1
00:00:00,499 --> 00:00:03,090
PROFESSOR: Last time we
discussed the differential

2
00:00:03,090 --> 00:00:03,690
equation.

3
00:00:03,690 --> 00:00:07,200
I'll be posting notes very soon.

4
00:00:07,200 --> 00:00:11,300
Probably this
afternoon, at some time.

5
00:00:11,300 --> 00:00:15,510
And last time, we solved
the differential equation,

6
00:00:15,510 --> 00:00:19,500
we found the energy
eigenstates, and then turned

7
00:00:19,500 --> 00:00:22,530
into an algebraic
analysis in which we

8
00:00:22,530 --> 00:00:25,050
factorized the Hamiltonian.

9
00:00:25,050 --> 00:00:26,820
Which meant,
essentially, that you

10
00:00:26,820 --> 00:00:30,080
could write the Hamiltonian--

11
00:00:30,080 --> 00:00:35,150
up to an overall constant that
doesn't complicate matters--

12
00:00:35,150 --> 00:00:39,356
as the product of an a dagger a.

13
00:00:39,356 --> 00:00:44,010
And that was very useful
to show, for example,

14
00:00:44,010 --> 00:00:47,960
that any energy eigenstate
would have to have energy

15
00:00:47,960 --> 00:00:52,070
greater than h omega over 2.

16
00:00:52,070 --> 00:00:55,280
We call this a
dagger a the number

17
00:00:55,280 --> 00:01:01,670
operator n, which is
a Hermitian operator.

18
00:01:01,670 --> 00:01:05,750
Recall that the dagger
of a product of operators

19
00:01:05,750 --> 00:01:10,740
is the reverse order product
of the daggered operators.

20
00:01:10,740 --> 00:01:15,655
So the dagger of a
dagger a is itself.

21
00:01:18,200 --> 00:01:24,140
And then a was related to x
and p, and so was a dagger.

22
00:01:24,140 --> 00:01:27,370
Recall that x and
p are Hermitian.

23
00:01:27,370 --> 00:01:30,710
And there are overall constants
here that were wrote last time,

24
00:01:30,710 --> 00:01:34,200
but now they're not that urgent.

25
00:01:34,200 --> 00:01:38,590
And a and a dagger, the
commutator is equal to one.

26
00:01:38,590 --> 00:01:40,830
That was very useful.

27
00:01:40,830 --> 00:01:46,130
Finally, we also show that
while the energy of any state

28
00:01:46,130 --> 00:01:50,600
would have to be greater
than h omega over 2,

29
00:01:50,600 --> 00:01:55,680
if you had a state that
is killed by a hat,

30
00:01:55,680 --> 00:01:58,040
it would have the
lowest allowed energy--

31
00:01:58,040 --> 00:02:01,730
which is h omega over 2.

32
00:02:01,730 --> 00:02:04,610
And, therefore, that
is the ground state.

33
00:02:04,610 --> 00:02:07,280
And we looked at this
differential equation,

34
00:02:07,280 --> 00:02:10,460
and we found this
Gaussian wave function.

35
00:02:10,460 --> 00:02:13,220
And it's a first order
differential equation.

36
00:02:13,220 --> 00:02:15,710
And, therefore, it
has just one solution.

37
00:02:15,710 --> 00:02:19,280
And, therefore, there is
just one ground state,

38
00:02:19,280 --> 00:02:20,510
and it's a bound state.

39
00:02:20,510 --> 00:02:23,300
And, of course, you wouldn't
expect more than one ground

40
00:02:23,300 --> 00:02:28,890
state, because there's no
degeneracies in the bound state

41
00:02:28,890 --> 00:02:32,570
spectrum of a
one-dimensional potential.

42
00:02:32,570 --> 00:02:38,580
So we found one ground state was
phi 0, and it's killed by a--

43
00:02:38,580 --> 00:02:42,690
which means that
it's killed by n-hat,

44
00:02:42,690 --> 00:02:45,710
because a is to
the right in n-hat.

45
00:02:45,710 --> 00:02:49,870
So the a finds phi
0 and just kills it.

46
00:02:52,440 --> 00:02:55,800
Now, the other thing to
note is that the Hamiltonian

47
00:02:55,800 --> 00:02:59,640
is really, pretty much, the
same thing as the number

48
00:02:59,640 --> 00:03:03,720
operator multiplied by
something with units of energy.

49
00:03:03,720 --> 00:03:08,070
The number operator has no
units, because a and a dagger

50
00:03:08,070 --> 00:03:10,520
have no units.

51
00:03:10,520 --> 00:03:12,870
And that's very useful.

52
00:03:12,870 --> 00:03:16,760
So it's like a dimensionless
version of the energy.

53
00:03:16,760 --> 00:03:21,260
And, certainly, if you
have an eigenstate of h

54
00:03:21,260 --> 00:03:23,960
it must be an eigenstate of n.

55
00:03:23,960 --> 00:03:31,610
And the eigenvalue of n--
if we call it capital n.

56
00:03:31,610 --> 00:03:34,760
Therefore, you can
imagine this equation

57
00:03:34,760 --> 00:03:36,920
acting on an eigenstate--

58
00:03:36,920 --> 00:03:39,770
which happens to be an
eigenstate of n or of h.

59
00:03:39,770 --> 00:03:42,880
On the left-hand side you
would read the energy,

60
00:03:42,880 --> 00:03:46,730
and on the right-hand side
you would read the eigenvalue

61
00:03:46,730 --> 00:03:50,010
of the n operator.

62
00:03:50,010 --> 00:03:53,820
So that gives you a very
nice simple expression.

63
00:03:53,820 --> 00:03:58,260
You see that the energy is the
number plus a 1/2 multiplied

64
00:03:58,260 --> 00:04:01,930
by h-bar [INAUDIBLE].

65
00:04:01,930 --> 00:04:04,270
So that's pretty
much the content

66
00:04:04,270 --> 00:04:06,510
of what we reached last time.

67
00:04:06,510 --> 00:04:10,110
And now we have to
complete the solution.

68
00:04:10,110 --> 00:04:13,660
And the plan for today is
to complete the solution,

69
00:04:13,660 --> 00:04:17,079
familiarize ourselves
with these operators,

70
00:04:17,079 --> 00:04:21,339
learn how to work with a
harmonic oscillator with them.

71
00:04:21,339 --> 00:04:25,390
And then we'll leave
the harmonic oscillator

72
00:04:25,390 --> 00:04:28,840
for the time being--

73
00:04:28,840 --> 00:04:31,480
let you do some
exercises with it--

74
00:04:31,480 --> 00:04:33,640
but turn to scattering states.

75
00:04:33,640 --> 00:04:36,910
So the second part
of today's lecture

76
00:04:36,910 --> 00:04:38,845
we'll be talking about
scattering states.

77
00:04:41,880 --> 00:04:42,650
OK.

78
00:04:42,650 --> 00:04:48,710
So when we look at this
thing and you have a number

79
00:04:48,710 --> 00:04:51,950
operator-- which encodes
the Hamiltonian--

80
00:04:51,950 --> 00:04:55,790
it's a good idea to
try to understand

81
00:04:55,790 --> 00:05:01,220
how it interacts with the other
operators that you have here.

82
00:05:01,220 --> 00:05:04,460
And a good question,
whenever you have operators,

83
00:05:04,460 --> 00:05:05,452
is the commutator.

84
00:05:05,452 --> 00:05:10,400
So you can ask, what is
the commutator of n with a?

85
00:05:14,657 --> 00:05:17,430
And this commutator
is going to show up.

86
00:05:17,430 --> 00:05:20,080
But it's basically
that kind of thing.

87
00:05:20,080 --> 00:05:24,040
If you have a and a dagger, you
ask, what is the commutator?

88
00:05:24,040 --> 00:05:27,910
If you have n, you ask,
what is the commutator

89
00:05:27,910 --> 00:05:28,870
with the other thing?

90
00:05:28,870 --> 00:05:36,790
So n with a would be the
commutator of a dagger with a--

91
00:05:36,790 --> 00:05:38,530
a like that.

92
00:05:38,530 --> 00:05:40,780
And sometimes I will
not write the hats

93
00:05:40,780 --> 00:05:43,430
to write things more quickly.

94
00:05:43,430 --> 00:05:48,400
Now, in this commutator,
you can move the a out,

95
00:05:48,400 --> 00:05:54,737
and you have a dagger a a.

96
00:05:54,737 --> 00:05:57,590
And a dagger a is minus 1.

97
00:05:57,590 --> 00:05:59,840
Because aa dagger is 1.

98
00:05:59,840 --> 00:06:04,690
So this is minus a.

99
00:06:04,690 --> 00:06:07,970
So that's pretty nice.

100
00:06:07,970 --> 00:06:09,940
It's simple.

101
00:06:09,940 --> 00:06:13,780
How about n with a dagger?

102
00:06:16,420 --> 00:06:20,110
Well, this would be a
dagger a with a dagger.

103
00:06:23,190 --> 00:06:27,000
A dagger with a dagger commute.

104
00:06:27,000 --> 00:06:38,350
So this a dagger can go
out, and you get aa dagger.

105
00:06:38,350 --> 00:06:40,395
And that's 1, so
you get a dagger.

106
00:06:43,700 --> 00:06:50,570
So it's a nice kind of
computation relation.

107
00:06:50,570 --> 00:06:52,910
You would have
commutation reserve x

108
00:06:52,910 --> 00:06:55,700
with p given a constant.

109
00:06:55,700 --> 00:07:00,170
Now n with a gives
a number times a.

110
00:07:00,170 --> 00:07:04,880
N commutated with a dagger
gives a number times a dagger.

111
00:07:04,880 --> 00:07:08,315
And those numbers are
pretty significant,

112
00:07:08,315 --> 00:07:10,640
so I'll write this again.

113
00:07:10,640 --> 00:07:16,314
N with a is minus a.

114
00:07:16,314 --> 00:07:23,305
And n with a dagger
is plus a dagger.

115
00:07:33,430 --> 00:07:35,380
This is part of the reason--

116
00:07:35,380 --> 00:07:36,670
as we will see soon--

117
00:07:36,670 --> 00:07:41,020
that the name of a--

118
00:07:41,020 --> 00:07:43,360
which we call
destruction operator,

119
00:07:43,360 --> 00:07:46,750
because it destroys the vacuum--

120
00:07:46,750 --> 00:07:49,450
it's sometimes called
lowering operator,

121
00:07:49,450 --> 00:07:53,320
because it comes with
a negative sign here.

122
00:07:53,320 --> 00:07:58,070
And we'll see a better
reason for that name.

123
00:07:58,070 --> 00:08:01,430
A dagger is sometimes
called the creation operator

124
00:08:01,430 --> 00:08:06,160
or the raising operator, because
it increases some number,

125
00:08:06,160 --> 00:08:07,430
as you will see.

126
00:08:07,430 --> 00:08:11,360
And here it's reflected
by these plots.

127
00:08:11,360 --> 00:08:16,070
But we need a little
more than that.

128
00:08:16,070 --> 00:08:18,950
We need a little more
commutators than this.

129
00:08:21,850 --> 00:08:28,180
So for example, if I would
have the commutator of a

130
00:08:28,180 --> 00:08:30,730
with a dagger to the k--

131
00:08:40,370 --> 00:08:42,020
you can imagine this.

132
00:08:45,110 --> 00:08:49,790
You have to become very
used and very comfortable

133
00:08:49,790 --> 00:08:51,470
with these
commutation relations.

134
00:08:51,470 --> 00:08:55,860
And sometimes the only way to
do that is to just do examples.

135
00:08:55,860 --> 00:08:59,390
So I'm doing this with a k here.

136
00:08:59,390 --> 00:09:02,610
Maybe-- when you
review this lecture--

137
00:09:02,610 --> 00:09:05,840
you should do it with k
equals 2 or with k equals 3,

138
00:09:05,840 --> 00:09:07,700
and do it a few times.

139
00:09:07,700 --> 00:09:09,680
Until you're comfortable
with these things,

140
00:09:09,680 --> 00:09:13,250
and you know what identities
you've been using.

141
00:09:13,250 --> 00:09:17,060
If this was a little
quick, then go more slowly

142
00:09:17,060 --> 00:09:19,100
and make absolutely
sure you know

143
00:09:19,100 --> 00:09:21,890
how to do those commutators.

144
00:09:21,890 --> 00:09:25,610
In here, I'm going
to say what happens.

145
00:09:25,610 --> 00:09:27,890
You have an a and
you have to move it

146
00:09:27,890 --> 00:09:32,270
across a string of a-hats.

147
00:09:32,270 --> 00:09:35,930
Now, moving an a
cross an a-hat--

148
00:09:38,490 --> 00:09:42,780
because of the commutator--
gives you a factor of 1,

149
00:09:42,780 --> 00:09:47,790
but it destroys the
a and the a-hat.

150
00:09:47,790 --> 00:09:50,490
As you move the a
across the a-hats--

151
00:09:50,490 --> 00:09:53,910
because this is a
with all the a-hats,

152
00:09:53,910 --> 00:10:01,840
here, minus the a-hats
times the a there.

153
00:10:01,840 --> 00:10:06,310
So if you could just move
the a all the way across--

154
00:10:06,310 --> 00:10:08,530
then you cancel with this.

155
00:10:08,530 --> 00:10:10,900
What you get is
what happens when

156
00:10:10,900 --> 00:10:13,620
you're moving it to across.

157
00:10:13,620 --> 00:10:15,930
And you're moving across
a string of those.

158
00:10:15,930 --> 00:10:20,140
So each time you try to
move on a across an a-hat,

159
00:10:20,140 --> 00:10:23,490
you get this factor of
1 and you kill the a

160
00:10:23,490 --> 00:10:27,040
and you kill one a dagger.

161
00:10:27,040 --> 00:10:32,120
So this answer
will not have an a,

162
00:10:32,120 --> 00:10:34,760
and it will have
one less a dagger.

163
00:10:34,760 --> 00:10:41,840
So a dagger to the k, minus 1.

164
00:10:41,840 --> 00:10:46,650
And then I would argue-- and
you should do it more slowly--

165
00:10:46,650 --> 00:10:51,970
that you have to go
across k of those.

166
00:10:51,970 --> 00:10:54,420
And each time you
get a factor of 1,

167
00:10:54,420 --> 00:10:56,820
and you lose the a
and the a dagger.

168
00:10:56,820 --> 00:10:58,550
So at the end you get a k.

169
00:11:06,180 --> 00:11:08,340
You should realize
that this is not

170
00:11:08,340 --> 00:11:12,810
all that different from the
kind of commutators you had.

171
00:11:12,810 --> 00:11:17,580
Like, with x to the n.

172
00:11:17,580 --> 00:11:19,140
This was very similar--

173
00:11:19,140 --> 00:11:23,250
it might be a good time to
review how that was done--

174
00:11:23,250 --> 00:11:27,570
in which that pretty much gives
you an x to the n minus 1,

175
00:11:27,570 --> 00:11:30,690
times a factor of n,
because p is a derivative.

176
00:11:30,690 --> 00:11:35,730
You could almost think of a
as the derivative with respect

177
00:11:35,730 --> 00:11:38,240
to a dagger.

178
00:11:38,240 --> 00:11:42,150
And then this
commutator would be 1.

179
00:11:42,150 --> 00:11:44,830
So this is true.

180
00:11:44,830 --> 00:11:46,250
And there is also--

181
00:11:46,250 --> 00:11:53,220
if you want-- an a dagger
with a to the n or a to the k.

182
00:11:57,190 --> 00:11:58,930
This would give you--

183
00:11:58,930 --> 00:12:02,380
if you had just one of them
you would get a minus sign,

184
00:12:02,380 --> 00:12:04,540
because a dagger with a is that.

185
00:12:04,540 --> 00:12:09,520
But the same thing holds, you're
going to get one less a-hat.

186
00:12:09,520 --> 00:12:13,980
So a-hat to the k minus 1.

187
00:12:13,980 --> 00:12:16,060
A factor of k--

188
00:12:16,060 --> 00:12:21,340
because k times you're going to
move an a dagger across an a.

189
00:12:21,340 --> 00:12:25,120
And a minus because you're
getting a dagger commutator

190
00:12:25,120 --> 00:12:28,810
with a, as opposed to a
commutator with a dagger--

191
00:12:28,810 --> 00:12:30,940
which is 1.

192
00:12:30,940 --> 00:12:35,380
So these are two very
nice and useful equations

193
00:12:35,380 --> 00:12:36,920
that you should be
comfortable with.

194
00:12:40,330 --> 00:12:46,790
Now, this implies that you can
do more with an n operator.

195
00:12:46,790 --> 00:13:09,820
So n with a-hat to the k, this
time will be minus k a-hat

196
00:13:09,820 --> 00:13:10,375
to the k.

197
00:13:13,060 --> 00:13:16,780
It doesn't change
the number of a-hats,

198
00:13:16,780 --> 00:13:21,950
because you're now making
commutators with a dagger a.

199
00:13:21,950 --> 00:13:26,730
So each time you have
this commuted with one

200
00:13:26,730 --> 00:13:31,790
a-hat, the a dagger
and the a give you 1,

201
00:13:31,790 --> 00:13:33,470
but you have another a back.

202
00:13:33,470 --> 00:13:36,900
So the power is the same.

203
00:13:36,900 --> 00:13:41,130
The sign comes from this sign.

204
00:13:41,130 --> 00:13:43,490
AUDIENCE: Shouldn't
the n there have a hat?

205
00:13:43,490 --> 00:13:45,065
PROFESSOR: Yes, it
should have a hat.

206
00:13:45,065 --> 00:13:46,255
I'm sorry.

207
00:13:46,255 --> 00:13:46,755
Yes.

208
00:13:51,020 --> 00:14:02,550
And, similarly, n
a-hat dagger to the k.

209
00:14:02,550 --> 00:14:08,100
This is k a-hat dagger to the k.

210
00:14:08,100 --> 00:14:16,100
So what happened before,
that n-hat operator leaves

211
00:14:16,100 --> 00:14:18,600
the a same but puts a number--

212
00:14:18,600 --> 00:14:21,880
leaves the a dagger the
same and puts a number.

213
00:14:21,880 --> 00:14:25,890
Here, you see it
happening again.

214
00:14:25,890 --> 00:14:30,840
N with a collection--
with a string of a-hats--

215
00:14:30,840 --> 00:14:34,770
gives you the same
string, but the number.

216
00:14:34,770 --> 00:14:37,230
And with a collection
of a daggers

217
00:14:37,230 --> 00:14:40,410
gives you the same collection
of a daggers with a number.

218
00:14:40,410 --> 00:14:45,090
And the number happens to be
the number of a's or the number

219
00:14:45,090 --> 00:14:46,890
of a daggers.

220
00:14:46,890 --> 00:14:51,610
So that's the reason it's
called the number operator,

221
00:14:51,610 --> 00:14:56,180
because the eigenvalues are the
number of creation operators

222
00:14:56,180 --> 00:14:59,580
or the number of
destruction operators.

223
00:14:59,580 --> 00:15:04,120
I was a little glib by
calling it the eigenvalue.

224
00:15:04,120 --> 00:15:07,300
But it almost looks like
an eigenvalue equation,

225
00:15:07,300 --> 00:15:10,590
which have an operator,
another operator, and a number

226
00:15:10,590 --> 00:15:12,490
times the second operator.

227
00:15:12,490 --> 00:15:15,480
It is not exactly an
eigenvalue equation, though,

228
00:15:15,480 --> 00:15:17,790
because with eigenvalues
you would just have

229
00:15:17,790 --> 00:15:21,040
this acting on the second one.

230
00:15:21,040 --> 00:15:25,660
But the fact that
this case appear here

231
00:15:25,660 --> 00:15:31,000
are the reason these
are number operators.

232
00:15:31,000 --> 00:15:35,110
So it was a little
quick for many of you.

233
00:15:35,110 --> 00:15:36,880
Some of you may have
seen this before.

234
00:15:36,880 --> 00:15:41,020
It was a little slow,
but the important thing

235
00:15:41,020 --> 00:15:43,130
is after a couple
of days from now,

236
00:15:43,130 --> 00:15:49,750
or by Friday, you find all
this very straightforward.