WEBVTT
00:00:00.570 --> 00:00:05.020
PROFESSOR: OK, so our
discussion at the beginning
00:00:05.020 --> 00:00:08.950
was based on just
taking the state,
00:00:08.950 --> 00:00:13.690
those instantaneous
energy island states,
00:00:13.690 --> 00:00:18.820
and calculating what phases
would make it satisfy
00:00:18.820 --> 00:00:20.500
the Schrodinger equation.
00:00:20.500 --> 00:00:22.540
And we found those
are the phases that
00:00:22.540 --> 00:00:25.810
came close to satisfying
the Schrodinger equation,
00:00:25.810 --> 00:00:27.430
but not quite.
00:00:27.430 --> 00:00:37.280
So in order to do this under a
more controlled approximation,
00:00:37.280 --> 00:00:43.850
let's do a calculation where
we put all the information in.
00:00:43.850 --> 00:00:50.230
So if you have a
state psi of t, we'll
00:00:50.230 --> 00:00:56.020
write it as a
superposition of states
00:00:56.020 --> 00:00:58.210
of instantaneous eigenstates.
00:01:01.890 --> 00:01:04.470
So this is a general solution.
00:01:10.190 --> 00:01:15.660
Maybe general [? an ?]
[? sets ?] for a solution.
00:01:15.660 --> 00:01:19.960
The wave function-- since
those instantaneous energy
00:01:19.960 --> 00:01:23.170
eigenstates are
complete orthonormal,
00:01:23.170 --> 00:01:29.650
this form a ON basis,
orthonormal basis at all
00:01:29.650 --> 00:01:31.300
times--
00:01:31.300 --> 00:01:35.500
at any time, it's an
orthonormal set of states--
00:01:35.500 --> 00:01:41.300
we should be able to write our
state as that superposition.
00:01:41.300 --> 00:01:43.840
So what we're going
to do is now kind
00:01:43.840 --> 00:01:47.770
of re-do the analysis of the
[INAUDIBLE] approximation
00:01:47.770 --> 00:01:52.240
more generally so that we see,
in fact, equations that show up
00:01:52.240 --> 00:01:54.710
that you can solve in general.
00:01:54.710 --> 00:02:01.460
So the Schrodinger equation
is i h bar dvt of psi
00:02:01.460 --> 00:02:05.140
is equal to H psi.
00:02:05.140 --> 00:02:08.419
So let's look at what
it gives us here.
00:02:08.419 --> 00:02:14.560
So we'll have i
h bar sum over n.
00:02:14.560 --> 00:02:17.050
And I have to
differentiate this state.
00:02:17.050 --> 00:02:19.480
So we get Cn dot--
00:02:19.480 --> 00:02:22.150
dot for time derivatives--
00:02:22.150 --> 00:02:29.620
psi n plus Cn psi n dot--
00:02:29.620 --> 00:02:31.830
this is a time derivative
of this state--
00:02:36.360 --> 00:02:38.700
is equal to H of psi--
00:02:38.700 --> 00:02:42.915
this is the sum over n Cn of t--
00:02:42.915 --> 00:02:51.570
H of psi n, is equal
to E n of t psi n of t.
00:02:58.420 --> 00:03:02.020
OK, so that's your equation.
00:03:02.020 --> 00:03:06.100
Now let's see in various
components what it gives you.
00:03:06.100 --> 00:03:12.790
So to see the various
components, we form an overlap
00:03:12.790 --> 00:03:15.440
with a psi k of t.
00:03:15.440 --> 00:03:19.360
So we'll bring in a psi k of t.
00:03:19.360 --> 00:03:22.630
And what do we get?
00:03:22.630 --> 00:03:24.940
Since these states
are orthonormal,
00:03:24.940 --> 00:03:28.570
psi k, when it comes here,
this is a function of time.
00:03:28.570 --> 00:03:30.080
It doesn't care.
00:03:30.080 --> 00:03:32.740
Psi k hits a psi n.
00:03:32.740 --> 00:03:34.660
That's a Kronecker delta.
00:03:34.660 --> 00:03:35.990
The sum disappears.
00:03:35.990 --> 00:03:43.780
And the only term that
is left here is Ck dot.
00:03:43.780 --> 00:03:49.380
So we get i h bar Ck
dot from this term.
00:03:55.360 --> 00:04:02.430
And let's put the second
term to the right hand side.
00:04:02.430 --> 00:04:07.650
So let's just write what we
get from the right hand side
00:04:07.650 --> 00:04:09.960
and from this term.
00:04:09.960 --> 00:04:12.090
So from the right
hand side, we have
00:04:12.090 --> 00:04:15.990
the psi k on that thing
that is on the right.
00:04:15.990 --> 00:04:21.899
That, again, hits this state
and produces a Kronecker delta.
00:04:21.899 --> 00:04:31.260
So we get Ck Ek of
t from the term that
00:04:31.260 --> 00:04:33.370
was on the right hand side.
00:04:33.370 --> 00:04:37.110
And here, however, we
don't get rid of the sum
00:04:37.110 --> 00:04:42.022
because psi k is not
orthonormal to psi n dot.
00:04:42.022 --> 00:04:45.670
Psi n dot is more complicated.
00:04:45.670 --> 00:04:47.530
So what do we get here?
00:04:47.530 --> 00:04:59.700
Minus i h bar the sum over n psi
k psi n dot inner product Cn.
00:05:08.120 --> 00:05:10.970
OK, that's pretty
close to what we want.
00:05:10.970 --> 00:05:16.060
But let's write it still in
a slightly different way.
00:05:16.060 --> 00:05:21.430
I want to isolate the Ck's.
00:05:21.430 --> 00:05:29.620
So from that sum, I will
separate the Ck part.
00:05:29.620 --> 00:05:34.480
So we'll have Ek of t.
00:05:34.480 --> 00:05:43.670
And there's going to be a term
here, when we have n equal k,
00:05:43.670 --> 00:05:58.820
so I'll bring it out there--
minus i h bar psi k psi k dot
00:05:58.820 --> 00:06:01.210
Ck.
00:06:01.210 --> 00:06:04.300
And the last term
now becomes i h bar
00:06:04.300 --> 00:06:16.800
the sum over n different
from k psi k psi n dot Cn.
00:06:16.800 --> 00:06:21.570
OK, so this is the form
of the equation that
00:06:21.570 --> 00:06:28.060
is nice and gives you a
little understanding of what's
00:06:28.060 --> 00:06:30.800
going on.
00:06:30.800 --> 00:06:34.550
That's a general
treatment of trying
00:06:34.550 --> 00:06:38.870
to make a solution from
instantaneous energy
00:06:38.870 --> 00:06:40.430
eigenstates.
00:06:40.430 --> 00:06:42.870
Here were your instantaneous
energy eigenstates.
00:06:42.870 --> 00:06:45.170
We tried to make a solution.
00:06:45.170 --> 00:06:48.980
That is the full equation.
00:06:48.980 --> 00:06:50.990
What did we do before?
00:06:50.990 --> 00:06:54.290
We used just one of them.
00:06:54.290 --> 00:06:58.680
We took one instantaneous
energy eigenstate
00:06:58.680 --> 00:07:02.930
and we tried to make a solution
by multiplying by one thing,
00:07:02.930 --> 00:07:04.100
and then we tried.
00:07:04.100 --> 00:07:06.980
But then it doesn't
work because when
00:07:06.980 --> 00:07:12.620
you have just one coefficient,
say k, with some fixed k,
00:07:12.620 --> 00:07:14.600
you have this equation.
00:07:14.600 --> 00:07:19.340
But then you couple to
all other coefficients
00:07:19.340 --> 00:07:23.720
where n is different from k.
00:07:23.720 --> 00:07:28.910
So what we did before
was essentially,
00:07:28.910 --> 00:07:35.440
by claiming that
this term is small,
00:07:35.440 --> 00:07:39.610
just focus on this
thing, and this
00:07:39.610 --> 00:07:42.730
is an easily solvable
equation that,
00:07:42.730 --> 00:07:48.070
in fact, gives the type
of solution we have there.
00:07:48.070 --> 00:07:52.090
When you have C
dot equal to this--
00:07:52.090 --> 00:07:56.480
I'll write it in our
previous approximation,
00:07:56.480 --> 00:08:05.712
so in the approximation where
the last term is negligible.
00:08:10.560 --> 00:08:13.990
And we would see why
it could be negligible.
00:08:13.990 --> 00:08:20.770
Then we get just i
h bar Ck dot equals
00:08:20.770 --> 00:08:31.180
Ek of t minus i h bar
psi k psi k dot Ck.
00:08:31.180 --> 00:08:40.150
And this thing is solved
by writing Ck of t
00:08:40.150 --> 00:08:46.930
is equal to e to the 1 over
i h bar integral from 0 to t
00:08:46.930 --> 00:08:56.090
of this whole thing Ek of t
prime minus i h bar psi k psi k
00:08:56.090 --> 00:09:07.260
dot of t prime dt
prime times Ck of 0.
00:09:10.730 --> 00:09:12.790
This is a differential equation.
00:09:12.790 --> 00:09:16.990
So i h bar times the
time derivative of this--
00:09:16.990 --> 00:09:20.650
if you apply a i h
bar time derivative,
00:09:20.650 --> 00:09:25.000
you differentiate with respect
to time, then exponent,
00:09:25.000 --> 00:09:29.110
you get 1 over i h bar
that cancels this i h bar.
00:09:29.110 --> 00:09:32.350
And the derivative of the
exponent is this factor--
00:09:32.350 --> 00:09:38.620
just this standard, first order,
time dependent differential
00:09:38.620 --> 00:09:39.710
equation.
00:09:39.710 --> 00:09:45.490
So last time we said we
ignored possible couplings
00:09:45.490 --> 00:09:49.990
between the different modes
represented by this term,
00:09:49.990 --> 00:09:52.850
and we just solved
this equation,
00:09:52.850 --> 00:09:56.710
which gave us this, which
is exactly what we've
00:09:56.710 --> 00:09:57.970
been writing here.
00:10:00.700 --> 00:10:05.240
e to the I theta of k
comes from the first term
00:10:05.240 --> 00:10:06.710
on that integral.
00:10:06.710 --> 00:10:10.280
And e to the i gamma of k
comes from the second term
00:10:10.280 --> 00:10:12.300
on that integral.
00:10:12.300 --> 00:10:16.290
These are the same things.
00:10:16.290 --> 00:10:21.840
So what is new here is
that there is a coupling.
00:10:21.840 --> 00:10:27.480
And you cannot assume that
just Ck evolves in time some
00:10:27.480 --> 00:10:30.610
particular k, and
the others don't.
00:10:30.610 --> 00:10:33.720
The others will get coupled.
00:10:33.720 --> 00:10:44.160
In particular, if you have
that at time equals 0--
00:10:44.160 --> 00:10:51.870
t equals 0-- some Ck
of 0 is equal to 1,
00:10:51.870 --> 00:10:56.730
but all the other
ones, Ck primes at 0,
00:10:56.730 --> 00:11:04.110
are equal to 0 for all k
prime different from k,
00:11:04.110 --> 00:11:09.660
so your initial condition is
you are in the state k at time
00:11:09.660 --> 00:11:12.570
equals 0.
00:11:12.570 --> 00:11:15.930
That's why you have Ck at time
equals [? 0 to ?] [? a 1. ?]
00:11:15.930 --> 00:11:20.440
And the others are 0
at different times.
00:11:20.440 --> 00:11:24.670
If you look at your
differential equation
00:11:24.670 --> 00:11:33.600
and try to see what happens
after a little time, well,
00:11:33.600 --> 00:11:38.250
we know Ck dot is
going to change,
00:11:38.250 --> 00:11:40.270
is going to have a
non-trivial value.
00:11:40.270 --> 00:11:41.770
This is going to happen.
00:11:41.770 --> 00:11:45.210
But these things are
not going to remain 0.
00:11:45.210 --> 00:11:48.730
Those other states are
going to get populated,
00:11:48.730 --> 00:11:55.500
in particular, i h
bar Ck prime dot.
00:11:55.500 --> 00:11:57.030
Look at the top equation.
00:11:57.030 --> 00:12:04.050
Apply for k equal to k prime
and look at time equals 0.
00:12:04.050 --> 00:12:06.960
What do you get
at time equals 0?
00:12:06.960 --> 00:12:09.730
Well, you would get
all these factor--
00:12:09.730 --> 00:12:15.740
Ek prime times Ck
prime at time equals 0.
00:12:15.740 --> 00:12:18.840
Well, that's 0.
00:12:18.840 --> 00:12:19.950
That's nothing.
00:12:19.950 --> 00:12:23.670
But from the last term,
what would we get?
00:12:23.670 --> 00:12:36.440
Minus i h bar sum over n
different from k prime of psi k
00:12:36.440 --> 00:12:44.520
prime psi n dot Cn
at time equals 0.
00:12:47.542 --> 00:12:53.260
And here, well, the only
one that is different from 0
00:12:53.260 --> 00:12:57.160
at time equals 0 is Ck.
00:12:57.160 --> 00:13:00.400
And k, when n is
equal to k, is allowed
00:13:00.400 --> 00:13:03.040
because we said k prime
is different from k.
00:13:03.040 --> 00:13:05.710
So there is one term here.
00:13:05.710 --> 00:13:15.670
This is minus i h bar
psi k prime psi k dot.
00:13:18.220 --> 00:13:23.560
Only when n equals to k
you get something here
00:13:23.560 --> 00:13:25.870
because this is the
only term that exists.
00:13:25.870 --> 00:13:27.520
And it's equal to 1.
00:13:27.520 --> 00:13:34.840
So immediately, at time equals
0, the other coefficients start
00:13:34.840 --> 00:13:35.830
changing.
00:13:35.830 --> 00:13:40.855
You start populating the
other instantaneous energy
00:13:40.855 --> 00:13:43.060
eigenstates.
00:13:43.060 --> 00:13:47.710
So there is real mixing
in this top thing that
00:13:47.710 --> 00:13:52.420
says it's not rigorous to claim
that you stay in that energy
00:13:52.420 --> 00:13:53.440
eigenstate.
00:13:53.440 --> 00:13:57.070
It starts to couple.
00:13:57.070 --> 00:14:00.990
And if it starts to
couple, then eventually you
00:14:00.990 --> 00:14:02.190
make transitions.
00:14:02.190 --> 00:14:07.760
The only hope, of course, is
that that term is really small.
00:14:07.760 --> 00:14:13.680
And basically, we can
argue how that term
00:14:13.680 --> 00:14:18.730
becomes a little small by
doing a little calculation.
00:14:18.730 --> 00:14:22.650
And then we can be rigorous
and take a long time
00:14:22.650 --> 00:14:25.770
to get to the conclusion,
or we can just state it.
00:14:25.770 --> 00:14:30.600
And that's what we're
going to do after analyzing
00:14:30.600 --> 00:14:32.720
that term a little more.