--

00:12:21.880 --> 00:12:29.160
so there's psi equal to psi
1 and psi 2, one for each--

00:12:29.160 --> 00:12:32.080
psi 1, by the linearity
of quantum mechanics,

00:12:32.080 --> 00:12:35.160
is going to develop
its own phase.

00:12:35.160 --> 00:12:38.260
Psi 2 is going to
develop its own phase.

00:12:38.260 --> 00:12:42.840
And now the relative
phase is observable.

00:12:42.840 --> 00:12:44.870
When you have a
superposition of states,

00:12:44.870 --> 00:12:47.940
the relative phase
is observable.

00:12:47.940 --> 00:12:53.070
And therefore, if the two states
evolve with different phases,

00:12:53.070 --> 00:12:56.100
you can have observable
consequences.

00:12:56.100 --> 00:12:59.500
So there is something
that can happen here,

00:12:59.500 --> 00:13:02.540
and something that
can be measured.