1
00:00:00,060 --> 00:00:01,670
The following
content is provided
2
00:00:01,670 --> 00:00:03,810
under a Creative
Commons license.
3
00:00:03,810 --> 00:00:06,540
Your support will help MIT
OpenCourseWare continue
4
00:00:06,540 --> 00:00:10,120
to offer high quality
educational resources for free.
5
00:00:10,120 --> 00:00:12,700
To make a donation or to
view additional materials
6
00:00:12,700 --> 00:00:16,600
from hundreds of MIT courses,
visit MIT OpenCourseWare
7
00:00:16,600 --> 00:00:17,305
at ocw.mit.edu.
8
00:00:21,740 --> 00:00:24,600
PROFESSOR: All right.
9
00:00:24,600 --> 00:00:29,660
Today we'll be talking a
little about angular momentum.
10
00:00:29,660 --> 00:00:33,220
Continuing the discussion
of those vector operators
11
00:00:33,220 --> 00:00:37,080
and their identities
that we had last time.
12
00:00:37,080 --> 00:00:42,190
So it will allow us to make
quite a bit of progress
13
00:00:42,190 --> 00:00:46,190
with those operators, and
understand them better.
14
00:00:46,190 --> 00:00:51,000
Then we'll go through
the algebraic analysis
15
00:00:51,000 --> 00:00:52,470
of the spectrum.
16
00:00:52,470 --> 00:00:56,100
This is something
that probably you've
17
00:00:56,100 --> 00:01:01,170
seen in some way or another,
perhaps in not so much detail.
18
00:01:01,170 --> 00:01:04,090
But you're probably
somewhat familiar,
19
00:01:04,090 --> 00:01:07,050
but it's good to see it again.
20
00:01:07,050 --> 00:01:12,210
And finally at the end
we'll discuss an application
21
00:01:12,210 --> 00:01:16,700
that is related to your last
problem in the homework.
22
00:01:16,700 --> 00:01:20,130
And it's a rather
mysterious thing
23
00:01:20,130 --> 00:01:22,680
that I think one
should appreciate
24
00:01:22,680 --> 00:01:26,990
how unusual the
result is, related
25
00:01:26,990 --> 00:01:30,180
to the two dimensional
harmonic oscillator.
26
00:01:30,180 --> 00:01:35,950
So I'll begin by reminding
you of a few things.
27
00:01:35,950 --> 00:01:40,460
We have L, which is r cross p.
28
00:01:40,460 --> 00:01:43,220
And we managed to
prove last time
29
00:01:43,220 --> 00:01:49,430
that that was equal to p
cross r, with a minus sign.
30
00:01:49,430 --> 00:01:51,840
And then part of the
problem's that you're
31
00:01:51,840 --> 00:01:55,720
solving with
angular momentum use
32
00:01:55,720 --> 00:01:59,320
the concept of a vector
and the rotations.
33
00:01:59,320 --> 00:02:11,320
So if u is a vector
under rotations--
34
00:02:11,320 --> 00:02:15,490
to say that something is a
vector under rotations means
35
00:02:15,490 --> 00:02:25,800
the following, means that if you
compute Li commutator with uj,
36
00:02:25,800 --> 00:02:26,810
you can put a hat.
37
00:02:26,810 --> 00:02:29,650
All these things are
operators, all these vectors.
38
00:02:29,650 --> 00:02:33,650
So maybe I won't put a hat
here on the blackboard.
39
00:02:33,650 --> 00:02:40,010
Then you're supposed to get
i, epsilon, ijk, ih bar.
40
00:02:40,010 --> 00:02:44,170
Epsilon, ijk, uk.
41
00:02:44,170 --> 00:02:47,610
So that's a definition
if you wish.
42
00:02:47,610 --> 00:02:53,990
Any object that does that
is a vector under rotations.
43
00:02:53,990 --> 00:02:59,880
And something that in the
homework you can verify
44
00:02:59,880 --> 00:03:07,910
is that r and p are
vectors under rotation.
45
00:03:07,910 --> 00:03:13,490
That is, if you put here xj,
you get this thing with xk.
46
00:03:13,490 --> 00:03:17,610
If you put here pj, you
get this thing with pk.
47
00:03:17,610 --> 00:03:20,230
If you compute the commutator.
48
00:03:20,230 --> 00:03:24,790
So r and p are vectors
under rotation.
49
00:03:24,790 --> 00:03:29,950
Then comes that little theorem,
that is awfully important,
50
00:03:29,950 --> 00:03:37,505
that shows that if u and v are
vectors under rotations-- u
51
00:03:37,505 --> 00:03:52,065
and v vectors under rotations--
then u dot v is a scalar.
52
00:03:54,870 --> 00:04:01,960
And u cross v is a vector.
53
00:04:01,960 --> 00:04:04,746
And in both cases,
under rotations.
54
00:04:08,840 --> 00:04:14,610
So this is something
you must prove,
55
00:04:14,610 --> 00:04:18,350
because if you know how u and
v commute with the angular
56
00:04:18,350 --> 00:04:22,412
momentum, you know how u
times v, in either the dot
57
00:04:22,412 --> 00:04:27,110
combination or the cross
combination, commute with j,
58
00:04:27,110 --> 00:04:31,840
with L. So to say that
something is a scalar,
59
00:04:31,840 --> 00:04:38,380
the translation is that
Li with u dot v will be 0.
60
00:04:38,380 --> 00:04:40,630
You don't have to
calculate it again.
61
00:04:40,630 --> 00:04:42,815
If you've shown that
u and v are vectors,
62
00:04:42,815 --> 00:04:46,850
that they transform like
that, this commutes with this.
63
00:04:46,850 --> 00:04:50,200
So r-- so what do you
conclude from this?
64
00:04:50,200 --> 00:04:59,840
That Li commutes with r squared,
commutes-- it's equal to p
65
00:04:59,840 --> 00:05:02,070
squared.
66
00:05:02,070 --> 00:05:07,140
And it's equal to Li, r dot p.
67
00:05:07,140 --> 00:05:10,320
They all are 0.
68
00:05:10,320 --> 00:05:13,040
Because r and p are
vectors under rotation,
69
00:05:13,040 --> 00:05:16,910
so you don't have to
compute those ones anymore.
70
00:05:16,910 --> 00:05:21,290
Li will commute with r squared,
with p squared r cross p.
71
00:05:21,290 --> 00:05:24,620
And also, the fact
that u cross v
72
00:05:24,620 --> 00:05:31,940
is a vector means that Li
commutated with u cross v,
73
00:05:31,940 --> 00:05:39,860
j-- the j component of u cross
v is ih bar, u cross v. I'm
74
00:05:39,860 --> 00:05:45,900
sorry-- epsilon,
ijk, u cross v, k.
75
00:05:49,160 --> 00:05:56,560
Which is to say that u cross
v is a vector under rotations.
76
00:05:56,560 --> 00:06:00,920
This has a lot of
important corollaries.
77
00:06:00,920 --> 00:06:04,080
The most important
perhaps is the commutation
78
00:06:04,080 --> 00:06:06,900
of angular momentum with itself.
79
00:06:06,900 --> 00:06:12,540
That is since you've shown
that r and p satisfy this,
80
00:06:12,540 --> 00:06:17,790
r cross p, which is
angular momentum,
81
00:06:17,790 --> 00:06:20,100
is also a vector under rotation.
82
00:06:20,100 --> 00:06:27,640
So here choosing u
equal r, and v equal p,
83
00:06:27,640 --> 00:06:37,485
you get that Li, Lj is equal
to ih bar, epsilon, ijk, Lk.
84
00:06:37,485 --> 00:06:39,310
And it's the end of the story.
85
00:06:39,310 --> 00:06:42,850
You got this commutation.
86
00:06:42,850 --> 00:06:46,020
The commutation you wanted.
87
00:06:46,020 --> 00:06:50,470
In earlier courses,
you probably found
88
00:06:50,470 --> 00:06:54,080
that this was a fairly
complicated calculation.
89
00:06:54,080 --> 00:06:57,690
Which you had to put the x's and
the p's, the x's and the p's,
90
00:06:57,690 --> 00:07:00,040
and start moving them.
91
00:07:00,040 --> 00:07:03,800
And it takes quite
a while to do it.
92
00:07:03,800 --> 00:07:06,560
So, that's important.
93
00:07:06,560 --> 00:07:09,480
Another property
that follows from all
94
00:07:09,480 --> 00:07:13,070
of this, which is sort of
interesting, that since L
95
00:07:13,070 --> 00:07:19,116
is now also a vector under
rotations, Li commutes with L
96
00:07:19,116 --> 00:07:19,615
squared.
97
00:07:27,720 --> 00:07:33,850
Because l squared is L dot
L, therefore it's a scalar.
98
00:07:33,850 --> 00:07:38,320
So Li commutes with L squared.
99
00:07:38,320 --> 00:07:44,100
And that property is
absolutely crucial.
100
00:07:44,100 --> 00:07:48,600
It's important that it's worth
checking that in fact, it
101
00:07:48,600 --> 00:07:50,920
follows just from this algebra.
102
00:07:56,290 --> 00:07:58,490
You see, the only
thing you need to know
103
00:07:58,490 --> 00:08:01,240
to compute the commutator
of Li with L squared
104
00:08:01,240 --> 00:08:03,850
is how L's commute.
105
00:08:03,850 --> 00:08:07,070
Therefore it should be
possible to calculate this
106
00:08:07,070 --> 00:08:09,750
based on this algebra.
107
00:08:09,750 --> 00:08:18,200
So this property is true just
because of this algebra, not
108
00:08:18,200 --> 00:08:21,460
because of anything
we've said before.
109
00:08:21,460 --> 00:08:24,120
And that's important
to realize it.
110
00:08:24,120 --> 00:08:28,140
Because you have
algebra like si,
111
00:08:28,140 --> 00:08:34,350
sj, ih bar, epsilon,
ijk, sk, which
112
00:08:34,350 --> 00:08:37,440
was the algebra of
spin angular momentum.
113
00:08:37,440 --> 00:08:41,039
And we claim that
for that same reason
114
00:08:41,039 --> 00:08:44,960
that this algebra
leads to this result,
115
00:08:44,960 --> 00:08:50,700
that si should commute
with s squared.
116
00:08:50,700 --> 00:08:55,540
And you may remember that
in the particular case
117
00:08:55,540 --> 00:08:59,910
we examined in this
course, s squared--
118
00:08:59,910 --> 00:09:06,520
that would be sx squared plus
sy squared plus sz squared-- was
119
00:09:06,520 --> 00:09:12,260
in fact h bar over 2 squared.
120
00:09:12,260 --> 00:09:15,810
And each matrix was
proportional to the identity.
121
00:09:15,810 --> 00:09:18,780
So there's a 3 in
the identity matrix.
122
00:09:18,780 --> 00:09:23,750
And s squared is really in the
way we represent that spin,
123
00:09:23,750 --> 00:09:29,050
by 2 by 2 matrices,
commutes with si.
124
00:09:29,050 --> 00:09:30,600
Because it is the identity.
125
00:09:30,600 --> 00:09:35,870
So it's no accident
that this thing is 0.
126
00:09:35,870 --> 00:09:41,220
Because this algebra,
whatever l is,
127
00:09:41,220 --> 00:09:47,940
implies that this with the
thing squared is equal to zero.
128
00:09:47,940 --> 00:09:51,550
So whenever we'll be
talking about spin angular
129
00:09:51,550 --> 00:09:56,400
momentum, orbital angular
momentum, total angular
130
00:09:56,400 --> 00:09:58,280
momentum, when we
add them, there's
131
00:09:58,280 --> 00:09:59,810
all kinds of angular momentum.
132
00:09:59,810 --> 00:10:05,610
And our another generic name
for angular momentum will be j.
133
00:10:05,610 --> 00:10:15,460
And we'll say that ji, jj,
equal ih bar, epsilon, ijk,
134
00:10:15,460 --> 00:10:20,485
jk is the algebra
of angular momentum.
135
00:10:25,590 --> 00:10:29,200
And by using j, you're
sending the signal
136
00:10:29,200 --> 00:10:31,906
that you may be talking about l.
137
00:10:31,906 --> 00:10:35,150
Or may be talking about
s, but it's not obvious
138
00:10:35,150 --> 00:10:36,650
which you're talking about.
139
00:10:36,650 --> 00:10:41,320
And you're focusing on those
properties of angular momentum
140
00:10:41,320 --> 00:10:49,320
that hold just because this
algebra is supposed to be true.
141
00:10:49,320 --> 00:10:54,460
So in this algebra, you will
have that ji commutes with j
142
00:10:54,460 --> 00:10:55,100
squared.
143
00:10:55,100 --> 00:10:56,720
And what is j squared?
144
00:10:56,720 --> 00:11:00,850
Of course, j squared
is j1 squared
145
00:11:00,850 --> 00:11:05,860
plus j2 squared plus j3 squared.
146
00:11:05,860 --> 00:11:08,560
Now this is so important,
and this derivation
147
00:11:08,560 --> 00:11:13,300
is a little bit indirect,
that I encourage you all
148
00:11:13,300 --> 00:11:16,230
to just do it.
149
00:11:16,230 --> 00:11:19,820
Without using any
formula, put the jx here,
150
00:11:19,820 --> 00:11:21,930
and compute this commutator.
151
00:11:21,930 --> 00:11:26,430
And it takes a couple of lines,
but just convince yourself
152
00:11:26,430 --> 00:11:29,460
that this is true.
153
00:11:29,460 --> 00:11:38,580
OK, now we did have a
little more discussion.
154
00:11:38,580 --> 00:11:41,760
And these are all things
that are basically related
155
00:11:41,760 --> 00:11:43,610
to what you've been
doing in the homework.
156
00:11:46,380 --> 00:11:51,330
Another fact is
that this algebra
157
00:11:51,330 --> 00:11:58,780
is translated into j
cross j equal ih bar, j.
158
00:12:03,030 --> 00:12:10,850
Another result in
transcription of equations
159
00:12:10,850 --> 00:12:24,370
is that the statement that u
is a vector under rotations
160
00:12:24,370 --> 00:12:26,900
corresponds to a
vector identity.
161
00:12:26,900 --> 00:12:29,060
Just the fact that
the algebra here
162
00:12:29,060 --> 00:12:35,460
is this, the fact
that l with u is this,
163
00:12:35,460 --> 00:12:37,640
implies the following algebra.
164
00:12:37,640 --> 00:12:51,276
j cross u plus u cross
j equal 2i h bar.
165
00:12:56,440 --> 00:12:59,776
So this is for a
vector under rotations.
166
00:13:04,450 --> 00:13:06,490
Under rotations.
167
00:13:06,490 --> 00:13:09,760
So this I think is in the notes.
168
00:13:09,760 --> 00:13:11,380
It's basically
saying that if you
169
00:13:11,380 --> 00:13:16,210
want to translate this equation
into vector form, which
170
00:13:16,210 --> 00:13:20,370
is a nice thing to have,
it reads like this.
171
00:13:20,370 --> 00:13:24,280
And the way to do that is to
just calculate the left hand
172
00:13:24,280 --> 00:13:24,780
side.
173
00:13:24,780 --> 00:13:27,500
Put and index, i.
174
00:13:27,500 --> 00:13:30,050
And just try to get
the right hand side.
175
00:13:30,050 --> 00:13:31,360
It will work out.
176
00:13:34,130 --> 00:13:34,710
OK.
177
00:13:34,710 --> 00:13:37,355
Any questions so far
with these identities?
178
00:13:48,710 --> 00:13:49,330
OK.
179
00:13:49,330 --> 00:13:52,370
So we move on to
another identity
180
00:13:52,370 --> 00:13:57,470
that you've been
working on, based
181
00:13:57,470 --> 00:14:04,540
on the calculation of what
is a cross b dot a cross b.
182
00:14:07,550 --> 00:14:11,540
If these things are
operators, there's
183
00:14:11,540 --> 00:14:14,020
corrections to the
classical formula
184
00:14:14,020 --> 00:14:21,390
for the answer of of what this
product is supposed to be.
185
00:14:21,390 --> 00:14:26,990
Actually, the classical
formula, so it's not
186
00:14:26,990 --> 00:14:33,820
equal to a squared, b squared,
minus a dot b squared.
187
00:14:33,820 --> 00:14:38,020
But it's actually equal
to this, plus dot dot dot.
188
00:14:38,020 --> 00:14:40,000
A few more things.
189
00:14:40,000 --> 00:14:41,960
Classically it's just that.
190
00:14:41,960 --> 00:14:44,640
You put 2 epsilons.
191
00:14:44,640 --> 00:14:46,240
Calculate the left hand side.
192
00:14:46,240 --> 00:14:49,080
And it's just these 2 terms.
193
00:14:49,080 --> 00:14:52,280
Since there are
more terms, let's
194
00:14:52,280 --> 00:14:55,590
look what they are for a
particular case of interest.
195
00:14:55,590 --> 00:15:01,060
So our case of interest is L
squared, that corresponds to r
196
00:15:01,060 --> 00:15:03,380
cross b, times r cross b.
197
00:15:08,100 --> 00:15:12,780
And indeed, it's
not just r squared,
198
00:15:12,780 --> 00:15:18,810
p squared, minus
r dot p squared.
199
00:15:18,810 --> 00:15:21,610
But there's a little extra.
200
00:15:21,610 --> 00:15:25,660
And perhaps you have computed
that little extra by now.
201
00:15:25,660 --> 00:15:29,258
It's ih bar r dot p.
202
00:15:32,680 --> 00:15:40,310
So that's a pretty
useful result.
203
00:15:40,310 --> 00:15:43,925
And from here, we typically
look for what is p squared.
204
00:15:46,720 --> 00:15:50,280
So for p squared-- so what we
do is pass these other terms
205
00:15:50,280 --> 00:15:52,180
to the other side.
206
00:15:52,180 --> 00:15:57,390
And therefore we have
1 over r squared,
207
00:15:57,390 --> 00:16:04,660
r dot p squared,
minus ih bar, r dot p.
208
00:16:08,600 --> 00:16:11,280
Yes.
209
00:16:11,280 --> 00:16:14,185
Plus 1 over r
squared, l squared.
210
00:16:17,890 --> 00:16:22,520
And, we've done this
with some prudence.
211
00:16:22,520 --> 00:16:25,620
The r squared is here in
front of the p squared.
212
00:16:25,620 --> 00:16:29,415
It may be fairly different from
having it to the other side.
213
00:16:31,930 --> 00:16:36,135
And therefore, when I apply
the inverse 1 over r squared,
214
00:16:36,135 --> 00:16:37,860
I apply it from the left.
215
00:16:37,860 --> 00:16:41,190
So I write it like that.
216
00:16:41,190 --> 00:16:42,970
And that's very
different for having
217
00:16:42,970 --> 00:16:45,220
the r squared on the other side.
218
00:16:45,220 --> 00:16:48,550
Could be completely different.
219
00:16:48,550 --> 00:16:52,010
Now, what is this?
220
00:16:52,010 --> 00:17:00,590
Well this is a
simple computation,
221
00:17:00,590 --> 00:17:08,899
when you remember that p vector
is h bar over i gradient.
222
00:17:11,839 --> 00:17:21,956
And r dot p, therefore
is h bar over i, r, dvr.
223
00:17:26,410 --> 00:17:29,980
Because r vector is
r magnitude times
224
00:17:29,980 --> 00:17:32,620
the unit vector in
the radial direction.
225
00:17:32,620 --> 00:17:36,550
And the radial direction
of gradient is dvr.
226
00:17:36,550 --> 00:17:38,920
So this can be simplified.
227
00:17:38,920 --> 00:17:42,300
I will not do it because
it's in the notes.
228
00:17:42,300 --> 00:17:49,280
And you get minus h squared, 1
over r, d second, d r squared,
229
00:17:49,280 --> 00:17:51,050
r.
230
00:17:51,050 --> 00:17:56,300
In a funny notation,
the r is on the right.
231
00:17:56,300 --> 00:17:58,750
And the 1 over r is on the left.
232
00:17:58,750 --> 00:18:03,990
And you would say, this
doesn't sound right.
233
00:18:03,990 --> 00:18:06,840
You have here all
this derivatives
234
00:18:06,840 --> 00:18:10,260
and what is an r doing to
the right of the derivatives.
235
00:18:10,260 --> 00:18:11,690
I see no r.
236
00:18:11,690 --> 00:18:14,775
But this is a kind of a
trick to rewrite everything
237
00:18:14,775 --> 00:18:17,010
in a short way.
238
00:18:17,010 --> 00:18:19,800
So if you want,
think of this being
239
00:18:19,800 --> 00:18:22,753
acting on some function of r.
240
00:18:22,753 --> 00:18:24,420
And see what it is.
241
00:18:24,420 --> 00:18:28,560
And then you put a function
of r here, and calculate it.
242
00:18:28,560 --> 00:18:30,980
And you will see,
you get the same.
243
00:18:30,980 --> 00:18:35,770
So it's a good
thing to try that.
244
00:18:35,770 --> 00:18:38,930
So p squared is given by this.
245
00:18:38,930 --> 00:18:42,480
There's another
formula for p squared.
246
00:18:42,480 --> 00:18:49,470
p squared is, of
course, the Laplacian.
247
00:18:52,890 --> 00:19:01,530
So p squared is also
equal to minus h squared
248
00:19:01,530 --> 00:19:04,730
times the Laplacian operator.
249
00:19:04,730 --> 00:19:10,600
And that's equal to minus
h squared times-- in fact,
250
00:19:10,600 --> 00:19:17,270
the Laplacian operator is 1
over r, d second, dr squared, r,
251
00:19:17,270 --> 00:19:23,680
plus 1 over r squared,
1 over sine theta, dd
252
00:19:23,680 --> 00:19:28,400
theta, sine theta, dd theta.
253
00:19:28,400 --> 00:19:30,380
It's a little bit messy.
254
00:19:30,380 --> 00:19:40,090
Plus 1 over sine squared
theta, d second, d phi squared,
255
00:19:40,090 --> 00:19:41,560
times closing this.
256
00:19:41,560 --> 00:19:43,955
So a few things
are there to learn.
257
00:19:47,220 --> 00:19:52,710
And the first thing is if you
compare these 2 expressions,
258
00:19:52,710 --> 00:19:55,130
you have a formula
for l squared.
259
00:19:57,640 --> 00:20:02,310
You have l squared is 1 over
r squared on the upper right.
260
00:20:02,310 --> 00:20:06,790
And here you have minus h
squared times this thing.
261
00:20:06,790 --> 00:20:12,165
So l squared, that
scalar operator
262
00:20:12,165 --> 00:20:18,830
is minus h squared,
1 over sine theta,
263
00:20:18,830 --> 00:20:26,380
dd theta, sine theta, dd
theta, plus 1 over sine
264
00:20:26,380 --> 00:20:30,223
squared theta, d
second, d phi squared.
265
00:20:35,720 --> 00:20:40,905
So in terms of functions of
3 variables, x, y, and z,
266
00:20:40,905 --> 00:20:45,700
L squared, which is a
very complicated object,
267
00:20:45,700 --> 00:20:49,080
has become just a function
of the angular variables.
268
00:20:49,080 --> 00:20:52,590
And that this a very
important intuitive fact.
269
00:20:52,590 --> 00:20:54,170
L squared.
270
00:20:54,170 --> 00:20:55,280
L is operator.
271
00:20:55,280 --> 00:20:56,570
That's rotation.
272
00:20:56,570 --> 00:21:00,790
So it shouldn't really affect
the r, shouldn't change r,
273
00:21:00,790 --> 00:21:02,710
modify r in any way.
274
00:21:02,710 --> 00:21:05,860
So it's a nice thing
to confirm here
275
00:21:05,860 --> 00:21:11,900
that this operator can be
thought as an operator acting
276
00:21:11,900 --> 00:21:14,050
on the angular variables.
277
00:21:14,050 --> 00:21:21,450
Or you could say, on functions,
on the units here for example.
278
00:21:21,450 --> 00:21:22,780
It's a good thing.
279
00:21:22,780 --> 00:21:24,660
The other thing that
you've learned here--
280
00:21:24,660 --> 00:21:27,620
so this is a very nice result.
281
00:21:27,620 --> 00:21:32,170
It's not all that easy to
get by direct computation.
282
00:21:32,170 --> 00:21:37,420
If you had to do Lx squared
plus Ly squared plus Lz
283
00:21:37,420 --> 00:21:41,950
squared, first all this
possible order-- well,
284
00:21:41,950 --> 00:21:43,770
there's no ordering
problems here.
285
00:21:43,770 --> 00:21:48,430
But you would have to write this
in terms of x, and py, and pz,
286
00:21:48,430 --> 00:21:52,260
and xy, and z, then pass
to angular variables.
287
00:21:52,260 --> 00:21:53,380
Simplify all that.
288
00:21:53,380 --> 00:21:57,530
It's a very bad way to do it.
289
00:21:57,530 --> 00:21:59,240
And it's painful.
290
00:21:59,240 --> 00:22:02,910
So the fact that we got
this like that is very nice.
291
00:22:02,910 --> 00:22:06,750
The other thing that we've
got is some understanding
292
00:22:06,750 --> 00:22:10,100
of the Hamiltonian for a
central potential, what
293
00:22:10,100 --> 00:22:13,360
we call a central
potential problem.
294
00:22:13,360 --> 00:22:14,305
v of r.
295
00:22:17,530 --> 00:22:21,710
Now, I will write
a v of r like this.
296
00:22:21,710 --> 00:22:24,690
But then we'll simplify it.
297
00:22:24,690 --> 00:22:29,060
In fact, let me just go to a
central potential case, which
298
00:22:29,060 --> 00:22:32,960
means that the potential just
depends on the magnitude of r.
299
00:22:32,960 --> 00:22:38,540
So r is the magnitude
of the vector r.
300
00:22:38,540 --> 00:22:44,220
So at this moment, you
have p squared over there.
301
00:22:44,220 --> 00:22:50,120
So this whole
Hamiltonian is minus h
302
00:22:50,120 --> 00:23:02,850
squared over 2m, 1 over r, d
second, dr squared, r, plus p
303
00:23:02,850 --> 00:23:04,080
squared over 2m.
304
00:23:04,080 --> 00:23:11,850
So 1 over 2m, r squared,
l squared plus v of r.
305
00:23:14,930 --> 00:23:19,506
So our Hamiltonian has
also been simplified.
306
00:23:23,960 --> 00:23:26,290
So this will be
the starting point
307
00:23:26,290 --> 00:23:30,350
for writing the
Schrodinger equation
308
00:23:30,350 --> 00:23:33,740
for central potentials.
309
00:23:33,740 --> 00:23:37,170
And you have the
operator l squared.
310
00:23:37,170 --> 00:23:41,930
And as far as we can, we'll
try to avoid computations
311
00:23:41,930 --> 00:23:45,060
in theta and phi
very explicitly,
312
00:23:45,060 --> 00:23:49,180
but try to do things
algebraically.
313
00:23:49,180 --> 00:23:53,500
So at this moment,
the last comment
314
00:23:53,500 --> 00:23:56,980
I want to make on this
subject is the issue
315
00:23:56,980 --> 00:24:01,790
of set of commuting observables.
316
00:24:01,790 --> 00:24:05,730
So if you have a
Hamiltonian like that,
317
00:24:05,730 --> 00:24:09,390
you can try to form a set of
commuting observables that
318
00:24:09,390 --> 00:24:13,180
are going to help you
understand the physics
319
00:24:13,180 --> 00:24:14,940
of your particular problem.
320
00:24:14,940 --> 00:24:17,140
So the first thing
that you would
321
00:24:17,140 --> 00:24:21,310
want to put in the list of
complete set of observables
322
00:24:21,310 --> 00:24:22,690
is the Hamiltonian.
323
00:24:22,690 --> 00:24:25,960
We really want to know the
energies of this thing.
324
00:24:25,960 --> 00:24:29,500
So what other
operators do I have?
325
00:24:29,500 --> 00:24:32,960
Well I have x1, x2, and x3.
326
00:24:37,820 --> 00:24:42,950
And well, can I add
them to the Hamiltonian
327
00:24:42,950 --> 00:24:45,880
to have a complete set
of commuting observables?
328
00:24:45,880 --> 00:24:47,915
Well, the x's commute
among themselves.
329
00:24:51,560 --> 00:24:52,720
So can I add them?
330
00:24:58,280 --> 00:24:59,930
Yes or no?
331
00:24:59,930 --> 00:25:00,590
No.
332
00:25:00,590 --> 00:25:03,450
No you can't add
them, because the x's
333
00:25:03,450 --> 00:25:05,820
don't commute with
the Hamiltonian.
334
00:25:05,820 --> 00:25:07,660
There's a p here.
335
00:25:07,660 --> 00:25:09,180
p doesn't commute with x's.
336
00:25:09,180 --> 00:25:11,750
So that's out of the question.
337
00:25:11,750 --> 00:25:15,500
They cannot be
added to our list.
338
00:25:15,500 --> 00:25:16,850
How about the p's?
339
00:25:16,850 --> 00:25:20,690
p1, p2, and p3.
340
00:25:20,690 --> 00:25:23,240
Not good either,
because they don't
341
00:25:23,240 --> 00:25:25,370
commune with the potential term.
342
00:25:25,370 --> 00:25:30,310
The potential has x dependents,
and will take a miracle for it
343
00:25:30,310 --> 00:25:30,970
to commute.
344
00:25:30,970 --> 00:25:32,370
In general, it won't commute.
345
00:25:32,370 --> 00:25:37,890
So no reason for it to commute,
unless the potential is 0.
346
00:25:37,890 --> 00:25:39,140
So this is not good.
347
00:25:41,760 --> 00:25:51,235
Nor is good to have r squared,
or p squared, or r dot p.
348
00:25:51,235 --> 00:25:56,870
r squared, p squared, r dot p.
349
00:25:56,870 --> 00:25:58,620
No good either.
350
00:25:58,620 --> 00:26:05,440
On the other hand, r
cross p is interesting.
351
00:26:05,440 --> 00:26:10,135
You have the angular
momentum, L1, L2, and L3.
352
00:26:14,660 --> 00:26:23,410
Well, the angular momentum
will commute, I think,
353
00:26:23,410 --> 00:26:24,950
with the Hamiltonian.
354
00:26:24,950 --> 00:26:27,600
You can see it here.
355
00:26:27,600 --> 00:26:34,990
You have p squared,
and Li's commute with p
356
00:26:34,990 --> 00:26:39,050
squared because p is a
vector under rotations.
357
00:26:39,050 --> 00:26:42,900
p doesn't communicate with
Li, but p squared does.
358
00:26:42,900 --> 00:26:44,580
Because that was a scalar.
359
00:26:44,580 --> 00:26:50,860
So this term commutes with
any angular momentum operator.
360
00:26:50,860 --> 00:26:54,350
Moreover, v or r, r is this.
361
00:26:54,350 --> 00:26:58,440
So a v of r is a
function of r squared.
362
00:26:58,440 --> 00:27:02,680
And r squared is the
vector r squared.
363
00:27:02,680 --> 00:27:07,280
So ultimately, anything that is
a function of r is a function
364
00:27:07,280 --> 00:27:10,650
of r squared that involves
the operator r squared,
365
00:27:10,650 --> 00:27:13,890
that also commutes
with all the Li's.
366
00:27:13,890 --> 00:27:17,170
So h commutes with all the Li's.
367
00:27:17,170 --> 00:27:19,490
And that's a great thing.
368
00:27:19,490 --> 00:27:23,010
So this is absolutely important.
369
00:27:23,010 --> 00:27:27,390
h commutes with all the Li's.
370
00:27:27,390 --> 00:27:32,780
That's angular
momentum conservation.
371
00:27:32,780 --> 00:27:41,310
As we've seen, the rate
of change of any operator
372
00:27:41,310 --> 00:27:44,450
is equal to expectation
value of the commutator
373
00:27:44,450 --> 00:27:47,460
of the operator with
the Hamiltonian.
374
00:27:47,460 --> 00:27:54,960
So if you put any Li,
this commutator is 0.
375
00:27:54,960 --> 00:27:58,250
And the operator is
conserved in the sense
376
00:27:58,250 --> 00:28:01,870
of expectation values.
377
00:28:01,870 --> 00:28:05,580
Now this conservation
law is great.
378
00:28:05,580 --> 00:28:08,890
You could add this operators
to the commuting set
379
00:28:08,890 --> 00:28:10,800
of observables.
380
00:28:10,800 --> 00:28:16,100
But this time, you have
a different problem.
381
00:28:16,100 --> 00:28:17,800
Yes, this commutes with h.
382
00:28:17,800 --> 00:28:18,945
This commutes with h.
383
00:28:18,945 --> 00:28:20,810
And this commutes with h.
384
00:28:20,810 --> 00:28:23,270
But these one's don't
commute with each other.
385
00:28:23,270 --> 00:28:26,090
So not quite good enough.
386
00:28:26,090 --> 00:28:27,515
You cannot add them all.
387
00:28:30,070 --> 00:28:33,750
So let's see how
many can we add.
388
00:28:33,750 --> 00:28:35,640
We can only add 1.
389
00:28:35,640 --> 00:28:39,360
Because once you have 2 of
them, they don't commute.
390
00:28:39,360 --> 00:28:46,000
So you're going to add 1, and
everybody has agreed to add L3.
391
00:28:46,000 --> 00:28:48,590
So we have H, L3.
392
00:28:51,420 --> 00:28:56,630
And happily we have
1 more is L squared.
393
00:28:56,630 --> 00:29:01,430
Remember, L squared commutes
with all the Li's, so that's
394
00:29:01,430 --> 00:29:02,240
another operator.
395
00:29:06,720 --> 00:29:10,010
And for a central
potential problem,
396
00:29:10,010 --> 00:29:18,094
this will be sufficient to
label all of our states some.
397
00:29:18,094 --> 00:29:21,976
AUDIENCE: So how do we know
that we need the L squared?
398
00:29:21,976 --> 00:29:23,805
How do we know
that we can't get--
399
00:29:23,805 --> 00:29:26,986
how do we know that just
H and L3 isn't already
400
00:29:26,986 --> 00:29:30,142
a complete set?
401
00:29:30,142 --> 00:29:34,010
PROFESSOR: I probably wouldn't
know now, but in a little bit,
402
00:29:34,010 --> 00:29:36,960
as we calculate
the kind of states
403
00:29:36,960 --> 00:29:40,040
that we get with
angular momentum,
404
00:29:40,040 --> 00:29:44,260
I will see that there are many
states with the same value
405
00:29:44,260 --> 00:29:51,260
of L3 that don't correspond
to the same value of the total
406
00:29:51,260 --> 00:29:53,920
or length of the
angular momentum.
407
00:29:53,920 --> 00:30:00,810
So it's almost like saying
that there are angular
408
00:30:00,810 --> 00:30:07,360
momenta-- here is--
let me draw a plane.
409
00:30:07,360 --> 00:30:12,100
Here is z component of
angular momentum, Lz.
410
00:30:12,100 --> 00:30:13,360
And here you got it.
411
00:30:13,360 --> 00:30:18,210
You can have an angular
momentum that is like that,
412
00:30:18,210 --> 00:30:20,400
and has this Lz.
413
00:30:20,400 --> 00:30:22,430
Or you can have an
angular momentum
414
00:30:22,430 --> 00:30:28,750
that is like this, L prime,
that has the same Lz.
415
00:30:28,750 --> 00:30:32,435
And then it will be difficult
to tell these 2 states apart.
416
00:30:32,435 --> 00:30:35,250
And they will correspond
to states of this angular
417
00:30:35,250 --> 00:30:38,390
momentum, or this angular
momentum, have the same Lz.
418
00:30:38,390 --> 00:30:45,480
Now drawing these arrows is
extraordinarily misleading.
419
00:30:45,480 --> 00:30:48,640
Hope you don't get
upset that I did it.
420
00:30:48,640 --> 00:30:53,120
It's misleading because this
vector you cannot measure
421
00:30:53,120 --> 00:30:55,080
simultaneously the 3 components.
422
00:30:55,080 --> 00:30:56,280
Because they don't commute.
423
00:30:56,280 --> 00:30:59,700
So what do I mean
by drawing an arrow?
424
00:30:59,700 --> 00:31:02,780
Nevertheless, the
intuition is sort of there.
425
00:31:02,780 --> 00:31:05,450
And it's not wrong,
the intuition.
426
00:31:05,450 --> 00:31:08,950
It will happen to be
the case that states
427
00:31:08,950 --> 00:31:16,320
that have same amount of Lz
will not be distinguished.
428
00:31:16,320 --> 00:31:21,530
But by the time we have this,
we will distinguish them.
429
00:31:21,530 --> 00:31:25,860
And that's also a peculiarity
of a result but we'll use.
430
00:31:25,860 --> 00:31:28,390
Even though we're talking
about 3 dimensions,
431
00:31:28,390 --> 00:31:33,040
the fact that the 1 dimensional
Schrodinger equation
432
00:31:33,040 --> 00:31:36,270
has non degenerate bound states.
433
00:31:36,270 --> 00:31:40,410
You say, what does that have
to do with 3 dimensions?
434
00:31:40,410 --> 00:31:44,600
What will happen is that the
3 dimensional Schrodinger
435
00:31:44,600 --> 00:31:48,180
equation will reduce to a 1
dimensional radial equation.
436
00:31:48,180 --> 00:31:53,100
And the fact that that
doesn't have degeneracies
437
00:31:53,100 --> 00:31:55,630
tells you that for
bound state problems,
438
00:31:55,630 --> 00:31:57,530
this will be enough to do it.
439
00:31:57,530 --> 00:31:59,950
So you will have
to wait a little
440
00:31:59,950 --> 00:32:01,970
to be sure that this will do it.
441
00:32:01,970 --> 00:32:05,120
But this is pretty much
the best we can do now.
442
00:32:05,120 --> 00:32:08,400
And I don't think you will
be able to add anything
443
00:32:08,400 --> 00:32:12,370
else to this at this stage.
444
00:32:12,370 --> 00:32:14,110
Now there's of
course funny things
445
00:32:14,110 --> 00:32:16,660
that you could add
like-- if there's spin,
446
00:32:16,660 --> 00:32:19,890
the particles have spin, well
we can add spin and things
447
00:32:19,890 --> 00:32:21,390
like that.
448
00:32:21,390 --> 00:32:26,930
But let's leave
it at that and now
449
00:32:26,930 --> 00:32:32,090
begin really our calculation,
algebraic calculation,
450
00:32:32,090 --> 00:32:36,260
of the angular momentum
representations.
451
00:32:36,260 --> 00:32:40,370
So at this moment,
we really want
452
00:32:40,370 --> 00:32:44,730
to make sure we work with this.
453
00:32:44,730 --> 00:32:49,075
Only this formula over here.
454
00:32:58,680 --> 00:33:04,470
And learn things about
the kind of states
455
00:33:04,470 --> 00:33:08,160
that can exist in a
system in which there
456
00:33:08,160 --> 00:33:10,570
are operators like that.
457
00:33:10,570 --> 00:33:12,360
So it's a funny thing.
458
00:33:12,360 --> 00:33:14,810
You're talking about
a vector space.
459
00:33:14,810 --> 00:33:17,570
And in fact, you don't
know almost anything
460
00:33:17,570 --> 00:33:20,200
about this vector space so far.
461
00:33:20,200 --> 00:33:23,780
But there is an action
of those operators.
462
00:33:23,780 --> 00:33:29,640
From that fact alone, and
one more important fact--
463
00:33:29,640 --> 00:33:33,435
the j's are Hermitian.
464
00:33:38,950 --> 00:33:42,690
From these 2 facts,
we're going to derive
465
00:33:42,690 --> 00:33:49,420
incredibly powerful results,
extremely powerful things.
466
00:33:49,420 --> 00:33:55,970
And as we'll see, they
have applications even
467
00:33:55,970 --> 00:33:59,120
in cases that you would
imagine they have nothing
468
00:33:59,120 --> 00:34:04,060
to do with angular momentum,
which is really surprising.
469
00:34:04,060 --> 00:34:08,949
So how do we proceed
with this stuff?
470
00:34:08,949 --> 00:34:10,639
Well, there's a hermeticity.
471
00:34:10,639 --> 00:34:13,750
And you immediately
introduce things
472
00:34:13,750 --> 00:34:22,290
called J plus minus, which
are J1 plus minus i J 2.
473
00:34:22,290 --> 00:34:26,278
Or Jx plus minus y Jc.
474
00:34:26,278 --> 00:34:33,409
Then you calculate
what is J plus J minus.
475
00:34:33,409 --> 00:34:40,960
Well J plus J minus will be
a J1 squared plus J2 squared.
476
00:34:40,960 --> 00:34:44,120
And then you have
the cross product
477
00:34:44,120 --> 00:34:45,502
that this doesn't cancel.
478
00:34:51,530 --> 00:35:01,561
So J plus times J minus would
be J1 plus i J2, J1 minus i J2.
479
00:35:01,561 --> 00:35:11,140
So the next term would
be minus i, J1, J2.
480
00:35:11,140 --> 00:35:15,400
And that's i h bar, J3.
481
00:35:15,400 --> 00:35:24,185
So this is J1 squared plus
J2 squared plus h bar J3.
482
00:35:27,050 --> 00:35:32,320
So that's a nice formula
for J plus, J minus.
483
00:35:32,320 --> 00:35:37,510
J minus, J plus would
be J1 squared plus J2
484
00:35:37,510 --> 00:35:42,700
squared minus h bar J3.
485
00:35:42,700 --> 00:35:52,740
These 2 formulas are summarized
by J plus, J minus-- minus,
486
00:35:52,740 --> 00:35:57,210
plus-- is equal to
J1 squared plus J2
487
00:35:57,210 --> 00:36:01,120
squared plus minus h bar J3.
488
00:36:08,260 --> 00:36:10,520
OK.
489
00:36:10,520 --> 00:36:12,215
Things to learn from this.
490
00:36:15,100 --> 00:36:18,440
Maybe I'll continue
here for a little while
491
00:36:18,440 --> 00:36:22,960
to use the blackboards,
up to here only.
492
00:36:22,960 --> 00:36:29,040
The commutator of
J plus and J minus
493
00:36:29,040 --> 00:36:31,540
can be obtained
from this equation.
494
00:36:31,540 --> 00:36:33,740
You just subtract them.
495
00:36:33,740 --> 00:36:37,820
And that's 2h bar, J3.
496
00:36:41,450 --> 00:36:47,190
And finally, one last
thing that we like to know
497
00:36:47,190 --> 00:36:51,600
is how to write J squared.
498
00:36:51,600 --> 00:37:01,930
So J squared is J1 squared
plus J2 squared plus J3
499
00:37:01,930 --> 00:37:06,500
squared, which
then show up here.
500
00:37:06,500 --> 00:37:10,970
So we might as well
add it and subtract it.
501
00:37:10,970 --> 00:37:16,440
So I add a J3 squared, and I
add it on the left hand side.
502
00:37:16,440 --> 00:37:19,400
And pass this term
to the other side.
503
00:37:19,400 --> 00:37:29,010
So J squared would be J plus,
J minus, plus J3 squared,
504
00:37:29,010 --> 00:37:32,890
minus h bar, J3.
505
00:37:32,890 --> 00:37:40,741
Or J minus, J plus, plus
J3 squared, plus h bar, J3.
506
00:37:44,800 --> 00:37:45,730
OK.
507
00:37:45,730 --> 00:37:47,110
So that's J squared.
508
00:37:54,770 --> 00:37:56,240
OK.
509
00:37:56,240 --> 00:37:59,680
So we're doing sort
of simple things.
510
00:37:59,680 --> 00:38:02,130
Basically at this
moment, we decided
511
00:38:02,130 --> 00:38:06,590
that we like better
J plus and J minus.
512
00:38:06,590 --> 00:38:09,390
And we tried to
figure out everything
513
00:38:09,390 --> 00:38:11,870
that we should know
about J plus, J minus.
514
00:38:11,870 --> 00:38:18,270
If we substitute Lx, and Jx,
and Jy for J plus and J minus,
515
00:38:18,270 --> 00:38:21,080
you better know what
is the commutator
516
00:38:21,080 --> 00:38:23,650
of J plus and J minus.
517
00:38:23,650 --> 00:38:28,000
And how to write J squared in
terms of J plus and J minus.
518
00:38:28,000 --> 00:38:30,640
And this is what
we've done here.
519
00:38:30,640 --> 00:38:35,910
And in particular, we have a
whole lot of nice formulas.
520
00:38:35,910 --> 00:38:39,515
So one more formula
is probably useful.
521
00:38:42,680 --> 00:38:48,790
And it's the formula for the
commutator of J plus and J
522
00:38:48,790 --> 00:38:51,050
minus with Jz.
523
00:38:51,050 --> 00:38:54,570
Because after all, the J
plus, J minus commutator,
524
00:38:54,570 --> 00:38:55,450
you've got it.
525
00:38:55,450 --> 00:38:58,700
So if you're systematic
about these things
526
00:38:58,700 --> 00:39:01,960
you should figure
out that at this I
527
00:39:01,960 --> 00:39:05,165
would like to know what is
the commutator of J plus and J
528
00:39:05,165 --> 00:39:06,250
minus with Jz.
529
00:39:06,250 --> 00:39:12,720
So I can do Jz, J plus.
530
00:39:12,720 --> 00:39:13,840
It's not hard.
531
00:39:13,840 --> 00:39:16,020
It's Jz.
532
00:39:16,020 --> 00:39:16,700
I'm sorry.
533
00:39:16,700 --> 00:39:17,990
I'm calling it 3.
534
00:39:17,990 --> 00:39:21,370
So, I think in the notes
I call them x, y, and z.
535
00:39:21,370 --> 00:39:24,550
But never mind.
536
00:39:24,550 --> 00:39:28,910
J1 plus i, J2.
537
00:39:28,910 --> 00:39:32,330
The plus is really
with a plus i.
538
00:39:32,330 --> 00:39:40,310
So J3 with J1 by the cyclic
ordering is ih bar, J2.
539
00:39:40,310 --> 00:39:52,670
And here you have plus i, and
J3 with J2 is minus ih bar, J1.
540
00:39:52,670 --> 00:40:02,770
So this is h bar, J1, plus i,
J2, which is h bar, J plus.
541
00:40:02,770 --> 00:40:09,290
So what you've learned
is that J3 with J plus
542
00:40:09,290 --> 00:40:12,340
is equal to h bar, J plus.
543
00:40:12,340 --> 00:40:16,190
And if you did it
with J minus, you'll
544
00:40:16,190 --> 00:40:20,580
find a minus, and
a plus minus here.
545
00:40:20,580 --> 00:40:25,120
So that is the complete result.
546
00:40:25,120 --> 00:40:30,970
And that should remind you
of the analogous relation
547
00:40:30,970 --> 00:40:35,130
in which you have in the
harmonic oscillator, N
548
00:40:35,130 --> 00:40:38,160
commutator, with a dagger.
549
00:40:38,160 --> 00:40:39,895
With a dagger.
550
00:40:39,895 --> 00:40:48,280
And N commutator
with a was minus a.
551
00:40:48,280 --> 00:40:52,120
Because of the fact that I
maybe didn't say it here,
552
00:40:52,120 --> 00:41:01,090
and I should have, that the
dagger of J plus is J minus.
553
00:41:01,090 --> 00:41:03,190
Because the operators
are Hermitians.
554
00:41:03,190 --> 00:41:06,890
So J plus and J minus are
daggers of each other,
555
00:41:06,890 --> 00:41:09,180
are adjoins of each other.
556
00:41:09,180 --> 00:41:11,770
And here you see a very
analogous situation.
557
00:41:11,770 --> 00:41:15,460
a and a dagger were
adjoins of each other.
558
00:41:15,460 --> 00:41:19,940
And with respect to N, a
counting number operator.
559
00:41:19,940 --> 00:41:21,160
One increased it.
560
00:41:21,160 --> 00:41:23,160
One decreased it.
561
00:41:23,160 --> 00:41:28,690
a dagger increased the
number eigenvalue of N. a
562
00:41:28,690 --> 00:41:31,580
decreased it, the same way
it's going to happen here.
563
00:41:31,580 --> 00:41:34,260
J plus is going
to increase the C
564
00:41:34,260 --> 00:41:36,290
component of angular momentum.
565
00:41:36,290 --> 00:41:38,320
And J minus is going
to decrease it.
566
00:41:40,930 --> 00:41:41,920
OK.
567
00:41:41,920 --> 00:41:47,380
So we've done most of the
calculations that we need.
568
00:41:47,380 --> 00:41:50,680
The rest is pretty easy work.
569
00:41:50,680 --> 00:41:52,920
Not that it was
difficult so far.
570
00:41:52,920 --> 00:41:57,970
But it took a little time.
571
00:41:57,970 --> 00:42:04,960
So what happens next
is the following.
572
00:42:04,960 --> 00:42:08,820
You must make a declaration.
573
00:42:08,820 --> 00:42:13,590
There should exist
states, basically.
574
00:42:13,590 --> 00:42:15,430
We have a vector space.
575
00:42:15,430 --> 00:42:17,370
It's very large.
576
00:42:17,370 --> 00:42:21,180
It's actually
infinite dimensional.
577
00:42:21,180 --> 00:42:24,360
Because they will be related
to all kinds of functions
578
00:42:24,360 --> 00:42:25,620
on the unit sphere.
579
00:42:25,620 --> 00:42:28,460
All these angular variables.
580
00:42:28,460 --> 00:42:30,400
So it's infinite dimensional.
581
00:42:30,400 --> 00:42:32,810
So it's a little scary.
582
00:42:32,810 --> 00:42:37,130
But let's not worry about that.
583
00:42:37,130 --> 00:42:40,510
Something very nice happens
with angular momentum.
584
00:42:40,510 --> 00:42:44,330
Something so nice that
it didn't happen actually
585
00:42:44,330 --> 00:42:47,370
with a and a dagger.
586
00:42:47,370 --> 00:42:49,650
With a and a dagger,
you build states
587
00:42:49,650 --> 00:42:51,090
in the harmonic oscillator.
588
00:42:51,090 --> 00:42:54,510
And you build
infinitely many ones.
589
00:42:54,510 --> 00:42:59,640
The operators x and p, you've
learned you cannot represent
590
00:42:59,640 --> 00:43:02,830
them by finite
dimensional matrices.
591
00:43:02,830 --> 00:43:07,110
So this is a lot more
complicated, you would say.
592
00:43:07,110 --> 00:43:11,080
And you would say, well,
this is just much harder.
593
00:43:11,080 --> 00:43:15,480
This algebra is so much
harder than this algebra.
594
00:43:18,650 --> 00:43:21,850
Nevertheless, this algebra
is the difficult one.
595
00:43:21,850 --> 00:43:25,182
Gives you infinite
dimensional representations.
596
00:43:25,182 --> 00:43:28,830
You can keep piling
the a daggers.
597
00:43:28,830 --> 00:43:31,670
Here, this is a
very dense algebra.
598
00:43:31,670 --> 00:43:36,190
Mathematicians would say this
is much simpler than this one.
599
00:43:36,190 --> 00:43:39,940
And we'll see the simplicity
of this one, in that you
600
00:43:39,940 --> 00:43:42,820
will manage to get
representations
601
00:43:42,820 --> 00:43:45,400
and matrices that are
finite dimensional
602
00:43:45,400 --> 00:43:47,550
to work these things out.
603
00:43:47,550 --> 00:43:51,170
So it's going to be
nicer in that sense.
604
00:43:51,170 --> 00:43:54,315
So what do we have?
605
00:43:54,315 --> 00:43:58,400
We have to think of our
commuting observables
606
00:43:58,400 --> 00:44:02,650
and the set of Hermitian
operators that commute.
607
00:44:02,650 --> 00:44:11,450
So we have J squared, and J3--
I call it Jz now, apologies.
608
00:44:11,450 --> 00:44:15,040
And we'll declare
that there are states.
609
00:44:15,040 --> 00:44:17,850
These are Hermitian,
and they commute.
610
00:44:17,850 --> 00:44:22,040
So they must be
diagonalized simultaneously.
611
00:44:22,040 --> 00:44:25,150
And there should
exist states that
612
00:44:25,150 --> 00:44:27,320
represent the diagonalization.
613
00:44:27,320 --> 00:44:31,720
In fact, since they commute,
and can be diagonalized
614
00:44:31,720 --> 00:44:35,500
simultaneously, the
vector space must
615
00:44:35,500 --> 00:44:37,930
break into a list of vectors.
616
00:44:37,930 --> 00:44:41,900
All of them eigenstates
of these 2 operators.
617
00:44:41,900 --> 00:44:44,930
And all of them
orthogonal to each other.
618
00:44:44,930 --> 00:44:46,775
Matthew, you had a question?
619
00:44:46,775 --> 00:44:48,472
AUDIENCE: I was
just wondering when
620
00:44:48,472 --> 00:44:53,080
we showed that Jz is Hermitian?
621
00:44:53,080 --> 00:44:54,660
PROFESSOR: We didn't show it.
622
00:44:54,660 --> 00:44:59,400
We postulated that J's
are Hermitian operators.
623
00:44:59,400 --> 00:45:03,550
So you know that when J
is L, yes it's Hermitian.
624
00:45:03,550 --> 00:45:08,760
You know when J is spin,
yes it's Hermitian.
625
00:45:08,760 --> 00:45:11,625
Whatever you're doing we'll
use Hermitian operators.
626
00:45:15,010 --> 00:45:21,040
So not only they can
diagonalize simultaneously,
627
00:45:21,040 --> 00:45:25,830
by our main theorem about
Hermitian operators,
628
00:45:25,830 --> 00:45:29,600
this should provide
an orthonormal basis
629
00:45:29,600 --> 00:45:33,770
for the full vector space.
630
00:45:33,770 --> 00:45:36,390
So the whole answer is
supposed to be here.
631
00:45:36,390 --> 00:45:37,380
Let's see.
632
00:45:37,380 --> 00:45:41,970
So I'll define
states, Jm, that are
633
00:45:41,970 --> 00:45:44,540
eigenstates of both
of these things.
634
00:45:44,540 --> 00:45:48,410
And I have 2 numbers to
declare those eigenvalues.
635
00:45:48,410 --> 00:45:52,300
You would say J squared.
636
00:45:52,300 --> 00:45:56,970
Now, any normal person
would put here maybe h
637
00:45:56,970 --> 00:45:59,830
squared, for units,
time J squared.
638
00:46:02,900 --> 00:46:03,890
And then Jm.
639
00:46:06,680 --> 00:46:09,360
Don't copy it yet.
640
00:46:09,360 --> 00:46:14,070
And Jz for Jm.
641
00:46:14,070 --> 00:46:15,800
It has units of
angular momentum.
642
00:46:15,800 --> 00:46:19,650
So an h, times m, times Jm.
643
00:46:22,830 --> 00:46:25,900
But that turns
out not to be very
644
00:46:25,900 --> 00:46:29,590
convenient to put
the J squared there.
645
00:46:29,590 --> 00:46:31,660
It ruins the algebra later.
646
00:46:31,660 --> 00:46:33,780
So we'll put something
different that we
647
00:46:33,780 --> 00:46:35,880
hope has the same effect.
648
00:46:35,880 --> 00:46:38,200
And I will discuss that.
649
00:46:38,200 --> 00:46:44,480
I'll put h squared,
J times J plus 1.
650
00:46:44,480 --> 00:46:47,200
It's a funny way
of declaring how
651
00:46:47,200 --> 00:46:48,760
you're going to
build the states.
652
00:46:48,760 --> 00:46:53,850
But it's a possible thing to do.
653
00:46:53,850 --> 00:46:56,910
So here are the states, J and m.
654
00:46:56,910 --> 00:46:59,660
And the only thing I
know at this moment
655
00:46:59,660 --> 00:47:02,430
is that since these are
Hermitian operators,
656
00:47:02,430 --> 00:47:04,820
their eigenvalues must be real.
657
00:47:04,820 --> 00:47:07,480
So J times J plus 1 is real.
658
00:47:07,480 --> 00:47:08,890
And m is real.
659
00:47:08,890 --> 00:47:11,470
So J and m belong to the reals.
660
00:47:18,670 --> 00:47:22,030
And they are
orthogonal to-- we can
661
00:47:22,030 --> 00:47:25,290
say they're orthonormal states.
662
00:47:25,290 --> 00:47:29,430
We will see very soon that
these things get quantized.
663
00:47:29,430 --> 00:47:34,590
But basically, the overlap of
a Jm with a J prime, m prime
664
00:47:34,590 --> 00:47:38,710
would be 0 whenever the J's
and the m's are different.
665
00:47:38,710 --> 00:47:45,460
As you know from our
theory, any 2 eigenstates
666
00:47:45,460 --> 00:47:48,070
with different eigenvalues
are orthonormal.
667
00:47:48,070 --> 00:47:51,020
And in fact, you
can choose a basis
668
00:47:51,020 --> 00:47:53,670
so that in fact,
everything is orthonormal.
669
00:47:53,670 --> 00:47:57,050
So there's no
question like that.
670
00:47:57,050 --> 00:48:00,250
So let's explain a little what's
happening with this thing.
671
00:48:00,250 --> 00:48:02,880
Why do we put this like that?
672
00:48:02,880 --> 00:48:05,640
Or why can we get
away with this?
673
00:48:05,640 --> 00:48:09,590
And the reason is the following.
674
00:48:09,590 --> 00:48:19,470
Let's consider
Jm, J squared, Jm.
675
00:48:19,470 --> 00:48:27,080
If I use this, J squared
on this is this number.
676
00:48:27,080 --> 00:48:31,980
And Jm with itself will be 1.
677
00:48:31,980 --> 00:48:34,400
And therefore I'll put
here h-- I'm sorry.
678
00:48:34,400 --> 00:48:35,950
This should be an h squared.
679
00:48:35,950 --> 00:48:38,610
J has units of angular momentum.
680
00:48:38,610 --> 00:48:44,020
h squared, J times J plus 1.
681
00:48:44,020 --> 00:48:47,930
And I'm assuming that
this will be discretized
682
00:48:47,930 --> 00:48:51,680
so I don't have to put the
delta function normalization.
683
00:48:51,680 --> 00:48:56,030
At any rate, this
thing is equal to this.
684
00:48:56,030 --> 00:48:59,300
And moreover, it's
equal to the following.
685
00:48:59,300 --> 00:49:12,990
Jm sum over i, Jm, Ji, Ji, Jm.
686
00:49:12,990 --> 00:49:18,950
But since J is Hermitian, this
is nothing but the sum over i
687
00:49:18,950 --> 00:49:25,300
of the norm squared of
Ji with J acting on Jm.
688
00:49:28,930 --> 00:49:31,250
The norm squared of this state.
689
00:49:31,250 --> 00:49:34,440
Because this times the
bra with Ji Hermitian
690
00:49:34,440 --> 00:49:35,760
is the norm squared.
691
00:49:35,760 --> 00:49:39,030
So this is greater
or equal than 0.
692
00:49:39,030 --> 00:49:43,370
Perhaps no surprise, this
is a vector operator,
693
00:49:43,370 --> 00:49:47,010
which is the sum of squares
of Hermitian operators.
694
00:49:47,010 --> 00:49:49,600
And therefore it
should be like that.
695
00:49:49,600 --> 00:50:00,250
Now, given that, we have
the following-- oops--
696
00:50:00,250 --> 00:50:07,200
the following fact that
L times L plus-- no.
697
00:50:07,200 --> 00:50:11,270
J times J plus 1 must be
greater or equal than 0.
698
00:50:11,270 --> 00:50:16,050
J times J plus 1 must be
greater or equal than 0.
699
00:50:16,050 --> 00:50:21,580
Well, plot it as a function
of J. It vanishes at 0.
700
00:50:21,580 --> 00:50:26,770
J times J plus 1 vanishes at
0, and vanishes at minus 1.
701
00:50:26,770 --> 00:50:29,440
It's a function like this.
702
00:50:29,440 --> 00:50:32,270
The function J times J plus 1.
703
00:50:32,270 --> 00:50:42,450
And this shows that all you need
is this thing to be positive.
704
00:50:42,450 --> 00:50:47,650
So to represent all the states
that have J times J plus 1
705
00:50:47,650 --> 00:50:52,680
positive, I could label them
with J's that are positive.
706
00:50:52,680 --> 00:50:56,380
Or J's that are
smaller than minus 1.
707
00:50:56,380 --> 00:51:00,510
So each way, I can label
uniquely those states.
708
00:51:00,510 --> 00:51:03,410
So if I get J times
J plus 1 equals 3,
709
00:51:03,410 --> 00:51:05,930
it may correspond
to a J of something
710
00:51:05,930 --> 00:51:08,720
and a J of some other thing.
711
00:51:08,720 --> 00:51:13,250
I will have just 1 state,
so I will choose J positive.
712
00:51:13,250 --> 00:51:17,080
So given that J times
J plus 1 is positive,
713
00:51:17,080 --> 00:51:28,320
I can label states
with J positive, or 0.
714
00:51:28,320 --> 00:51:32,720
So it allows you to do this.
715
00:51:32,720 --> 00:51:37,900
Whatever value of this
quantity that is positive
716
00:51:37,900 --> 00:51:41,540
corresponds to some J positive
that you can put in here.
717
00:51:41,540 --> 00:51:42,990
A unique J positive.
718
00:51:42,990 --> 00:51:48,450
So this is a fine
parametrization of the problem.
719
00:51:48,450 --> 00:51:49,460
OK.
720
00:51:49,460 --> 00:51:54,255
Now what's next?
721
00:51:56,925 --> 00:52:00,840
Next, we have to understand
what the J plus operators and J
722
00:52:00,840 --> 00:52:04,360
minus operators
do to the states.
723
00:52:04,360 --> 00:52:15,790
So, first thing is that J plus
and J minus commute with J
724
00:52:15,790 --> 00:52:16,290
squared.
725
00:52:19,400 --> 00:52:21,970
That should not be a surprise.
726
00:52:21,970 --> 00:52:24,160
J1 and J2 commute.
727
00:52:24,160 --> 00:52:26,660
Every J commutes with J squared.
728
00:52:26,660 --> 00:52:29,970
So J plus and J minus
commute with J squared.
729
00:52:29,970 --> 00:52:36,540
What this means in words
is that J plus and J minus
730
00:52:36,540 --> 00:52:42,220
do not change the eigenvalue
of J squared on a state.
731
00:52:42,220 --> 00:52:49,330
That is, if I would have J
squared on J plus or minus
732
00:52:49,330 --> 00:53:00,270
on Jm-- since I can move the J
squared up across the J plus,
733
00:53:00,270 --> 00:53:02,220
minus-- it hits here.
734
00:53:02,220 --> 00:53:07,690
Then I have J plus
minus, J squared, Jm.
735
00:53:07,690 --> 00:53:12,330
And that's there for h
squared, J times J plus 1,
736
00:53:12,330 --> 00:53:15,080
times J plus minus on Jm.
737
00:53:18,020 --> 00:53:22,590
So this state is also a
state with the same value
738
00:53:22,590 --> 00:53:25,040
of J squared.
739
00:53:25,040 --> 00:53:39,410
Therefore, it must have the
same value of J. In other words,
740
00:53:39,410 --> 00:53:46,670
this state J plus minus
of Jm must be proportional
741
00:53:46,670 --> 00:53:50,350
to a state with J and maybe
some different value of m,
742
00:53:50,350 --> 00:53:52,390
but the same value of J.
743
00:53:52,390 --> 00:53:53,860
J cannot have changed.
744
00:53:56,380 --> 00:53:57,570
J must be the same.
745
00:54:01,950 --> 00:54:10,280
Then we have to
see who changes m,
746
00:54:10,280 --> 00:54:14,220
or how does J plus
minus changes m.
747
00:54:14,220 --> 00:54:17,770
So here comes a little
bit of a same calculation.
748
00:54:17,770 --> 00:54:21,900
You want to see what is
the m value of this thing.
749
00:54:21,900 --> 00:54:26,850
So you have J plus minus on Jm.
750
00:54:26,850 --> 00:54:32,630
And you act with it with
a Jz, to see what it is.
751
00:54:32,630 --> 00:54:38,030
And then, you put, well,
the commutator first.
752
00:54:38,030 --> 00:54:46,830
Jz, J plus minus, plus J
plus minus, Jz on the state.
753
00:54:46,830 --> 00:54:49,370
The commutator,
you've calculated it
754
00:54:49,370 --> 00:54:53,770
before, was Jz with J
plus minus is there,
755
00:54:53,770 --> 00:55:01,490
is plus minus h
bar, J plus minus.
756
00:55:01,490 --> 00:55:04,360
And this Jz already act.
757
00:55:04,360 --> 00:55:16,900
So this is plus h bar
m, J plus minus on Jm.
758
00:55:16,900 --> 00:55:20,350
So we can get the
J plus minus out.
759
00:55:20,350 --> 00:55:32,180
And this h bar m plus
minus 1, j plus minus, Jm.
760
00:55:32,180 --> 00:55:34,220
So look what you got.
761
00:55:34,220 --> 00:55:43,940
Jz acting on this state is
h bar, m plus minus 1, Jm.
762
00:55:43,940 --> 00:55:51,520
So this state has m equal to
either m plus 1, or m minus 1.
763
00:55:51,520 --> 00:55:52,875
Something that we can write.
764
00:55:56,170 --> 00:56:02,510
Clearly-- oops--
in this way, we'll
765
00:56:02,510 --> 00:56:08,370
say that J plus minus,
Jm-- we know already
766
00:56:08,370 --> 00:56:13,150
it's a state with J
and m plus minus 1.
767
00:56:13,150 --> 00:56:14,970
So it raises m.
768
00:56:14,970 --> 00:56:19,630
Just like what we said
that the a's and a daggers
769
00:56:19,630 --> 00:56:21,860
raise or lower the number.
770
00:56:21,860 --> 00:56:27,420
J plus and J minus
raise and lower Jz.
771
00:56:27,420 --> 00:56:30,940
Therefore, it's this is
proportional to this state.
772
00:56:30,940 --> 00:56:33,430
But there's a constant
of proportionality
773
00:56:33,430 --> 00:56:36,080
that we have to figure out.
774
00:56:36,080 --> 00:56:41,140
And we'll call it
the constant C, Jm.
775
00:56:41,140 --> 00:56:41,890
To be calculated.
776
00:56:51,000 --> 00:56:55,350
So the way to calculate
this constant-- and that
777
00:56:55,350 --> 00:56:59,870
will bring us almost pretty
close to what we need--
778
00:56:59,870 --> 00:57:03,130
is to take inner products.
779
00:57:03,130 --> 00:57:07,980
So we must take the
dagger of this equation.
780
00:57:07,980 --> 00:57:16,544
So take the dagger, and you get
Jm, the adjoin, J minus plus.
781
00:57:16,544 --> 00:57:19,280
And hit it with this equation.
782
00:57:22,690 --> 00:57:26,600
So you'll have here--
well maybe I'll write it.
783
00:57:26,600 --> 00:57:28,380
The dagger of this
equation would
784
00:57:28,380 --> 00:57:34,239
be C plus minus star of Jm.
785
00:57:34,239 --> 00:57:37,545
Jm plus minus 1.
786
00:57:41,490 --> 00:57:46,390
And now, sandwich
this with that.
787
00:57:46,390 --> 00:57:54,400
So you have Jm, J minus
plus, J plus minus,
788
00:57:54,400 --> 00:58:05,310
Jm equals to norm
of C plus minus Jm.
789
00:58:05,310 --> 00:58:08,900
And then you have this state
times this state, but that's 1.
790
00:58:08,900 --> 00:58:11,910
Because it's J, J,
m plus 1, m plus 1.
791
00:58:11,910 --> 00:58:13,910
So this is an orthonormal basis.
792
00:58:13,910 --> 00:58:15,430
So we have just 1.
793
00:58:15,430 --> 00:58:16,795
And I don't have to write more.
794
00:58:19,470 --> 00:58:24,770
Well the left hand
side can be calculated.
795
00:58:24,770 --> 00:58:26,925
We have still that formula here.
796
00:58:30,110 --> 00:58:32,480
So let's calculate it.
797
00:58:37,210 --> 00:58:43,070
The left hand side,
I'll write it like this.
798
00:58:43,070 --> 00:58:50,140
I will have C plus
minus, Jm squared,
799
00:58:50,140 --> 00:58:56,500
which is equal to the norm
squared of J plus minus, Jm.
800
00:58:59,960 --> 00:59:01,196
It's equal to what?
801
00:59:01,196 --> 00:59:04,300
Whatever this is,
where you substitute
802
00:59:04,300 --> 00:59:07,300
that for this formula.
803
00:59:07,300 --> 00:59:10,270
So you'll put here Jm.
804
00:59:10,270 --> 00:59:17,470
And you'll have--
well, I want actually
805
00:59:17,470 --> 00:59:20,040
the formula I just erased.
806
00:59:20,040 --> 00:59:25,300
Because I actually would
prefer to have J squared.
807
00:59:25,300 --> 00:59:34,020
So I would have this is equal
to J squared, minus J3 squared,
808
00:59:34,020 --> 00:59:39,030
plus minus h, J3.
809
00:59:39,030 --> 00:59:40,350
So let's see.
810
00:59:40,350 --> 00:59:42,820
I have the sign minus
plus, plus minus.
811
00:59:42,820 --> 00:59:45,560
So I should change
the signs there.
812
00:59:45,560 --> 00:59:58,490
So it should be J squared,
minus J3 squared, minus plus J3,
813
00:59:58,490 --> 01:00:05,010
and Jm, minus plus
h bar, J3, Jm.
814
01:00:05,010 --> 01:00:11,540
So this is equal to h bar
squared, J times J plus 1,
815
01:00:11,540 --> 01:00:18,000
minus an m squared,
and a minus plus.
816
01:00:18,000 --> 01:00:21,635
So minus, plus, minus here.
817
01:00:26,380 --> 01:00:29,000
I think I have it here correct.
818
01:00:29,000 --> 01:00:31,680
Plus minus 1.
819
01:00:31,680 --> 01:00:34,060
And that's it.
820
01:00:34,060 --> 01:00:37,280
J squared is h squared this.
821
01:00:37,280 --> 01:00:39,430
J3 squared would give that.
822
01:00:39,430 --> 01:00:43,830
And the minus plus here is
correctly with this one.
823
01:00:43,830 --> 01:00:46,080
So m should be here.
824
01:00:46,080 --> 01:00:49,420
Plus minus m.
825
01:00:49,420 --> 01:00:55,410
So this is h squared,
J times J plus 1,
826
01:00:55,410 --> 01:00:59,445
minus m, times m plus minus 1.
827
01:01:05,611 --> 01:01:06,110
OK.
828
01:01:10,980 --> 01:01:15,980
So the C's have
been already found.
829
01:01:15,980 --> 01:01:21,730
And you can take
their square roots.
830
01:01:21,730 --> 01:01:26,400
In fact, we can ideally
just take the square roots,
831
01:01:26,400 --> 01:01:31,150
because these things
better be positive numbers
832
01:01:31,150 --> 01:01:32,990
because they're norms squared.
833
01:01:32,990 --> 01:01:36,270
So whenever we'll
be able to do this,
834
01:01:36,270 --> 01:01:39,360
these things better
be positive, being
835
01:01:39,360 --> 01:01:41,270
the square of some states.
836
01:01:41,270 --> 01:01:47,940
And therefore the C plus
minus is-- C plus minus of Jm
837
01:01:47,940 --> 01:01:53,980
can be simply taken to be
h bar, square root of J
838
01:01:53,980 --> 01:02:01,120
times J plus 1, minus
m, times m plus 1.
839
01:02:01,120 --> 01:02:05,290
And it's because of
this thing, this m times
840
01:02:05,290 --> 01:02:11,980
m plus 1, that it was convenient
to have J times J plus 1.
841
01:02:11,980 --> 01:02:16,040
So that we can compare
J's and m's better.
842
01:02:16,040 --> 01:02:20,310
Otherwise it would have
been pretty disastrous.
843
01:02:20,310 --> 01:02:26,390
So, OK, we're almost done
now with the calculation
844
01:02:26,390 --> 01:02:27,330
of the spectrum.
845
01:02:27,330 --> 01:02:32,110
You will say, well, we seem
to be getting no where.
846
01:02:32,110 --> 01:02:34,600
Learned all these
properties, these states,
847
01:02:34,600 --> 01:02:37,150
and now you're just
manipulating the states.
848
01:02:37,150 --> 01:02:39,180
But the main thing
is that we need
849
01:02:39,180 --> 01:02:42,830
these things to be positive.
850
01:02:42,830 --> 01:02:45,070
And that will give us
the whole condition.
851
01:02:45,070 --> 01:02:50,220
So, for example, we need
1, that the states J
852
01:02:50,220 --> 01:02:56,750
plus, Jm, their norm
squareds be positive.
853
01:02:56,750 --> 01:03:02,360
So for the plus sign-- so you
should have J times J plus 1,
854
01:03:02,360 --> 01:03:07,480
minus m, times m
plus 1 be positive.
855
01:03:07,480 --> 01:03:18,380
Or m times m plus 1 be
smaller then J times J plus 1.
856
01:03:21,150 --> 01:03:26,960
The best way for my mind to
solve these kind of things
857
01:03:26,960 --> 01:03:29,880
is to just plot them.
858
01:03:29,880 --> 01:03:34,820
So here is m.
859
01:03:37,520 --> 01:03:40,025
And here is m times m plus 1.
860
01:03:42,730 --> 01:03:45,010
So you plot this function.
861
01:03:45,010 --> 01:03:53,810
And you want it to be less than
some value of J times J plus 1.
862
01:03:53,810 --> 01:03:57,050
So here's J times J
plus 1, some value.
863
01:03:57,050 --> 01:04:00,150
So this is 0 here.
864
01:04:00,150 --> 01:04:02,400
This function is 0 at minus 1.
865
01:04:02,400 --> 01:04:07,260
So it will be
something like this.
866
01:04:07,260 --> 01:04:15,400
And there's 2 values at which
m becomes equal to this thing.
867
01:04:15,400 --> 01:04:22,670
And one is clearly J.
When m is equal to J,
868
01:04:22,670 --> 01:04:25,390
it's saturates an inequality.
869
01:04:25,390 --> 01:04:28,755
And the other one
is minus J, minus 1.
870
01:04:31,950 --> 01:04:34,910
If m is minus J,
minus 1, you will
871
01:04:34,910 --> 01:04:38,280
have minus J, minus
1 here, and minus J
872
01:04:38,280 --> 01:04:41,920
here, which would
be equal to this.
873
01:04:41,920 --> 01:04:49,350
So, in order for these states
to be good, the value of m
874
01:04:49,350 --> 01:04:56,130
must be in between J
and minus J, minus 1.
875
01:04:59,270 --> 01:05:11,280
Then the other case is that
J plus on-- J minus on Jm.
876
01:05:11,280 --> 01:05:13,760
If you produce those
states, they also
877
01:05:13,760 --> 01:05:15,380
must have positive norms.
878
01:05:15,380 --> 01:05:22,260
So J times J plus 1, minus
m, times m minus 1 this time,
879
01:05:22,260 --> 01:05:23,940
must be greater than 0.
880
01:05:23,940 --> 01:05:28,900
So m times m minus
1 must be less than
881
01:05:28,900 --> 01:05:33,990
or equal then J times J plus 1.
882
01:05:33,990 --> 01:05:39,550
And again, we try to
do it geometrically.
883
01:05:39,550 --> 01:05:43,420
So here it is.
884
01:05:43,420 --> 01:05:45,880
Here is m.
885
01:05:45,880 --> 01:05:48,280
And what values do you have?
886
01:05:48,280 --> 01:05:52,610
Well, if you plot here
m times m minus 1.
887
01:05:56,260 --> 01:05:58,410
And that should be
equal to some value
888
01:05:58,410 --> 01:06:02,845
that you get fixed, which is
the value J times J plus 1.
889
01:06:06,370 --> 01:06:10,630
So you think in terms of
m's, how far can they go?
890
01:06:10,630 --> 01:06:15,900
So if you take m equals
J plus 1 that hits it.
891
01:06:15,900 --> 01:06:18,890
So this is 0 here, at 1, at 0.
892
01:06:18,890 --> 01:06:21,930
So it's some function like this.
893
01:06:21,930 --> 01:06:26,820
And here you have J plus 1.
894
01:06:26,820 --> 01:06:30,800
And here you have
minus J. Both are
895
01:06:30,800 --> 01:06:36,570
the places for m equal
J plus 1, and minus J
896
01:06:36,570 --> 01:06:41,230
that you get the states.
897
01:06:41,230 --> 01:06:42,630
You get the saturation.
898
01:06:42,630 --> 01:06:45,670
So you can run m
from this range.
899
01:06:45,670 --> 01:06:51,210
Now, m can go less than
or equal to J plus 1,
900
01:06:51,210 --> 01:06:57,000
and greater than or
equal to minus J.
901
01:06:57,000 --> 01:07:00,520
But these 2 inequalities
must hold at the same time.
902
01:07:00,520 --> 01:07:05,780
You cannot allow either one to
go wrong for any set of states.
903
01:07:05,780 --> 01:07:13,120
So if both must hold at the
same time for any state,
904
01:07:13,120 --> 01:07:18,620
because both things have to
happen, you get constrained.
905
01:07:18,620 --> 01:07:24,900
This time for the upper range,
this is the stronger value.
906
01:07:24,900 --> 01:07:28,220
For the lower range, this
is the stronger value.
907
01:07:28,220 --> 01:07:37,085
So m must go between J and
minus J for both to hold.
908
01:07:40,470 --> 01:07:41,840
Oops.
909
01:07:41,840 --> 01:07:42,340
To hold.
910
01:07:46,950 --> 01:07:49,450
Now look what happens.
911
01:07:49,450 --> 01:07:54,890
Funny things happen
if-- this is reasonable
912
01:07:54,890 --> 01:07:58,410
that the strongest value
comes from this equation.
913
01:07:58,410 --> 01:08:01,800
Because J plus increases m.
914
01:08:01,800 --> 01:08:04,530
So at some point
you run into trouble
915
01:08:04,530 --> 01:08:06,680
if you increase m too much.
916
01:08:06,680 --> 01:08:08,370
How much can you increase it?
917
01:08:08,370 --> 01:08:12,610
You cannot go beyond J,
and that makes sense.
918
01:08:12,610 --> 01:08:14,250
In some sense, your
intuition should
919
01:08:14,250 --> 01:08:18,330
be that J is the
length of J squared.
920
01:08:18,330 --> 01:08:20,899
And m is mz.
921
01:08:20,899 --> 01:08:26,810
So m should not go beyond J.
And that's reasonable here.
922
01:08:26,810 --> 01:08:32,930
And in fact, when m is equal to
J, this whole thing vanishes.
923
01:08:32,930 --> 01:08:37,350
So if you reach that state
when m is equal to J,
924
01:08:37,350 --> 01:08:42,920
only then for m equal to J,
or for this state, you get 0.
925
01:08:42,920 --> 01:08:45,790
So you cannot raise
the state anymore.
926
01:08:45,790 --> 01:08:55,700
So actually, you see if you
choose some J over here,
927
01:08:55,700 --> 01:08:58,390
we need a few things to happen.
928
01:08:58,390 --> 01:09:03,620
You choose some J, and some m.
929
01:09:03,620 --> 01:09:07,460
Well you're going to
be shifting the m's.
930
01:09:07,460 --> 01:09:12,830
And if you keep adding
J pluses, eventually you
931
01:09:12,830 --> 01:09:15,500
will go beyond this point.
932
01:09:15,500 --> 01:09:18,899
The only way not to
go beyond this point
933
01:09:18,899 --> 01:09:23,840
is if m reaches the value J.
Because if m reaches the value
934
01:09:23,840 --> 01:09:27,029
J, the state is killed.
935
01:09:27,029 --> 01:09:32,229
So m should reach the value
J over here at some stage.
936
01:09:32,229 --> 01:09:36,819
So you fix J, and you try
to think what m can be.
937
01:09:36,819 --> 01:09:40,069
And m has to reach
the value J. So
938
01:09:40,069 --> 01:09:43,939
m at some point,
whatever m is, you add 1.
939
01:09:43,939 --> 01:09:44,689
You add 1.
940
01:09:44,689 --> 01:09:45,240
You add 1.
941
01:09:45,240 --> 01:09:50,260
And eventually you must
reach the value J. Reach
942
01:09:50,260 --> 01:09:52,370
with some m prime.
943
01:09:52,370 --> 01:09:55,150
m here.
944
01:09:55,150 --> 01:09:57,530
You should reach the
value J, so that you
945
01:09:57,530 --> 01:10:00,110
don't produce another
state that is higher.
946
01:10:00,110 --> 01:10:05,480
If you reach something before
that, that state is not killed.
947
01:10:05,480 --> 01:10:07,720
This number is not equal to 0.
948
01:10:07,720 --> 01:10:12,130
You produce a state and it's
a bad state of bad norm.
949
01:10:12,130 --> 01:10:14,080
So you must reach this one.
950
01:10:14,080 --> 01:10:16,826
On the other hand,
you can lower things.
951
01:10:16,826 --> 01:10:22,320
And if you go below minus
J, you produce bad states.
952
01:10:22,320 --> 01:10:25,700
So you must also,
when you decrease m,
953
01:10:25,700 --> 01:10:27,370
you must reach this point.
954
01:10:27,370 --> 01:10:31,700
Because if you didn't, and you
stop half a unit away from it,
955
01:10:31,700 --> 01:10:34,570
the next state that
you produce is bad.
956
01:10:34,570 --> 01:10:36,190
And that can't be.
957
01:10:36,190 --> 01:10:40,900
So you must reach this one too.
958
01:10:40,900 --> 01:10:44,120
And that's the key logical
part of the argument
959
01:10:44,120 --> 01:10:57,881
in which this distance
2J plus 1-- no.
960
01:10:57,881 --> 01:10:58,380
I'm sorry.
961
01:10:58,380 --> 01:11:03,490
This 2J must be equal
to some integer.
962
01:11:09,510 --> 01:11:11,880
And that's the key
thing that must happen,
963
01:11:11,880 --> 01:11:15,200
because you must reach this
and you must reach here.
964
01:11:15,200 --> 01:11:18,190
And m just varies by integers.
965
01:11:18,190 --> 01:11:21,630
So the distance between
this J and minus J
966
01:11:21,630 --> 01:11:23,750
must be twice an integer.
967
01:11:23,750 --> 01:11:28,110
And you've discovered
something remarkable by getting
968
01:11:28,110 --> 01:11:30,980
to that point,
because now you see
969
01:11:30,980 --> 01:11:33,230
that if this has to
be an integer, well
970
01:11:33,230 --> 01:11:37,230
it may be 0, 1, 2, 3.
971
01:11:37,230 --> 01:11:41,440
And when J-- then J-- this
integer is equal to 0,
972
01:11:41,440 --> 01:11:43,910
then J is equal to 0.
973
01:11:43,910 --> 01:11:46,760
1/2, 1, 3/2.
974
01:11:46,760 --> 01:11:51,620
And you get all these spins
with-- consider particles
975
01:11:51,620 --> 01:11:53,790
without spin having spin 0.
976
01:11:53,790 --> 01:11:55,720
Particles with spin 1/2.
977
01:11:55,720 --> 01:12:00,110
Particles of spin 1,
or angular momentum 1,
978
01:12:00,110 --> 01:12:03,690
orbital angular momentum 1.
979
01:12:03,690 --> 01:12:07,630
And both these things
have a reason for you.
980
01:12:07,630 --> 01:12:13,550
Now if you have 2J being
an integer, the values of m
981
01:12:13,550 --> 01:12:19,310
go from J to J minus
1, up to minus J.
982
01:12:19,310 --> 01:12:21,730
And there are two
J plus 1 values.
983
01:12:27,110 --> 01:12:29,630
And in fact, that
is the main result
984
01:12:29,630 --> 01:12:33,080
of the theory of
angular momentum.
985
01:12:33,080 --> 01:12:37,500
The values of the angular
momentum are 0, 1, 1/2, 3/2.
986
01:12:37,500 --> 01:12:42,500
So for J equals 0,
there's just one state.
987
01:12:42,500 --> 01:12:46,090
m is equal to 0.
988
01:12:46,090 --> 01:12:49,750
For J equals to 1,
there's two states.
989
01:12:49,750 --> 01:12:52,710
I'm sorry for 1/2, two states.
990
01:12:52,710 --> 01:12:55,490
One with m equals 1/2.
991
01:12:55,490 --> 01:12:58,990
And m equals minus 1/2.
992
01:12:58,990 --> 01:13:01,890
J equals 1, there's
three states.
993
01:13:01,890 --> 01:13:05,570
M equals 1, 0, and minus 1.
994
01:13:05,570 --> 01:13:06,995
And so on.
995
01:13:10,320 --> 01:13:12,440
OK.
996
01:13:12,440 --> 01:13:13,880
This is a great result.
997
01:13:13,880 --> 01:13:19,360
Let me give you an application
in the last 10 minutes.
998
01:13:19,360 --> 01:13:22,690
It's a remarkable application.
999
01:13:22,690 --> 01:13:27,260
Now actually, you
would say, so what
1000
01:13:27,260 --> 01:13:31,740
do you get-- what vector
space were we talking about?
1001
01:13:31,740 --> 01:13:34,770
And what's sort of
the punchline here
1002
01:13:34,770 --> 01:13:38,802
is that the vector space
was infinite dimensional
1003
01:13:38,802 --> 01:13:45,050
and it breaks down into
states with J equals 0.
1004
01:13:45,050 --> 01:13:47,120
States was J equal 1/2.
1005
01:13:47,120 --> 01:13:48,660
States with J equal 1.
1006
01:13:48,660 --> 01:13:51,580
States with J equal 3/2.
1007
01:13:51,580 --> 01:13:53,650
All these things
are possibilities.
1008
01:13:53,650 --> 01:13:56,750
They can all be present
in your vector space.
1009
01:13:56,750 --> 01:13:58,420
Maybe some are present.
1010
01:13:58,420 --> 01:13:59,560
Some are not.
1011
01:13:59,560 --> 01:14:03,560
That is part of figuring
out what's going on.
1012
01:14:03,560 --> 01:14:08,980
When we do central
potentials, 0, 1, 2, 4
1013
01:14:08,980 --> 01:14:13,270
will be present for the
angular momentum theory.
1014
01:14:13,270 --> 01:14:16,430
When we do spins, we have 1/2.
1015
01:14:16,430 --> 01:14:20,260
And when we do other things,
we can get some funny things
1016
01:14:20,260 --> 01:14:21,550
as well.
1017
01:14:21,550 --> 01:14:25,170
So let's do a case where
you get something funny.
1018
01:14:25,170 --> 01:14:27,390
So the 2D, SHO.
1019
01:14:31,240 --> 01:14:41,790
You have ax's, and ay's, and
a daggers, and ay daggers.
1020
01:14:41,790 --> 01:14:45,100
And this should
seem very strange.
1021
01:14:45,100 --> 01:14:48,710
What are we talking about
2 dimensional oscillators
1022
01:14:48,710 --> 01:14:51,410
after talking about
3 dimensional angular
1023
01:14:51,410 --> 01:14:53,730
momentum and all that?
1024
01:14:53,730 --> 01:14:55,390
Doesn't make any sense.
1025
01:14:55,390 --> 01:15:00,160
Well, what's going to happen
now is something more magical
1026
01:15:00,160 --> 01:15:04,090
than when a magician takes
a bunny out of a hat.
1027
01:15:04,090 --> 01:15:08,780
Out of this problem, an angular
momentum, a 3 dimensional
1028
01:15:08,780 --> 01:15:12,060
angular momentum,
is going to pop out.
1029
01:15:12,060 --> 01:15:17,210
No reason whatsoever there
should be there at first sight.
1030
01:15:17,210 --> 01:15:18,210
But it's there.
1031
01:15:18,210 --> 01:15:21,140
And it's an abstract
angular momentum,
1032
01:15:21,140 --> 01:15:23,760
but it's a full
angular momentum.
1033
01:15:23,760 --> 01:15:24,570
Let's see.
1034
01:15:29,450 --> 01:15:31,670
Let's look at the spectrum.
1035
01:15:31,670 --> 01:15:32,870
Ground state.
1036
01:15:32,870 --> 01:15:36,420
First excited
state is isotropic.
1037
01:15:36,420 --> 01:15:43,010
So 2 states
degenerate in energy.
1038
01:15:43,010 --> 01:15:44,080
Next state.
1039
01:15:44,080 --> 01:15:48,110
ax dagger, ax dagger.
1040
01:15:48,110 --> 01:15:51,210
ax, ay.
1041
01:15:51,210 --> 01:15:53,924
ay, ay.
1042
01:15:53,924 --> 01:15:57,620
3 states, degenerate.
1043
01:15:57,620 --> 01:16:04,780
Go up to ax dagger
to the n, up to ax--
1044
01:16:04,780 --> 01:16:07,570
no ax, or ax dagger to the 0.
1045
01:16:07,570 --> 01:16:10,075
And ay dagger to the n.
1046
01:16:12,880 --> 01:16:18,880
And that's n a daggers up to 0
a daggers, so n plus 1 states.
1047
01:16:22,150 --> 01:16:27,440
3 states, 2 states, 1 state.
1048
01:16:27,440 --> 01:16:32,650
And you'll come here
and say, that's strange.
1049
01:16:32,650 --> 01:16:37,042
1 state, 2 states,
3 states, 4 states.
1050
01:16:37,042 --> 01:16:41,790
Does that have
anything to do with it?
1051
01:16:41,790 --> 01:16:44,370
Well, the surprise is it
has something to do with it.
1052
01:16:44,370 --> 01:16:45,900
Let's think about it.
1053
01:16:49,290 --> 01:16:56,470
Well, first thing is to put
these aR's and aL oscillators--
1054
01:16:56,470 --> 01:17:02,650
these were 1/2, 1 over square
root of 2, ax plus iay.
1055
01:17:05,330 --> 01:17:12,960
And a left was 1 over square
root of 2, ax minus iay.
1056
01:17:12,960 --> 01:17:16,940
I may have-- no,
the signs are wrong.
1057
01:17:16,940 --> 01:17:20,500
Plus and minus.
1058
01:17:20,500 --> 01:17:22,730
And we had number operators.
1059
01:17:22,730 --> 01:17:26,960
n right, which were a
right dagger, a right.
1060
01:17:26,960 --> 01:17:33,798
And n left, which was
a left dagger, a left.
1061
01:17:33,798 --> 01:17:37,350
And they don't mix a
lefts and a rights.
1062
01:17:37,350 --> 01:17:42,890
And now, we could build a
state the following way.
1063
01:17:42,890 --> 01:17:44,720
0.
1064
01:17:44,720 --> 01:17:50,900
a right dagger on 0.
a left dagger on 0.
1065
01:17:50,900 --> 01:17:55,580
A right dagger squared on 0.
1066
01:17:55,580 --> 01:17:59,730
a right, a left on 0.
1067
01:17:59,730 --> 01:18:04,764
and a left dagger,
a left dagger on 0.
1068
01:18:04,764 --> 01:18:10,060
Up to a right dagger
to the n on 0.
1069
01:18:10,060 --> 01:18:18,840
Up to a left dagger
to the n on 0.
1070
01:18:18,840 --> 01:18:21,820
And this is completely
analogous to what we had.
1071
01:18:24,990 --> 01:18:30,540
Now here comes the real thing.
1072
01:18:30,540 --> 01:18:35,440
You did compute the angular
momentum in the z direction.
1073
01:18:41,140 --> 01:18:45,360
And the angular momentum
in the z direction was Lz.
1074
01:18:45,360 --> 01:18:50,330
And you could compute
this. xpy minus ypx.
1075
01:18:50,330 --> 01:18:52,670
And this was all legal.
1076
01:18:52,670 --> 01:19:00,763
And the answer was h
bar, N right, minus NL.
1077
01:19:04,630 --> 01:19:09,320
That was the Lz component
of angular momentum.
1078
01:19:09,320 --> 01:19:15,730
So, let's see what
Lz's those states have.
1079
01:19:15,730 --> 01:19:22,660
This one has no n rights, or
n lefts, so has Lz equals 0.
1080
01:19:22,660 --> 01:19:28,270
This state has Nz equal h bar.
1081
01:19:28,270 --> 01:19:32,580
And this has minus h bar.
1082
01:19:32,580 --> 01:19:33,280
OK.
1083
01:19:33,280 --> 01:19:35,450
h bar and minus h bar.
1084
01:19:35,450 --> 01:19:39,330
That doesn't quite
seem to fit here,
1085
01:19:39,330 --> 01:19:42,750
because the z component
of angular momentum
1086
01:19:42,750 --> 01:19:46,060
is 1/2 of h bar, and
minus 1/2 of h bar.
1087
01:19:46,060 --> 01:19:48,950
That's-- something went wrong.
1088
01:19:48,950 --> 01:19:50,770
OK.
1089
01:19:50,770 --> 01:19:51,810
You go here.
1090
01:19:51,810 --> 01:19:54,796
You say, well, what is Lz?
1091
01:19:54,796 --> 01:19:59,990
Lz here was h bar, minus h bar.
1092
01:19:59,990 --> 01:20:06,140
Here is 2h bar, 0,
and minus 2h bar.
1093
01:20:06,140 --> 01:20:11,210
And you look there, and say, no,
that's not quite right either.
1094
01:20:11,210 --> 01:20:14,990
This-- if you would
say these 3 states
1095
01:20:14,990 --> 01:20:18,350
should correspond
to angular momentum,
1096
01:20:18,350 --> 01:20:24,160
they should have m equal plus 1,
plus h bar, 0, and minus h bar.
1097
01:20:24,160 --> 01:20:25,490
So it's not right.
1098
01:20:28,680 --> 01:20:29,400
OK.
1099
01:20:29,400 --> 01:20:35,180
Well one other thing maybe
we can make sense of this.
1100
01:20:35,180 --> 01:20:41,640
If we had L plus, should
be the kind of thing
1101
01:20:41,640 --> 01:20:44,640
that you can't annihilate.
1102
01:20:44,640 --> 01:20:46,670
That you annihilate
the top state.
1103
01:20:46,670 --> 01:20:51,840
Remember L plus, or J
plus, kept increasing
1104
01:20:51,840 --> 01:20:54,885
so it should annihilate
the top state.
1105
01:20:54,885 --> 01:20:57,510
And I could try to
devise something
1106
01:20:57,510 --> 01:20:59,780
that annihilates the top state.
1107
01:20:59,780 --> 01:21:04,860
And it would be something
like aR dagger, a left.
1108
01:21:04,860 --> 01:21:05,980
Why?
1109
01:21:05,980 --> 01:21:12,460
Because if aR dagger, a
left, goes to the top state,
1110
01:21:12,460 --> 01:21:15,730
the top state has
no a left daggers,
1111
01:21:15,730 --> 01:21:20,330
so the a left just zooms in,
and hits the 0 and kills it.
1112
01:21:20,330 --> 01:21:21,450
Kills it here.
1113
01:21:21,450 --> 01:21:27,020
So actually I do have
something like an L plus.
1114
01:21:27,020 --> 01:21:30,080
And I would have the
dagger-- would be something
1115
01:21:30,080 --> 01:21:34,730
like an L minus-- would
be aL dagger, a right.
1116
01:21:34,730 --> 01:21:37,630
And this one should
annihilate the bottom one.
1117
01:21:37,630 --> 01:21:38,820
And it does.
1118
01:21:38,820 --> 01:21:41,692
Because the bottom
state has no aR's,
1119
01:21:41,692 --> 01:21:45,160
and therefore has no aR daggers.
1120
01:21:45,160 --> 01:21:48,350
And therefore, the aR comes
there, and hits the state,
1121
01:21:48,350 --> 01:21:49,820
and kills it.
1122
01:21:49,820 --> 01:21:52,930
So we seem to have more
or less everything,
1123
01:21:52,930 --> 01:21:56,370
but nothing is working.
1124
01:21:56,370 --> 01:21:59,500
So we have to do a
last conceptual step.
1125
01:21:59,500 --> 01:22:05,150
And say-- you see, this
is moving in a plane.
1126
01:22:05,150 --> 01:22:08,600
There's no 3 dimensional
angular momentum.
1127
01:22:08,600 --> 01:22:12,470
You are fooling
yourself with this.
1128
01:22:12,470 --> 01:22:17,730
But what could exist is an
abstract angular momentum.
1129
01:22:17,730 --> 01:22:22,350
And for that, in
order to-- it's time
1130
01:22:22,350 --> 01:22:26,920
to change the
letter from L to J.
1131
01:22:26,920 --> 01:22:30,770
That means some kind of
abstract angular momentum.
1132
01:22:30,770 --> 01:22:36,020
And I'll put a 1/2
here, now a definition.
1133
01:22:36,020 --> 01:22:41,670
If this is what I called Jz,
oh well, then thing's may
1134
01:22:41,670 --> 01:22:42,800
look good.
1135
01:22:42,800 --> 01:22:52,570
Because this one for Jz has now
angular momentum 1/2 of h bar,
1136
01:22:52,570 --> 01:22:55,480
and minus a half of h bar.
1137
01:22:55,480 --> 01:23:01,940
And that fits with
this, these 2 states.
1138
01:23:01,940 --> 01:23:07,180
And with the 1/2, the
other ones, the Jz's, also
1139
01:23:07,180 --> 01:23:09,310
have something here.
1140
01:23:09,310 --> 01:23:15,220
So Jz here now becomes
h bar, minus h bar,
1141
01:23:15,220 --> 01:23:16,225
and it looks right.
1142
01:23:19,030 --> 01:23:22,880
And now you put the
1/2 here, and in fact,
1143
01:23:22,880 --> 01:23:26,360
if you tried to
make these things
1144
01:23:26,360 --> 01:23:29,230
J-- call it J plus and J minus.
1145
01:23:29,230 --> 01:23:32,920
Now you put a number
here, and a number here.
1146
01:23:32,920 --> 01:23:36,160
If you would have
put a number here,
1147
01:23:36,160 --> 01:23:39,470
if you try to enforce
that the algebra be
1148
01:23:39,470 --> 01:23:42,850
the algebra of angular
momentum, the number
1149
01:23:42,850 --> 01:23:45,380
would have come out to be 1/2.
1150
01:23:45,380 --> 01:23:50,040
But now we claim that in this
2 dimensional oscillator,
1151
01:23:50,040 --> 01:23:52,660
there is-- because
there's a number here
1152
01:23:52,660 --> 01:23:54,370
that works with this 1/2.
1153
01:23:54,370 --> 01:23:56,900
Something you have to calculate.
1154
01:23:56,900 --> 01:24:03,570
And with this number, you
have some sort of Jx, Jy, Jz,
1155
01:24:03,570 --> 01:24:07,260
where this is like 1/2 of Lz.
1156
01:24:07,260 --> 01:24:11,310
And those have come
out of thin air.
1157
01:24:11,310 --> 01:24:14,590
But they form an algebra
of angular momentum.
1158
01:24:14,590 --> 01:24:16,640
And what have we
learned today, if you
1159
01:24:16,640 --> 01:24:18,960
have an algebra of
angular momentum,
1160
01:24:18,960 --> 01:24:23,600
the states must
organize themselves
1161
01:24:23,600 --> 01:24:28,040
into representations
of angular momentum.
1162
01:24:28,040 --> 01:24:34,910
So the whole spectrum of the 2
dimensional harmonic oscillator
1163
01:24:34,910 --> 01:24:38,680
has in fact all spin
representations.
1164
01:24:38,680 --> 01:24:40,530
J equals 0.
1165
01:24:40,530 --> 01:24:42,640
J equals 1/2.
1166
01:24:42,640 --> 01:24:44,400
J equals 1.
1167
01:24:44,400 --> 01:24:46,690
J equals 2.
1168
01:24:46,690 --> 01:24:49,510
J equals n, and all of them.
1169
01:24:49,510 --> 01:24:54,500
So the best example of
all the representations
1170
01:24:54,500 --> 01:24:57,370
of angular momentum
are in the states
1171
01:24:57,370 --> 01:25:00,960
of the 2 dimensional
simple harmonic oscillator.
1172
01:25:00,960 --> 01:25:05,270
It's an abstract angular
momentum, but it's very useful.
1173
01:25:05,270 --> 01:25:09,360
The one step I didn't do
here for you is to check.
1174
01:25:09,360 --> 01:25:13,560
Although you check that
all of these Ji commute
1175
01:25:13,560 --> 01:25:15,920
with the Hamiltonian.
1176
01:25:15,920 --> 01:25:18,360
Simple calculation to do it.
1177
01:25:18,360 --> 01:25:21,620
In fact, the Hamiltonian
is NL plus N right,
1178
01:25:21,620 --> 01:25:23,130
and you can check it.
1179
01:25:23,130 --> 01:25:27,470
Since they commute with them,
these operators act in states
1180
01:25:27,470 --> 01:25:29,600
and don't change the energy.
1181
01:25:29,600 --> 01:25:32,180
And they're a symmetry
of the problem.
1182
01:25:32,180 --> 01:25:35,270
So that's why they fell
into representations.
1183
01:25:35,270 --> 01:25:39,540
So this is our first example
of a hidden symmetry.
1184
01:25:39,540 --> 01:25:42,890
A problem that there
was no reason a priori
1185
01:25:42,890 --> 01:25:47,090
to expect an angular
momentum to exist,
1186
01:25:47,090 --> 01:25:51,080
but it's there, and helps
explain the degeneracies.
1187
01:25:51,080 --> 01:25:53,880
These degeneracies you could
have said they're accidental.
1188
01:25:53,880 --> 01:25:56,180
But by the time
you know they have
1189
01:25:56,180 --> 01:25:59,760
to fall into angular
momentum representations,
1190
01:25:59,760 --> 01:26:02,180
you have great
control over them.
1191
01:26:02,180 --> 01:26:05,100
You couldn't have
found different number
1192
01:26:05,100 --> 01:26:08,090
of degenerate states
at any level here.
1193
01:26:08,090 --> 01:26:11,270
This was in fact discovered
by Julian Schwinger
1194
01:26:11,270 --> 01:26:13,535
in a very famous paper.
1195
01:26:13,535 --> 01:26:16,910
And is a classic example
of angular momentum.
1196
01:26:16,910 --> 01:26:17,500
All right.
1197
01:26:17,500 --> 01:26:18,650
That's it for today.
1198
01:26:18,650 --> 01:26:25,580
See you on Wednesday if
you come I'll be here.