1
00:00:00,050 --> 00:00:01,770
The following
content is provided

2
00:00:01,770 --> 00:00:04,010
under a Creative
Commons license.

3
00:00:04,010 --> 00:00:06,860
Your support will help MIT
OpenCourseWare continue

4
00:00:06,860 --> 00:00:10,720
to offer high-quality
educational resources for free.

5
00:00:10,720 --> 00:00:13,330
To make a donation, or
view additional materials

6
00:00:13,330 --> 00:00:17,205
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:17,205 --> 00:00:17,830
at ocw.mit.edu.

8
00:00:25,920 --> 00:00:31,730
PROFESSOR: The last topic
we discussed on Monday it

9
00:00:31,730 --> 00:00:35,400
was the situation of the
Landau-Zener transition

10
00:00:35,400 --> 00:00:38,280
that you sweep for resonance.

11
00:00:38,280 --> 00:00:41,000
You've all seen the
Landau-Zener formula,

12
00:00:41,000 --> 00:00:46,080
you all know that in crossing
turns into an avoided crossing.

13
00:00:46,080 --> 00:00:51,050
But I tried to at least
provide you additional insight

14
00:00:51,050 --> 00:00:55,310
by emphasizing that the whole
process is absolute coherent.

15
00:00:55,310 --> 00:00:59,230
That it's full-phase
coherent throughout.

16
00:00:59,230 --> 00:01:07,010
And what happens is there is a
coherent transfer of amplitude

17
00:01:07,010 --> 00:01:10,170
from the state 1 to the state 2.

18
00:01:10,170 --> 00:01:14,150
This is nothing else than
Schrodinger's equation.

19
00:01:14,150 --> 00:01:18,030
But I want to point out
that for the short times

20
00:01:18,030 --> 00:01:22,040
you sweep through, there
is no T2 independence here.

21
00:01:22,040 --> 00:01:28,670
In other words, when
I discussed what

22
00:01:28,670 --> 00:01:32,580
is the effective
time during which

23
00:01:32,580 --> 00:01:36,140
the transfer of
population takes place,

24
00:01:36,140 --> 00:01:39,790
at the time here is so
short that the detuning

25
00:01:39,790 --> 00:01:41,180
doesn't matter.

26
00:01:41,180 --> 00:01:45,260
Actually, the
criterion which leads

27
00:01:45,260 --> 00:01:49,540
to this effective time during
which the transfer takes place

28
00:01:49,540 --> 00:01:52,390
is actually exactly
the time window

29
00:01:52,390 --> 00:01:55,410
where the detuning
is small enough

30
00:01:55,410 --> 00:01:58,780
that-- to say it
loosely, it doesn't

31
00:01:58,780 --> 00:02:01,670
make a difference whether your
in resonance or slightly away.

32
00:02:01,670 --> 00:02:04,270
The atom experiences
the same tri field.

33
00:02:07,790 --> 00:02:13,900
And so based on
this criterion, we

34
00:02:13,900 --> 00:02:19,730
discussed that we can understand
the Landau-Zener probability

35
00:02:19,730 --> 00:02:23,240
in the perturbative limit
as a coherent process

36
00:02:23,240 --> 00:02:26,590
where we transfer population,
we transfer amplitude

37
00:02:26,590 --> 00:02:30,180
with a Rabi frequency during
this effective time, delta t.

38
00:02:33,647 --> 00:02:35,480
It's not the only way
how we can look at it,

39
00:02:35,480 --> 00:02:39,090
but it's one way which
I think is insightful.

40
00:02:41,700 --> 00:02:47,270
Any questions about this or
what we discussed on Monday?

41
00:02:52,800 --> 00:03:03,080
If not, I would like to take
the topic one step further

42
00:03:03,080 --> 00:03:05,695
and discuss the density
matrix formalism.

43
00:03:29,370 --> 00:03:35,720
So we have so far
discussed purely

44
00:03:35,720 --> 00:03:38,000
Hamiltonian unitary evolution.

45
00:03:38,000 --> 00:03:41,520
Namely, the
Schrodinger equation.

46
00:03:41,520 --> 00:03:45,940
And of course, unitary
evolution leaves the system

47
00:03:45,940 --> 00:03:48,600
which is in a pure
state in a pure state.

48
00:03:48,600 --> 00:03:52,620
It's just that the
quantum state evolves.

49
00:03:52,620 --> 00:03:56,500
However, that means we cannot
describe processes like

50
00:03:56,500 --> 00:04:04,870
decoherence or some losses
away from the two levels we are

51
00:04:04,870 --> 00:04:08,230
focusing on.

52
00:04:08,230 --> 00:04:11,280
And so now we want to
use the density operator,

53
00:04:11,280 --> 00:04:14,705
the density operator
formalism to have

54
00:04:14,705 --> 00:04:17,399
a description of two-level
system which goes beyond that.

55
00:04:20,279 --> 00:04:31,770
So let me just-- so Schrodinger
equation deals only with pure

56
00:04:31,770 --> 00:04:44,767
states, cannot describe loss
of particles, loss of photons,

57
00:04:44,767 --> 00:04:45,433
and decoherence.

58
00:04:48,460 --> 00:04:51,040
Well, there is one exception.

59
00:04:51,040 --> 00:04:57,880
If the decoherence process is
merely a state-dependent loss

60
00:04:57,880 --> 00:05:00,280
of atoms to a third
state, then you

61
00:05:00,280 --> 00:05:04,450
can still use the wave
function formalism.

62
00:05:04,450 --> 00:05:05,655
So this is the exception.

63
00:05:09,260 --> 00:05:17,535
If you have two states,
that's the excited state.

64
00:05:21,870 --> 00:05:25,610
If you have two states
and all of what happens

65
00:05:25,610 --> 00:05:31,410
is that you have some
loss to some other levels

66
00:05:31,410 --> 00:05:42,480
and their rate
coefficients, then one

67
00:05:42,480 --> 00:05:55,470
can still use a
Hamiltonain description,

68
00:05:55,470 --> 00:06:06,770
but you have to replace the
eigenvalues by complex numbers.

69
00:06:06,770 --> 00:06:17,230
In other words, you have to add
an imaginary part to the energy

70
00:06:17,230 --> 00:06:17,840
levels.

71
00:06:17,840 --> 00:06:19,215
And that means
the time evolution

72
00:06:19,215 --> 00:06:22,520
is exponentially dent.

73
00:06:22,520 --> 00:06:27,770
So that's as much as you
can incorporate decoherence

74
00:06:27,770 --> 00:06:31,080
and losses into a wave
function formalism.

75
00:06:38,870 --> 00:06:57,340
However, many other processes
require the formalism

76
00:06:57,340 --> 00:06:59,410
of the density matrix.

77
00:06:59,410 --> 00:07:05,880
And the simplest process
where wave function formalism

78
00:07:05,880 --> 00:07:09,600
is absolutely inadequate
is the process

79
00:07:09,600 --> 00:07:14,160
of spontaneous emission.

80
00:07:14,160 --> 00:07:17,940
When you have a loss
in the excited state,

81
00:07:17,940 --> 00:07:20,370
you could still describe
the excited state

82
00:07:20,370 --> 00:07:23,100
with a complex
energy eigenvalue.

83
00:07:23,100 --> 00:07:34,840
But the fact that whatever is
lost from the excited state

84
00:07:34,840 --> 00:07:36,770
is added to the ground state.

85
00:07:36,770 --> 00:07:38,600
There is no wave
function formalism

86
00:07:38,600 --> 00:07:39,600
which can describe that.

87
00:07:46,895 --> 00:07:54,100
So for those processes and
for decoherence in general,

88
00:07:54,100 --> 00:07:57,850
we require the use of
the density operator.

89
00:08:01,870 --> 00:08:04,730
So I know that most
of you have seen

90
00:08:04,730 --> 00:08:08,450
the density operator in
statistical mechanics

91
00:08:08,450 --> 00:08:10,920
or some advanced course
in quantum mechanics.

92
00:08:10,920 --> 00:08:15,060
So therefore, I only spend
about five minutes on it.

93
00:08:15,060 --> 00:08:18,960
So I want to kind of just
remind you or give you

94
00:08:18,960 --> 00:08:20,230
a very short introduction.

95
00:08:20,230 --> 00:08:22,890
So for those of you who
have never heard about it,

96
00:08:22,890 --> 00:08:26,010
I probably say enough
that you understand

97
00:08:26,010 --> 00:08:27,530
the following discussion.

98
00:08:27,530 --> 00:08:30,670
And for those of you who know
already everything about it,

99
00:08:30,670 --> 00:08:35,020
five minutes of recapitulation
is hopefully not too boring.

100
00:08:35,020 --> 00:08:38,110
So my way of introducing
the density operator

101
00:08:38,110 --> 00:08:42,949
is first introduce it formally.

102
00:08:42,949 --> 00:08:48,700
Write down a few
equations for pure state.

103
00:08:48,700 --> 00:08:52,880
But then in a moment,
add something to it.

104
00:08:52,880 --> 00:08:58,270
So if you have a time-dependent
wave function, which

105
00:08:58,270 --> 00:09:07,590
we expand into eigenfunctions,
then we can, in these spaces,

106
00:09:07,590 --> 00:09:10,420
define arbitrary
operators by matrices.

107
00:09:20,880 --> 00:09:24,610
We want to describe our
system by finding out

108
00:09:24,610 --> 00:09:27,390
measurable observables,
expectation

109
00:09:27,390 --> 00:09:34,330
values of operators, which
of course, depend on time.

110
00:09:34,330 --> 00:09:39,760
And this is, of course, nothing
else than the expectation value

111
00:09:39,760 --> 00:09:42,000
taken with a time-dependent
wave function.

112
00:09:45,130 --> 00:09:53,810
But now we can expand
it into the bases m, n

113
00:09:53,810 --> 00:10:08,030
and we can then rewrite
it as a matrix, which

114
00:10:08,030 --> 00:10:09,205
is a density matrix.

115
00:10:12,650 --> 00:10:17,190
Or, simply as the
trace of the density

116
00:10:17,190 --> 00:10:19,330
matrix with the operator.

117
00:10:22,450 --> 00:10:31,180
And what I introduced here
as the density matrix can

118
00:10:31,180 --> 00:10:36,320
be written as psi
of t, psi of t.

119
00:10:40,140 --> 00:10:48,750
And the matrix element
given by this combination

120
00:10:48,750 --> 00:10:53,140
of amplitudes when we
expand the wave function

121
00:10:53,140 --> 00:10:54,515
psi into its basis.

122
00:10:57,040 --> 00:11:02,510
So this density matrix his
diagonal and off-diagonal

123
00:11:02,510 --> 00:11:04,250
matrix elements.

124
00:11:04,250 --> 00:11:08,320
The diagram matrix elements
are called the populations,

125
00:11:08,320 --> 00:11:11,780
the populations in state n
and the off-diagonal matrix

126
00:11:11,780 --> 00:11:14,140
elements are called coherences.

127
00:11:18,830 --> 00:11:22,720
OK, so this is just rewriting
Schrodinger's equation

128
00:11:22,720 --> 00:11:25,660
expectation value in
a matrix formalism.

129
00:11:25,660 --> 00:11:26,854
Yes, please.

130
00:11:26,854 --> 00:11:30,340
AUDIENCE: Why are you
starring the coefficients?

131
00:11:30,340 --> 00:11:32,340
PROFESSOR: Oh, there's
one star too many.

132
00:11:32,340 --> 00:11:33,300
Thank you.

133
00:11:33,300 --> 00:11:34,770
AUDIENCE: That makes sense.

134
00:11:34,770 --> 00:11:36,540
PROFESSOR: But the
reason why I wrote it

135
00:11:36,540 --> 00:11:43,850
is that we want to now add
some probability to it.

136
00:11:43,850 --> 00:11:49,400
We do not know for sure that
the system is in a pure state.

137
00:11:49,400 --> 00:12:03,230
We have probabilities
P i that the system

138
00:12:03,230 --> 00:12:07,810
is in a quantum state psi i.

139
00:12:07,810 --> 00:12:10,800
So we add another index to it.

140
00:12:10,800 --> 00:12:21,110
And when we perform
the expectation value--

141
00:12:21,110 --> 00:12:23,560
there's also one star too many.

142
00:12:23,560 --> 00:12:26,320
When we perform the
expectation value,

143
00:12:26,320 --> 00:12:29,200
we sort of do it for
each quantum state

144
00:12:29,200 --> 00:12:33,690
with a probability P i.

145
00:12:33,690 --> 00:12:36,220
So we are actually-- and this
is what I wanted to point out.

146
00:12:36,220 --> 00:12:38,310
This was the purpose of
this short discussion,

147
00:12:38,310 --> 00:12:41,370
that we are now actually
performing two averages.

148
00:12:41,370 --> 00:12:45,110
One can be regarded
as the normal quantum

149
00:12:45,110 --> 00:12:49,450
mechanical average when
you find the average value

150
00:12:49,450 --> 00:12:51,970
or the expectation
value for quantum state.

151
00:12:51,970 --> 00:12:55,090
So this is sort of the
statistics or the averaging,

152
00:12:55,090 --> 00:12:57,670
which is inherent
in quantum physics.

153
00:12:57,670 --> 00:13:00,070
But then in addition,
there may simply

154
00:13:00,070 --> 00:13:04,180
be another probabilistic
average because you have not

155
00:13:04,180 --> 00:13:06,210
prepared the system
in a pure state,

156
00:13:06,210 --> 00:13:09,400
or the system may undergo
some stochastic forces

157
00:13:09,400 --> 00:13:11,840
and wind up in different states.

158
00:13:11,840 --> 00:13:14,930
So there are two kinds of
averages which are performed.

159
00:13:14,930 --> 00:13:18,790
And the advantage of the
density matrix formalism

160
00:13:18,790 --> 00:13:23,500
is that both kinds of averages
can be done simultaneously

161
00:13:23,500 --> 00:13:26,300
in a very compact formalism.

162
00:13:26,300 --> 00:13:30,440
So therefore, I put this
probabilistic average in now

163
00:13:30,440 --> 00:13:34,845
into the definition
of the density matrix.

164
00:13:37,850 --> 00:13:46,270
Or I can write the density
matrix in this way.

165
00:13:50,180 --> 00:13:55,140
And with this extended
definition of the density

166
00:13:55,140 --> 00:13:59,540
matrix, both kinds
of averages are

167
00:13:59,540 --> 00:14:03,580
done when I determine the
expectation value and operator

168
00:14:03,580 --> 00:14:06,730
by performing the trace
with the density matrix.

169
00:14:19,770 --> 00:14:23,100
A lot of properties
of the density matrix,

170
00:14:23,100 --> 00:14:27,010
I think are-- you're
familiar with many of those.

171
00:14:27,010 --> 00:14:32,730
For instance, that Schrodinger's
equation for the density matrix

172
00:14:32,730 --> 00:14:38,680
becomes this following equation.

173
00:14:38,680 --> 00:14:43,640
You can derive that by-- if you
take the Schrodinger equation

174
00:14:43,640 --> 00:14:49,270
and you apply the Schrodinger
equation to each state psi i.

175
00:14:53,260 --> 00:14:56,860
And then you do the averaging
with a probability P i.

176
00:14:56,860 --> 00:14:59,910
You find that the Schrodinger
equation for each state

177
00:14:59,910 --> 00:15:03,240
psi i turns into this equation
for the density matrix.

178
00:15:07,560 --> 00:15:10,540
Let me just write down
that the purpose now

179
00:15:10,540 --> 00:15:12,890
is we have two averages here.

180
00:15:12,890 --> 00:15:16,580
One is the quantum
mechanic average and one

181
00:15:16,580 --> 00:15:25,510
is sort of an ensemble average
with the probabilities P i.

182
00:15:29,840 --> 00:15:31,440
I want to discuss
in a few moments

183
00:15:31,440 --> 00:15:35,000
the density matrix
for two-level system.

184
00:15:35,000 --> 00:15:39,000
So I have to remind
you of two properties,

185
00:15:39,000 --> 00:15:43,240
that the density matrix
is normalized to unity.

186
00:15:43,240 --> 00:15:45,900
So there's probability of
unity to find the system

187
00:15:45,900 --> 00:15:47,790
in one of the states.

188
00:15:50,410 --> 00:15:53,590
When we look at the
square of the density

189
00:15:53,590 --> 00:15:56,260
matrix, a trace of
rho square, this

190
00:15:56,260 --> 00:16:00,370
is simply the probability-- the
sum of the probability squared.

191
00:16:00,370 --> 00:16:02,990
And this is smaller than 1.

192
00:16:02,990 --> 00:16:07,550
And the only way that it
is one is for pure state.

193
00:16:10,480 --> 00:16:12,840
So pure state is
characterized by the fact

194
00:16:12,840 --> 00:16:16,470
that there is only-- that
we can find the basis where

195
00:16:16,470 --> 00:16:19,840
only one of the probabilities,
P i is non-vanishing.

196
00:16:19,840 --> 00:16:24,020
And then, of course, almost
trivially the trace rho is 1

197
00:16:24,020 --> 00:16:25,836
and the trace rho square is 1.

198
00:16:38,000 --> 00:16:41,370
So, so far I've presented
you the density matrix

199
00:16:41,370 --> 00:16:45,680
just as an elegant way
of integrating the two

200
00:16:45,680 --> 00:16:48,120
averages into one formalism.

201
00:16:48,120 --> 00:16:50,200
And in essence,
this is what it is.

202
00:16:50,200 --> 00:16:53,710
But you can now also
use the density matrix

203
00:16:53,710 --> 00:16:57,860
if the whole system undergoes
a time evolution, which

204
00:16:57,860 --> 00:16:59,170
is no longer unitary.

205
00:16:59,170 --> 00:17:01,660
No longer described by
a Hamilton operator.

206
00:17:01,660 --> 00:17:03,240
Because you're
interested in the time

207
00:17:03,240 --> 00:17:05,630
evolution or a
small system which

208
00:17:05,630 --> 00:17:07,720
is part of a bigger system.

209
00:17:07,720 --> 00:17:10,400
The bigger system is always
described by unitary time

210
00:17:10,400 --> 00:17:15,359
evolution, but a smaller
system is usually not described

211
00:17:15,359 --> 00:17:17,099
by unitary time evolution.

212
00:17:17,099 --> 00:17:20,020
And that's when the density
matrix becomes crucial.

213
00:17:20,020 --> 00:17:22,640
Of course, you can see this is
just you describe the smaller

214
00:17:22,640 --> 00:17:25,480
system and you do some
probabilistic average what

215
00:17:25,480 --> 00:17:27,410
the other part of
the system does.

216
00:17:27,410 --> 00:17:31,250
And therefore, it's just another
version of doing two averages.

217
00:17:31,250 --> 00:17:34,480
But this is sort of why
we want to use the density

218
00:17:34,480 --> 00:17:37,810
matrix in general.

219
00:17:40,600 --> 00:17:52,143
So we want to use the density
matrix for non-unitary time

220
00:17:52,143 --> 00:17:52,643
evolution.

221
00:17:56,960 --> 00:18:03,160
And the keyword here
is that this is often

222
00:18:03,160 --> 00:18:07,720
the situation for
open systems where

223
00:18:07,720 --> 00:18:10,480
we are interested
in a small system,

224
00:18:10,480 --> 00:18:12,630
but it is open to
a bigger system.

225
00:18:12,630 --> 00:18:15,120
Like, we're interested
to describe an atom,

226
00:18:15,120 --> 00:18:18,080
but the atom can
spontaneously emit photons

227
00:18:18,080 --> 00:18:19,705
into other parts of
[INAUDIBLE] space.

228
00:18:19,705 --> 00:18:21,580
And we're not interested
in those other parts

229
00:18:21,580 --> 00:18:22,455
of [INAUDIBLE] space.

230
00:18:31,580 --> 00:18:33,660
So an open system
for this purpose

231
00:18:33,660 --> 00:18:43,750
is where we limit
our description

232
00:18:43,750 --> 00:18:49,300
to a small part of
a larger system.

233
00:18:54,650 --> 00:18:58,560
Again, an atom interacting
with all the modes

234
00:18:58,560 --> 00:19:00,840
of the electromagnetic
field, but we simply

235
00:19:00,840 --> 00:19:02,540
want to describe the atom.

236
00:19:02,540 --> 00:19:06,190
And then, we cannot use
a wave function anymore.

237
00:19:06,190 --> 00:19:10,140
We have to use the
density matrix.

238
00:19:10,140 --> 00:19:11,490
OK

239
00:19:11,490 --> 00:19:17,920
After these
preliminaries, I want

240
00:19:17,920 --> 00:19:28,502
to now use the density
matrix formalism

241
00:19:28,502 --> 00:19:32,795
for arbitrary two-level systems.

242
00:19:50,030 --> 00:19:59,280
So what is the most
general Hamiltonian

243
00:19:59,280 --> 00:20:03,030
for the most general
two-level system?

244
00:20:03,030 --> 00:20:06,480
Well, the most
general Hamiltonian

245
00:20:06,480 --> 00:20:08,330
is the most general
Hamiltonian we

246
00:20:08,330 --> 00:20:11,550
can construct with
2 by 2 matrices.

247
00:20:11,550 --> 00:20:14,940
And the base is
set to expand the 2

248
00:20:14,940 --> 00:20:18,160
by 2 matrices are
the Pauli matrices.

249
00:20:18,160 --> 00:20:28,240
So if you expand the Hamiltonian
into the unity matrix, sigma x,

250
00:20:28,240 --> 00:20:37,290
sigma y, and sigma z, we
have four coefficients,

251
00:20:37,290 --> 00:20:41,210
four amplitudes, which are
complex in general-- omega

252
00:20:41,210 --> 00:20:44,710
1, omega 2, omega 3.

253
00:20:44,710 --> 00:20:48,110
And here is something which
I've called omega bar.

254
00:20:58,930 --> 00:21:05,290
By appropriately shifting
what is the 0 point of energy,

255
00:21:05,290 --> 00:21:07,300
we can always get rid of this.

256
00:21:07,300 --> 00:21:11,130
So this is just this
definitional character.

257
00:21:11,130 --> 00:21:16,600
So therefore, the most
general Hamiltonian

258
00:21:16,600 --> 00:21:20,260
for any two-level system
can be written in this very

259
00:21:20,260 --> 00:21:25,660
compact way that it is the
scalar product of the vector

260
00:21:25,660 --> 00:21:30,950
omega-- omega 1, omega 2,
omega 3-- with the vector

261
00:21:30,950 --> 00:21:34,800
sigma of the three Pauli
matrices-- sigma x, sigma y,

262
00:21:34,800 --> 00:21:35,430
sigma z.

263
00:21:43,380 --> 00:21:48,230
OK, so this is a
way to write down

264
00:21:48,230 --> 00:21:52,045
the most general Hamiltonian
for a two-level system.

265
00:21:54,700 --> 00:21:59,100
Now, we describe two-level
systems by a density matrix,

266
00:21:59,100 --> 00:22:04,230
by statistical operator,
which is also a 2 by 2 matrix.

267
00:22:04,230 --> 00:22:08,820
And the most general
density matrix

268
00:22:08,820 --> 00:22:16,730
can also be expanded into
the its four components.

269
00:22:16,730 --> 00:22:19,670
Sort of the basis
set of matrices

270
00:22:19,670 --> 00:22:25,600
is the unitary matrix and
the three Pauli matrices.

271
00:22:25,600 --> 00:22:31,740
So 1, 2, 3.

272
00:22:36,290 --> 00:22:39,310
Of course, this time we cannot
throw away the unity matrix

273
00:22:39,310 --> 00:22:42,402
because otherwise the density
matrix would have no trace

274
00:22:42,402 --> 00:22:44,110
and there would be no
probability to find

275
00:22:44,110 --> 00:22:44,651
the particle.

276
00:22:48,850 --> 00:22:56,130
But we can, again, write
it in a compact form

277
00:22:56,130 --> 00:23:01,580
that it is 1/2-- yes,
I'm using the fact now

278
00:23:01,580 --> 00:23:07,290
that the trace of rho is r0.

279
00:23:07,290 --> 00:23:12,820
And this is, by definition, or
by conservation of probability,

280
00:23:12,820 --> 00:23:13,930
is 1.

281
00:23:13,930 --> 00:23:15,910
So therefore, r0 is
not a free parameter.

282
00:23:15,910 --> 00:23:19,260
It's just the sum of
all the probabilities

283
00:23:19,260 --> 00:23:24,390
to find the system in any state.

284
00:23:24,390 --> 00:23:30,080
And the non-trivial part
is then the scalar product

285
00:23:30,080 --> 00:23:35,470
of this vector r-- rx, ry, rz--
with the vector of the three

286
00:23:35,470 --> 00:23:36,135
Pauli matrices.

287
00:23:39,900 --> 00:23:48,650
Well, so we have our
most general Hamiltonian.

288
00:23:48,650 --> 00:23:51,820
We have our most
general density matrix.

289
00:23:51,820 --> 00:23:56,460
And now we can insert
this into the equation

290
00:23:56,460 --> 00:24:00,330
of motion for the
density matrix.

291
00:24:00,330 --> 00:24:05,840
Which, as I said before,
is just a reformulation

292
00:24:05,840 --> 00:24:07,820
of Schrodinger's equation.

293
00:24:07,820 --> 00:24:11,870
And if you insert the
Hamiltonian and the density

294
00:24:11,870 --> 00:24:14,710
matrix into this
equation, we find actually

295
00:24:14,710 --> 00:24:19,640
something which is very simple.

296
00:24:19,640 --> 00:24:26,220
It says that this
vector r, which

297
00:24:26,220 --> 00:24:30,890
we call the Bloch vector-- the
derivative of the Bloch vector

298
00:24:30,890 --> 00:24:38,020
is given by the cross product
of the vector omega, which

299
00:24:38,020 --> 00:24:44,185
were the coefficients with which
we parametrized the Hamiltonian

300
00:24:44,185 --> 00:24:45,150
cross r.

301
00:24:49,330 --> 00:24:50,746
The derivation is
straightforward.

302
00:24:53,270 --> 00:24:58,630
And you will be asked to do
that on your homework assignment

303
00:24:58,630 --> 00:24:59,940
number 1.

304
00:24:59,940 --> 00:25:01,960
But it has a very
powerful meaning.

305
00:25:01,960 --> 00:25:07,150
It tells us that an
arbitrary two-level system

306
00:25:07,150 --> 00:25:10,000
with an arbitrary
Hamiltonian can

307
00:25:10,000 --> 00:25:15,230
be regarded as a system
where we have a vector

308
00:25:15,230 --> 00:25:17,530
R which undergoes precession.

309
00:25:17,530 --> 00:25:19,630
This is the time
evolution of the system.

310
00:25:22,690 --> 00:25:31,670
So this is a powerful
generalization

311
00:25:31,670 --> 00:25:40,840
from the result we
discussed previously

312
00:25:40,840 --> 00:25:44,950
where we found that if
you have an arbitrary

313
00:25:44,950 --> 00:25:50,470
quantum-mechanical spin,
the time derivative can

314
00:25:50,470 --> 00:25:53,920
be written in that way.

315
00:25:53,920 --> 00:25:57,060
So previously, we found
it for a pure state,

316
00:25:57,060 --> 00:25:58,910
but now we find
it-- that it's even

317
00:25:58,910 --> 00:26:03,060
valid for a general density
matrix and its time evolution.

318
00:26:08,930 --> 00:26:25,730
So what I've derived for
you is-- Classroom Files.

319
00:26:34,060 --> 00:26:38,060
Is a famous theorem,
which is traced back

320
00:26:38,060 --> 00:26:41,650
to Feynman, Vernon,
and Hellwarth.

321
00:26:41,650 --> 00:26:45,160
It's sort of a
famous paper which--

322
00:26:57,230 --> 00:27:00,650
So this famous
theorem-- and I've

323
00:27:00,650 --> 00:27:02,420
summarized it here
for you-- says

324
00:27:02,420 --> 00:27:06,550
that the time evolution of the
density matrix for the most

325
00:27:06,550 --> 00:27:13,230
general two-level system is
isomorphic to pure precession.

326
00:27:13,230 --> 00:27:16,890
And that means it's
isomorphic to the behavior

327
00:27:16,890 --> 00:27:20,870
of a classical moment,
classical magnetic moment,

328
00:27:20,870 --> 00:27:24,259
in a suitable time-dependent
magnetic field.

329
00:27:24,259 --> 00:27:25,800
So when you have a
Hamiltonian, which

330
00:27:25,800 --> 00:27:28,430
is characterized by-- the
most general Hamiltonian

331
00:27:28,430 --> 00:27:34,260
is characterized by the three
coefficients-- w1, w2, w3.

332
00:27:34,260 --> 00:27:37,140
But if you would create
a classical system

333
00:27:37,140 --> 00:27:45,300
where w1, w2, and w3 are the
time-dependent components,

334
00:27:45,300 --> 00:27:49,410
xyz-component of a
magnetic field, then

335
00:27:49,410 --> 00:27:52,350
the precession of
a magnetic moment

336
00:27:52,350 --> 00:27:55,670
would be exactly the same
as the time evolution

337
00:27:55,670 --> 00:27:57,500
of a quantum-mechanical
density matrix.

338
00:28:06,340 --> 00:28:07,175
Any question?

339
00:28:13,860 --> 00:28:16,190
So in other words,
we've started out

340
00:28:16,190 --> 00:28:18,270
with rotating
frames and rotation

341
00:28:18,270 --> 00:28:20,830
and now we've gone
as far as I will go.

342
00:28:20,830 --> 00:28:23,740
Namely, I've in a
way told you that

343
00:28:23,740 --> 00:28:26,930
an arbitrary quantum-mechanical
two-level system, the time

344
00:28:26,930 --> 00:28:29,050
evolution is just precession.

345
00:28:29,050 --> 00:28:30,960
It's rotation.

346
00:28:30,960 --> 00:28:33,570
There is nothing more
complicated possible.

347
00:28:40,190 --> 00:28:42,890
Well, unless we talk
about decoherence.

348
00:28:47,350 --> 00:28:54,040
If we have such a Hamiltonian,
we know, of course,

349
00:28:54,040 --> 00:29:02,480
that a pure state will
stay pure forever.

350
00:29:05,580 --> 00:29:11,890
And you can immediately
verify that if you

351
00:29:11,890 --> 00:29:14,230
look at the trace of rho square.

352
00:29:18,240 --> 00:29:22,390
If the trace of rho square
is 1, we have a pure state.

353
00:29:22,390 --> 00:29:25,450
And now we have parametrized
the density matrix

354
00:29:25,450 --> 00:29:30,550
with the Bloch vector
component-- r1, r2, r3.

355
00:29:30,550 --> 00:29:37,970
So in those components,
the trace of rho square

356
00:29:37,970 --> 00:29:39,770
can be written in this way.

357
00:29:43,090 --> 00:29:46,140
And of course, r0
square was constant.

358
00:29:46,140 --> 00:29:48,740
This was our normalization of 1.

359
00:29:48,740 --> 00:29:51,890
So the question is now when
we have an arbitrary time

360
00:29:51,890 --> 00:29:54,990
evolution, which we
know now according

361
00:29:54,990 --> 00:30:00,656
to the Feynman, Vernon,
Hellwarth theorem.

362
00:30:00,656 --> 00:30:03,860
The arbitrary time evolution
of the Bloch vector

363
00:30:03,860 --> 00:30:07,590
can be written as omega cross r.

364
00:30:07,590 --> 00:30:11,310
So this equation
tells us immediately

365
00:30:11,310 --> 00:30:15,470
that the length of the
vector r is constant

366
00:30:15,470 --> 00:30:19,680
because r dot is
always orthogonal to r.

367
00:30:19,680 --> 00:30:23,260
And therefore, the lengths of
the vector r is not changing.

368
00:30:23,260 --> 00:30:27,280
So what we have derived says
that with the most general

369
00:30:27,280 --> 00:30:33,170
Hamiltonian, the lengths of
the vector r will be constant.

370
00:30:33,170 --> 00:30:37,800
And therefore, the trace of
rho square will be constant.

371
00:30:42,160 --> 00:30:50,330
This is constant because r
dot is perpendicular to r.

372
00:30:50,330 --> 00:30:53,090
So this will tell us that
a pure state will just

373
00:30:53,090 --> 00:30:55,810
precess with the constant
lengths of its Bloch vector

374
00:30:55,810 --> 00:30:57,420
forever.

375
00:30:57,420 --> 00:31:01,840
However, we know that in real
life some coherences are lost

376
00:31:01,840 --> 00:31:03,760
and now we have to
introduce something else.

377
00:31:08,640 --> 00:31:19,055
So this does not describe
loss of coherence.

378
00:31:24,340 --> 00:31:28,140
So now we are just
one tiny step away

379
00:31:28,140 --> 00:31:31,705
from introducing
the Bloch equations.

380
00:31:31,705 --> 00:31:36,320
We will fully feature the
optical Bloch equations

381
00:31:36,320 --> 00:31:38,660
in 8.422.

382
00:31:38,660 --> 00:31:42,990
but since we have discussed
two-level systems to quite some

383
00:31:42,990 --> 00:31:48,160
extent, I cannot resist
to show you now in three,

384
00:31:48,160 --> 00:31:50,830
four minutes what the
Bloch equations are.

385
00:31:50,830 --> 00:31:53,810
And then when you take the
second part of the course,

386
00:31:53,810 --> 00:31:57,000
you will already be
familiar with it.

387
00:31:57,000 --> 00:32:00,160
So let me just now
tell you what has

388
00:32:00,160 --> 00:32:07,930
to be added to do this step
from the previous formalism

389
00:32:07,930 --> 00:32:10,950
to the Bloch equations.

390
00:32:10,950 --> 00:32:13,280
And this is the one
step you have to do.

391
00:32:13,280 --> 00:32:17,605
We have to include relaxation
processes into the description.

392
00:32:22,870 --> 00:32:34,120
So my less than five-minute
way to now derive the Bloch

393
00:32:34,120 --> 00:32:36,140
equations for you
goes as follows.

394
00:32:38,850 --> 00:32:42,180
I first remind you
that everything

395
00:32:42,180 --> 00:32:43,700
has to come to
thermal equilibrium.

396
00:32:48,500 --> 00:32:51,740
In other words, if you
have an atomic system,

397
00:32:51,740 --> 00:32:53,890
if you have a quantum
computer, whatever system you

398
00:32:53,890 --> 00:32:56,180
have and you prepared
in a pure state,

399
00:32:56,180 --> 00:32:59,920
you know if you
will wait forever,

400
00:32:59,920 --> 00:33:04,020
the system will be described
by a density matrix, which

401
00:33:04,020 --> 00:33:07,780
is the density matrix of
thermal equilibrium, which

402
00:33:07,780 --> 00:33:11,260
has only diagonal
matrix elements.

403
00:33:11,260 --> 00:33:17,580
The populations follow
the [INAUDIBLE] factor.

404
00:33:20,200 --> 00:33:25,450
And everything is normalized
by the partition function.

405
00:33:25,450 --> 00:33:30,070
So we know that this will
happen after long, long times.

406
00:33:30,070 --> 00:33:42,530
So no matter with what
density matrix we start out,

407
00:33:42,530 --> 00:33:44,960
if we start with a
density matrix rho

408
00:33:44,960 --> 00:33:49,510
pure state, for instance,
there will be inevitably

409
00:33:49,510 --> 00:33:55,905
some relaxation process which
will restore rho to rho t,

410
00:33:55,905 --> 00:33:57,030
to the thermal equilibrium.

411
00:34:00,130 --> 00:34:02,515
Now, how this happens
can be formulated

412
00:34:02,515 --> 00:34:04,500
in a microscopic way.

413
00:34:04,500 --> 00:34:07,370
And we will go through
a beautiful derivation

414
00:34:07,370 --> 00:34:09,830
of a master equation
and really provide

415
00:34:09,830 --> 00:34:12,870
some insight what
causes relaxation.

416
00:34:12,870 --> 00:34:17,460
But here for the
purpose of this course,

417
00:34:17,460 --> 00:34:21,800
I want to say, well,
there is relaxation.

418
00:34:21,800 --> 00:34:27,170
And I want to introduce now,
in a phenomenological way,

419
00:34:27,170 --> 00:34:29,969
damping and damping times.

420
00:34:29,969 --> 00:34:50,920
So the phenomenological
way to introduce damping

421
00:34:50,920 --> 00:34:52,280
goes as follows.

422
00:34:52,280 --> 00:34:56,070
Our equation of motion
for the density matrix

423
00:34:56,070 --> 00:35:00,130
was that this is a unitary
evolution described

424
00:35:00,130 --> 00:35:06,420
by-- the Schrodinger
equation was that the density

425
00:35:06,420 --> 00:35:09,216
matrix evolves according
to the commutative

426
00:35:09,216 --> 00:35:10,090
with the Hamiltonian.

427
00:35:13,110 --> 00:35:17,730
But now-- and I have to pick
big quotation marks around it

428
00:35:17,730 --> 00:35:20,790
because this is not a
mathematically exact way

429
00:35:20,790 --> 00:35:21,950
of writing it.

430
00:35:21,950 --> 00:35:26,750
But now I want to
introduce some term which

431
00:35:26,750 --> 00:35:35,890
will damp the density matrix to
the thermal equilibrium density

432
00:35:35,890 --> 00:35:39,360
matrix with some
equilibration time, Te.

433
00:35:49,490 --> 00:35:51,300
I mean, this is
what you can always

434
00:35:51,300 --> 00:35:53,450
do if you know the
system is damped.

435
00:35:53,450 --> 00:35:55,630
You have some
coherent evolution,

436
00:35:55,630 --> 00:35:58,390
but eventually you
added a damping term

437
00:35:58,390 --> 00:36:01,720
and you make the damping term--
you formulate in such a way

438
00:36:01,720 --> 00:36:03,440
that asymptotically
the system will

439
00:36:03,440 --> 00:36:05,850
be damped to the
thermal equilibrium.

440
00:36:05,850 --> 00:36:07,410
In other words, the
damping term will

441
00:36:07,410 --> 00:36:09,160
have no effect on the
dynamics once you've

442
00:36:09,160 --> 00:36:10,430
reached equilibrium.

443
00:36:10,430 --> 00:36:11,780
So it does all the right things.

444
00:36:15,345 --> 00:36:17,220
Of course, we have to
be a little bit careful

445
00:36:17,220 --> 00:36:21,130
because everything is either
an operator or matrix.

446
00:36:21,130 --> 00:36:23,370
And I was just adding
the damping term

447
00:36:23,370 --> 00:36:26,780
as you would probably do it
to a one-dimensional equation.

448
00:36:26,780 --> 00:36:30,720
So therefore, let me be a
little bit more specific.

449
00:36:30,720 --> 00:36:37,886
In many cases, you
will find that there

450
00:36:37,886 --> 00:36:45,960
are two distinctly
different relaxation times.

451
00:36:45,960 --> 00:36:51,140
In other words, the system
will have usually at least two

452
00:36:51,140 --> 00:36:55,920
physically distinct
relaxation times.

453
00:36:55,920 --> 00:37:00,290
They are traditionally
called T1 and T2.

454
00:37:00,290 --> 00:37:06,835
T1 is the damping time for
population differences.

455
00:37:17,930 --> 00:37:21,260
So this is the damping
time to shovel population

456
00:37:21,260 --> 00:37:24,250
from some inverted state
or some other state

457
00:37:24,250 --> 00:37:27,140
into the equilibrium state.

458
00:37:27,140 --> 00:37:34,400
That usually
involves the removal

459
00:37:34,400 --> 00:37:36,490
of energy out of the system.

460
00:37:36,490 --> 00:37:39,740
So it's an energy decay time.

461
00:37:39,740 --> 00:37:44,120
And if you would inspect our
parameterization of the Bloch

462
00:37:44,120 --> 00:37:47,260
vector, population or
population differences

463
00:37:47,260 --> 00:37:50,740
are described by
the z-component,

464
00:37:50,740 --> 00:37:52,934
the third component
of the Bloch vector.

465
00:37:57,290 --> 00:38:01,620
Well, we have other
components of the Bloch vector

466
00:38:01,620 --> 00:38:03,350
which correspond to coherences.

467
00:38:06,190 --> 00:38:09,950
The off-diagonal matrix
element of the density matrix.

468
00:38:09,950 --> 00:38:14,590
And they're only nonzero if
you have two states populated

469
00:38:14,590 --> 00:38:17,540
with a value-defined
relative phase.

470
00:38:17,540 --> 00:38:20,170
When the system, quantum
mechanical system,

471
00:38:20,170 --> 00:38:24,890
loses its memory of the
phase, the r1 and r2 component

472
00:38:24,890 --> 00:38:27,910
of the Bloch vector go to 0.

473
00:38:27,910 --> 00:38:33,120
So therefore, the
time T2 is a time

474
00:38:33,120 --> 00:38:36,930
which describes the loss of
coherences, the dephasing

475
00:38:36,930 --> 00:38:38,150
times.

476
00:38:38,150 --> 00:38:42,800
And in most situations,
well, if you lose energy,

477
00:38:42,800 --> 00:38:44,860
you've also lost--
if you lose energy

478
00:38:44,860 --> 00:38:47,450
because you quench
a quantum state,

479
00:38:47,450 --> 00:38:49,120
you've also lost the phase.

480
00:38:49,120 --> 00:38:54,110
So therefore in general,
T2 is smaller than T1.

481
00:38:57,281 --> 00:39:01,370
Often by a lot.

482
00:39:01,370 --> 00:39:06,590
So with those remarks about
the two damping times,

483
00:39:06,590 --> 00:39:10,260
I can now go back to the
equation at the top, which

484
00:39:10,260 --> 00:39:12,690
was sort of written
with quotation marks,

485
00:39:12,690 --> 00:39:19,510
and write it in a more accurate
way as a matrix equation

486
00:39:19,510 --> 00:39:25,090
for the damping of the
components of the density

487
00:39:25,090 --> 00:39:28,280
matrix expressed by
the Bloch vector.

488
00:39:28,280 --> 00:39:30,500
In other words, the
equation of motion

489
00:39:30,500 --> 00:39:33,050
for the z-component
of the Bloch vector,

490
00:39:33,050 --> 00:39:35,570
which is describing
the population,

491
00:39:35,570 --> 00:39:41,880
has a coherent part, which is
this generalized precession.

492
00:39:41,880 --> 00:39:48,932
And then, it has a damping part,
which damps the populations

493
00:39:48,932 --> 00:39:54,700
to the equilibrium value
with a damping time T1.

494
00:39:54,700 --> 00:39:59,500
And then we have the
corresponding equations

495
00:39:59,500 --> 00:40:03,230
for the x and y, or the 1 or 2
component of the optical Bloch

496
00:40:03,230 --> 00:40:03,730
vector.

497
00:40:06,790 --> 00:40:13,250
We just replace the z index by
x and y from the equation above,

498
00:40:13,250 --> 00:40:17,952
but then we divide by a
different relaxation time, T2.

499
00:40:21,880 --> 00:40:27,320
So what we have
found here, these

500
00:40:27,320 --> 00:40:38,100
are the famous Bloch
equations, which

501
00:40:38,100 --> 00:40:43,100
were introduced
by Bloch in 1946.

502
00:40:43,100 --> 00:40:47,000
Introduced first for
magnetic resonance,

503
00:40:47,000 --> 00:40:50,190
but they're also valid
in the optical domain.

504
00:40:50,190 --> 00:40:53,160
For magnetic resonance, you
have a two-level system,

505
00:40:53,160 --> 00:40:54,370
spin up and spin down.

506
00:40:54,370 --> 00:40:57,610
In the optical domain, you have
a ground and excited state.

507
00:40:57,610 --> 00:40:59,330
In the latter
case, they're often

508
00:40:59,330 --> 00:41:02,260
referred to as the
optical Bloch equations.

509
00:41:09,580 --> 00:41:12,870
Any questions about that?

510
00:41:12,870 --> 00:41:14,232
Yes, please.

511
00:41:14,232 --> 00:41:15,940
AUDIENCE: So what
determines [INAUDIBLE]?

512
00:41:21,900 --> 00:41:25,620
PROFESSOR: Well, that's
a long discussion.

513
00:41:25,620 --> 00:41:29,320
We spent a long time in 8.422
discussing various processes.

514
00:41:29,320 --> 00:41:33,610
But just to give you an example,
if you have a gas of atoms

515
00:41:33,610 --> 00:41:37,490
and there are slightly
inhomogeneous magnetic field,

516
00:41:37,490 --> 00:41:41,250
that would mean that each atom,
if you look at it as precession

517
00:41:41,250 --> 00:41:44,330
motion, precesses at
slightly differently rates.

518
00:41:44,330 --> 00:41:47,965
And the atoms will decohere.

519
00:41:47,965 --> 00:41:51,170
They all will eventually wind
up with a different phase,

520
00:41:51,170 --> 00:41:54,740
that if you look at the average
of coherence, it's equal to 0.

521
00:41:54,740 --> 00:41:57,970
So any form of
inhomogeneity, which

522
00:41:57,970 --> 00:42:00,460
is not quenching a quantum
state, which is not

523
00:42:00,460 --> 00:42:03,620
creating any form of
de-activation of the excited

524
00:42:03,620 --> 00:42:07,280
state, can actually
decohere the phase.

525
00:42:07,280 --> 00:42:09,460
And these are
contributions to T2.

526
00:42:09,460 --> 00:42:11,630
So often, contributions
to T2 come

527
00:42:11,630 --> 00:42:15,310
from inhomogeneous
environment, but they are not

528
00:42:15,310 --> 00:42:18,130
changing the
population of states.

529
00:42:18,130 --> 00:42:21,850
Whereas, what contributes
to T1 are often collisions.

530
00:42:21,850 --> 00:42:24,460
Collisions which, when an
atom in an excited state

531
00:42:24,460 --> 00:42:27,517
collides with the
buffer gas atom,

532
00:42:27,517 --> 00:42:29,850
it undergoes a transition
from the excited to the ground

533
00:42:29,850 --> 00:42:31,750
state.

534
00:42:31,750 --> 00:42:35,840
So these are two distinctly
different processes.

535
00:42:35,840 --> 00:42:40,440
One is really a collision
and energy transfer.

536
00:42:40,440 --> 00:42:45,530
Each atom has to change
its quantum state.

537
00:42:45,530 --> 00:42:49,020
Whereas, decoherence
can simply happen

538
00:42:49,020 --> 00:42:52,110
that there is a small
pertubation of the energy

539
00:42:52,110 --> 00:42:55,200
levels due to external fields.

540
00:42:55,200 --> 00:43:00,360
And then, the system as an
ensemble loses its phase.

541
00:43:00,360 --> 00:43:02,470
In the simplest
way, you can assume

542
00:43:02,470 --> 00:43:03,990
inhomogeneous broadening.

543
00:43:03,990 --> 00:43:07,580
But you can also assume,
if the whole ensemble

544
00:43:07,580 --> 00:43:12,490
is subject to
fluctuating fields,

545
00:43:12,490 --> 00:43:17,430
then since you don't know
how the fields exactly

546
00:43:17,430 --> 00:43:21,550
fluctuate after characteristic
time, you no longer have

547
00:43:21,550 --> 00:43:23,400
a phased coherent system.

548
00:43:23,400 --> 00:43:27,900
Rather, phase at a later time
is deterministically related

549
00:43:27,900 --> 00:43:30,110
to the phase at which
you prepared it.

550
00:43:30,110 --> 00:43:33,170
And that would mean the
system has dephased.

551
00:43:33,170 --> 00:43:37,340
And this dephasing time
is called the T2 time.

552
00:43:37,340 --> 00:43:38,261
Nancy.

553
00:43:38,261 --> 00:43:39,760
AUDIENCE: I think
I have two things.

554
00:43:39,760 --> 00:43:42,733
First, you said that
it's generally true

555
00:43:42,733 --> 00:43:45,098
that T2 is less than T1.

556
00:43:45,098 --> 00:43:49,287
Is it ever true that
it's not the case?

557
00:43:49,287 --> 00:43:49,870
PROFESSOR: Oh.

558
00:43:55,580 --> 00:43:57,570
There is one exception.

559
00:43:57,570 --> 00:44:00,070
And that's the following.

560
00:44:00,070 --> 00:44:04,960
Let me put it this way, every
process which contributes to T1

561
00:44:04,960 --> 00:44:07,290
will also contribute to T2.

562
00:44:07,290 --> 00:44:11,130
But there are lots of processes
which only contribute to T2.

563
00:44:11,130 --> 00:44:14,870
So therefore, in general,
T2 is much faster

564
00:44:14,870 --> 00:44:18,740
because many more processes
can contribute to it.

565
00:44:18,740 --> 00:44:21,747
However, now if you ask
me, is it always true?

566
00:44:21,747 --> 00:44:22,830
Well, there is one glitch.

567
00:44:22,830 --> 00:44:24,500
And this is the following.

568
00:44:24,500 --> 00:44:28,030
T1 is the time to
damp populations.

569
00:44:28,030 --> 00:44:30,950
And that's the
damping of psi square.

570
00:44:30,950 --> 00:44:34,260
T2 is due to the
damping of the phase.

571
00:44:34,260 --> 00:44:37,090
And this is actually more
a damping time of the wave

572
00:44:37,090 --> 00:44:38,690
function itself.

573
00:44:38,690 --> 00:44:41,740
And if you have a wave
function psi which

574
00:44:41,740 --> 00:44:44,660
is damped with a
damping time tau.

575
00:44:44,660 --> 00:44:49,240
Psi square is damped with
twice the damping time.

576
00:44:49,240 --> 00:44:52,420
So if the only process
you have is, for instance,

577
00:44:52,420 --> 00:44:55,910
spontaneous emission,
then you find out

578
00:44:55,910 --> 00:44:58,610
that the damping rate
for population is gamma.

579
00:44:58,610 --> 00:45:01,280
This is the definition of the
spontaneous emission rate.

580
00:45:01,280 --> 00:45:08,070
But the damping rate 1
over T2 is 1/2 gamma.

581
00:45:08,070 --> 00:45:11,650
But because simply the way
how we have defined it,

582
00:45:11,650 --> 00:45:14,000
one involves the square
of the wave function.

583
00:45:14,000 --> 00:45:17,060
The other one involves
simply the wave function.

584
00:45:17,060 --> 00:45:19,440
So there is this
factor of 2 which

585
00:45:19,440 --> 00:45:26,780
can make-- by just a factor
of 2-- T1 faster than T2.

586
00:45:26,780 --> 00:45:30,120
But apart from this
factor of 2, if T2

587
00:45:30,120 --> 00:45:32,790
would be defined in a way
which would incorporate

588
00:45:32,790 --> 00:45:38,654
the factor of 2, then T2 would
always be faster than T1.

589
00:45:38,654 --> 00:45:42,056
AUDIENCE: Yeah, it
makes sense [INAUDIBLE].

590
00:45:42,056 --> 00:45:47,402
I can't imagine if the
system has a smaller T1,

591
00:45:47,402 --> 00:45:50,804
then it still has any
coherence left in it.

592
00:45:55,000 --> 00:45:57,740
PROFESSOR: So maybe to
be absolutely correct,

593
00:45:57,740 --> 00:45:59,180
I should say this.

594
00:45:59,180 --> 00:46:05,040
T1 is much larger than-- is
larger or equal than T2 over 2.

595
00:46:05,040 --> 00:46:08,930
In general, we have
even the situation

596
00:46:08,930 --> 00:46:14,960
that T1 is much,
much larger than T2.

597
00:46:14,960 --> 00:46:17,080
But with this factor
of 2, I've incorporated

598
00:46:17,080 --> 00:46:20,080
this subtlety of the definition.

599
00:46:20,080 --> 00:46:20,770
Other questions?

600
00:46:20,770 --> 00:46:21,876
Yes.

601
00:46:21,876 --> 00:46:25,820
AUDIENCE: Just a question
about real motivation

602
00:46:25,820 --> 00:46:29,764
of using Bloch
equation [INAUDIBLE].

603
00:46:33,190 --> 00:46:36,900
I understand that [INAUDIBLE].

604
00:46:36,900 --> 00:46:38,274
But you mentioned
before that you

605
00:46:38,274 --> 00:46:41,570
can't describe spontaneous
emission with a Hamiltonian

606
00:46:41,570 --> 00:46:42,070
formalism.

607
00:46:42,070 --> 00:46:42,695
PROFESSOR: Yes.

608
00:46:42,695 --> 00:46:45,022
AUDIENCE: But couldn't
you use-- [INAUDIBLE].

609
00:46:59,066 --> 00:47:01,849
Don't you still get spontaneous
emission out of the coupling

610
00:47:01,849 --> 00:47:02,640
into the continuum?

611
00:47:02,640 --> 00:47:06,480
The emission into
the different modes?

612
00:47:06,480 --> 00:47:08,360
You don't necessarily
need [INAUDIBLE].

613
00:47:11,480 --> 00:47:18,730
PROFESSOR: Yes, but let me
kind of remind you of this.

614
00:47:18,730 --> 00:47:21,650
If you are interested in
a quantum state and it

615
00:47:21,650 --> 00:47:23,240
decays to a level.

616
00:47:23,240 --> 00:47:25,340
But we're not really
interested what this level is

617
00:47:25,340 --> 00:47:27,215
and we're not keeping
track of the population

618
00:47:27,215 --> 00:47:32,630
here, when we can describe the
time evolution of the excited

619
00:47:32,630 --> 00:47:38,810
state with a Hamiltonian
because of the imaginary part,

620
00:47:38,810 --> 00:47:41,350
the Hamiltonian is
no longer imaginary.

621
00:47:41,350 --> 00:47:46,470
And this is what Victor
Weisskopf theory does.

622
00:47:46,470 --> 00:47:48,760
It looks at a system
in the excited state

623
00:47:48,760 --> 00:47:51,630
and looks at the time
evolution of the excited state.

624
00:47:51,630 --> 00:47:54,250
But if you want to include
in this description what

625
00:47:54,250 --> 00:47:58,570
happens in the ground state, you
are not having this situation.

626
00:47:58,570 --> 00:48:01,540
You have this situation.

627
00:48:01,540 --> 00:48:04,320
And what eventually
will happen is

628
00:48:04,320 --> 00:48:07,220
you can look at a pure
state which decays.

629
00:48:07,220 --> 00:48:09,770
And this is what is done
in Victor Weisskopf theory.

630
00:48:09,770 --> 00:48:12,700
But if you want to know now what
happens in the ground state,

631
00:48:12,700 --> 00:48:16,830
well, I'm speaking loosely,
but that's what really happens.

632
00:48:16,830 --> 00:48:19,950
Every spontaneous emission adds
something to the ground state,

633
00:48:19,950 --> 00:48:21,610
but in incoherent way.

634
00:48:21,610 --> 00:48:24,240
So what is being built
up in the ground state

635
00:48:24,240 --> 00:48:25,760
is not a wave function.

636
00:48:25,760 --> 00:48:28,850
It's just population which has
to be described with a density

637
00:48:28,850 --> 00:48:29,810
matrix.

638
00:48:29,810 --> 00:48:32,690
Or in other words, if you have
a coherent superposition between

639
00:48:32,690 --> 00:48:36,250
excited and ground state, you
cannot just say spontaneous

640
00:48:36,250 --> 00:48:40,730
emission is now increasing the
amplitude to be in the ground

641
00:48:40,730 --> 00:48:41,300
state.

642
00:48:41,300 --> 00:48:43,820
It really does something
fundamentally different.

643
00:48:43,820 --> 00:48:47,870
It puts population into the
ground state with-- I'm loosely

644
00:48:47,870 --> 00:48:50,110
speaking now, but
with a random phase.

645
00:48:50,110 --> 00:48:52,470
And this can only be
described probabilistically

646
00:48:52,470 --> 00:48:54,820
by using the density matrix.

647
00:48:54,820 --> 00:48:57,562
But what you are talking
about is actually,

648
00:48:57,562 --> 00:48:59,520
for the Victor Weisskopf
theory, is pretty much

649
00:48:59,520 --> 00:49:00,980
this part of the diagram.

650
00:49:00,980 --> 00:49:03,770
We prepare an excited
state, and we study it

651
00:49:03,770 --> 00:49:06,330
with all its glorious
details, with the many modes

652
00:49:06,330 --> 00:49:09,353
of the electromagnetic field
how the excited state decays.

653
00:49:17,280 --> 00:49:17,780
OK.

654
00:49:20,330 --> 00:49:31,010
Actually with that, we have
finished one big chapter

655
00:49:31,010 --> 00:49:34,380
of our course, which is
the general discussion

656
00:49:34,380 --> 00:49:39,220
of resonance, classical
resonance, and our discussion

657
00:49:39,220 --> 00:49:41,800
of two-level systems.

658
00:49:41,800 --> 00:49:45,450
AUDIENCE: But
[INAUDIBLE], wouldn't you

659
00:49:45,450 --> 00:49:50,184
have to do a sum over every
single mode [INAUDIBLE]?

660
00:49:50,184 --> 00:49:53,356
Which would be the
exact same thing

661
00:49:53,356 --> 00:49:58,000
you do when you do a partial
trace over the environment.

662
00:49:58,000 --> 00:50:01,000
Isn't the end result
sort of the same thing

663
00:50:01,000 --> 00:50:03,488
that you have to do some
[INAUDIBLE] infinite

664
00:50:03,488 --> 00:50:05,440
sum and integral or
all the [INAUDIBLE]?

665
00:50:09,807 --> 00:50:11,140
PROFESSOR: You need a sum, but--

666
00:50:11,140 --> 00:50:12,550
AUDIENCE: That's where the
decoherence comes from?

667
00:50:12,550 --> 00:50:13,180
PROFESSOR: Yes.

668
00:50:13,180 --> 00:50:16,940
But if you're interested in only
the decay of an excited state,

669
00:50:16,940 --> 00:50:19,280
it can decay in
many, many modes,

670
00:50:19,280 --> 00:50:25,560
but all these different
modes provide a contribution

671
00:50:25,560 --> 00:50:27,250
to the decay rate gamma.

672
00:50:27,250 --> 00:50:31,100
So at the end of the day, you
have a Hamiltonian evolution

673
00:50:31,100 --> 00:50:32,670
with a damping time gamma.

674
00:50:32,670 --> 00:50:38,280
And this damping time gamma is
the sum of the other states.

675
00:50:38,280 --> 00:50:45,850
So in other words, the loss
of population from the excited

676
00:50:45,850 --> 00:50:50,660
state, you just incorporate
it by adding a damping time

677
00:50:50,660 --> 00:50:54,920
to the Schrodinger
equation because you're not

678
00:50:54,920 --> 00:50:59,930
keeping track of the other
modes where the population goes.

679
00:50:59,930 --> 00:51:01,550
You're not keeping track.

680
00:51:01,550 --> 00:51:04,860
You just say, excited
state is lost.

681
00:51:04,860 --> 00:51:07,040
You're not interested
whether the atoms are now

682
00:51:07,040 --> 00:51:08,990
in the ground state
or some other state.

683
00:51:08,990 --> 00:51:13,750
All you are describing the loss
rate from the excited state.

684
00:51:13,750 --> 00:51:17,930
And this is possible
by simply doing--

685
00:51:17,930 --> 00:51:20,120
by adding damping terms to
the Schrodinger equation.

686
00:51:25,410 --> 00:51:28,230
In other words, what I'm saying
is actually fairly simple.

687
00:51:28,230 --> 00:51:30,500
If you have a coherent
state and you lose it,

688
00:51:30,500 --> 00:51:31,630
you just lose amplitude.

689
00:51:31,630 --> 00:51:33,200
what is left is coherent.

690
00:51:33,200 --> 00:51:35,200
When it's gone, it's gone.

691
00:51:35,200 --> 00:51:38,290
You have a smaller amplitude,
smaller probability.

692
00:51:38,290 --> 00:51:40,070
And that's simple to describe.

693
00:51:40,070 --> 00:51:44,110
What is harder to describe
is if you accumulate

694
00:51:44,110 --> 00:51:47,420
population in the ground
state and the population

695
00:51:47,420 --> 00:51:49,960
arrives in incoherent pieces.

696
00:51:49,960 --> 00:51:53,480
How to treat that, this
is more complicated.

697
00:51:53,480 --> 00:51:55,960
But simply the decay
of a pure state,

698
00:51:55,960 --> 00:51:58,530
it's just-- you have e
to the i omega t, which

699
00:51:58,530 --> 00:52:01,460
is a coherent evolution, and
then you add an imaginary part

700
00:52:01,460 --> 00:52:03,940
and this is a damping time.

701
00:52:03,940 --> 00:52:07,220
So what I'm saying,
it's sort of subtle

702
00:52:07,220 --> 00:52:10,171
but it's also very trivial.

703
00:52:10,171 --> 00:52:12,045
I don't know if this
addresses your question.

704
00:52:15,900 --> 00:52:19,330
In the end, in general
you need a density matrix.

705
00:52:19,330 --> 00:52:21,450
I just wanted to
sort of emphasize

706
00:52:21,450 --> 00:52:24,300
that there is a little bit of
decoherence where you can still

707
00:52:24,300 --> 00:52:26,160
get away with a wave
function description.

708
00:52:26,160 --> 00:52:28,500
And actually, Victor
Weisskopf theory

709
00:52:28,500 --> 00:52:32,060
is the wonderful example.

710
00:52:32,060 --> 00:52:34,870
OK, so we have
discussed resonance.

711
00:52:34,870 --> 00:52:38,280
Arizonans have discussed in
particular two-level systems.

712
00:52:38,280 --> 00:52:44,620
And if I wanted, we could now
continue with two-level systems

713
00:52:44,620 --> 00:52:46,980
and talk about the
wonderful things

714
00:52:46,980 --> 00:52:48,780
you can do with
two-level systems.

715
00:52:48,780 --> 00:52:51,850
Absorbing photons, emitting
photons, and all that.

716
00:52:51,850 --> 00:52:58,252
But let's put that on
hold for a few weeks.

717
00:52:58,252 --> 00:52:59,710
And I think what
we should first do

718
00:52:59,710 --> 00:53:04,680
is realize, where do
those levels come from?

719
00:53:04,680 --> 00:53:11,990
And we discuss where
those levels come form

720
00:53:11,990 --> 00:53:13,514
in our discussion of atoms.

721
00:53:23,900 --> 00:53:30,480
So our big next chapter is
now atoms or atomic structure.

722
00:53:30,480 --> 00:53:35,715
And we build it up
in several stages.

723
00:53:38,610 --> 00:53:40,930
Well, first things first.

724
00:53:40,930 --> 00:53:44,750
And the first things are
the big chunks of energy

725
00:53:44,750 --> 00:53:47,050
which define the
electronic structure.

726
00:53:50,460 --> 00:53:54,170
We discuss electronic structure
for one electron and two

727
00:53:54,170 --> 00:53:57,430
electron atoms,
hydrogen and helium.

728
00:53:57,430 --> 00:54:00,150
We don't go higher in
the periodic table.

729
00:54:04,010 --> 00:54:08,450
But then we talk about
other contributions

730
00:54:08,450 --> 00:54:12,860
to the energy of atoms, other
contributions to the level

731
00:54:12,860 --> 00:54:15,140
structure of atoms.

732
00:54:15,140 --> 00:54:25,542
And this will start with fine
structure, the Lamb shift.

733
00:54:25,542 --> 00:54:29,180
We bring in properties
of the nucleus

734
00:54:29,180 --> 00:54:32,990
by discussing
hyperfine structure.

735
00:54:32,990 --> 00:54:37,970
And then as a next
big chapter, we

736
00:54:37,970 --> 00:54:42,310
will learn how external
fields, magnetic fields,

737
00:54:42,310 --> 00:54:48,180
electric fields, and
electromagnetic fields

738
00:54:48,180 --> 00:54:52,010
will modify the level
structure of atoms.

739
00:54:52,010 --> 00:54:54,570
So by going through all
those different layers,

740
00:54:54,570 --> 00:55:00,030
we will arrive at a rather
complete description.

741
00:55:00,030 --> 00:55:02,060
If you have an atom
in the laboratory,

742
00:55:02,060 --> 00:55:05,070
what determines its energy
level and the transitions

743
00:55:05,070 --> 00:55:06,920
between those energy levels?

744
00:55:06,920 --> 00:55:09,550
So this is our agenda for
the next few lectures.

745
00:55:13,310 --> 00:55:17,920
Today, we start with single
electron atom with a hydrogen

746
00:55:17,920 --> 00:55:19,120
atom.

747
00:55:19,120 --> 00:55:24,560
And I cannot resist to start
with some quotes from Dan

748
00:55:24,560 --> 00:55:27,950
Kleppner, who I sometimes
call Mr. Hydrogen.

749
00:55:53,820 --> 00:55:59,680
So there is some beautiful
piece of writing in a reference

750
00:55:59,680 --> 00:56:04,720
frame in Physics Today, "The
Yin and Yang of Hydrogen."

751
00:56:04,720 --> 00:56:07,200
I mean, those of you who
know Dan Kleppner know

752
00:56:07,200 --> 00:56:11,540
that he's always said hydrogen
is the only atom, other atom he

753
00:56:11,540 --> 00:56:12,460
wants to work with.

754
00:56:12,460 --> 00:56:14,370
Other atoms are too complicated.

755
00:56:14,370 --> 00:56:17,740
And he studied--
actually, hydrogen

756
00:56:17,740 --> 00:56:20,470
was-- he did a little bit
on alkali atoms, of course,

757
00:56:20,470 --> 00:56:24,460
but hydrogen was
really the central part

758
00:56:24,460 --> 00:56:26,400
of his scientific work.

759
00:56:26,400 --> 00:56:28,220
Whether he studied
Rydberg states

760
00:56:28,220 --> 00:56:32,440
in hydrogen or Bose-Einstein
condensation in hydrogen.

761
00:56:32,440 --> 00:56:38,090
And this column in
Physics Today, he

762
00:56:38,090 --> 00:56:39,860
talks about the yin and yang.

763
00:56:39,860 --> 00:56:41,940
The simplicity of hydrogen.

764
00:56:41,940 --> 00:56:44,180
It's the simplest atom.

765
00:56:44,180 --> 00:56:45,840
But if you want to
work with hydrogen,

766
00:56:45,840 --> 00:56:48,260
you need vacuum UV
because the step

767
00:56:48,260 --> 00:56:52,020
from the 1s to the 2p
transition is-- Lyman-alpha

768
00:56:52,020 --> 00:56:54,810
is vacuum UV at 121 nanometer.

769
00:56:54,810 --> 00:56:58,650
So it's simple, but challenging.

770
00:56:58,650 --> 00:57:01,490
And hydrogen is the
most pristine atom.

771
00:57:01,490 --> 00:57:04,900
But for those of you who do
Bose-Einstein condensation,

772
00:57:04,900 --> 00:57:07,590
it's the hardest atom
to Bose condense.

773
00:57:07,590 --> 00:57:10,260
Because the physical
properties of hydrogen,

774
00:57:10,260 --> 00:57:12,170
it's simple in its structure.

775
00:57:12,170 --> 00:57:14,410
But the properties
of hydrogen, in

776
00:57:14,410 --> 00:57:16,500
particular the
collision cross-section,

777
00:57:16,500 --> 00:57:18,730
which is important for
evaporative cooling,

778
00:57:18,730 --> 00:57:21,140
is very, very unfavorable.

779
00:57:21,140 --> 00:57:24,450
So that's why he talks about the
yin and the yang of hydrogen.

780
00:57:27,450 --> 00:57:35,830
Let me just show you the
first sentence of this paper,

781
00:57:35,830 --> 00:57:36,940
of this reference frame.

782
00:57:48,836 --> 00:57:49,336
Oops.

783
00:57:55,350 --> 00:58:01,078
Just a technical problem to
make this fit the screen.

784
00:58:01,078 --> 00:58:02,915
I think I select it.

785
00:58:22,240 --> 00:58:24,450
What's going on?

786
00:58:24,450 --> 00:58:26,181
Yep.

787
00:58:26,181 --> 00:58:29,340
So now it's smaller.

788
00:58:29,340 --> 00:58:31,262
I can move it over there.

789
00:58:41,340 --> 00:58:42,840
Well, why don't we
read it together?

790
00:58:42,840 --> 00:58:47,650
It's a tribute to hydrogen,
a tribute to famous people.

791
00:58:47,650 --> 00:58:50,960
Viki Weisskopf was on
the faculty at MIT.

792
00:58:50,960 --> 00:58:53,690
I met him, but he was already
retired at this point.

793
00:58:53,690 --> 00:58:56,200
But then, Kleppner
interacted with him.

794
00:58:56,200 --> 00:58:59,520
And you see the first quote,
"To understand hydrogen

795
00:58:59,520 --> 00:59:02,320
is to understand
all of physics."

796
00:59:02,320 --> 00:59:05,460
Well, it simply says
that if you understand

797
00:59:05,460 --> 00:59:08,570
some of this paradigmatic
systems in physics,

798
00:59:08,570 --> 00:59:10,540
you understand all of physics.

799
00:59:10,540 --> 00:59:12,230
I would actually
say, well, you really

800
00:59:12,230 --> 00:59:14,085
have to understand the
harmonic oscillator,

801
00:59:14,085 --> 00:59:17,929
the two-level
system, and hydrogen.

802
00:59:17,929 --> 00:59:19,970
And maybe a little bit
about three-level systems.

803
00:59:19,970 --> 00:59:24,940
But if you understand,
really, those simple systems--

804
00:59:24,940 --> 00:59:25,940
they're not so simple.

805
00:59:25,940 --> 00:59:28,640
But if you understand those
so-called simple system

806
00:59:28,640 --> 00:59:32,230
very well in all
its glorious detail,

807
00:59:32,230 --> 00:59:34,210
then you have really
understood, maybe

808
00:59:34,210 --> 00:59:38,070
not all of physics, but a
hell of a lot of physics.

809
00:59:38,070 --> 00:59:41,330
And this quote goes on that,
"To understand hydrogen

810
00:59:41,330 --> 00:59:44,090
is to understand
all of physics."

811
00:59:44,090 --> 00:59:46,720
But then Viki Weisskopf
said, "Well, I

812
00:59:46,720 --> 00:59:49,690
wish I had understood
all of hydrogen."

813
00:59:49,690 --> 00:59:55,020
And this is sort of Dan's
Kleppner's wise words.

814
00:59:55,020 --> 00:59:59,040
For me, hydrogen holds an
almost mystical attraction.

815
00:59:59,040 --> 01:00:01,650
Probably because I'm among
the small band of physicists

816
01:00:01,650 --> 01:00:05,137
who actually confront
it, more or less, daily.

817
01:00:07,760 --> 01:00:11,600
So that's what we are starting
out now to talk about hydrogen.

818
01:00:14,890 --> 01:00:20,360
I know that a discussion of
the hydrogen atom, the solution

819
01:00:20,360 --> 01:00:22,820
of the Schrodinger equation
for the hydrogen atom

820
01:00:22,820 --> 01:00:24,670
is in all quantum
mechanics textbooks.

821
01:00:24,670 --> 01:00:26,300
I'm not doing it here.

822
01:00:26,300 --> 01:00:29,840
I rather want to give you
a few insightful comments

823
01:00:29,840 --> 01:00:36,040
about the structure of hydrogen,
some scaling of length scales

824
01:00:36,040 --> 01:00:37,860
and energy levels,
because this is

825
01:00:37,860 --> 01:00:40,690
something we need
later in the course.

826
01:00:40,690 --> 01:00:43,450
So in other words, I want to
highlight a few things which

827
01:00:43,450 --> 01:00:46,790
are often not emphasized
in the textbook.

828
01:00:46,790 --> 01:00:48,879
So let's talk about
the hydrogen atom.

829
01:00:58,460 --> 01:01:03,630
So the energy levels
of the hydrogen atom

830
01:01:03,630 --> 01:01:05,400
are described by
the Rydberg formula.

831
01:01:09,670 --> 01:01:15,100
This actually follows already
from the simple Bohr model.

832
01:01:15,100 --> 01:01:19,170
But of course, also from
the Schrodinger equation.

833
01:01:19,170 --> 01:01:29,150
And it says that
the energy levels--

834
01:01:29,150 --> 01:01:31,070
let me write it in
the following way.

835
01:01:33,620 --> 01:01:39,100
It depends on the electron
mass, the electron charge, h bar

836
01:01:39,100 --> 01:01:40,820
square.

837
01:01:40,820 --> 01:01:48,890
It has a reduced
mass correction.

838
01:01:48,890 --> 01:01:52,500
And then, n is the
principal quantum number.

839
01:01:52,500 --> 01:01:54,250
It scales as 1 over n squared.

840
01:01:57,610 --> 01:02:05,770
So this here is the
reduced mass factor.

841
01:02:05,770 --> 01:02:11,740
This here is called the
Rydberg constant R, sometimes

842
01:02:11,740 --> 01:02:16,400
with the index infinity
because it is the Rydberg

843
01:02:16,400 --> 01:02:19,840
constant which describes the
spectrum of a hydrogen atom

844
01:02:19,840 --> 01:02:23,880
where the nucleus
has infinite mass.

845
01:02:23,880 --> 01:02:30,790
If you include the
reduced mass correction

846
01:02:30,790 --> 01:02:35,650
for the mass of the
proton, then this factor

847
01:02:35,650 --> 01:02:38,230
which determines the
spectrum of hydrogen

848
01:02:38,230 --> 01:02:41,890
is called the Rydberg constant
with an index H for hydrogen.

849
01:02:54,820 --> 01:03:00,752
You find the electronic
eigenfunctions

850
01:03:00,752 --> 01:03:05,040
as the solution of
Schrodinger's equation.

851
01:03:05,040 --> 01:03:12,050
And the eigenfunctions
have a simple angular part,

852
01:03:12,050 --> 01:03:14,205
which are the
spherical harmonics.

853
01:03:14,205 --> 01:03:16,410
We are not talking about that.

854
01:03:16,410 --> 01:03:19,590
But there is a radial
part, radial wave function.

855
01:03:30,950 --> 01:03:36,120
So if you solve it, if you
find those wave functions,

856
01:03:36,120 --> 01:03:40,240
there are a number of
noteworthy results.

857
01:03:40,240 --> 01:03:45,180
One is in short form the
spectrum is the Rydberg

858
01:03:45,180 --> 01:03:47,020
constant divided by n squared.

859
01:03:50,750 --> 01:03:56,230
I want to talk to you
about your intuition

860
01:03:56,230 --> 01:03:58,330
for the size of
the hydrogen atom,

861
01:03:58,330 --> 01:04:01,910
or for the size of
hydrogen-like atoms.

862
01:04:01,910 --> 01:04:07,290
So what I want to discuss
is several important aspects

863
01:04:07,290 --> 01:04:10,380
about the radius or
the expectation value

864
01:04:10,380 --> 01:04:13,420
of the position of the electron.

865
01:04:13,420 --> 01:04:16,620
And it's important
to distinguish

866
01:04:16,620 --> 01:04:19,530
between the expectation
value for the radius

867
01:04:19,530 --> 01:04:21,160
and the inverse radius.

868
01:04:25,210 --> 01:04:27,830
The expectation
value for the radius

869
01:04:27,830 --> 01:04:34,400
is, well, a little
bit more complicated.

870
01:04:34,400 --> 01:04:41,070
1/2 1 minus l times l
plus 1 over n squared.

871
01:04:44,200 --> 01:04:54,000
Whereas, the result for the
inverse radius is very simple.

872
01:04:54,000 --> 01:04:58,480
What I've introduced here
is the natural length scale

873
01:04:58,480 --> 01:05:04,950
for the hydrogen atom
which is the Bohr radius.

874
01:05:04,950 --> 01:05:11,322
And just to be general,
mu is the reduced mass.

875
01:05:11,322 --> 01:05:12,780
So it's close to
the electron mass.

876
01:05:21,560 --> 01:05:26,400
Well, the one thing I
want to discuss with you--

877
01:05:26,400 --> 01:05:34,020
we will need leader for the
discussion of quantum defects,

878
01:05:34,020 --> 01:05:37,430
for field ionization
and other processes,

879
01:05:37,430 --> 01:05:41,610
we have to know what the
size of the wave function is.

880
01:05:41,610 --> 01:05:44,900
And so usually, if
you wave your hands,

881
01:05:44,900 --> 01:05:48,260
you would say the
expectation value of 1/r

882
01:05:48,260 --> 01:05:50,920
is 1 over the
expectation value or r.

883
01:05:50,920 --> 01:05:54,680
But there are now some
important differences.

884
01:05:54,680 --> 01:05:56,500
I first want to
sort of ask you, why

885
01:05:56,500 --> 01:05:59,160
is the expectation
value of 1/r, why

886
01:05:59,160 --> 01:06:01,178
does it have this
very, very simple form?

887
01:06:05,825 --> 01:06:06,866
AUDIENCE: Virial theorem?

888
01:06:06,866 --> 01:06:08,900
PROFESSOR: The Virial theorem.

889
01:06:08,900 --> 01:06:10,590
Yes.

890
01:06:10,590 --> 01:06:15,360
We know that there is a fairly
simple form for the energy

891
01:06:15,360 --> 01:06:16,090
eigenvalues.

892
01:06:16,090 --> 01:06:18,460
It's 1 over n squared.

893
01:06:18,460 --> 01:06:23,810
Well, Coulomb energy
e square over r.

894
01:06:23,810 --> 01:06:26,880
So if the only energy
of the hydrogen atom

895
01:06:26,880 --> 01:06:29,250
were Coulomb energy,
it's very clear

896
01:06:29,250 --> 01:06:32,700
that 1/r, which is proportional
to the Coulomb energy,

897
01:06:32,700 --> 01:06:37,780
has to have the same simple
form as the energy eigenvalue.

898
01:06:37,780 --> 01:06:39,860
Well, there is a
second contribution

899
01:06:39,860 --> 01:06:42,130
to the energy in addition
to Coulomb energy.

900
01:06:42,130 --> 01:06:43,620
This is kinetic energy.

901
01:06:43,620 --> 01:06:47,240
But due to the Virial
theorem, the kinetic energy

902
01:06:47,240 --> 01:06:50,030
is actually proportional
to the Coulomb energy.

903
01:06:50,030 --> 01:06:54,570
And therefore, the total
energy is proportional to 1/r.

904
01:06:54,570 --> 01:07:00,690
And therefore, 1/r has to
scale exactly as the energy.

905
01:07:00,690 --> 01:07:05,990
Since the energy until we
introduce fine structure is

906
01:07:05,990 --> 01:07:09,530
independent of l, only depends
on the principal quantum number

907
01:07:09,530 --> 01:07:12,730
n, we find there's
only an n-dependence.

908
01:07:12,730 --> 01:07:15,510
But if you would ask, what
is the expectation value

909
01:07:15,510 --> 01:07:16,600
for the radius?

910
01:07:16,600 --> 01:07:18,750
You find an l-dependence
because you're

911
01:07:18,750 --> 01:07:20,710
talking about a very
different quantity.

912
01:07:23,640 --> 01:07:29,830
So let me just summarize
what we just discussed.

913
01:07:29,830 --> 01:07:35,680
We have the Virial
theorem, which

914
01:07:35,680 --> 01:07:40,350
in general is of
the following form.

915
01:07:40,350 --> 01:07:42,630
If you have potential
energy which

916
01:07:42,630 --> 01:07:48,990
is proportional to
radius to the n,

917
01:07:48,990 --> 01:07:55,010
then the expectation value
for the kinetic energy

918
01:07:55,010 --> 01:08:00,540
is n/2 times the expectation
value for the potential energy.

919
01:08:00,540 --> 01:08:02,800
The most famous
example is n equals

920
01:08:02,800 --> 01:08:04,120
2, the harmonic oscillator.

921
01:08:04,120 --> 01:08:07,710
You have an equal contribution
to potential energy

922
01:08:07,710 --> 01:08:10,430
of the spring and
kinetic energy.

923
01:08:10,430 --> 01:08:12,710
Well, here for the
Coulomb problem,

924
01:08:12,710 --> 01:08:17,020
we discuss n equals minus 1.

925
01:08:17,020 --> 01:08:20,390
And therefore,
the kinetic energy

926
01:08:20,390 --> 01:08:24,386
is minus 1/2 times
the potential energy.

927
01:08:32,670 --> 01:08:41,420
So this factor of 2 appears
now in a number of relations

928
01:08:41,420 --> 01:08:46,670
and that's as follows.

929
01:08:46,670 --> 01:08:50,670
If you take the
Rydberg constant,

930
01:08:50,670 --> 01:08:54,970
the Rydberg constant
in CGS units

931
01:08:54,970 --> 01:09:03,210
is-- well, that's the Coulomb
energy at the Bohr radius.

932
01:09:03,210 --> 01:09:05,970
But the Rydberg
constant is 1/2 of it.

933
01:09:09,069 --> 01:09:22,069
So the Rydberg constant is
1/2 of another quantity,

934
01:09:22,069 --> 01:09:26,050
which is called 1 Hartree.

935
01:09:26,050 --> 01:09:27,990
We'll talk, probably
not , today,

936
01:09:27,990 --> 01:09:33,040
but on Monday
about atomic units,

937
01:09:33,040 --> 01:09:36,710
about sort of fundamental
system of units.

938
01:09:36,710 --> 01:09:40,750
And the fundamental way--
the fundamental energy

939
01:09:40,750 --> 01:09:44,490
of the hydrogen atom, the
fundamental unit of energy

940
01:09:44,490 --> 01:09:49,609
is whatever energy
you can construct

941
01:09:49,609 --> 01:09:53,590
using the electron mass, the
electron charge, and h bar.

942
01:09:53,590 --> 01:09:56,670
And what you get is 1 Hartree.

943
01:09:56,670 --> 01:10:02,180
If you ever wondered why the
Rydberg is 1/2 Hartree, what

944
01:10:02,180 --> 01:10:05,350
happens is in the ground
state of hydrogen,

945
01:10:05,350 --> 01:10:10,280
you have 1 Hartree
worth of Coulomb energy.

946
01:10:10,280 --> 01:10:14,720
But then because of the Virial
theorem, you have minus 1/2

947
01:10:14,720 --> 01:10:16,870
of it as kinetic energy.

948
01:10:16,870 --> 01:10:19,767
And therefore, the
binding energy in the n

949
01:10:19,767 --> 01:10:23,770
equals 1 ground state,
which is 1 Rydberg,

950
01:10:23,770 --> 01:10:25,810
is 1/2 of the Hartree.

951
01:10:25,810 --> 01:10:28,620
So this factor of 1/2
of the Virial theorem

952
01:10:28,620 --> 01:10:32,165
is responsible for this factor
of 2 for those two energies.

953
01:10:35,580 --> 01:10:39,120
I usually prefer SI units
for all calculations,

954
01:10:39,120 --> 01:10:46,710
but there's certain relations
where we should use CGS units.

955
01:10:46,710 --> 01:10:50,550
Just as a side remark, if
you want to go to SI units,

956
01:10:50,550 --> 01:10:53,860
you simply replace
the electron charge e

957
01:10:53,860 --> 01:10:57,440
squared by e squared
divided by 4 pi epsilon0.

958
01:11:10,851 --> 01:11:11,350
OK.

959
01:11:11,350 --> 01:11:15,010
So I've discussed
the hydrogen atom.

960
01:11:15,010 --> 01:11:18,860
It's also insightful
and you should actually

961
01:11:18,860 --> 01:11:22,950
remember that or be able to
re-derive it for yourself.

962
01:11:22,950 --> 01:11:29,210
How do things depend on
the nuclear charge z?

963
01:11:29,210 --> 01:11:32,120
Well, if you have
a nuclear charge z,

964
01:11:32,120 --> 01:11:39,430
the Coulomb energy
goes up by-- well,

965
01:11:39,430 --> 01:11:41,020
if you have a
stronger attraction.

966
01:11:41,020 --> 01:11:48,570
If you would go to helium
nucleus or even more

967
01:11:48,570 --> 01:11:51,110
highly-charged nucleus and
put one electron in it.

968
01:11:51,110 --> 01:11:54,240
Because of the stronger
Coulomb attraction,

969
01:11:54,240 --> 01:11:58,315
all the length scales
are divided by z.

970
01:12:00,890 --> 01:12:04,590
So everything is smaller
by a factor of z.

971
01:12:04,590 --> 01:12:06,750
So what does that now
imply for the energy?

972
01:12:09,340 --> 01:12:13,260
Well, you have a Coulomb field
which is z times stronger,

973
01:12:13,260 --> 01:12:17,330
but you probe it now at
a z times smaller radius.

974
01:12:17,330 --> 01:12:22,469
So therefore, the energies
scale with z squared.

975
01:12:45,000 --> 01:12:50,510
Let me formulate a question
because we need that later on.

976
01:12:50,510 --> 01:12:57,790
So if you have a
hydrogen-like atom

977
01:12:57,790 --> 01:13:00,850
and the electron is in a state
with principal quantum number

978
01:13:00,850 --> 01:13:02,040
n.

979
01:13:02,040 --> 01:13:04,080
And let's assume there
is no angular momentum.

980
01:13:07,280 --> 01:13:11,330
So what I'm writing down
for you is the probability

981
01:13:11,330 --> 01:13:14,470
for the electron to
be at the nucleus.

982
01:13:14,470 --> 01:13:16,170
This will be very
important later on

983
01:13:16,170 --> 01:13:19,880
when ewe discuss hyperfine
structure because hyperfine

984
01:13:19,880 --> 01:13:23,600
is responsible-- for
hyperfine structure, what

985
01:13:23,600 --> 01:13:28,630
is responsible is the fact
that the electron can overlap

986
01:13:28,630 --> 01:13:29,850
with the nucleus.

987
01:13:29,850 --> 01:13:32,280
So this factor will
appear in our discussion

988
01:13:32,280 --> 01:13:34,570
of hyperfine structure.

989
01:13:34,570 --> 01:13:43,740
And what I want to ask you is,
how does this quantity depend

990
01:13:43,740 --> 01:13:48,800
on the principal quantum
number n and on z?

991
01:13:54,190 --> 01:13:56,470
And I want to give
you four choices.

992
01:14:18,760 --> 01:14:21,570
Of course, for
dimensional reasons,

993
01:14:21,570 --> 01:14:23,480
everything is 1
over the Bohr radius

994
01:14:23,480 --> 01:14:25,970
cubed because it's a density.

995
01:14:25,970 --> 01:14:30,250
But you cannot use
dimensional analysis to guess,

996
01:14:30,250 --> 01:14:36,550
how do things scale
with z and with n?

997
01:14:36,550 --> 01:14:42,480
So here are your four choices.

998
01:14:42,480 --> 01:14:46,290
Does it scale with z,
z squared, z cubed?

999
01:14:46,290 --> 01:14:50,712
Does it scale with n
squared, n cubed, n to the 6?

1000
01:15:03,320 --> 01:15:05,884
If you don't know it,
just make your best guess.

1001
01:15:19,540 --> 01:15:23,240
OK, one part should be
relatively straightforward.

1002
01:15:23,240 --> 01:15:25,700
And this is the scaling with z.

1003
01:15:25,700 --> 01:15:27,440
Let me just stop it.

1004
01:15:31,930 --> 01:15:49,040
So the exact answer is that z n
00 at the origin squared is pi

1005
01:15:49,040 --> 01:15:50,960
a0 cubed.

1006
01:15:50,960 --> 01:15:55,840
And its c cubed over n cubed.

1007
01:15:55,840 --> 01:15:59,745
So the correct
answer is this one.

1008
01:16:02,430 --> 01:16:11,190
Let me first say-- OK,
I gave you four choices

1009
01:16:11,190 --> 01:16:16,460
and it's difficult to
distinguish all of them.

1010
01:16:16,460 --> 01:16:20,200
But the first one you should
have gotten rather simply,

1011
01:16:20,200 --> 01:16:23,590
and this is the z-scaling.

1012
01:16:23,590 --> 01:16:30,436
Because the scaling
with z is the following,

1013
01:16:30,436 --> 01:16:35,120
that everything-- if you write
down the Schrodinger equation,

1014
01:16:35,120 --> 01:16:40,230
if you have z, you replace
e squared by z e squared.

1015
01:16:40,230 --> 01:16:43,340
And I actually just mentioned
it five minutes ago,

1016
01:16:43,340 --> 01:16:48,850
that all length scales, the
Bohr radius is h bar squared

1017
01:16:48,850 --> 01:16:51,970
over electron mass
times e squared.

1018
01:16:51,970 --> 01:16:55,290
It actually scales with 1/z.

1019
01:16:55,290 --> 01:16:58,990
So if all length
scales go as 1/z,

1020
01:16:58,990 --> 01:17:05,410
the density goes with z cubed.

1021
01:17:05,410 --> 01:17:08,155
So therefore, one should
have immediately narrowed

1022
01:17:08,155 --> 01:17:09,015
down the choice.

1023
01:17:11,520 --> 01:17:13,150
It should be A or
C because they have

1024
01:17:13,150 --> 01:17:15,450
the correct scaling with z.

1025
01:17:15,450 --> 01:17:17,580
The scaling with
n is more subtle

1026
01:17:17,580 --> 01:17:20,980
and there was something
surprising I learned about it.

1027
01:17:20,980 --> 01:17:25,710
And this is what
I want to present

1028
01:17:25,710 --> 01:17:28,780
to you in the last
three or four minutes.

1029
01:17:28,780 --> 01:17:32,280
So the z-scaling, just remember
that the length scaling

1030
01:17:32,280 --> 01:17:34,450
is the length scales as 1/z.

1031
01:17:34,450 --> 01:17:36,140
Therefore, density
scale was z cubed.

1032
01:17:40,010 --> 01:17:42,460
The interesting thing
about the length scaling

1033
01:17:42,460 --> 01:17:44,990
is-- and I just want to draw
your attention to it because it

1034
01:17:44,990 --> 01:17:49,460
can be confusing,
that in high torsion

1035
01:17:49,460 --> 01:17:55,240
we have not only one length
scale, but two length scales.

1036
01:17:55,240 --> 01:17:57,380
We have mentioned
one of it already,

1037
01:17:57,380 --> 01:18:01,380
which is the energetic
length scale 1/r.

1038
01:18:01,380 --> 01:18:05,680
1/r is the Coulomb energy.

1039
01:18:05,680 --> 01:18:07,450
Because of the
Virial theorem, it's

1040
01:18:07,450 --> 01:18:09,611
proportional to
the total energy.

1041
01:18:09,611 --> 01:18:12,110
And that's what you know, what
you remember when you wake up

1042
01:18:12,110 --> 01:18:14,450
in the middle of the
night out of deep sleep,

1043
01:18:14,450 --> 01:18:20,030
that the energy of high
torsion is 1 over n squared.

1044
01:18:20,030 --> 01:18:25,850
So therefore, this
is a0 over n squared.

1045
01:18:25,850 --> 01:18:33,440
However, if you look at the
wave function of hydrogen,

1046
01:18:33,440 --> 01:18:35,880
you factor out.

1047
01:18:35,880 --> 01:18:38,120
When you solve the
radial equation,

1048
01:18:38,120 --> 01:18:39,790
you factor out an exponential.

1049
01:18:39,790 --> 01:18:41,870
There's sort of
polynomial and then

1050
01:18:41,870 --> 01:18:44,430
there is an exponential decay.

1051
01:18:44,430 --> 01:19:03,080
And the characteristic lengths
in the exponential decay

1052
01:19:03,080 --> 01:19:13,540
of the wave function
is n e0 over z.

1053
01:19:18,010 --> 01:19:22,200
So therefore, when we
talk about wave functions

1054
01:19:22,200 --> 01:19:26,260
with principal quantum number
n, there are two length scale.

1055
01:19:26,260 --> 01:19:32,140
1 over r n l scales
with n squared.

1056
01:19:32,140 --> 01:19:34,740
But the characteristic
length scale

1057
01:19:34,740 --> 01:19:36,880
in the exponential
part of the radial wave

1058
01:19:36,880 --> 01:19:43,760
function scales with n
and not with n squared.

1059
01:19:48,450 --> 01:19:52,330
And it is this exponential
part of the wave function

1060
01:19:52,330 --> 01:20:00,690
which scales with n which is
responsible for the probability

1061
01:20:00,690 --> 01:20:02,850
to find the electron
as the nucleus.

1062
01:20:02,850 --> 01:20:06,850
Which, as I said before,
the z-scaling is simple

1063
01:20:06,850 --> 01:20:09,630
but the n-scaling
is not n to the 6.

1064
01:20:09,630 --> 01:20:11,130
It's n cubed.

1065
01:20:18,630 --> 01:20:22,680
And this is really important.

1066
01:20:22,680 --> 01:20:29,680
And this describes the
scaling with n for everything

1067
01:20:29,680 --> 01:20:34,600
which depends on the presence
of the electron as a nucleus.

1068
01:20:34,600 --> 01:20:37,510
One is the quantum
defect and the other one

1069
01:20:37,510 --> 01:20:39,140
is the hyperfine structure.

1070
01:20:43,940 --> 01:20:47,670
Let me just give you
one more scaling.

1071
01:20:47,670 --> 01:20:50,200
I've discussed now what
happens for 0 angular

1072
01:20:50,200 --> 01:20:54,030
momentum, for finite
angular momentum states,

1073
01:20:54,030 --> 01:20:57,620
psi is proportional
to r to the l.

1074
01:20:57,620 --> 01:21:03,640
So therefore, if you
ask, what is psi square,

1075
01:21:03,640 --> 01:21:06,450
it scales with 2l.

1076
01:21:06,450 --> 01:21:11,100
And at least for
large n, the n-scaling

1077
01:21:11,100 --> 01:21:13,996
is, again, 1 over n cubed.

1078
01:21:20,190 --> 01:21:22,747
OK, that's what I wanted
to present you today.

1079
01:21:22,747 --> 01:21:23,330
Any questions?

1080
01:21:30,830 --> 01:21:36,720
OK, so we meet again
on Wednesday next week.