1
00:00:00,950 --> 00:00:03,260
PROFESSOR: So this is
our adiabatic change.

2
00:00:03,260 --> 00:00:08,890
So now we can say
several things.

3
00:00:08,890 --> 00:00:14,340
OK, if omega is changing slowly,
the energy is changing slowly,

4
00:00:14,340 --> 00:00:20,420
but do we have something that
changes even more slowly,

5
00:00:20,420 --> 00:00:26,390
something that really
almost doesn't change?

6
00:00:26,390 --> 00:00:29,990
What you need here
is basically--

7
00:00:29,990 --> 00:00:32,254
this was a very
important discovery

8
00:00:32,254 --> 00:00:34,130
in classical mechanics.

9
00:00:34,130 --> 00:00:38,240
You need like two
things that change.

10
00:00:38,240 --> 00:00:40,070
Everything is going
to change slowly,

11
00:00:40,070 --> 00:00:44,180
but then there's going to be
one thing that changes slowly

12
00:00:44,180 --> 00:00:46,580
and another thing
that changes slowly,

13
00:00:46,580 --> 00:00:50,600
and they change kind of in
the same way in such a way

14
00:00:50,600 --> 00:00:53,840
that the ratio or some
combination of them

15
00:00:53,840 --> 00:00:58,400
doesn't change almost at all.

16
00:00:58,400 --> 00:00:59,930
That's what we're trying to get.

17
00:01:03,070 --> 00:01:09,700
Anybody knows in classical
mechanics what quantity here

18
00:01:09,700 --> 00:01:11,035
doesn't change much?

19
00:01:27,840 --> 00:01:28,340
Nobody.

20
00:01:30,890 --> 00:01:33,220
No clue?

21
00:01:39,330 --> 00:01:42,990
It's not obvious what
doesn't change much,

22
00:01:42,990 --> 00:01:46,260
but here is the claim.

23
00:01:46,260 --> 00:01:54,570
Claim is that the quantity
that doesn't change much

24
00:01:54,570 --> 00:01:57,945
is, in fact, the energy
divided by omega.

25
00:02:00,710 --> 00:02:03,680
The energy will change slowly.

26
00:02:03,680 --> 00:02:05,690
Omega will change slowly.

27
00:02:05,690 --> 00:02:10,560
But the ratio is almost
not going to change at all.

28
00:02:10,560 --> 00:02:11,780
So here is the claim.

29
00:02:11,780 --> 00:02:17,840
There is an I of t called
adiabatic invariant, which

30
00:02:17,840 --> 00:02:24,090
is basically H of t
divided by omega of t,

31
00:02:24,090 --> 00:02:28,690
and it's almost constant.

32
00:02:35,040 --> 00:02:38,520
And this quantity has the
units of energy times time.

33
00:02:52,300 --> 00:02:55,160
I don't want to give
away the whole story.

34
00:02:55,160 --> 00:02:57,760
But I think it's good
if you, at this moment,

35
00:02:57,760 --> 00:03:03,190
think a second, well, what
could it mean, or do I even have

36
00:03:03,190 --> 00:03:06,310
a clue why this could happen?

37
00:03:06,310 --> 00:03:09,280
And you think oh,
quantum mechanics.

38
00:03:09,280 --> 00:03:12,040
The harmonic oscillator,
what happened?

39
00:03:12,040 --> 00:03:18,130
The energy was equal to
h omega times the level.

40
00:03:22,250 --> 00:03:27,160
So kind of energy
divided by omega

41
00:03:27,160 --> 00:03:29,090
is kind of a nice quantity.

42
00:03:29,090 --> 00:03:32,020
It's a quantum number.

43
00:03:32,020 --> 00:03:35,290
Quantum numbers are
quantized, and they

44
00:03:35,290 --> 00:03:39,250
don't like to
change, because how

45
00:03:39,250 --> 00:03:41,950
could an integer change slowly?

46
00:03:41,950 --> 00:03:45,290
As soon as it changes,
it changes big.

47
00:03:45,290 --> 00:03:49,810
So a little bit of
what we're getting at

48
00:03:49,810 --> 00:03:55,300
is the resistance of a system
to change quantum level.

49
00:03:55,300 --> 00:03:59,050
When something is quantized,
it cannot change slowly,

50
00:03:59,050 --> 00:04:05,290
and the adiabatic invariant is
exploiting in classical physics

51
00:04:05,290 --> 00:04:09,940
that quantum
property, if you wish.

52
00:04:09,940 --> 00:04:12,740
So let's look at that.

53
00:04:12,740 --> 00:04:18,730
So the claim is that the name
i is for adiabatic invariant,

54
00:04:18,730 --> 00:04:24,130
and we can verify it, and get
some intuition as to why those

55
00:04:24,130 --> 00:04:25,130
very slowly.

56
00:04:25,130 --> 00:04:29,050
Now, I cannot prove that
thing doesn't change.

57
00:04:29,050 --> 00:04:33,150
That would be too much, but it's
going to change very slowly.

58
00:04:33,150 --> 00:04:34,960
You will appreciate that.

59
00:04:34,960 --> 00:04:36,220
Let's see.

60
00:04:36,220 --> 00:04:40,780
Let's compute the
derivative, di dt.

61
00:04:40,780 --> 00:04:42,250
So it's a ratio.

62
00:04:42,250 --> 00:04:46,630
So I have omega squared.

63
00:04:46,630 --> 00:04:47,320
Omega.

64
00:04:47,320 --> 00:04:49,330
I'm going to use
dots, and I'm going

65
00:04:49,330 --> 00:04:53,770
to stop writing the
factor, the key dependents.

66
00:04:53,770 --> 00:05:00,730
Omega H dot minus H omega dot.

67
00:05:04,050 --> 00:05:05,910
So what do we have here?

68
00:05:05,910 --> 00:05:11,790
Omega, H dot was calculated
up there, m omega,

69
00:05:11,790 --> 00:05:21,810
omega dot x squared minus
H p squared over 2m minus--

70
00:05:21,810 --> 00:05:23,220
no, minus.

71
00:05:23,220 --> 00:05:38,330
Plus 1/2 m omega squared
x squared times omega dot

72
00:05:38,330 --> 00:05:39,730
over omega squared.

73
00:05:39,730 --> 00:05:45,490
And well, I still remember
when I first saw that.

74
00:05:45,490 --> 00:05:49,770
I probably wanted the numerator
to cancel and to do something

75
00:05:49,770 --> 00:05:52,830
very nice and simplify a lot.

76
00:05:52,830 --> 00:05:55,850
But it doesn't happen.

77
00:05:55,850 --> 00:05:58,545
So let's see what
really happens.

78
00:06:01,350 --> 00:06:04,650
Well, you have this
term, omega squared,

79
00:06:04,650 --> 00:06:08,000
omega dot, x squared
m, omega squared,

80
00:06:08,000 --> 00:06:09,650
omega dot, x squared m.

81
00:06:09,650 --> 00:06:13,700
But the factors of 2
don't make it cancel.

82
00:06:13,700 --> 00:06:16,130
So it's there.

83
00:06:16,130 --> 00:06:21,925
So let me write what we
get when we simplify this.

84
00:06:21,925 --> 00:06:30,080
di dt is equal to omega dot
over omega squared times

85
00:06:30,080 --> 00:06:36,680
1/2 m omega squared x squared
minus p squared over 2m.

86
00:06:43,220 --> 00:06:44,870
That term is clear.

87
00:06:44,870 --> 00:06:51,080
The p squared is that, and
here, we cancel the 1, partially

88
00:06:51,080 --> 00:06:52,700
with a 1/2.

89
00:06:52,700 --> 00:06:54,940
So we've got this.

90
00:06:54,940 --> 00:07:00,110
OK, so it doesn't look
like it wants to be 0.

91
00:07:00,110 --> 00:07:01,780
But it's still very good.

92
00:07:01,780 --> 00:07:06,940
Let's see why that
result is nice.

93
00:07:06,940 --> 00:07:12,200
Well, one thing you realize
here is that it actually

94
00:07:12,200 --> 00:07:15,680
gave you kind of
back the Hamiltonian

95
00:07:15,680 --> 00:07:18,410
with a different sine there.

96
00:07:18,410 --> 00:07:21,090
This is negative, and
this will remain positive.

97
00:07:21,090 --> 00:07:28,360
So let's write this, this
omega dot omega squared.

98
00:07:28,360 --> 00:07:32,420
And this is the kinetic energy
minus the potential energy,

99
00:07:32,420 --> 00:07:37,330
well, the potential
energy v of t

100
00:07:37,330 --> 00:07:44,860
minus the kinetic energy k of
t in the harmonic oscillator.

101
00:07:44,860 --> 00:07:53,730
Moreover, this quantity
is already very small.

102
00:07:53,730 --> 00:07:56,880
So this thing is very
small, but the fact

103
00:07:56,880 --> 00:08:02,940
that the adiabatic
invariant is adiabatic

104
00:08:02,940 --> 00:08:07,290
that it's really good,
should go beyond this.

105
00:08:07,290 --> 00:08:11,490
There should be something
suppressing about this factor,

106
00:08:11,490 --> 00:08:14,640
because you know, this
came from just the fact

107
00:08:14,640 --> 00:08:19,260
that things vary with omega dot.

108
00:08:19,260 --> 00:08:21,090
So what is happening?

109
00:08:21,090 --> 00:08:26,805
This is small and
slowly varying.

110
00:08:36,780 --> 00:08:44,120
This is neither small, nor
slowly varying, in fact.

111
00:08:44,120 --> 00:08:46,120
Why?

112
00:08:46,120 --> 00:08:48,610
Potential minus kinetic energy.

113
00:08:48,610 --> 00:08:51,400
The potential energy
in an oscillator

114
00:08:51,400 --> 00:08:55,450
goes up when the
kinetic energy is 0.

115
00:08:55,450 --> 00:08:58,130
I see the oscillator goes
to the end, stretches

116
00:08:58,130 --> 00:09:01,900
[INAUDIBLE] potential energy is
large, the kinetic energy is 0.

117
00:09:01,900 --> 00:09:04,810
As it goes through the
center, the equilibrium point,

118
00:09:04,810 --> 00:09:06,950
the kinetic energy is larger.

119
00:09:06,950 --> 00:09:09,280
So this is oscillating.

120
00:09:09,280 --> 00:09:16,030
And it's very large,
but now, you probably

121
00:09:16,030 --> 00:09:22,330
remember this fact about
the harmonic oscillators.

122
00:09:22,330 --> 00:09:26,740
While the potential and
kinetic energies oscillate,

123
00:09:26,740 --> 00:09:30,350
their averages are the same.

124
00:09:30,350 --> 00:09:34,240
So that's how this term
is going to help you.

125
00:09:34,240 --> 00:09:41,230
The average of this quantity
is roughly 0 over any period.

126
00:09:41,230 --> 00:09:46,660
And a period over a period,
this quantity changes little.

127
00:09:46,660 --> 00:09:50,500
So this is going to help us.

128
00:09:50,500 --> 00:09:53,800
Let me remind you here, suppose
you have an oscillation,

129
00:09:53,800 --> 00:10:07,770
an x equals sine omega t, then
the momentum would be m x dot,

130
00:10:07,770 --> 00:10:17,730
so m omega cosine omega
t, and the kinetic energy

131
00:10:17,730 --> 00:10:21,900
minus the potential energy, if
you do this little calculation,

132
00:10:21,900 --> 00:10:28,710
will go like omega
squared cosine of 2

133
00:10:28,710 --> 00:10:34,200
omega t, the v minus k.

134
00:10:34,200 --> 00:10:36,680
I leave for you that
little calculation.

135
00:10:36,680 --> 00:10:42,450
But it will go like cosine of
2 omega t, twice the period.

136
00:10:42,450 --> 00:10:45,790
And that thing tends
to have a 0 average.

137
00:10:45,790 --> 00:10:48,660
So let's see what happens now.

138
00:10:48,660 --> 00:10:52,110
The idt to see
what happens to it.

139
00:10:52,110 --> 00:11:04,900
Let's calculate I at t plus
the period minus I at some t.

140
00:11:08,720 --> 00:11:13,410
So let's see how much
I changes in a period.

141
00:11:13,410 --> 00:11:15,810
So from here, we
have the derivative.

142
00:11:15,810 --> 00:11:27,830
So we must do the integral
from t to t plus t of the dI dt

143
00:11:27,830 --> 00:11:31,240
prime dt prime.

144
00:11:31,240 --> 00:11:35,340
So this will be the
integral from t to t

145
00:11:35,340 --> 00:11:42,250
plus capital T of this whole
thing, omega dot over omega

146
00:11:42,250 --> 00:11:47,610
squared of t times v of t--

147
00:11:47,610 --> 00:11:54,075
it's all t prime, actually-- vt
prime minus kt prime dt prime.

148
00:11:57,630 --> 00:12:00,550
Let's see that.

149
00:12:00,550 --> 00:12:03,070
You have the
derivative of I, so you

150
00:12:03,070 --> 00:12:07,400
can calculate the change in I by
integrating with the derivative

151
00:12:07,400 --> 00:12:09,580
of I over mep.

152
00:12:09,580 --> 00:12:14,650
We've done that, and we've
asked how much does this thing

153
00:12:14,650 --> 00:12:17,020
change over a period.

154
00:12:17,020 --> 00:12:24,220
Then we have that I of
t plus t minus I of t,

155
00:12:24,220 --> 00:12:27,460
we have an integral
over a period.

156
00:12:27,460 --> 00:12:33,110
We set this quantity very slowly
and very little over a period,

157
00:12:33,110 --> 00:12:40,060
so roughly speaking, this is
equal to omega dot over omega

158
00:12:40,060 --> 00:12:44,330
squared at t.

159
00:12:44,330 --> 00:12:47,900
It didn't change much
over the integral.

160
00:12:47,900 --> 00:12:53,080
And then we have the
integral over a period

161
00:12:53,080 --> 00:12:57,340
of the potential energy
minus the kinetic energy.

162
00:13:03,000 --> 00:13:08,430
And for a normal oscillator
that is time independent,

163
00:13:08,430 --> 00:13:12,690
this quantity is strictly 0.

164
00:13:12,690 --> 00:13:18,780
If omega was not changing,
this would be identically 0.

165
00:13:18,780 --> 00:13:24,630
So if omega is changing
slowly, this quantity

166
00:13:24,630 --> 00:13:27,750
must be very close to 0.

167
00:13:27,750 --> 00:13:31,660
It's identically 0
when it doesn't change.

168
00:13:31,660 --> 00:13:36,190
Therefore, you see you got
an extra suppression factor.

169
00:13:36,190 --> 00:13:41,650
The change in the adiabatic,
so-called adiabatic invariant

170
00:13:41,650 --> 00:13:48,460
over time was already small,
because everything goes slow.

171
00:13:48,460 --> 00:13:51,280
But there is an
extra suppression

172
00:13:51,280 --> 00:13:55,060
due to the fact that
these two energies have

173
00:13:55,060 --> 00:13:58,940
the same average over a period.

174
00:13:58,940 --> 00:14:01,480
So you gain something.

175
00:14:01,480 --> 00:14:08,180
If the energy changes slowly,
this energy over omega

176
00:14:08,180 --> 00:14:13,340
changes even much
more slowly than that.

177
00:14:13,340 --> 00:14:36,820
So this is really exactly 0
for time independent omega,

178
00:14:36,820 --> 00:14:49,470
approximately 0 for slow
omega, slowly changing omega.

179
00:14:49,470 --> 00:14:52,500
So that's the extra
suppression factor,

180
00:14:52,500 --> 00:14:57,870
and that's what makes this an
adiabatic invariant, something

181
00:14:57,870 --> 00:15:03,030
that really changes
dramatically slower

182
00:15:03,030 --> 00:15:08,300
in a system in which everything
is already changing slowly.