1
00:00:00,917 --> 00:00:01,500
PROFESSOR: OK.

2
00:00:01,500 --> 00:00:14,020
Let me then say a little
more about that extra term.

3
00:00:14,020 --> 00:00:18,020
Let's rewrite it in a
slightly different way

4
00:00:18,020 --> 00:00:22,180
so that we get the
feeling of what

5
00:00:22,180 --> 00:00:24,120
it has to do with Hamiltonians.

6
00:00:26,960 --> 00:00:30,720
So this is a coupling term.

7
00:00:30,720 --> 00:00:39,060
So we try to understand, what is
that coupling psi k, psi n dot?

8
00:00:39,060 --> 00:00:40,720
How could we write it?

9
00:00:40,720 --> 00:00:47,200
So what is psi k.

10
00:00:47,200 --> 00:00:48,705
Psi n dot?

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00:01:00,240 --> 00:01:04,379
To figure this out, we
have no option, probably,

12
00:01:04,379 --> 00:01:09,330
except looking at the
Schrodinger equation--

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00:01:09,330 --> 00:01:11,040
not really the
Schrodinger equation--

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00:01:11,040 --> 00:01:17,830
the time-instantaneous
state conditions--

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00:01:17,830 --> 00:01:23,030
En of t, psi n of t.

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00:01:23,030 --> 00:01:26,810
So you look at this
instantaneous state condition

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00:01:26,810 --> 00:01:30,390
for n and say, OK, I'm
going to differentiate,

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00:01:30,390 --> 00:01:34,080
and then I'm going
to make the overlap.

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00:01:34,080 --> 00:01:37,110
What else could you do?

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00:01:37,110 --> 00:01:40,860
This is the equation
that tells you

21
00:01:40,860 --> 00:01:44,220
how the instantaneous
energy eigenstate is

22
00:01:44,220 --> 00:01:45,940
supposed to change in time.

23
00:01:45,940 --> 00:01:49,920
So it must have the information
of how to compute this,

24
00:01:49,920 --> 00:01:53,100
or how to rewrite it in
terms of other things

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00:01:53,100 --> 00:01:54,090
that are interesting.

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00:01:54,090 --> 00:01:57,900
So let's differentiate
with respect to time.

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00:01:57,900 --> 00:02:03,550
Differentiate-- respect to time.

28
00:02:03,550 --> 00:02:06,280
And we use dots, as well.

29
00:02:06,280 --> 00:02:19,750
So let's write this as H dot
times psi n plus H psi n dot

30
00:02:19,750 --> 00:02:30,895
is equal to En dot, psi
n plus En, psi n dot.

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00:02:33,730 --> 00:02:35,460
And then we do what--

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00:02:38,310 --> 00:02:43,330
we want psi k here
from the left.

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00:02:43,330 --> 00:03:01,430
So we'll have a psi
k, H dot, psi n plus--

34
00:03:01,430 --> 00:03:05,390
Now you're going to
put the psi k here.

35
00:03:05,390 --> 00:03:08,450
The H will act on the left.

36
00:03:08,450 --> 00:03:19,410
So we will have an
Ek psi k, psi n dot.

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That's all that is
left from this term.

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00:03:22,280 --> 00:03:27,600
The psi k came from the
left and was acted by the H

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00:03:27,600 --> 00:03:29,640
to give you an Ek times this.

40
00:03:33,720 --> 00:03:42,720
On this state, I have the psi
k already hitting the psi n.

41
00:03:42,720 --> 00:03:46,920
Now, we are interested
in this term

42
00:03:46,920 --> 00:03:49,110
when k is different from n.

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00:03:52,250 --> 00:03:59,900
These are the couplings to
other states in the analysis

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00:03:59,900 --> 00:04:02,420
of the adiabatic approximation.

45
00:04:02,420 --> 00:04:05,930
So let's take k
different from n.

46
00:04:05,930 --> 00:04:10,490
In which case, this
term will give 0.

47
00:04:10,490 --> 00:04:14,630
Because k is different from
n and the state just hits it.

48
00:04:14,630 --> 00:04:21,622
And here we have, finally,
En times psi k, psi n dot.

49
00:04:25,070 --> 00:04:26,070
So good.

50
00:04:26,070 --> 00:04:33,650
I think we can solve
for this state in terms

51
00:04:33,650 --> 00:04:39,760
of a matrix element of the time
derivative of the Hamiltonian.

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00:04:45,730 --> 00:04:46,925
So what do we have?

53
00:04:52,390 --> 00:05:04,280
We have that psi k,
psi n dot is equal to 1

54
00:05:04,280 --> 00:05:20,030
over En minus Ek times psi
k, H dot, psi n, which is--

55
00:05:20,030 --> 00:05:25,910
we can use the
notation H dot kn.

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00:05:25,910 --> 00:05:32,780
The kn matrix element
of H dot is not

57
00:05:32,780 --> 00:05:39,140
equal to the time derivative
of the kn matrix element of H.

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00:05:39,140 --> 00:05:42,680
You calculate the H dot.

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00:05:42,680 --> 00:05:44,720
And you put the kn if you wish.

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00:05:47,420 --> 00:05:52,460
It really is this thing--
the matrix element of H

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00:05:52,460 --> 00:05:58,770
dot kn over En minus Ek.

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The nice thing of
this representation

63
00:06:03,900 --> 00:06:06,450
is that it now
begins to give you

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00:06:06,450 --> 00:06:13,080
a feeling of why we care
even about slow changes.

65
00:06:13,080 --> 00:06:16,710
We have, in this coupling term--

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00:06:20,290 --> 00:06:24,990
this is the coupling term
that will take you away

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00:06:24,990 --> 00:06:28,200
from an instantaneous
energy eigenstate.

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00:06:28,200 --> 00:06:32,980
This is the term that can ruin
your adiabatic approximation.

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00:06:32,980 --> 00:06:36,600
And that term,
now, you understand

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00:06:36,600 --> 00:06:42,210
that it has to do with the rate
of change of the Hamiltonian,

71
00:06:42,210 --> 00:06:47,910
which, after all, is the
thing that drives the physics.

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00:06:47,910 --> 00:06:50,070
You could say that
term must be small,

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00:06:50,070 --> 00:06:54,780
because the instantaneous
energy eigenstates must be slow.

74
00:06:54,780 --> 00:06:56,610
And that's true.

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00:06:56,610 --> 00:07:03,120
But this equation
makes it clear that you

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00:07:03,120 --> 00:07:09,540
have here a matrix element or
some information about the time

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00:07:09,540 --> 00:07:11,430
rate of change of
your Hamiltonian.

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00:07:11,430 --> 00:07:15,480
And if that is slow,
then this term is small.

79
00:07:15,480 --> 00:07:19,320
And presumably, a small term
in a differential equation

80
00:07:19,320 --> 00:07:21,600
has little effect.

81
00:07:21,600 --> 00:07:29,220
So probably, it's a good
idea to just represent

82
00:07:29,220 --> 00:07:31,700
a particular case
of a Hamiltonian

83
00:07:31,700 --> 00:07:35,020
to just complete
this discussion.

84
00:07:35,020 --> 00:07:38,930
So imagine you have
a Hamiltonian that--

85
00:07:38,930 --> 00:07:42,320
it's of this kind, varies a bit.

86
00:07:42,320 --> 00:07:46,190
Maybe it doesn't vary for
time equal less than 0.

87
00:07:46,190 --> 00:07:51,930
It varies for a while,
and then it stops varying.

88
00:07:51,930 --> 00:07:57,920
And let's call the time capital
T-- the time for the variation.

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00:08:00,870 --> 00:08:03,665
This is the typical
thing that people do

90
00:08:03,665 --> 00:08:05,600
in the adiabatic approximation.

91
00:08:05,600 --> 00:08:10,100
So that is the
language of things--

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00:08:10,100 --> 00:08:17,010
nothing, variation during a
time T, nothing after all.

93
00:08:17,010 --> 00:08:21,230
So what is the goal,
the result here?

94
00:08:21,230 --> 00:08:23,780
What is the real
statement at the end

95
00:08:23,780 --> 00:08:27,950
of the day is that
the errors that

96
00:08:27,950 --> 00:08:30,500
appear in the
adiabatic approximation

97
00:08:30,500 --> 00:08:35,490
are all proportional
to 1 over t.

98
00:08:35,490 --> 00:08:38,400
So that is a
mathematical statement.

99
00:08:38,400 --> 00:08:43,299
The 1 over this capital T
controls all the errors.

100
00:08:43,299 --> 00:08:46,230
So if the change
happens very slowly,

101
00:08:46,230 --> 00:08:51,180
the errors go to
0, like 1 over t.

102
00:08:51,180 --> 00:08:53,430
And we can see the
beginning of that,

103
00:08:53,430 --> 00:08:58,100
because I could make a
model of this Hamiltonian

104
00:08:58,100 --> 00:09:00,020
that is a little different.

105
00:09:00,020 --> 00:09:05,230
It just may be a line
growing here, for example.

106
00:09:05,230 --> 00:09:09,180
So this is a
Hamiltonian H of t that

107
00:09:09,180 --> 00:09:18,040
begins as H0 plus t
over capital T times V.

108
00:09:18,040 --> 00:09:23,950
That is for t between
0 and capital T,

109
00:09:23,950 --> 00:09:35,580
and then H0 plus V for t
greater than that capital T.

110
00:09:35,580 --> 00:09:37,920
So here is H0.

111
00:09:37,920 --> 00:09:44,280
Here's H0 plus V.

112
00:09:44,280 --> 00:09:49,680
So if you have this
thing here, you

113
00:09:49,680 --> 00:09:58,290
will have that H dot
is equal to V over T--

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00:09:58,290 --> 00:10:01,350
capital T. And that's intuitive.

115
00:10:01,350 --> 00:10:05,550
The more time you take to
make this finite transition,

116
00:10:05,550 --> 00:10:09,780
the smaller the value
of the derivative.

117
00:10:09,780 --> 00:10:17,970
And therefore, an H
dot matrix element kn

118
00:10:17,970 --> 00:10:22,830
will be 1 over capital
T, the matrix element

119
00:10:22,830 --> 00:10:31,140
kn of the operator V.

120
00:10:31,140 --> 00:10:41,460
This term, this overlap, will
be proportional to 1 over T.

121
00:10:41,460 --> 00:10:45,390
And this error term in
the differential equation

122
00:10:45,390 --> 00:10:51,700
will be proportional
to 1 over capital T.

123
00:10:51,700 --> 00:10:57,790
That takes still some effort
to show that when you integrate

124
00:10:57,790 --> 00:11:02,710
the differential equation, the
error remains 1 over capital T.

125
00:11:02,710 --> 00:11:04,660
It doesn't get increased.

126
00:11:04,660 --> 00:11:09,160
But that is precisely
the statement

127
00:11:09,160 --> 00:11:11,560
of the adiabatic theorem.

128
00:11:11,560 --> 00:11:14,945
The adiabatic
theorem says that--

129
00:11:14,945 --> 00:11:23,000
and I'll quote it,
maybe, this way.

130
00:11:39,340 --> 00:11:44,590
If you have the norm
of a wave function,

131
00:11:44,590 --> 00:11:50,710
you can define the square root
of the overlap of the wave

132
00:11:50,710 --> 00:11:52,670
function.

133
00:11:52,670 --> 00:11:56,620
And the way people
precisely state

134
00:11:56,620 --> 00:12:01,350
the adiabatic theorem is that.

135
00:12:01,350 --> 00:12:17,580
The state psi, during a time
t minus the psi adiabatic--

136
00:12:17,580 --> 00:12:22,320
the [? n sets ?] for the
adiabatic state of time t.

137
00:12:22,320 --> 00:12:23,870
The true-- the different--

138
00:12:23,870 --> 00:12:24,430
I'm sorry.

139
00:12:24,430 --> 00:12:27,580
The difference
between the true state

140
00:12:27,580 --> 00:12:31,780
psi of t and the
adiabatic approximation t,

141
00:12:31,780 --> 00:12:41,020
for t between 0
and T, is less than

142
00:12:41,020 --> 00:12:49,690
or equal than a
number of times T.

143
00:12:49,690 --> 00:12:55,090
So the adiabatic approximation
that we've written,

144
00:12:55,090 --> 00:12:57,850
the state with all
these phases, is

145
00:12:57,850 --> 00:13:02,890
as close to the real
Schrodinger equation solution--

146
00:13:02,890 --> 00:13:07,250
the error is less
than 1 over T, where

147
00:13:07,250 --> 00:13:15,140
T is this time in which you have
nothing here, nothing there.

148
00:13:15,140 --> 00:13:20,000
And this, in fact, gets even
better for larger times.

149
00:13:20,000 --> 00:13:24,050
So after larger times, the
error doesn't increase.

150
00:13:24,050 --> 00:13:25,670
So you can keep--

151
00:13:25,670 --> 00:13:30,950
the change has already
happened, and it doesn't matter.

152
00:13:30,950 --> 00:13:36,920
So basically, we've
argued intuitively

153
00:13:36,920 --> 00:13:38,930
what the adiabatic
approximation should

154
00:13:38,930 --> 00:13:42,830
be by tracking an
instantaneous eigenstate.

155
00:13:42,830 --> 00:13:48,570
We've now done it with explicit
differential equations.

156
00:13:48,570 --> 00:13:52,460
If you have an explicit
problem, you can, after all,

157
00:13:52,460 --> 00:13:56,660
solve this and see
what things happen,

158
00:13:56,660 --> 00:14:01,970
and therefore, reach the
conclusion that then there are

159
00:14:01,970 --> 00:14:05,310
a good number of conditions.

160
00:14:05,310 --> 00:14:07,800
Many situations,
they're slowly varying.

161
00:14:07,800 --> 00:14:10,980
The adiabatic [? n sets ?]
will give a good solution

162
00:14:10,980 --> 00:14:14,290
to the Schrodinger equation.

163
00:14:14,290 --> 00:14:17,740
So that completes our
first part of the analysis.

164
00:14:17,740 --> 00:14:22,400
Now we're going to turn into
a little different story.