1
00:00:00,870 --> 00:00:02,490
PROFESSOR: So,
adiabatic results.

2
00:00:02,490 --> 00:00:07,380
So last time we just
followed and tried

3
00:00:07,380 --> 00:00:10,560
to use an instantaneous
eigenstate

4
00:00:10,560 --> 00:00:13,890
to construct the solution
of the Schrodinger equation.

5
00:00:13,890 --> 00:00:16,140
Our result was that
we couldn't quite

6
00:00:16,140 --> 00:00:19,510
construct the solution of
the Schrodinger equation.

7
00:00:19,510 --> 00:00:24,210
What we wrote didn't exactly
solve the Schrodinger equation.

8
00:00:24,210 --> 00:00:28,320
But we claimed that it was
important and interesting.

9
00:00:28,320 --> 00:00:32,250
And therefore let me
remind you of what we said.

10
00:00:32,250 --> 00:00:39,990
So we showed that the
wave function, psi of t,

11
00:00:39,990 --> 00:00:48,720
could be written as a constant
here, e to the i theta of t, e

12
00:00:48,720 --> 00:00:57,170
to the i gamma of t
times this wave function.

13
00:00:57,170 --> 00:01:01,540
And that wave
function here was what

14
00:01:01,540 --> 00:01:06,460
we called an instantaneous
eigenstate of the Hamiltonian.

15
00:01:06,460 --> 00:01:19,036
So H of t acting on psi of t
was, at any instant of time,

16
00:01:19,036 --> 00:01:20,310
an eigenstate.

17
00:01:24,190 --> 00:01:27,750
Nevertheless, we said that
these eigenstates are not

18
00:01:27,750 --> 00:01:30,600
solutions of the Schrodinger
equation in general.

19
00:01:30,600 --> 00:01:37,500
They solve this funny time--

20
00:01:37,500 --> 00:01:41,880
inspired by the time-independent
Schrodinger equation,

21
00:01:41,880 --> 00:01:44,400
this instantaneous
eigenstate condition.

22
00:01:44,400 --> 00:01:48,030
But when you try to solve
the Schrodinger equation,

23
00:01:48,030 --> 00:01:50,950
this psi of t
would not solve it.

24
00:01:50,950 --> 00:01:53,490
So we tried to
add things, and we

25
00:01:53,490 --> 00:01:57,780
constructed what seemed
to be close to a solution.

26
00:01:57,780 --> 00:02:00,420
And there it is.

27
00:02:00,420 --> 00:02:03,850
What was theta of t?

28
00:02:03,850 --> 00:02:07,190
It was a dynamical
phase, it's called,

29
00:02:07,190 --> 00:02:14,560
and is defined by this integral.

30
00:02:14,560 --> 00:02:21,580
If E of t was a constant, it
would be a minus E times t

31
00:02:21,580 --> 00:02:23,420
over H bar.

32
00:02:23,420 --> 00:02:26,410
And this would be the
familiar phase with which

33
00:02:26,410 --> 00:02:29,620
you evolve energy eigenstates.

34
00:02:29,620 --> 00:02:34,260
The second term, this gamma of
t, was a little more intricate.

35
00:02:34,260 --> 00:02:36,980
We define an
intermediate thing called

36
00:02:36,980 --> 00:02:43,720
nu of t, which was i
times the inner product

37
00:02:43,720 --> 00:02:47,980
of the instantaneous
eigenstate with the derivative

38
00:02:47,980 --> 00:02:51,790
with respect to time of the
instantaneous eigenstate.

39
00:02:51,790 --> 00:02:56,920
We argued that this factor
over here was imaginary.

40
00:02:56,920 --> 00:03:01,600
Therefore we put an
i so that nu is real.

41
00:03:01,600 --> 00:03:05,680
And the gamma of t-- so
these were definitions.

42
00:03:05,680 --> 00:03:11,170
It's defined as 0 to t
dt prime nu of t prime.

43
00:03:17,230 --> 00:03:18,850
OK.

44
00:03:18,850 --> 00:03:25,620
So the claim was
that if you happen

45
00:03:25,620 --> 00:03:31,350
to have a wave function
that, at time equals 0,

46
00:03:31,350 --> 00:03:35,190
it's in one of those
instantaneous eigenstates,

47
00:03:35,190 --> 00:03:40,980
it would remain in such
instantaneous eigenstate up

48
00:03:40,980 --> 00:03:45,421
to phases that are calculable.

49
00:03:45,421 --> 00:03:45,920
OK.

50
00:03:45,920 --> 00:03:51,580
So let's make this a
little more explicit,

51
00:03:51,580 --> 00:03:54,740
in the sense of an
adiabatic theorem.

52
00:03:54,740 --> 00:04:02,280
So I'll also put here that
this is not an exact solution.

53
00:04:02,280 --> 00:04:05,920
So I would say the
wave function at time t

54
00:04:05,920 --> 00:04:08,470
is roughly given
by that quantity.

55
00:04:08,470 --> 00:04:12,310
It's not exactly, because
it's not an exact solution

56
00:04:12,310 --> 00:04:14,450
of the Schrodinger equation.

57
00:04:14,450 --> 00:04:18,310
So if we want to state the
adiabatic theorem a little more

58
00:04:18,310 --> 00:04:27,340
precisely, we consider a set
of instantaneous eigenstates.

59
00:04:27,340 --> 00:04:32,065
Instantaneous eigenstates.

60
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.

61
00:04:52,340 --> 00:04:58,700
And here n will go from 1
up to possibly infinity.

62
00:04:58,700 --> 00:05:00,980
It just doesn't have to stop.

63
00:05:00,980 --> 00:05:04,340
It can go on forever.

64
00:05:04,340 --> 00:05:04,840
OK.

65
00:05:04,840 --> 00:05:07,170
Equipped with the
instantaneous eigenstates,

66
00:05:07,170 --> 00:05:10,500
you might decide
that you initially

67
00:05:10,500 --> 00:05:12,990
want to be in a state.

68
00:05:12,990 --> 00:05:24,090
So psi at t equals 0
is given by psi K at 0.

69
00:05:24,090 --> 00:05:27,040
That's your initial condition.

70
00:05:27,040 --> 00:05:29,140
Now, you look at
this and say, OK,

71
00:05:29,140 --> 00:05:32,340
when is the adiabatic
result that you're

72
00:05:32,340 --> 00:05:35,450
going to stay roughly in
that energy eigenstate going

73
00:05:35,450 --> 00:05:36,810
to be true?

74
00:05:36,810 --> 00:05:44,640
This is going to require
that, as you have the energy E

75
00:05:44,640 --> 00:05:50,240
K as a function of time--

76
00:05:50,240 --> 00:05:54,730
here is the energy E K
as a function of time.

77
00:05:54,730 --> 00:05:56,541
0.

78
00:05:56,541 --> 00:05:57,600
So I'll write it here.

79
00:06:00,110 --> 00:06:03,500
You should have that
this doesn't get mixed,

80
00:06:03,500 --> 00:06:07,070
or doesn't coincide,
with some other energy,

81
00:06:07,070 --> 00:06:10,430
like something like
that, the crossing,

82
00:06:10,430 --> 00:06:13,610
or that they touch
with another one.

83
00:06:13,610 --> 00:06:15,710
You should keep them separate.

84
00:06:15,710 --> 00:06:18,020
You know that the
general perturbation

85
00:06:18,020 --> 00:06:20,610
theory makes life complicated.

86
00:06:20,610 --> 00:06:25,080
And if you find another
state that gets very close,

87
00:06:25,080 --> 00:06:28,400
the adiabatic theorem or
the adiabatic approximation

88
00:06:28,400 --> 00:06:29,490
might go wrong.

89
00:06:29,490 --> 00:06:32,640
So we want other
energy eigenstates.

90
00:06:32,640 --> 00:06:40,790
Here is E K minus 1,
and here is E K plus 1.

91
00:06:40,790 --> 00:06:45,590
So you have all these
energy eigenvalues

92
00:06:45,590 --> 00:06:47,390
as a function of time.

93
00:06:47,390 --> 00:06:51,480
And you want that,
for example, E K--

94
00:06:51,480 --> 00:06:56,300
you order all your energies,
and you want that E K of t

95
00:06:56,300 --> 00:06:59,690
is definitely less
than E K plus 1

96
00:06:59,690 --> 00:07:04,860
of t, which is less than
or equal to the other ones.

97
00:07:04,860 --> 00:07:08,660
And this is definitely
greater than E K minus 1

98
00:07:08,660 --> 00:07:12,690
of t, which is greater or
equal than the other one.

99
00:07:12,690 --> 00:07:16,700
So the instantaneous
energy eigenstates

100
00:07:16,700 --> 00:07:20,870
that you are focused in
should be well separated

101
00:07:20,870 --> 00:07:21,950
from the other ones.

102
00:07:21,950 --> 00:07:24,530
Otherwise you could
run into difficulties.

103
00:07:24,530 --> 00:07:31,710
So that's definitely
necessary for this to hold.

104
00:07:31,710 --> 00:07:37,530
Then the adiabatic
theorem says that psi of t

105
00:07:37,530 --> 00:07:43,450
will be approximately
equal to e to the i theta

106
00:07:43,450 --> 00:07:52,780
K of t, e to the i gamma
K of t times psi K of t.

107
00:07:55,550 --> 00:07:58,240
So I'm keeping a little
difference of notation,

108
00:07:58,240 --> 00:08:00,200
trying to be careful.

109
00:08:00,200 --> 00:08:06,310
These wave functions, at all
times, I put them with a line

110
00:08:06,310 --> 00:08:08,020
below the psi.

111
00:08:08,020 --> 00:08:12,070
And these instantaneous energy
eigenstates, I don't put that.

112
00:08:12,070 --> 00:08:17,470
So a little difference that
should help you determine this.

113
00:08:17,470 --> 00:08:21,400
Now, what are these
theta K or gamma K?

114
00:08:21,400 --> 00:08:25,360
Are just the same quantities.

115
00:08:25,360 --> 00:08:28,240
I might as well write
them for completeness.

116
00:08:28,240 --> 00:08:31,600
Theta K of t is
what you had before.

117
00:08:31,600 --> 00:08:34,960
Minus 1 over H bar 0 to t.

118
00:08:34,960 --> 00:08:40,630
But this time E
K of the t prime.

119
00:08:40,630 --> 00:08:43,990
Then you have nu of t.

120
00:08:43,990 --> 00:08:54,610
Nu K is equal to i psi
K of t psi K dot of t.

121
00:08:54,610 --> 00:09:01,150
And finally the gamma K of
t is the integral from 0

122
00:09:01,150 --> 00:09:09,670
to t of nu of t prime dt
prime K. Everything with K,

123
00:09:09,670 --> 00:09:13,000
corresponding to
the Kth eigenstate.

124
00:09:13,000 --> 00:09:16,240
So this is a more
precise version

125
00:09:16,240 --> 00:09:25,120
already of the statement of
the adiabatic approximation.

126
00:09:25,120 --> 00:09:29,890
You want to make it
even more precise?

127
00:09:29,890 --> 00:09:35,680
We'll say something about
the error in this formula.

128
00:09:35,680 --> 00:09:40,630
In a sense, the hard work in
the adiabatic approximation

129
00:09:40,630 --> 00:09:44,530
is telling you how
much error there is.

130
00:09:44,530 --> 00:09:49,400
So we will not go through
proof of the error.

131
00:09:49,400 --> 00:09:51,910
We'll just motivate it.

132
00:09:51,910 --> 00:09:54,970
And perhaps this will be
discussed in recitation.

133
00:09:54,970 --> 00:09:59,200
It's an interesting
subject, and goes back

134
00:09:59,200 --> 00:10:03,430
to work of Born, Max
Born, the one that

135
00:10:03,430 --> 00:10:06,040
gave the probabilistic
interpretation of quantum

136
00:10:06,040 --> 00:10:10,210
mechanics, and Fock,
of Fock space states

137
00:10:10,210 --> 00:10:12,730
of the harmonic oscillator.

138
00:10:12,730 --> 00:10:14,890
They did the first
work doing that.

139
00:10:14,890 --> 00:10:19,740
Then a Japanese fellow,
Kato, in the middle '50s--

140
00:10:19,740 --> 00:10:23,530
so Born and Fock
was 1928, I believe.

141
00:10:23,530 --> 00:10:28,180
And in the '50s, Kato
improved the analysis

142
00:10:28,180 --> 00:10:29,970
of the adiabatic theorem.

143
00:10:29,970 --> 00:10:35,890
And this whole thing
has many applications

144
00:10:35,890 --> 00:10:41,560
to the theory of molecules
and spin states and systems.

145
00:10:41,560 --> 00:10:46,190
We will begin to see
some of this today.

146
00:10:46,190 --> 00:10:52,420
And had the big revival
with the discovery of Berry

147
00:10:52,420 --> 00:10:58,840
that there's quite a lot
of physics in this phase.

148
00:10:58,840 --> 00:11:01,330
This phase here--
this is a normal phase

149
00:11:01,330 --> 00:11:04,100
that you know for
energy eigenstates,

150
00:11:04,100 --> 00:11:05,560
slightly generalized.

151
00:11:05,560 --> 00:11:07,820
But this phase is
very different.

152
00:11:07,820 --> 00:11:12,200
And that's what Barry
elaborated and explained

153
00:11:12,200 --> 00:11:16,360
and showed that it could be
observable in some cases.

154
00:11:16,360 --> 00:11:19,420
And it was a pretty
nice discovery.

155
00:11:19,420 --> 00:11:24,640
So basically, the
adiabatic statement here

156
00:11:24,640 --> 00:11:29,790
says that you don't jump to
another instantaneous energy

157
00:11:29,790 --> 00:11:34,030
eigenstate, you remain in
that instantaneous energy

158
00:11:34,030 --> 00:11:34,810
eigenstate.

159
00:11:39,460 --> 00:11:40,530
OK.

160
00:11:40,530 --> 00:11:45,240
Another thing that is important
is that sometimes you say,

161
00:11:45,240 --> 00:11:47,880
well, these are phases.

162
00:11:47,880 --> 00:11:50,940
When you have states,
the phases are

163
00:11:50,940 --> 00:11:54,120
unobservable over all
phases of a state.

164
00:11:54,120 --> 00:11:56,650
You calculate the
state after a while.

165
00:11:56,650 --> 00:11:57,480
It has a phase.

166
00:11:57,480 --> 00:12:00,180
Well, that phase of
your physical state

167
00:12:00,180 --> 00:12:03,360
doesn't make any difference.

168
00:12:03,360 --> 00:12:04,500
Well, that's true.

169
00:12:04,500 --> 00:12:07,950
The overall phase of a state
doesn't make any difference.

170
00:12:07,950 --> 00:12:13,650
But if you have a system that
is in a superposition of two

171
00:12:13,650 --> 00:12:16,470
quantum states,
one corresponding

172
00:12:16,470 --> 00:12:19,980
to one instantaneous
energy eigenstate and one

173
00:12:19,980 --> 00:12:21,880
corresponding to another--

174
00:12:21,880 --> 00:12:29,160
so there's psi equal to psi
1 and psi 2, one for each--

175
00:12:29,160 --> 00:12:32,080
psi 1, by the linearity
of quantum mechanics,

176
00:12:32,080 --> 00:12:35,160
is going to develop
its own phase.

177
00:12:35,160 --> 00:12:38,260
Psi 2 is going to
develop its own phase.

178
00:12:38,260 --> 00:12:42,840
And now the relative
phase is observable.

179
00:12:42,840 --> 00:12:44,870
When you have a
superposition of states,

180
00:12:44,870 --> 00:12:47,940
the relative phase
is observable.

181
00:12:47,940 --> 00:12:53,070
And therefore, if the two states
evolve with different phases,

182
00:12:53,070 --> 00:12:56,100
you can have observable
consequences.

183
00:12:56,100 --> 00:12:59,500
So there is something
that can happen here,

184
00:12:59,500 --> 00:13:02,540
and something that
can be measured.