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PROFESSOR: Today we
have plenty to do.
9
00:00:23,770 --> 00:00:29,780
We really begin
in all generality
10
00:00:29,780 --> 00:00:32,580
the addition of
angular momentum.
11
00:00:32,580 --> 00:00:37,970
But we will do it in the set
up of a physical problem.
12
00:00:37,970 --> 00:00:40,020
The problem of
computing the spin
13
00:00:40,020 --> 00:00:46,960
orbit interactions of
electrons with the nucleus.
14
00:00:46,960 --> 00:00:51,940
So this is a rather interesting
and complicated interaction.
15
00:00:51,940 --> 00:00:54,610
So we'll spend a
little time telling you
16
00:00:54,610 --> 00:00:57,520
about the physics
of this interaction.
17
00:00:57,520 --> 00:01:01,230
And then once the
physics is clear,
18
00:01:01,230 --> 00:01:04,720
it will become
more obvious why we
19
00:01:04,720 --> 00:01:08,980
have to do these mathematical
contortions of adding angular
20
00:01:08,980 --> 00:01:13,960
momentum in order to solve
this physical problem.
21
00:01:13,960 --> 00:01:17,790
So it's a sophisticated problem
that requires several steps.
22
00:01:17,790 --> 00:01:22,420
The first step is
something that is
23
00:01:22,420 --> 00:01:26,480
a result in perturbation theory.
24
00:01:26,480 --> 00:01:30,460
Feynman Hellman Theorem
of perturbation theory.
25
00:01:30,460 --> 00:01:32,270
And that's where we begin.
26
00:01:32,270 --> 00:01:35,770
So it's called Feynman
Hellman Theorem.
27
00:01:35,770 --> 00:01:41,900
It's a very simple result.
28
00:01:46,605 --> 00:01:47,105
Theorem.
29
00:01:50,580 --> 00:01:54,530
And we'll need it in
order to understand
30
00:01:54,530 --> 00:01:57,990
how a small perturbation
to the Hamiltonian
31
00:01:57,990 --> 00:02:00,240
changes the energy spectrum.
32
00:02:00,240 --> 00:02:12,110
So we have H of lambda be a
Hamiltonian with a parameter
33
00:02:12,110 --> 00:02:13,252
in lambda.
34
00:02:17,190 --> 00:02:19,540
Lambda.
35
00:02:19,540 --> 00:02:30,640
And psi n of lambda
being normalized energy
36
00:02:30,640 --> 00:02:40,550
eigenstate with
energy, En of lambda.
37
00:02:44,460 --> 00:02:48,890
So that's the whole
assumption of the theorem.
38
00:02:48,890 --> 00:02:51,130
We have a Hamiltonian.
39
00:02:51,130 --> 00:02:55,080
It depends on some parameter
that we're going to vary.
40
00:02:55,080 --> 00:03:01,530
And suppose we consider now an
eigenstate of this Hamiltonian
41
00:03:01,530 --> 00:03:04,140
that depends on lambda,
so the eigenstate also
42
00:03:04,140 --> 00:03:06,050
depends on lambda.
43
00:03:06,050 --> 00:03:09,770
And it has an
energy, En of lambda.
44
00:03:09,770 --> 00:03:11,860
So the purpose of
this theorem is
45
00:03:11,860 --> 00:03:15,130
to relate these
various quantities.
46
00:03:15,130 --> 00:03:22,530
And the claim is that
the rate of change
47
00:03:22,530 --> 00:03:26,810
of the energy with
respect to lambda
48
00:03:26,810 --> 00:03:34,900
can be computed pretty much by
evaluating the rate of change
49
00:03:34,900 --> 00:03:38,140
of the Hamiltonian on
the relevant states.
50
00:03:50,740 --> 00:03:51,660
So that's the claim.
51
00:04:01,010 --> 00:04:05,740
And it's a pretty nice result.
52
00:04:05,740 --> 00:04:08,940
It's useful in
many circumstances.
53
00:04:08,940 --> 00:04:13,630
And for us will be a way to
discuss a little perturbation
54
00:04:13,630 --> 00:04:14,470
theory.
55
00:04:14,470 --> 00:04:19,240
Perturbation theory is the
subject of 806 in all details.
56
00:04:19,240 --> 00:04:21,970
And it's a very
sophisticated subject.
57
00:04:21,970 --> 00:04:25,970
Even today we were going
to be finding that it's not
58
00:04:25,970 --> 00:04:30,140
all that easy to carry it out.
59
00:04:30,140 --> 00:04:32,100
So how does this begin?
60
00:04:32,100 --> 00:04:32,890
Well, proof.
61
00:04:36,290 --> 00:04:39,410
You begin by saying
that En of lambda
62
00:04:39,410 --> 00:04:42,860
is the energy eigenstate,
is nothing else
63
00:04:42,860 --> 00:04:47,200
but psi n of lambda.
64
00:04:47,200 --> 00:04:49,950
H of lambda.
65
00:04:49,950 --> 00:04:51,810
Psi n of lambda.
66
00:04:55,730 --> 00:05:01,860
And the reason is, of
course, that H and psi n
67
00:05:01,860 --> 00:05:07,400
is En of lambda times
psi n of lambda.
68
00:05:07,400 --> 00:05:09,385
And this is a number goes out.
69
00:05:09,385 --> 00:05:12,400
And the inner product
of this things
70
00:05:12,400 --> 00:05:16,140
is 1, because the
state is normalized.
71
00:05:16,140 --> 00:05:18,980
So this is a good
starting point.
72
00:05:18,980 --> 00:05:24,780
And the funny thing that you see
already is that, in some sense,
73
00:05:24,780 --> 00:05:27,840
you just get the
middle term when
74
00:05:27,840 --> 00:05:30,290
you take the derivative
with respect to lambda.
75
00:05:30,290 --> 00:05:32,320
You don't get anything
from these two.
76
00:05:34,870 --> 00:05:36,540
And it's simple in fact.
77
00:05:36,540 --> 00:05:38,310
So let me just do it.
78
00:05:38,310 --> 00:05:43,880
V, En, V lambda
would be the term
79
00:05:43,880 --> 00:05:49,140
that Feynman and Hellman gave.
80
00:05:49,140 --> 00:05:55,400
V, H, V lambda, psi n of lambda.
81
00:05:55,400 --> 00:05:58,470
Plus one term in which we
differentiate this one.
82
00:05:58,470 --> 00:06:05,500
V, d lambda of the
state psi n of lambda.
83
00:06:09,340 --> 00:06:17,130
Times H of lambda,
psi n of lambda.
84
00:06:17,130 --> 00:06:21,380
Plus the other term in which
you differentiate the ket.
85
00:06:21,380 --> 00:06:32,610
So psi n of lambda, H of lambda,
d, d lambda of psi n of lambda.
86
00:06:38,880 --> 00:06:46,710
And the reason these
terms are going to vanish
87
00:06:46,710 --> 00:06:53,020
is that you can now act
with H again on psi n.
88
00:06:53,020 --> 00:06:57,620
H is supposed to be Hermitian,
so it can act on the left.
89
00:06:57,620 --> 00:07:07,220
And therefore, these two
terms give you En of lambda,
90
00:07:07,220 --> 00:07:19,400
times d d lambda of psi n of
lambda-- psi n of lambda--
91
00:07:19,400 --> 00:07:23,750
plus the other term, which
would be psi n of lambda,
92
00:07:23,750 --> 00:07:27,395
the bra times the
derivative of the ket.
93
00:07:35,450 --> 00:07:43,550
But this is nothing
else than the derivative
94
00:07:43,550 --> 00:07:44,730
of the inner product.
95
00:07:55,480 --> 00:07:57,250
In the inner product
to differentiate--
96
00:07:57,250 --> 00:08:00,020
the inner product
differentiates the bra,
97
00:08:00,020 --> 00:08:01,920
it differentiates the ket.
98
00:08:01,920 --> 00:08:04,280
And do it.
99
00:08:04,280 --> 00:08:08,220
And this thing is equal to
1, because it's normalized.
100
00:08:08,220 --> 00:08:10,890
So this is 0.
101
00:08:10,890 --> 00:08:12,650
End of proof.
102
00:08:12,650 --> 00:08:16,320
These two terms vanish.
103
00:08:16,320 --> 00:08:20,860
And the result holds.
104
00:08:20,860 --> 00:08:21,652
Yes?
105
00:08:21,652 --> 00:08:23,527
AUDIENCE: How do you
know it stays normalized
106
00:08:23,527 --> 00:08:24,505
when you vary lambda?
107
00:08:24,505 --> 00:08:28,440
PROFESSOR: It's an assumption.
108
00:08:28,440 --> 00:08:31,770
The state is normalized
for all values of n.
109
00:08:31,770 --> 00:08:34,925
So if you have a state
that you've constructed,
110
00:08:34,925 --> 00:08:39,669
that is normalized, you
can have this result.
111
00:08:39,669 --> 00:08:41,030
So it's an assumption.
112
00:08:41,030 --> 00:08:42,845
You have to keep the
state normalized.
113
00:08:46,900 --> 00:08:50,900
Now this is a baby version
of perturbation theory.
114
00:08:50,900 --> 00:08:55,320
It's a result I think that
Feynman did as an undergrad.
115
00:08:55,320 --> 00:08:59,290
And as you can see,
it's very simple.
116
00:08:59,290 --> 00:09:03,290
Calling it a theorem
is a little too much.
117
00:09:03,290 --> 00:09:06,540
But still, the fact
is that it's useful.
118
00:09:06,540 --> 00:09:10,670
And so we'll just
go ahead and use it.
119
00:09:10,670 --> 00:09:14,910
Now I want to rewrite
it in another way.
120
00:09:14,910 --> 00:09:19,250
So, suppose you have
a Hamiltonian, H,
121
00:09:19,250 --> 00:09:24,755
which has a term
H0, plus lambda, H1.
122
00:09:27,380 --> 00:09:41,770
So, the parameter lambda, H of
lambda, is given in this way.
123
00:09:41,770 --> 00:09:45,180
And that's a
reasonable H of lambda.
124
00:09:45,180 --> 00:09:49,860
Sometimes, this could be
written as H0 plus something
125
00:09:49,860 --> 00:09:52,670
that we will call the
change in the Hamiltonian.
126
00:09:52,670 --> 00:09:57,540
And we usually think
of it as a small thing.
127
00:10:01,300 --> 00:10:05,190
So what do we have
from this theorem?
128
00:10:08,410 --> 00:10:18,020
From this here we would
have the d, En, d lambda
129
00:10:18,020 --> 00:10:29,380
is equal to psi n of
lambda, H1, psi n of lambda.
130
00:10:34,100 --> 00:10:37,400
Now, we can be
particularly interested
131
00:10:37,400 --> 00:10:41,250
in the evaluation of this
thing at lambda equals 0.
132
00:10:41,250 --> 00:10:44,830
So what is d En of lambda?
133
00:10:44,830 --> 00:10:55,330
d lambda at lambda equals
0 would be psi n at zero,
134
00:10:55,330 --> 00:11:00,970
H1, psi n at 0.
135
00:11:00,970 --> 00:11:07,870
And therefore, you would say
that the En of lambda energies
136
00:11:07,870 --> 00:11:19,790
would be the energies
at 0, plus lambda, d En
137
00:11:19,790 --> 00:11:25,510
of lambda, d lambda
at lambda equals 0,
138
00:11:25,510 --> 00:11:28,093
plus order lambda squared.
139
00:11:35,610 --> 00:11:38,740
I'm doing just the
Taylor expansion
140
00:11:38,740 --> 00:11:46,450
of En's of lambda
from lambda equals 0.
141
00:11:46,450 --> 00:11:56,480
So this thing tells you that
En of lambda is equal En of 0,
142
00:11:56,480 --> 00:12:08,302
plus-- this derivative you can
write it as psi n, lambda, H1,
143
00:12:08,302 --> 00:12:13,008
psi n, all at 0.
144
00:12:13,008 --> 00:12:13,980
Like that.
145
00:12:13,980 --> 00:12:16,955
Plus order lambda squared.
146
00:12:23,650 --> 00:12:28,600
So in this step, I just use the
evaluation that we did here.
147
00:12:28,600 --> 00:12:32,380
I substituted that
and put the lambda in.
148
00:12:32,380 --> 00:12:45,470
So that I recognize now that En
of lambda is equal to En of 0,
149
00:12:45,470 --> 00:12:53,990
plus psi n of 0-- and I can
write this as delta H-- psi
150
00:12:53,990 --> 00:13:00,803
n of 0, plus order
delta H squared.
151
00:13:09,260 --> 00:13:12,780
It's nice to write it this
way, because you appreciate
152
00:13:12,780 --> 00:13:15,540
more the power of the theorem.
153
00:13:15,540 --> 00:13:20,640
The theorem here doesn't assume
which value of lambda you have.
154
00:13:20,640 --> 00:13:23,070
And you have to have
normalized eigenstates.
155
00:13:23,070 --> 00:13:26,320
And you wonder what is
it helping you with,
156
00:13:26,320 --> 00:13:30,820
if finding the states for every
value of lambda is complicated.
157
00:13:30,820 --> 00:13:37,160
Well, it certainly
helps you to figure out
158
00:13:37,160 --> 00:13:42,960
how the energy of the state
varies by a simple calculation.
159
00:13:42,960 --> 00:13:46,725
Suppose you know the states
of the simple Hamiltonian.
160
00:13:49,980 --> 00:13:55,630
Those are the psi's, n, 0.
161
00:13:55,630 --> 00:13:59,870
So if you have the
psi n 0 over here,
162
00:13:59,870 --> 00:14:03,120
you can do the following step.
163
00:14:03,120 --> 00:14:07,930
If you want to figure out
how it's energy has varied,
164
00:14:07,930 --> 00:14:14,260
use this formula in which
you compute the expectation
165
00:14:14,260 --> 00:14:20,370
value of the change in the
Hamiltonian on that state.
166
00:14:20,370 --> 00:14:26,310
And that is the first correction
to the energy of the state.
167
00:14:26,310 --> 00:14:29,010
So you have this state.
168
00:14:29,010 --> 00:14:31,710
You compute the
expectation value
169
00:14:31,710 --> 00:14:34,880
of the extra piece
in the Hamiltonian.
170
00:14:34,880 --> 00:14:39,270
And that's the
correction to the energy.
171
00:14:39,270 --> 00:14:41,300
It's a little more
complicated of course
172
00:14:41,300 --> 00:14:43,600
to compute the
correction to the state.
173
00:14:43,600 --> 00:14:47,570
But that's a subject
of perturbation theory.
174
00:14:47,570 --> 00:14:51,790
And that's not what we
care about right now.
175
00:14:51,790 --> 00:14:57,410
So the reason we're doing this
is because actually whatever
176
00:14:57,410 --> 00:15:01,010
we're going to have
with spin orbit coupling
177
00:15:01,010 --> 00:15:04,100
represents an addition
to the hydrogen
178
00:15:04,100 --> 00:15:06,680
Hamiltonian of a new term.
179
00:15:06,680 --> 00:15:10,060
Therefore, you want to know what
happens to the energy levels.
180
00:15:10,060 --> 00:15:11,970
And the best thing
to think about them
181
00:15:11,970 --> 00:15:15,360
is to-- if you know the
energy levels of this one,
182
00:15:15,360 --> 00:15:17,980
well, a formula of
this type can let
183
00:15:17,980 --> 00:15:20,082
you know what happens
to the energy levels
184
00:15:20,082 --> 00:15:21,040
after the perturbation.
185
00:15:23,850 --> 00:15:25,790
There will be an
extra complication
186
00:15:25,790 --> 00:15:28,500
in that the energy levels
that we're going to deal with
187
00:15:28,500 --> 00:15:29,930
are going to be degenerate.
188
00:15:29,930 --> 00:15:32,980
But let's wait for that
complication until it appears.
189
00:15:32,980 --> 00:15:34,650
So any questions?
190
00:15:37,960 --> 00:15:38,870
Yes?
191
00:15:38,870 --> 00:15:40,820
AUDIENCE: So I would
imagine that this
192
00:15:40,820 --> 00:15:42,704
would work just
as well for time.
193
00:15:42,704 --> 00:15:45,974
Because time [INAUDIBLE] a
parameter in quantum mechanics.
194
00:15:45,974 --> 00:15:48,404
So [INAUDIBLE]
195
00:15:48,404 --> 00:15:50,830
PROFESSOR: Time dependent
perturbation theory
196
00:15:50,830 --> 00:15:52,930
is a bit more complicated.
197
00:15:52,930 --> 00:15:56,860
I'd rather not get into it now.
198
00:15:56,860 --> 00:16:01,450
So let's leave it here, in
which we don't have time.
199
00:16:01,450 --> 00:16:03,280
And the Schrodinger
equation is something
200
00:16:03,280 --> 00:16:07,617
like H psi equal [INAUDIBLE]
psi, that's all we care.
201
00:16:07,617 --> 00:16:11,250
And leave it for that moment.
202
00:16:11,250 --> 00:16:11,933
Other questions?
203
00:16:21,496 --> 00:16:22,460
OK.
204
00:16:22,460 --> 00:16:28,530
So let's proceed with
addition of angular momentum.
205
00:16:28,530 --> 00:16:34,160
So first, let me give you
the fundamental result
206
00:16:34,160 --> 00:16:36,010
of addition of angular momentum.
207
00:16:36,010 --> 00:16:39,880
It's a little abstract,
but it's what we really
208
00:16:39,880 --> 00:16:41,725
mean by addition of
angular momentum.
209
00:16:45,197 --> 00:16:48,763
Of angular momentum.
210
00:16:51,520 --> 00:16:53,960
And the main result
is the following.
211
00:16:53,960 --> 00:17:00,450
Suppose you have a set
of operators, J, i,
212
00:17:00,450 --> 00:17:04,929
1, that have the algebra
of angular momentum.
213
00:17:08,200 --> 00:17:11,250
Of angular momentum.
214
00:17:11,250 --> 00:17:21,609
Which is to say
Ji1, JJ1, is equal
215
00:17:21,609 --> 00:17:25,712
i, h bar, epsilon iJK, JK1.
216
00:17:29,870 --> 00:17:34,180
And this algebra is realized
on some state space.
217
00:17:34,180 --> 00:17:41,330
On some vector space, V1.
218
00:17:41,330 --> 00:17:44,760
And suppose you have
another operator,
219
00:17:44,760 --> 00:17:48,180
J-- set of operators actually.
220
00:17:48,180 --> 00:17:51,395
Ji2, which have the algebra
of angular momentum.
221
00:17:51,395 --> 00:17:53,470
I will not write that.
222
00:17:53,470 --> 00:17:56,280
On some V2.
223
00:17:59,530 --> 00:18:00,640
OK.
224
00:18:00,640 --> 00:18:03,340
Angular momentum,
some sets of states.
225
00:18:03,340 --> 00:18:07,570
Angular momentum on some
other set of states.
226
00:18:07,570 --> 00:18:10,090
Here comes the thing.
227
00:18:10,090 --> 00:18:13,620
There is a new angular
momentum, which
228
00:18:13,620 --> 00:18:24,970
is the sum Ji defined
as Ji1, added with Ji2.
229
00:18:24,970 --> 00:18:30,560
Now, soon enough you will
just write Ji1, plus Ji2.
230
00:18:30,560 --> 00:18:34,590
But let me be a little
more careful now.
231
00:18:34,590 --> 00:18:45,290
This sum is Ji1,
plus 1, tensor Ji2.
232
00:18:45,290 --> 00:18:48,080
So i is the same index.
233
00:18:48,080 --> 00:18:53,230
But here, we're
having this operator
234
00:18:53,230 --> 00:18:57,140
that we're being defined
that we call it the sum.
235
00:18:57,140 --> 00:19:01,730
Now how do you sum two operators
that act in different spaces?
236
00:19:01,730 --> 00:19:04,680
Well, the only thing
that you can actually do
237
00:19:04,680 --> 00:19:07,120
is sum them in the
tensor product.
238
00:19:07,120 --> 00:19:19,136
So the claim is that this is an
angular momentum in V1 tensor
239
00:19:19,136 --> 00:19:19,635
V2.
240
00:19:25,810 --> 00:19:28,190
That is an operator.
241
00:19:28,190 --> 00:19:29,690
You see, you have to sum them.
242
00:19:29,690 --> 00:19:34,490
So you have to create a
space where both can act,
243
00:19:34,490 --> 00:19:36,180
and you can sum them.
244
00:19:36,180 --> 00:19:40,320
You cannot sum a thing, an
operator that acts on one
245
00:19:40,320 --> 00:19:43,420
vector space to an operator that
acts on another vector space.
246
00:19:43,420 --> 00:19:47,850
You have to create one
vector space where both act.
247
00:19:47,850 --> 00:19:51,250
And then you can define
the sum of the operators.
248
00:19:51,250 --> 00:19:53,750
Sum of operators
is a simple thing.
249
00:19:53,750 --> 00:19:56,460
So you form the tensor product.
250
00:19:56,460 --> 00:20:00,680
In here, this
operator gets upgraded
251
00:20:00,680 --> 00:20:05,390
in this way, in which in the
tensor product it has a 1
252
00:20:05,390 --> 00:20:06,970
for the second input.
253
00:20:06,970 --> 00:20:09,290
This one gets
upgrade to this way.
254
00:20:09,290 --> 00:20:10,250
And this is the sum.
255
00:20:12,950 --> 00:20:17,270
So this is a claim--
this is a definition.
256
00:20:17,270 --> 00:20:18,640
And this is a claim.
257
00:20:18,640 --> 00:20:21,430
So this has to be proven.
258
00:20:21,430 --> 00:20:23,370
So let me prove it.
259
00:20:26,140 --> 00:20:28,910
Ji, JJ.
260
00:20:28,910 --> 00:20:30,730
I compute this commutator.
261
00:20:30,730 --> 00:20:32,930
So I don't have to
do the following.
262
00:20:32,930 --> 00:20:45,530
I have to do Ji1, tensor
1, plus 1 tensor Ji2.
263
00:20:45,530 --> 00:20:56,620
And then the JJ would be JJ1,
tensor 1, plus 1, tensor JJ2.
264
00:20:59,670 --> 00:21:03,320
Have to compute this commutator.
265
00:21:03,320 --> 00:21:06,880
Now, an important
fact about this
266
00:21:06,880 --> 00:21:09,320
result that I'm not
trying to generalize,
267
00:21:09,320 --> 00:21:12,700
if you had put a minus
here, it wouldn't work out.
268
00:21:12,700 --> 00:21:15,480
If you would have put a 2
here, it wouldn't work out.
269
00:21:15,480 --> 00:21:19,130
If you would have put a 1/2
here, it won't work out.
270
00:21:19,130 --> 00:21:22,060
This is pretty much
the only way you
271
00:21:22,060 --> 00:21:26,370
can have two angular momenta,
and create a third angular
272
00:21:26,370 --> 00:21:27,620
momentum.
273
00:21:27,620 --> 00:21:29,240
So look at this.
274
00:21:32,730 --> 00:21:34,820
It looks like we're going
to have to work hard,
275
00:21:34,820 --> 00:21:38,490
but that's not true.
276
00:21:38,490 --> 00:21:40,650
Consider this commutator.
277
00:21:40,650 --> 00:21:42,970
The commutator of this
term with this term.
278
00:21:45,510 --> 00:21:47,430
That's 0 actually.
279
00:21:47,430 --> 00:21:51,950
Because if you multiply them
in this order, this times that,
280
00:21:51,950 --> 00:21:57,080
you get Ji1 times Ji2,
because the ones do nothing.
281
00:21:57,080 --> 00:21:59,710
You multiply them in
the reverse order,
282
00:21:59,710 --> 00:22:03,860
you get again, Ji1 times Ji2.
283
00:22:03,860 --> 00:22:08,895
This is to say that the
operators that originally lived
284
00:22:08,895 --> 00:22:13,510
in the different
vector spaces commute.
285
00:22:13,510 --> 00:22:15,260
Yes?
286
00:22:15,260 --> 00:22:19,320
AUDIENCE: Since the cross
terms between those two
287
00:22:19,320 --> 00:22:22,900
are 0-- like you just said,
the cross terms are 0.
288
00:22:22,900 --> 00:22:27,600
And if you put a minus sign
in there, it will cancel.
289
00:22:27,600 --> 00:22:29,494
But when you do
the multiplications
290
00:22:29,494 --> 00:22:33,318
with the second ones, why can't
you put a minus sign in there?
291
00:22:33,318 --> 00:22:34,170
[INAUDIBLE]
292
00:22:34,170 --> 00:22:35,420
PROFESSOR: In the whole thing?
293
00:22:35,420 --> 00:22:37,790
In this definition,
a minus sign?
294
00:22:37,790 --> 00:22:38,720
AUDIENCE: Yeah.
295
00:22:38,720 --> 00:22:41,900
PROFESSOR: Well, here
if I put a minus--
296
00:22:41,900 --> 00:22:45,270
it's like I'm going to
prove that this works.
297
00:22:45,270 --> 00:22:48,860
So if-- I'm going to
get an angular momentum.
298
00:22:48,860 --> 00:22:51,750
If I put a minus sign
to angular momentum,
299
00:22:51,750 --> 00:22:54,420
I ruin the algebra here.
300
00:22:54,420 --> 00:22:56,810
I put a minus minus, it cancels.
301
00:22:56,810 --> 00:23:00,010
But then I get a
minus sign here.
302
00:23:00,010 --> 00:23:03,940
So I cannot really
even change a sign.
303
00:23:03,940 --> 00:23:09,690
So any way, these are operators
acting on different spaces.
304
00:23:09,690 --> 00:23:11,320
They commute.
305
00:23:11,320 --> 00:23:13,375
It's clear they commute.
306
00:23:13,375 --> 00:23:15,700
You just multiply
them, and see that.
307
00:23:15,700 --> 00:23:18,860
These one's commute as well.
308
00:23:18,860 --> 00:23:23,730
The only ones that don't
commute are this with this.
309
00:23:23,730 --> 00:23:25,310
And that with that.
310
00:23:25,310 --> 00:23:27,520
So let me just write them.
311
00:23:27,520 --> 00:23:38,430
Ji1, tensor 1,
with JJ1, tensor 1.
312
00:23:38,430 --> 00:23:50,347
Plus this one, 1 tensor
Ji2, 1 tensor JJ2.
313
00:23:58,730 --> 00:24:01,870
OK, next step is to
realize that actually
314
00:24:01,870 --> 00:24:07,280
the 1 is a spectator here.
315
00:24:07,280 --> 00:24:10,180
Therefore, this
commutator is nothing
316
00:24:10,180 --> 00:24:19,130
but the commutator Ji1
with JJ1, tensor 1.
317
00:24:23,370 --> 00:24:24,510
You can do it.
318
00:24:24,510 --> 00:24:27,530
If you prefer to
write it, write it.
319
00:24:27,530 --> 00:24:32,930
This product is Ji
times JJ, tensor 1.
320
00:24:32,930 --> 00:24:37,240
And the other product
is JJ, Ji, tensor 1.
321
00:24:37,240 --> 00:24:41,040
So the tensor 1 factors out.
322
00:24:41,040 --> 00:24:44,980
Here the tensor 1
also factors out.
323
00:24:44,980 --> 00:24:50,470
And you get an honest
commutator, Ji2, JJ2.
324
00:24:53,300 --> 00:24:57,290
So one last step.
325
00:24:57,290 --> 00:25:01,920
This is i, h bar, epsilon, iJK.
326
00:25:01,920 --> 00:25:04,750
I'll put a big parentheses.
327
00:25:04,750 --> 00:25:12,990
JK1, tensor 1,
for the first one.
328
00:25:12,990 --> 00:25:17,530
Because J1 forms an
angular momentum algebra.
329
00:25:17,530 --> 00:25:21,750
And here, 1 tensor JK2.
330
00:25:30,670 --> 00:25:36,280
And this thing is i,
h bar, epsilon, iJK.
331
00:25:36,280 --> 00:25:41,880
The total angular
momentum, K. And you've
332
00:25:41,880 --> 00:25:43,990
shown the algebra works out.
333
00:25:48,370 --> 00:25:52,620
Now most people after
a little practice,
334
00:25:52,620 --> 00:26:02,890
they just say, oh, Ji is
J1 plus J2, J1 plus J2.
335
00:26:02,890 --> 00:26:04,910
J1 and J2 don't commute.
336
00:26:04,910 --> 00:26:07,870
J2 and J1-- I'm sorry.
337
00:26:07,870 --> 00:26:09,390
J1 and J2 commute.
338
00:26:09,390 --> 00:26:11,110
J2 and J1 commute.
339
00:26:11,110 --> 00:26:14,470
Therefore you get
this 2, like that.
340
00:26:14,470 --> 00:26:18,760
And this gives you--
J1 and J1 gives you J1.
341
00:26:18,760 --> 00:26:22,230
J2 and J2 gives you J2,
so the sum works out.
342
00:26:22,230 --> 00:26:25,530
So most people after
a little practice
343
00:26:25,530 --> 00:26:28,100
just don't put all
these tensor things.
344
00:26:28,100 --> 00:26:32,320
But at the beginning it's
nice to just make sure
345
00:26:32,320 --> 00:26:36,700
that you understand what
these tensor things do.
346
00:26:36,700 --> 00:26:37,280
All right.
347
00:26:37,280 --> 00:26:40,700
So that's our main
theorem-- that you
348
00:26:40,700 --> 00:26:45,180
start with one angular
momentum on a state space.
349
00:26:45,180 --> 00:26:49,090
Another angular momentum that
has nothing to do perhaps
350
00:26:49,090 --> 00:26:53,170
with the first on
another vector space.
351
00:26:53,170 --> 00:26:58,390
And on the tensor product you
have another angular momentum,
352
00:26:58,390 --> 00:26:59,420
which is the sum.
353
00:27:01,940 --> 00:27:02,670
All right.
354
00:27:02,670 --> 00:27:05,590
So now, we do spin
orbit coupling
355
00:27:05,590 --> 00:27:08,660
to try to apply these ideas.
356
00:27:08,660 --> 00:27:25,060
So for spin orbit coupling, we
will consider the hydrogen atom
357
00:27:25,060 --> 00:27:28,000
coupling.
358
00:27:28,000 --> 00:27:36,610
And the new term in the
Hamiltonian, mu dot B.
359
00:27:36,610 --> 00:27:41,390
The kind of term that we've
done so much in this semester.
360
00:27:41,390 --> 00:27:43,880
We've looked over magnetic ones.
361
00:27:43,880 --> 00:27:46,700
So which magnetic
moment at which B?
362
00:27:46,700 --> 00:27:50,050
There was no B in
the hydrogen atom.
363
00:27:50,050 --> 00:27:54,850
Well, there's no
B to begin with.
364
00:27:54,850 --> 00:27:58,980
But here is one where you can
think there is a B. First,
365
00:27:58,980 --> 00:28:06,280
this will be the
electron dipole moment.
366
00:28:06,280 --> 00:28:07,790
Magnetic dipole moment.
367
00:28:07,790 --> 00:28:11,350
So we have a formula for it.
368
00:28:11,350 --> 00:28:17,370
The formula for it is the mu of
the electron is minus E over m,
369
00:28:17,370 --> 00:28:19,960
times the spin of the electron.
370
00:28:19,960 --> 00:28:25,090
And I actually will use a
little different formula
371
00:28:25,090 --> 00:28:28,130
that is valued in
Gaussian units.
372
00:28:28,130 --> 00:28:35,550
ge over mC, S, in
Gaussian units.
373
00:28:38,590 --> 00:28:42,070
And g is the g factor of
the electron, which is 2.
374
00:28:42,070 --> 00:28:42,770
I'm sorry.
375
00:28:42,770 --> 00:28:44,850
There's a 2 here.
376
00:28:44,850 --> 00:28:45,440
OK.
377
00:28:45,440 --> 00:28:46,990
So look what I've written.
378
00:28:46,990 --> 00:28:49,540
I don't want to distract
you with this too much.
379
00:28:49,540 --> 00:28:53,270
But you know that the magnetic
dipole of the electron
380
00:28:53,270 --> 00:28:55,310
is given by this quantity.
381
00:28:55,310 --> 00:28:57,930
Now, you could put a
2 up, and a 2 down.
382
00:28:57,930 --> 00:29:02,120
And that's why people
actually classically
383
00:29:02,120 --> 00:29:04,410
there seems to be a 2 down.
384
00:29:04,410 --> 00:29:08,220
But there's a 2 up, because
it's an effect of the electron.
385
00:29:08,220 --> 00:29:09,640
And you have this formula.
386
00:29:09,640 --> 00:29:11,650
The only thing I've
added in that formula
387
00:29:11,650 --> 00:29:16,390
is a factor of C that is
because of Gaussian units.
388
00:29:16,390 --> 00:29:20,250
And it allows you to estimate
terms a little more easily.
389
00:29:20,250 --> 00:29:22,610
So that's the mu
of the electron.
390
00:29:22,610 --> 00:29:28,600
But the electron apparently
would feel no magnetic field.
391
00:29:28,600 --> 00:29:31,320
You didn't put an
external magnetic field.
392
00:29:31,320 --> 00:29:37,070
Except that here
you go in this way
393
00:29:37,070 --> 00:29:40,840
of thinking-- you think
suppose you are the electron.
394
00:29:40,840 --> 00:29:45,340
You see a proton, which is
a nucleus going around you.
395
00:29:45,340 --> 00:29:49,750
And a proton going around you
is a current going around you.
396
00:29:49,750 --> 00:29:52,280
It generates a magnetic field.
397
00:29:52,280 --> 00:29:54,840
And therefore, you
see a magnetic field
398
00:29:54,840 --> 00:30:00,260
created by the proton
going around you.
399
00:30:00,260 --> 00:30:02,400
So there is a magnetic field.
400
00:30:02,400 --> 00:30:04,720
And there's a magnetic
field experienced
401
00:30:04,720 --> 00:30:11,440
by the electron--
felt by electron.
402
00:30:14,300 --> 00:30:19,570
So you can think of
this, the electron.
403
00:30:19,570 --> 00:30:22,100
Here is the proton
with the plus charge,
404
00:30:22,100 --> 00:30:25,190
and here's the electron.
405
00:30:25,190 --> 00:30:29,060
And the electron is
going around the proton.
406
00:30:29,060 --> 00:30:31,730
Now, from the viewpoint
of the electron,
407
00:30:31,730 --> 00:30:35,690
the proton is going around him.
408
00:30:35,690 --> 00:30:38,000
So here is the proton.
409
00:30:38,000 --> 00:30:40,330
Here is the electron
going like that.
410
00:30:40,330 --> 00:30:42,040
From the viewpoint
of the electron,
411
00:30:42,040 --> 00:30:44,760
the proton is going like this.
412
00:30:44,760 --> 00:30:48,350
Also, from the viewpoint
of the electron,
413
00:30:48,350 --> 00:30:51,335
the proton would be
going in this direction
414
00:30:51,335 --> 00:30:54,150
and creating a
magnetic field up.
415
00:30:59,990 --> 00:31:06,570
And the magnetic field
up corresponds actually
416
00:31:06,570 --> 00:31:11,775
to the idea that the angular
momentum of the electron
417
00:31:11,775 --> 00:31:16,970
is also up-- L of the
electron is also up.
418
00:31:16,970 --> 00:31:20,870
So the whole point
of this thing is
419
00:31:20,870 --> 00:31:25,670
that somehow this magnetic field
is proportional to the angular
420
00:31:25,670 --> 00:31:27,100
momentum.
421
00:31:27,100 --> 00:31:29,900
And then, L will come here.
422
00:31:29,900 --> 00:31:31,770
And here, you have
S. So you have
423
00:31:31,770 --> 00:31:39,440
L dot S. That's the
fine structure coupling.
424
00:31:39,440 --> 00:31:45,300
Now let me do a little
of this so that we just
425
00:31:45,300 --> 00:31:49,080
get a bit more feeling,
although it's unfortunately
426
00:31:49,080 --> 00:31:53,000
a somewhat frustrating exercise.
427
00:31:53,000 --> 00:31:56,790
So let me tell you
what's going on.
428
00:31:56,790 --> 00:32:00,836
So consider the electron.
429
00:32:00,836 --> 00:32:05,600
At some point, look at
it and draw a plane.
430
00:32:05,600 --> 00:32:09,000
So the electron-- let's
assume it's going down.
431
00:32:09,000 --> 00:32:10,090
Here is the proton.
432
00:32:10,090 --> 00:32:11,670
It's going around in circles.
433
00:32:11,670 --> 00:32:14,660
So here, it's going down.
434
00:32:14,660 --> 00:32:16,280
The electron is going down.
435
00:32:16,280 --> 00:32:20,900
Electron, its velocity of
the electron is going down.
436
00:32:20,900 --> 00:32:24,380
The proton is over here.
437
00:32:24,380 --> 00:32:28,740
And the electron is
going around like that.
438
00:32:28,740 --> 00:32:32,350
The proton would produce an
electric field of this form.
439
00:32:35,770 --> 00:32:42,080
Now, in relativity, the
electric and magnetic fields
440
00:32:42,080 --> 00:32:44,920
seen by different
observers are different.
441
00:32:44,920 --> 00:32:48,210
So there is this electric
field that we see.
442
00:32:48,210 --> 00:32:52,260
We sit here, and we
see in our rest frame
443
00:32:52,260 --> 00:32:56,330
this proton creates
an electric field.
444
00:32:56,330 --> 00:33:00,760
And then, from the
viewpoint of the electron,
445
00:33:00,760 --> 00:33:02,280
the electron is moving.
446
00:33:02,280 --> 00:33:04,840
And there is an electric field.
447
00:33:04,840 --> 00:33:09,020
But whenever you are moving
inside an electric field,
448
00:33:09,020 --> 00:33:12,560
you also see a magnetic field
generated by the motion,
449
00:33:12,560 --> 00:33:14,910
by relativistic effects.
450
00:33:14,910 --> 00:33:20,010
The magnetic field
that you see is roughly
451
00:33:20,010 --> 00:33:27,560
given to first order in
relativity by V cross E over c.
452
00:33:30,350 --> 00:33:37,120
So V cross E, VE
V cross E over c
453
00:33:37,120 --> 00:33:39,900
up-- change sign
because of this.
454
00:33:39,900 --> 00:33:42,740
And the magnetic
field consistently,
455
00:33:42,740 --> 00:33:45,860
as we would expect,
goes in this direction.
456
00:33:45,860 --> 00:33:48,210
So it's consistent
with the picture
457
00:33:48,210 --> 00:33:51,840
that we developed that if you
were the electron, the proton,
458
00:33:51,840 --> 00:33:53,940
would be going around
in circles like that
459
00:33:53,940 --> 00:33:55,790
and the magnetic
field would be up.
460
00:33:58,430 --> 00:34:07,730
Now here I can change the sign
by doing E cross V over c.
461
00:34:07,730 --> 00:34:11,783
So this is the magnetic
field seen by the electron.
462
00:34:23,389 --> 00:34:30,010
OK, so we need a little more
work on that magnetic field
463
00:34:30,010 --> 00:34:33,610
by calculating the
electric field.
464
00:34:33,610 --> 00:34:36,219
Now, what is the electric field?
465
00:34:36,219 --> 00:34:39,865
Well, the scalar potential
for the hydrogen atom,
466
00:34:39,865 --> 00:34:44,670
we write it as minus
e squared over r.
467
00:34:44,670 --> 00:34:47,159
It's actually not quite
the scalar potential.
468
00:34:47,159 --> 00:34:49,980
But it is the potential energy.
469
00:34:49,980 --> 00:34:55,440
It has one factor of e more than
what the scalar potential is.
470
00:34:55,440 --> 00:34:57,940
Remember, the scalar
potential in electromagnetism
471
00:34:57,940 --> 00:35:00,260
is charge divided by r.
472
00:35:00,260 --> 00:35:04,000
So it has one factor of e more.
473
00:35:04,000 --> 00:35:07,100
What is the derivative
of this potential?
474
00:35:07,100 --> 00:35:12,370
With respect to r, it's
e squared over r squared.
475
00:35:12,370 --> 00:35:16,990
So the electric field goes
like e over r squared.
476
00:35:16,990 --> 00:35:29,410
So the electric field is
equal to dV dr divided by e.
477
00:35:32,220 --> 00:35:34,500
That's the magnitude
of the electric field.
478
00:35:34,500 --> 00:35:40,150
And its direction is radial from
the viewpoint of the proton.
479
00:35:40,150 --> 00:35:42,935
The electric field is here.
480
00:35:47,790 --> 00:35:51,920
So this can be written
as r vector divided by r.
481
00:35:58,880 --> 00:36:07,000
Therefore, the magnetic field
will-- [INAUDIBLE] this.
482
00:36:07,000 --> 00:36:10,320
The magnetic field
now can be calculated.
483
00:36:10,320 --> 00:36:16,400
And we'll see what
we claimed was
484
00:36:16,400 --> 00:36:18,930
the relation with
angular momentum.
485
00:36:18,930 --> 00:36:25,850
Because B prime
is now E cross V.
486
00:36:25,850 --> 00:36:37,170
So you have 1 over
ec 1 over r dV dr.
487
00:36:37,170 --> 00:36:38,960
I've taken care of this.
488
00:36:38,960 --> 00:36:44,770
And now I just have
r cross V. Well,
489
00:36:44,770 --> 00:36:51,240
r cross V is your angular
momentum if you had p here.
490
00:36:51,240 --> 00:36:56,970
So we borrow a factor of
the mass of the electron,
491
00:36:56,970 --> 00:37:06,570
ecm 1 over r dV dr
L, L of the electron.
492
00:37:11,710 --> 00:37:16,380
p equals mv.
493
00:37:16,380 --> 00:37:22,180
So we have a nice formula
for B. And then, we
494
00:37:22,180 --> 00:37:28,460
can go and calculate delta
H. Delta H would then
495
00:37:28,460 --> 00:37:45,380
be minus mu dot B. And that
would be ge over 2mc spin
496
00:37:45,380 --> 00:38:01,670
dot L-- mu was given here--
S dot L 1 over r dV dr.
497
00:38:01,670 --> 00:38:07,150
And that is the split
spin orbit interaction.
498
00:38:07,150 --> 00:38:10,150
Now, the downside
of this derivation
499
00:38:10,150 --> 00:38:13,015
is that it has a
relativistic error.
500
00:38:16,740 --> 00:38:20,370
There's a phenomenon
called Thomas precession
501
00:38:20,370 --> 00:38:23,870
that affects this result.
502
00:38:23,870 --> 00:38:26,330
We didn't waste our time.
503
00:38:26,330 --> 00:38:32,250
The true result is that you
must subtract from this g 1.
504
00:38:32,250 --> 00:38:37,730
So g must really be
replaced by g minus 1.
505
00:38:37,730 --> 00:38:41,040
Since g is approximately
2 for the electron,
506
00:38:41,040 --> 00:38:46,080
the true result is
really 1/2 of this thing.
507
00:38:46,080 --> 00:38:49,760
So this should not
be in parentheses,
508
00:38:49,760 --> 00:38:54,200
but true result is this.
509
00:38:54,200 --> 00:38:59,770
And the mistake that is done
in calculating this spin orbit
510
00:38:59,770 --> 00:39:04,970
coupling is that this
spin orbit coupling
511
00:39:04,970 --> 00:39:08,510
affects precession rates.
512
00:39:08,510 --> 00:39:12,260
All these interactions
of magnetic dipoles
513
00:39:12,260 --> 00:39:15,540
with magnetic fields
affect precession rates.
514
00:39:15,540 --> 00:39:18,020
And you have to be a
little more careful here
515
00:39:18,020 --> 00:39:24,200
that the system where you've
worked, the electron rest frame
516
00:39:24,200 --> 00:39:26,370
is not quite an inertial system.
517
00:39:26,370 --> 00:39:29,330
Because it's doing
circular motion.
518
00:39:29,330 --> 00:39:33,420
So there's an extra correction
that has to be done.
519
00:39:33,420 --> 00:39:37,350
Thomas precession or Thomas
correction it's called.
520
00:39:37,350 --> 00:39:41,170
And it would be a
detour of about one hour
521
00:39:41,170 --> 00:39:43,900
in special relativity
to do it right.
522
00:39:43,900 --> 00:39:47,676
So Griffiths doesn't do it.
523
00:39:47,676 --> 00:39:51,010
I don't think Shankar does it.
524
00:39:51,010 --> 00:39:53,280
Pretty much graduate
books do it.
525
00:39:56,088 --> 00:39:59,990
So we will not try to do better.
526
00:39:59,990 --> 00:40:02,550
I mentioned that
fact that this really
527
00:40:02,550 --> 00:40:06,260
should be reduced to
one half of its value.
528
00:40:06,260 --> 00:40:08,780
And it's an interesting
system to analyze.
529
00:40:08,780 --> 00:40:15,170
So Thomas precession is
that relativistic correction
530
00:40:15,170 --> 00:40:18,540
to precession rates when the
object that is precessing
531
00:40:18,540 --> 00:40:21,820
is in an accelerated frame.
532
00:40:21,820 --> 00:40:24,670
And any rotating
frame is accelerated.
533
00:40:24,670 --> 00:40:31,200
So this result needs correction.
534
00:40:31,200 --> 00:40:36,450
OK, but let's take this result
as it is-- instead of g,
535
00:40:36,450 --> 00:40:37,640
g minus 1.
536
00:40:37,640 --> 00:40:39,810
Let's not worry
too much about it.
537
00:40:39,810 --> 00:40:44,686
And let's just estimate
how big this effect is.
538
00:40:44,686 --> 00:40:50,470
It's the last thing I want
to do as a way of motivating
539
00:40:50,470 --> 00:40:51,460
this subject.
540
00:40:51,460 --> 00:40:53,850
So delta H is this.
541
00:40:53,850 --> 00:40:55,770
Let's estimate it.
542
00:40:55,770 --> 00:41:03,090
Now for estimates, a couple of
things are useful to remember,
543
00:41:03,090 --> 00:41:08,640
that Bohr radius is h
squared over me squared.
544
00:41:08,640 --> 00:41:10,740
We did that last time.
545
00:41:10,740 --> 00:41:13,680
And there's this constant
that is very useful,
546
00:41:13,680 --> 00:41:18,620
the fine structure constant,
which is e squared over hc.
547
00:41:18,620 --> 00:41:21,885
And it's about 1 over 137.
548
00:41:21,885 --> 00:41:27,090
And it helps you estimate
all kinds of things.
549
00:41:27,090 --> 00:41:31,580
Because it's a rather
complicated number to evaluate,
550
00:41:31,580 --> 00:41:36,450
you need all kinds of
units and things like that.
551
00:41:36,450 --> 00:41:43,180
So the charge of the electron
divided by hc being 1 over 137
552
00:41:43,180 --> 00:41:46,680
is quite nice.
553
00:41:46,680 --> 00:41:53,050
So let's estimate
delta H. Well, g
554
00:41:53,050 --> 00:41:58,160
we won't worry-- 2,
1, doesn't matter.
555
00:41:58,160 --> 00:42:07,530
e mc-- so far, that
is kind of simple.
556
00:42:07,530 --> 00:42:13,036
Then we have S dot L. Well,
how do I estimate S dot L?
557
00:42:13,036 --> 00:42:15,740
I don't do too much.
558
00:42:15,740 --> 00:42:18,850
S spin is multiples of h bar.
559
00:42:18,850 --> 00:42:23,960
L for an atomic state will be
1, 2, 3, so multiples of h bar.
560
00:42:23,960 --> 00:42:29,430
So h bar squared,
that's it for S dot L.
561
00:42:29,430 --> 00:42:33,500
1 over r is 1 over r.
562
00:42:33,500 --> 00:42:37,570
dV dr is e squared
over r squared.
563
00:42:37,570 --> 00:42:39,280
And that's it.
564
00:42:39,280 --> 00:42:43,760
But here, instead of r, I
should put the typical length
565
00:42:43,760 --> 00:42:46,950
of the hydrogen
atom, which is a0.
566
00:42:46,950 --> 00:42:48,210
So what do I get?
567
00:42:53,230 --> 00:42:57,300
I'm sorry, I made
a mistake here.
568
00:42:57,300 --> 00:42:59,407
AUDIENCE: Yeah, it's up there.
569
00:42:59,407 --> 00:43:03,079
PROFESSOR: Oh, I
made a mistake here
570
00:43:03,079 --> 00:43:09,910
in that I didn't put
this factor, 1 over ecm.
571
00:43:09,910 --> 00:43:11,980
So the e cancels.
572
00:43:11,980 --> 00:43:15,920
And this is the result
here-- g over 2m
573
00:43:15,920 --> 00:43:39,900
squared c squared S dot L 1 over
r dV dr. So let me start again.
574
00:43:39,900 --> 00:43:49,680
1 over m squared c squared h
bar squared 1 over r dV dr--
575
00:43:49,680 --> 00:43:51,550
that much I got right.
576
00:43:51,550 --> 00:43:58,810
So this is roughly
1 over [INAUDIBLE]
577
00:43:58,810 --> 00:44:02,220
of the electron
squared c squared e
578
00:44:02,220 --> 00:44:08,780
squared over a0 cubed
h squared-- still quite
579
00:44:08,780 --> 00:44:12,610
messy, but not that terrible.
580
00:44:12,610 --> 00:44:16,700
So in order to get an
idea of how big this is,
581
00:44:16,700 --> 00:44:19,720
the ground state energy
of the hydrogen atom
582
00:44:19,720 --> 00:44:23,400
was e squared over 2a0.
583
00:44:23,400 --> 00:44:30,610
So let's divide delta H over
the ground state energy.
584
00:44:30,610 --> 00:44:32,220
And that's how much?
585
00:44:32,220 --> 00:44:37,850
Well, we have all this
quantity, 1 over m
586
00:44:37,850 --> 00:44:44,490
squared c squared e
squared a0 cubed h squared.
587
00:44:44,490 --> 00:44:53,110
And now, we must divide by
e squared over a0 like this.
588
00:44:59,160 --> 00:45:02,370
Well, the e squareds cancel.
589
00:45:02,370 --> 00:45:11,800
And we get h squared over m
squared c squared a0 squared.
590
00:45:11,800 --> 00:45:13,950
You need to know what a0 is.
591
00:45:13,950 --> 00:45:16,640
Let's just boil it down
to the simplest thing,
592
00:45:16,640 --> 00:45:20,760
so h squared m
squared c squared.
593
00:45:20,760 --> 00:45:26,120
a0 squared would be h to
the fourth m squared e
594
00:45:26,120 --> 00:45:28,540
to the fourth.
595
00:45:28,540 --> 00:45:33,840
So this is actually e
to the fourth over h
596
00:45:33,840 --> 00:45:41,200
squared c squared, or e squared
over hc squared, which is alpha
597
00:45:41,200 --> 00:45:41,700
squared.
598
00:45:41,700 --> 00:45:45,510
Whew-- lots of work to
get something very nice.
599
00:45:48,030 --> 00:45:56,910
The ratio of the spin orbit
coupling to the ground state
600
00:45:56,910 --> 00:46:00,850
energy is 1 over alpha squared.
601
00:46:00,850 --> 00:46:05,310
It's alpha squared, which
is 1 over 137 squared.
602
00:46:05,310 --> 00:46:08,040
So it's a pretty small thing.
603
00:46:08,040 --> 00:46:14,250
It's about 1 over 19,000.
604
00:46:14,250 --> 00:46:19,850
So when this is called fine
structure of the hydrogen atom,
605
00:46:19,850 --> 00:46:24,300
it means that it's in
the level in your page
606
00:46:24,300 --> 00:46:30,040
that you use a few inches
to plot the 13.6 electron
607
00:46:30,040 --> 00:46:35,450
volts-- well, you're talking
about 20,000 times smaller,
608
00:46:35,450 --> 00:46:38,820
something that you don't see.
609
00:46:38,820 --> 00:46:41,410
But of course, it's a
pretty important thing.
610
00:46:41,410 --> 00:46:48,960
So all in all, in the
conventions of-- this
611
00:46:48,960 --> 00:46:53,900
is done in Gaussian units.
612
00:46:53,900 --> 00:47:01,140
In SI units, which is
what Griffiths uses,
613
00:47:01,140 --> 00:47:08,560
delta H is e squared over
8 pi epsilon 0 1 over m
614
00:47:08,560 --> 00:47:17,450
squared c squared r cubed S
dot L. That's for reference.
615
00:47:17,450 --> 00:47:18,275
This is Griffiths.
616
00:47:22,300 --> 00:47:24,310
But this is correct as well.
617
00:47:24,310 --> 00:47:27,440
This is the correct value.
618
00:47:27,440 --> 00:47:30,960
This is the correct
value already taking
619
00:47:30,960 --> 00:47:33,340
into account the
relativistic correction.
620
00:47:33,340 --> 00:47:37,270
So here, you're supposed
to let g go to g minus 1.
621
00:47:37,270 --> 00:47:42,490
So you can put the 1 there,
and it's pretty accurate.
622
00:47:42,490 --> 00:47:45,590
All right, so what is
the physics question
623
00:47:45,590 --> 00:47:49,500
we want to answer with
this spin orbit coupling?
624
00:47:49,500 --> 00:47:54,640
So here it comes.
625
00:47:54,640 --> 00:47:58,920
You have the hydrogen
atom spectrum.
626
00:47:58,920 --> 00:48:01,460
And that spectrum you know.
627
00:48:01,460 --> 00:48:05,050
At L equals 0, you
have one state here.
628
00:48:05,050 --> 00:48:09,890
Then, that's n
equals 1, n equals 2.
629
00:48:12,470 --> 00:48:16,580
You have one state here and
one state here at L equals 1.
630
00:48:16,580 --> 00:48:21,260
Then n equals 3, they start
getting very close together.
631
00:48:21,260 --> 00:48:25,980
n equals 4 is like that.
632
00:48:25,980 --> 00:48:31,370
Let's consider if you want
to have spin orbit coupling,
633
00:48:31,370 --> 00:48:35,880
we must have angular momentum.
634
00:48:35,880 --> 00:48:42,330
And that's L. And therefore,
let's consider this state here.
635
00:48:42,330 --> 00:48:47,420
l equals 1, n equals 1--
n equals 2, I'm sorry.
636
00:48:51,410 --> 00:48:56,520
What happens to those
states, is the question.
637
00:48:56,520 --> 00:48:59,110
First, how many states
do you have there
638
00:48:59,110 --> 00:49:02,120
and how should
you think of them?
639
00:49:02,120 --> 00:49:06,430
Well actually, we know that
an l equals 1 corresponds
640
00:49:06,430 --> 00:49:07,970
to three states.
641
00:49:07,970 --> 00:49:13,820
So you'd have lm
with l equals 1.
642
00:49:13,820 --> 00:49:18,310
And then m can be
1, 0, or minus 1.
643
00:49:18,310 --> 00:49:20,730
So you have three states.
644
00:49:20,730 --> 00:49:23,840
But there's not
really three states.
645
00:49:23,840 --> 00:49:26,420
Because the electron
can have spin.
646
00:49:26,420 --> 00:49:31,770
So here it is, a tensor product
that appears in your face
647
00:49:31,770 --> 00:49:36,150
because there is
more than angular
648
00:49:36,150 --> 00:49:37,430
momentum to the electron.
649
00:49:37,430 --> 00:49:38,630
There's spin.
650
00:49:38,630 --> 00:49:42,130
And it's a totally different
vector space, the same particle
651
00:49:42,130 --> 00:49:45,780
but another vector
space, the spin space.
652
00:49:45,780 --> 00:49:50,140
So here, you have the possible
spins of the electron.
653
00:49:50,140 --> 00:49:53,130
So that's another
angular momentum.
654
00:49:53,130 --> 00:49:57,525
And well, you could have the
plus/minus states, for example.
655
00:50:00,730 --> 00:50:05,290
So you have three states
here and two states here.
656
00:50:05,290 --> 00:50:16,440
So this is really six
states, so six states
657
00:50:16,440 --> 00:50:20,700
whose fate we would
like to understand
658
00:50:20,700 --> 00:50:23,355
due to this spin orbit coupling.
659
00:50:27,600 --> 00:50:35,610
So to use the language
of angular momentum,
660
00:50:35,610 --> 00:50:37,740
instead of writing
plus/minus, you
661
00:50:37,740 --> 00:50:49,990
could write Smz, if you will--
ms I will call, spin of s.
662
00:50:49,990 --> 00:50:59,780
You have here spin of 1/2
and states 1/2 or minus 1/2.
663
00:50:59,780 --> 00:51:01,040
This is the up.
664
00:51:01,040 --> 00:51:07,040
When the z component of the spin
that we always call m-- m now
665
00:51:07,040 --> 00:51:09,760
corresponds to the z
component of angular momentum.
666
00:51:09,760 --> 00:51:12,846
So in general, even
for spin, we use m.
667
00:51:12,846 --> 00:51:16,410
And we have that our two
spin states of the electron
668
00:51:16,410 --> 00:51:21,880
are spin 1/2 particle with
plus spin in the z direction,
669
00:51:21,880 --> 00:51:26,550
spin 1/2 particle with minus
spin in the z direction.
670
00:51:26,550 --> 00:51:29,380
We usually never
put this 1/2 here.
671
00:51:29,380 --> 00:51:32,700
But now you have here
really three states--
672
00:51:32,700 --> 00:51:39,950
1, 1, 1, 0, 1, minus 1,
the first telling you
673
00:51:39,950 --> 00:51:42,810
about the total
angular momentum.
674
00:51:42,810 --> 00:51:45,480
Here, the total spin is 1/2.
675
00:51:45,480 --> 00:51:47,860
But it happens to be
either up or down.
676
00:51:47,860 --> 00:51:50,550
Here, the total
angular momentum is 1.
677
00:51:50,550 --> 00:51:56,010
But it happens to be plus
1, 0, or minus 1 here.
678
00:51:56,010 --> 00:51:58,830
So these are our six states.
679
00:51:58,830 --> 00:52:01,600
You can combine this with this,
this with that, this with this,
680
00:52:01,600 --> 00:52:02,320
this with that.
681
00:52:02,320 --> 00:52:03,870
You make all the products.
682
00:52:03,870 --> 00:52:06,610
And these are the six
states of the hydrogen
683
00:52:06,610 --> 00:52:08,090
atom at this level.
684
00:52:08,090 --> 00:52:13,100
And we wish to know
what happens to them.
685
00:52:13,100 --> 00:52:17,910
Now, this correction is small.
686
00:52:17,910 --> 00:52:22,750
So it fits our understanding
of the perturbation theory
687
00:52:22,750 --> 00:52:25,310
of Feynman-Hellman
in which we try
688
00:52:25,310 --> 00:52:29,660
to find the corrections
to these things.
689
00:52:29,660 --> 00:52:35,130
Our difficulty now is a
little serious, however.
690
00:52:35,130 --> 00:52:37,290
It's the fact that
Feynman-Hellman
691
00:52:37,290 --> 00:52:39,870
assumed that you had a state.
692
00:52:39,870 --> 00:52:43,600
And it was an eigenstate of
the corrected Hamiltonian
693
00:52:43,600 --> 00:52:46,930
as you moved along.
694
00:52:46,930 --> 00:52:51,220
And then, you could compute
how its energy changes.
695
00:52:51,220 --> 00:52:56,260
Here, unfortunately, we have a
much more difficult situation.
696
00:52:56,260 --> 00:52:59,580
These six states that
I'm not listing yet,
697
00:52:59,580 --> 00:53:07,170
but I will list very soon,
are not obviously eigenstates
698
00:53:07,170 --> 00:53:12,340
of delta H. In fact, they are
not eigenstates of delta H.
699
00:53:12,340 --> 00:53:15,790
They're degenerate states,
six degenerate states,
700
00:53:15,790 --> 00:53:19,440
that are not eigenstates
of delta H. Therefore,
701
00:53:19,440 --> 00:53:24,360
I cannot use the Feynman-Hellman
theorem until I find what are
702
00:53:24,360 --> 00:53:28,499
the combinations that
are eigenstates of this
703
00:53:28,499 --> 00:53:29,040
perturbation.
704
00:53:32,680 --> 00:53:35,380
So we are a little
bit in trouble.
705
00:53:35,380 --> 00:53:43,010
Because we have a perturbation
for which these product
706
00:53:43,010 --> 00:53:52,110
states-- we call them uncoupled
bases-- are not eigenstates.
707
00:53:52,110 --> 00:53:56,850
Now, we've written this
operator a little naively.
708
00:53:56,850 --> 00:54:01,270
What does this operator
really mean, S dot L?
709
00:54:06,150 --> 00:54:15,030
In our tensor products,
it means S1 tensor L1.
710
00:54:15,030 --> 00:54:19,180
Actually, I'll use
L dot S. I'll always
711
00:54:19,180 --> 00:54:24,740
put the L information first and
the S information afterward.
712
00:54:24,740 --> 00:54:29,850
So L dot S is clearly
an operator that
713
00:54:29,850 --> 00:54:33,910
must be thought to act
on the tensor product.
714
00:54:33,910 --> 00:54:35,830
Because both have to act.
715
00:54:35,830 --> 00:54:37,990
S has to act and L has to act.
716
00:54:37,990 --> 00:54:39,800
So it only lives in
the tensor product.
717
00:54:39,800 --> 00:54:41,400
So what does it mean?
718
00:54:41,400 --> 00:54:55,960
It means this-- S2 L2 plus S3
L3, or sum over i Si tensor Li.
719
00:54:55,960 --> 00:54:59,860
So this is the kind of thing
that you need to understand--
720
00:54:59,860 --> 00:55:05,080
how do you find for this
operator's eigenstates here?
721
00:55:07,940 --> 00:55:16,400
So that is our difficulty.
722
00:55:16,400 --> 00:55:19,910
And that's what
we have to solve.
723
00:55:19,910 --> 00:55:23,770
We're going to solve it
in the next half hour.
724
00:55:23,770 --> 00:55:31,140
So it's a complicated operator,
L dot S. But on the other hand,
725
00:55:31,140 --> 00:55:34,780
we have to use our ideas
that we've learned already
726
00:55:34,780 --> 00:55:40,090
about summing angular momenta.
727
00:55:40,090 --> 00:55:49,310
What if I define
J to be L plus S,
728
00:55:49,310 --> 00:55:58,545
which really means L
tensor 1 plus 1 tensor S?
729
00:56:03,940 --> 00:56:09,130
So this is what I really
mean by this operator.
730
00:56:12,170 --> 00:56:17,440
J, as we've demonstrated,
will be an angular momentum,
731
00:56:17,440 --> 00:56:20,510
because this satisfies the
algebra of angular momentum
732
00:56:20,510 --> 00:56:23,820
and this satisfies the
algebra of angular momentum.
733
00:56:23,820 --> 00:56:28,280
So this thing satisfies the
algebra of angular momentum.
734
00:56:28,280 --> 00:56:32,310
And why do we look at that term?
735
00:56:32,310 --> 00:56:36,340
Because of the following reason.
736
00:56:36,340 --> 00:56:39,590
We can square it-- JiJi.
737
00:56:43,930 --> 00:56:48,870
Now we would have to
square this thing.
738
00:56:48,870 --> 00:56:50,510
How do you square this thing?
739
00:56:50,510 --> 00:56:52,590
Well, there's two ways.
740
00:56:52,590 --> 00:56:56,060
Naively-- L squared
plus L squared plus 2L
741
00:56:56,060 --> 00:56:59,285
dot S-- basically correct.
742
00:56:59,285 --> 00:57:01,750
But you can do it a
little more slowly.
743
00:57:01,750 --> 00:57:07,120
If you square this term,
you get L squared tensor 1.
744
00:57:07,120 --> 00:57:14,000
If you square this term,
you get 1 tensor S squared.
745
00:57:14,000 --> 00:57:16,210
But when you do
the mixed products,
746
00:57:16,210 --> 00:57:20,070
you just must take the
i's here and the i's here
747
00:57:20,070 --> 00:57:21,250
and multiply them.
748
00:57:21,250 --> 00:57:28,530
So actually, you do get two i's,
the sum over i Li tensor Si.
749
00:57:35,710 --> 00:57:37,320
This is sum over i.
750
00:57:37,320 --> 00:57:38,500
This is J squared.
751
00:57:42,500 --> 00:57:50,420
So basically, what I'm saying
is that J squared naively
752
00:57:50,420 --> 00:57:56,100
is L squared plus S squared
plus our interaction
753
00:57:56,100 --> 00:58:01,425
2L dot S defined property.
754
00:58:04,110 --> 00:58:16,900
So L dot S is equal to 1/2
of J squared minus L squared
755
00:58:16,900 --> 00:58:19,690
minus S squared.
756
00:58:19,690 --> 00:58:26,020
And that tells you all kinds of
interesting things about L dot
757
00:58:26,020 --> 00:58:28,470
S.
758
00:58:28,470 --> 00:58:34,180
Basically, we can trade
L dot S for J squared,
759
00:58:34,180 --> 00:58:36,490
L squared, and S squared.
760
00:58:36,490 --> 00:58:39,590
L squared is very
simple, and S squared
761
00:58:39,590 --> 00:58:42,190
is extremely simple as well.
762
00:58:42,190 --> 00:58:46,520
Remember, L squared
commutes with any Li.
763
00:58:46,520 --> 00:58:51,340
So L squared with
any Li is equal to 0.
764
00:58:51,340 --> 00:58:55,740
S squared with any
Si is equal to 0.
765
00:58:55,740 --> 00:58:58,180
And Li's and Si's commute.
766
00:58:58,180 --> 00:58:59,780
They live in different worlds.
767
00:58:59,780 --> 00:59:03,280
So L squared and Si's commute.
768
00:59:03,280 --> 00:59:06,720
S squareds and Li's commute.
769
00:59:06,720 --> 00:59:11,230
These things are
pretty nice and simple.
770
00:59:11,230 --> 00:59:15,600
So let's think now
of our Hamiltonian
771
00:59:15,600 --> 00:59:22,260
and what is happening to it.
772
00:59:22,260 --> 00:59:31,260
Whenever we had
the hydrogen atom,
773
00:59:31,260 --> 00:59:39,410
we had a set of commuting
observables H, L squared,
774
00:59:39,410 --> 00:59:40,120
and Lz.
775
00:59:44,390 --> 00:59:48,470
It's a complete set of
commuting observables.
776
00:59:48,470 --> 00:59:53,020
Now, in the hydrogen
atom, you could add to it
777
00:59:53,020 --> 00:59:57,620
S squared and Sz.
778
00:59:57,620 --> 01:00:00,160
We didn't talk about
spin at the beginning,
779
01:00:00,160 --> 01:00:02,760
because we just considered
a particle going
780
01:00:02,760 --> 01:00:04,520
around the hydrogen atom.
781
01:00:04,520 --> 01:00:09,330
But if you have spin, the
hydrogen atom Hamiltonian,
782
01:00:09,330 --> 01:00:13,820
the original one, doesn't
involve spin in any way.
783
01:00:13,820 --> 01:00:16,890
So certainly, Hamiltonian
commutes with spin,
784
01:00:16,890 --> 01:00:18,740
with spin z.
785
01:00:18,740 --> 01:00:21,880
L and S don't talk, so
this is the complete set
786
01:00:21,880 --> 01:00:24,870
of commuting observables.
787
01:00:24,870 --> 01:00:27,215
But what happens to this list?
788
01:00:27,215 --> 01:00:35,210
This is our problem for
H0, the hydrogen atom,
789
01:00:35,210 --> 01:00:43,450
plus delta H that
has the S dot L.
790
01:00:43,450 --> 01:00:49,100
Well, what are complete set
of commuting observables?
791
01:00:49,100 --> 01:00:51,550
This is a very
important question.
792
01:00:51,550 --> 01:00:53,560
Because this is
what tells you how
793
01:00:53,560 --> 01:00:56,050
you're going to try to
organize the spectrum.
794
01:00:56,050 --> 01:01:01,880
So we could have H,
the total, H total.
795
01:01:05,800 --> 01:01:08,960
And what else?
796
01:01:08,960 --> 01:01:14,630
Well, can I still
have L squared here?
797
01:01:18,450 --> 01:01:23,135
Can I include L squared and say
it commutes with the total H?
798
01:01:27,229 --> 01:01:30,170
A little worrisome,
but actually,
799
01:01:30,170 --> 01:01:35,500
you know that L squared commutes
with the original Hamiltonian.
800
01:01:35,500 --> 01:01:38,260
Now, the question is
whether L squared commutes
801
01:01:38,260 --> 01:01:40,320
with this extra piece.
802
01:01:40,320 --> 01:01:44,110
Well, but L squared
commutes with any Li.
803
01:01:44,110 --> 01:01:47,730
And it doesn't even talk
to S. So L squared is safe.
804
01:01:47,730 --> 01:01:50,400
L squared we can keep.
805
01:01:50,400 --> 01:01:55,660
OK, S squared-- can
we keep S squared?
806
01:01:55,660 --> 01:01:57,480
Well, S squared was here.
807
01:01:57,480 --> 01:02:01,420
So it commuted with the
Hamiltonian, and that was good.
808
01:02:01,420 --> 01:02:06,000
S squared commutes with any
Si, and it doesn't talk to L.
809
01:02:06,000 --> 01:02:07,730
So S squared can stay.
810
01:02:11,040 --> 01:02:13,550
But that's not good enough.
811
01:02:13,550 --> 01:02:16,950
We won't be able to solve
the problem with this still.
812
01:02:16,950 --> 01:02:17,790
We need more.
813
01:02:21,100 --> 01:02:22,166
How about Lz?
814
01:02:22,166 --> 01:02:24,885
It was here, so
let's try our luck.
815
01:02:29,010 --> 01:02:32,340
Any opinions on Lz--
can we keep it or not?
816
01:02:37,840 --> 01:02:38,838
Yes.
817
01:02:38,838 --> 01:02:40,182
AUDIENCE: I don't think so.
818
01:02:40,182 --> 01:02:44,117
Because in the J term, we
have Lx's and Ly's, which
819
01:02:44,117 --> 01:02:45,340
don't commute with Lz.
820
01:02:45,340 --> 01:02:47,980
PROFESSOR: Right,
it can't be kept.
821
01:02:47,980 --> 01:02:57,540
Here, this term has SxLx
plus SyLy plus SzLz.
822
01:02:57,540 --> 01:03:02,370
And Lz doesn't
commute with this one.
823
01:03:02,370 --> 01:03:04,995
So no, you can't
keep Lz-- no good.
824
01:03:09,680 --> 01:03:14,496
On the other hand, let's
think about J squared.
825
01:03:17,640 --> 01:03:22,670
J squared is here.
826
01:03:22,670 --> 01:03:26,685
And J squared commutes with
L squared and with S squared.
827
01:03:29,690 --> 01:03:36,600
J squared, therefore, is--
well, let me say it this way.
828
01:03:36,600 --> 01:03:42,460
Here is L dot S, which
is our extra interaction.
829
01:03:42,460 --> 01:03:44,920
Here we have this thing.
830
01:03:44,920 --> 01:03:49,700
I would like to say
on behalf of J squared
831
01:03:49,700 --> 01:03:56,200
that we can include
it here, J squared,
832
01:03:56,200 --> 01:04:00,840
because J squared is
really pretty much the same
833
01:04:00,840 --> 01:04:05,860
as L dot S up to this L
squared and S squared.
834
01:04:05,860 --> 01:04:09,640
But J squared commutes with
L squared and S squared.
835
01:04:09,640 --> 01:04:11,400
I should probably
write it there.
836
01:04:15,120 --> 01:04:22,350
J squared commutes
with L squared.
837
01:04:22,350 --> 01:04:29,010
And J squared communicates with
S squared that we have here.
838
01:04:29,010 --> 01:04:38,300
And moreover, we have over
here that J squared therefore
839
01:04:38,300 --> 01:04:41,960
will commute, or it's
pretty much the same,
840
01:04:41,960 --> 01:04:45,600
as L dot S. J squared
with L dot S would
841
01:04:45,600 --> 01:04:50,810
be J squared times
this thing, which is 0.
842
01:04:50,810 --> 01:04:55,840
So J squared commutes
with this term.
843
01:04:55,840 --> 01:04:59,210
And it commutes with
the Hamiltonian,
844
01:04:59,210 --> 01:05:02,410
your original
hydrogen Hamiltonian.
845
01:05:02,410 --> 01:05:06,240
So J squared can be added here.
846
01:05:09,670 --> 01:05:13,980
J square is a good
operator to have.
847
01:05:13,980 --> 01:05:19,110
And now we can get one more
kind of free from here.
848
01:05:19,110 --> 01:05:21,290
It's Jz.
849
01:05:21,290 --> 01:05:23,280
Z
850
01:05:23,280 --> 01:05:29,630
Because Jz commutes
with J squared.
851
01:05:29,630 --> 01:05:32,140
Jz commutes with these things.
852
01:05:32,140 --> 01:05:38,090
And Jz, which is a symmetry
of the original Hamiltonian,
853
01:05:38,090 --> 01:05:44,342
also commutes with our new
interaction, the L dot S,
854
01:05:44,342 --> 01:05:47,770
which is proportional
to J squared.
855
01:05:47,770 --> 01:05:53,340
So you have to go
through this yourselves
856
01:05:53,340 --> 01:05:56,980
probably even a little
more slowly than I've gone.
857
01:05:56,980 --> 01:05:59,660
Just check that
everything that I'm
858
01:05:59,660 --> 01:06:02,480
saying about whatever
commutes commutes.
859
01:06:02,480 --> 01:06:06,350
So for example, when I say
that J squared commutes
860
01:06:06,350 --> 01:06:09,700
with L dot S, it's
because I can put
861
01:06:09,700 --> 01:06:12,590
instead of L dot S all of this.
862
01:06:12,590 --> 01:06:15,920
And go slowly through this.
863
01:06:15,920 --> 01:06:22,030
So this is actually the complete
set of committing observables.
864
01:06:22,030 --> 01:06:25,510
And it's basically
saying to us, try
865
01:06:25,510 --> 01:06:31,755
to diagonalize this thing
with total angular momentum.
866
01:06:34,590 --> 01:06:38,290
So it's about time
to really do it.
867
01:06:38,290 --> 01:06:39,790
We haven't done it yet.
868
01:06:39,790 --> 01:06:43,930
But now the part that
we have to do now,
869
01:06:43,930 --> 01:06:46,470
it's kind of a nice exercise.
870
01:06:46,470 --> 01:06:48,970
And it's fun.
871
01:06:48,970 --> 01:06:52,180
Now, there's one problem
in the homework set
872
01:06:52,180 --> 01:06:55,710
that sort of uses
this kind of thing.
873
01:06:55,710 --> 01:07:01,850
And I will suggest there to
Will and Aram that tomorrow,
874
01:07:01,850 --> 01:07:07,120
they spend some time discussing
it and helping you with it.
875
01:07:07,120 --> 01:07:10,930
The last problem
in the homework set
876
01:07:10,930 --> 01:07:15,060
would've been better if you
had a little more time for it
877
01:07:15,060 --> 01:07:17,930
and you had more time to
digest what I'm doing today.
878
01:07:17,930 --> 01:07:21,920
But nevertheless,
go to recitation,
879
01:07:21,920 --> 01:07:23,490
learn more about the problem.
880
01:07:23,490 --> 01:07:26,740
It will not be all
that difficult.
881
01:07:26,740 --> 01:07:31,900
OK, so we're trying
now to finally form
882
01:07:31,900 --> 01:07:33,890
another basis of states.
883
01:07:33,890 --> 01:07:36,970
We had these six states.
884
01:07:36,970 --> 01:07:39,020
And we're going to
try to organize them
885
01:07:39,020 --> 01:07:45,480
in a better way-- as eigenstates
of the total angular momentum L
886
01:07:45,480 --> 01:07:51,210
plus S. So I'm going to
write them here in this way.
887
01:07:51,210 --> 01:07:58,830
Here is one of the
states of this L equals
888
01:07:58,830 --> 01:08:04,330
1 electron, the 1, 1
coupled to the 1/2, 1/2.
889
01:08:04,330 --> 01:08:20,740
Here are two more states- 1, 0,
1/2, 1/2, 1, 1, 1/2, minus 1/2,
890
01:08:20,740 --> 01:08:27,850
so the 1, 0 with the top,
the 1, 1 with the bottom.
891
01:08:27,850 --> 01:08:35,020
Here are two more
states-- 1, 0 with 1/2,
892
01:08:35,020 --> 01:08:43,620
minus 1/2 and 1,
minus 1 with 1/2, 1/2.
893
01:08:47,790 --> 01:08:56,100
And here is the last state--
1, minus 1 with 1/2, minus 1.
894
01:09:02,520 --> 01:09:05,832
These are our six states.
895
01:09:05,832 --> 01:09:09,654
And I've organized them
in a nice way actually.
896
01:09:12,760 --> 01:09:14,990
I've organized
them in such a way
897
01:09:14,990 --> 01:09:21,750
that you can read what is
the value of Jz over h bar.
898
01:09:21,750 --> 01:09:28,563
Remember, Jz is 1
over h bar Lz plus Sz.
899
01:09:34,080 --> 01:09:35,479
So what is it?
900
01:09:35,479 --> 01:09:39,760
These are, I claim,
eigenstates of Jz.
901
01:09:39,760 --> 01:09:40,899
Why?
902
01:09:40,899 --> 01:09:42,569
Because let's act on them.
903
01:09:42,569 --> 01:09:45,510
Suppose I act with
Jz on this state.
904
01:09:45,510 --> 01:09:47,930
The Lz comes here and says, 1.
905
01:09:47,930 --> 01:09:50,890
The Sz comes here and says, 1/2.
906
01:09:50,890 --> 01:09:56,337
So the sum of them give you
Jz over h bar equal to 3/2.
907
01:10:01,080 --> 01:10:05,240
And that's why I organized
these states in such a way
908
01:10:05,240 --> 01:10:10,510
that these second things add up
to the same value-- 0 and 1/2,
909
01:10:10,510 --> 01:10:12,590
1 and minus 1/2.
910
01:10:12,590 --> 01:10:16,360
So if you act with
Jz on this state,
911
01:10:16,360 --> 01:10:19,330
it's an eigenstate with Jz.
912
01:10:19,330 --> 01:10:22,470
Here, 0 contribution, here 1/2.
913
01:10:22,470 --> 01:10:26,700
So this is with plus 1/2.
914
01:10:26,700 --> 01:10:31,130
Here, you have 0 and minus 1/2,
minus 1, and that is minus 1/2.
915
01:10:34,330 --> 01:10:36,930
And here you have minus 3/2.
916
01:10:40,990 --> 01:10:43,943
OK, questions.
917
01:10:48,980 --> 01:10:50,330
We've written the states.
918
01:10:50,330 --> 01:10:55,240
And I'm evaluating the total z
component of angular momentum.
919
01:10:55,240 --> 01:10:58,470
And these two states
are like that.
920
01:10:58,470 --> 01:11:02,150
So what does our theorem
guarantee for us?
921
01:11:02,150 --> 01:11:06,670
Our theorem guarantees that we
have-- in this tensor product,
922
01:11:06,670 --> 01:11:10,770
there is an algebra of angular
momentum of the Jz operators.
923
01:11:10,770 --> 01:11:14,270
And the states have to
fall into representations
924
01:11:14,270 --> 01:11:15,930
of those operators.
925
01:11:15,930 --> 01:11:19,920
So you must have angular
momentum multiplets.
926
01:11:19,920 --> 01:11:23,710
So at this moment,
you can figure out
927
01:11:23,710 --> 01:11:30,240
what angular momentum you're
going to get for the result.
928
01:11:30,240 --> 01:11:36,770
Here we obtained a
maximum Jz of 3/2.
929
01:11:36,770 --> 01:11:42,005
So we must get a J
equals 3/2 multiplet.
930
01:11:46,100 --> 01:11:48,660
Because a J equaling
3/2 multiplets
931
01:11:48,660 --> 01:11:55,650
has Jz 3/2, 1/2,
minus 1/2, and 0.
932
01:11:55,650 --> 01:12:01,510
So actually, this state must be
the top state of the multiplet.
933
01:12:01,510 --> 01:12:05,500
This state must be the bottom
state of the multiplet.
934
01:12:05,500 --> 01:12:10,160
I don't know which one is the
middle state of the multiplet
935
01:12:10,160 --> 01:12:12,250
and which one is here.
936
01:12:12,250 --> 01:12:15,670
But we have four states
here, four states.
937
01:12:18,560 --> 01:12:22,940
So one linear combination
of these two states
938
01:12:22,940 --> 01:12:29,190
must be, then, that Jz equals
1/2 state of the multiplet.
939
01:12:29,190 --> 01:12:31,600
And one inner combination
of these two states
940
01:12:31,600 --> 01:12:35,220
must be that Jz equals minus
1/2 state of the multiplet.
941
01:12:35,220 --> 01:12:36,750
Which one is it?
942
01:12:36,750 --> 01:12:38,180
I don't know.
943
01:12:38,180 --> 01:12:40,010
But we can figure it out.
944
01:12:40,010 --> 01:12:41,930
We'll figure it out in a second.
945
01:12:41,930 --> 01:12:44,640
Once you get this
J 3/2 multiplet,
946
01:12:44,640 --> 01:12:48,270
there will be one linear
combination here left over
947
01:12:48,270 --> 01:12:51,500
and one linear combination
here left over.
948
01:12:51,500 --> 01:12:56,300
Those are two state, one with
Jz plus 1/2 and one with Jz
949
01:12:56,300 --> 01:12:58,060
equals minus 1/2.
950
01:12:58,060 --> 01:13:01,740
So you also get a J
equals 1/2 multiplet.
951
01:13:07,670 --> 01:13:11,290
So the whole tensor
product of six
952
01:13:11,290 --> 01:13:18,260
states-- it was the
tensor product of a spin 1
953
01:13:18,260 --> 01:13:21,620
with a spin 1/2.
954
01:13:21,620 --> 01:13:24,560
So we write it like this.
955
01:13:24,560 --> 01:13:32,940
The tensor product of a
spin 1 with a spin 1/2
956
01:13:32,940 --> 01:13:43,660
will give you a total
spin 3/2 plus total spin
957
01:13:43,660 --> 01:13:52,940
1/2-- funny formula.
958
01:13:52,940 --> 01:13:56,820
Here is the tensor
product, the tensor
959
01:13:56,820 --> 01:14:01,710
product of these three
states with these two states.
960
01:14:01,710 --> 01:14:08,720
This can be written as 3
times 2 is equal to 4 plus 2
961
01:14:08,720 --> 01:14:11,870
in terms of number of states.
962
01:14:11,870 --> 01:14:16,020
The tensor product of
this spin 1 and spin 1/2
963
01:14:16,020 --> 01:14:21,040
gives you a spin 3/2
multiplet with four states
964
01:14:21,040 --> 01:14:23,520
and a spin 1/2 multiplet
with two states.
965
01:14:27,700 --> 01:14:33,040
So how do you calculate what
are the states themselves?
966
01:14:35,890 --> 01:14:38,380
So the states themselves
are the following.
967
01:14:47,100 --> 01:14:51,280
All right, here I have them.
968
01:14:51,280 --> 01:14:57,560
I claim that the J
equals 3/2 states,
969
01:14:57,560 --> 01:15:02,610
m equals 3/2 states, the
top state of that multiplet
970
01:15:02,610 --> 01:15:11,860
can only be the state here,
the 1, 1 tensor 1/2, 1/2.
971
01:15:11,860 --> 01:15:18,490
And there's no way any other
state can be put on the right.
972
01:15:18,490 --> 01:15:22,020
Because there's no other
state with total z component
973
01:15:22,020 --> 01:15:24,160
of angular momentum equals 3/2.
974
01:15:24,160 --> 01:15:26,490
So that must be the state.
975
01:15:26,490 --> 01:15:33,000
Similarly, the J
equals 3/2, m equals
976
01:15:33,000 --> 01:15:40,980
minus 3/2 state must be the
bottom one-- 1, minus 1, 1/2,
977
01:15:40,980 --> 01:15:44,220
minus 1/2.
978
01:15:44,220 --> 01:15:46,490
The one that we
wish to figure out
979
01:15:46,490 --> 01:15:53,710
is the next state here, which is
the J equals 3/2, m equals 1/2.
980
01:15:53,710 --> 01:15:57,200
It's a linear
combination of these two.
981
01:15:57,200 --> 01:15:58,390
But which one?
982
01:16:01,060 --> 01:16:05,250
That is kind of the last
thing we want to do.
983
01:16:05,250 --> 01:16:08,520
Because it will pretty much
solve the rest of the problem.
984
01:16:13,880 --> 01:16:17,540
So how do we solve for this?
985
01:16:17,540 --> 01:16:25,270
Well, we had this
basic relation that we
986
01:16:25,270 --> 01:16:36,470
know how to lower or raise
states of angular momentum--
987
01:16:36,470 --> 01:16:43,530
m times m plus/minus 1 J--
I should have written it
988
01:16:43,530 --> 01:16:52,510
J plus/minus Jm equals
h bar square root.
989
01:16:52,510 --> 01:16:56,790
More space for everybody
to see this-- J times
990
01:16:56,790 --> 01:17:02,180
J plus 1 minus m
times m plus/minus 1.
991
01:17:02,180 --> 01:17:08,120
Close the square
root-- Jm plus/minus 1.
992
01:17:08,120 --> 01:17:14,020
So what I should try to
do is lower this state,
993
01:17:14,020 --> 01:17:19,630
try to find this state
by acting with J minus.
994
01:17:19,630 --> 01:17:22,660
So let me try to
lower the state, so
995
01:17:22,660 --> 01:17:31,170
J minus on this state, on
J equals 3/2, m equals 3/2.
996
01:17:31,170 --> 01:17:39,770
I can go to that formula and
write it as h bar square root.
997
01:17:39,770 --> 01:17:47,260
J is 3/2, so 3/2 times
5/2 minus m, which is 3/2,
998
01:17:47,260 --> 01:17:49,960
times m minus 1, 1/2.
999
01:17:49,960 --> 01:17:57,570
We're doing the minus--
times the state 3/2, 1/2.
1000
01:17:57,570 --> 01:18:00,030
So the state we want is here.
1001
01:18:00,030 --> 01:18:03,570
And it's obtained by
doing J minus on that.
1002
01:18:03,570 --> 01:18:05,320
But we want the number here.
1003
01:18:05,320 --> 01:18:08,340
So that's why I did
all these square roots.
1004
01:18:08,340 --> 01:18:16,650
And that just gives h bar
square root of 3, 3/2, 1/2.
1005
01:18:16,650 --> 01:18:19,670
Well, that still doesn't
calculate it for me.
1006
01:18:19,670 --> 01:18:21,510
But it comes very close.
1007
01:18:29,490 --> 01:18:33,250
So you have it there.
1008
01:18:33,250 --> 01:18:39,580
Now I want to do this but
using the right hand side.
1009
01:18:39,580 --> 01:18:41,560
So look at the right hand side.
1010
01:18:41,560 --> 01:18:52,570
We want to do J minus, but
on 1, 1 tensor 1/2, 1/2.
1011
01:18:52,570 --> 01:18:55,820
So I applied J minus
to the left hand side.
1012
01:18:55,820 --> 01:19:01,130
Now we have to apply J minus
to the right hand side.
1013
01:19:01,130 --> 01:19:19,680
But J minus is L minus plus S
minus on 1, 1 tensor 1/2, 1/2.
1014
01:19:19,680 --> 01:19:22,750
When this acts, it
acts on the first.
1015
01:19:22,750 --> 01:19:31,570
So you get L minus on
1, 1 tensor 1/2, 1/2.
1016
01:19:31,570 --> 01:19:38,280
And in the second term,
you get plus 1, 1 tensor S
1017
01:19:38,280 --> 01:19:40,520
minus on 1/2, 1/2.
1018
01:19:44,420 --> 01:19:47,540
Now, what is L minus on 1, 1?
1019
01:19:47,540 --> 01:19:50,330
You can use the same formula.
1020
01:19:50,330 --> 01:19:51,450
It's 1, 1.
1021
01:19:51,450 --> 01:19:53,250
And it's an angular momentum.
1022
01:19:53,250 --> 01:20:00,350
So it just goes on and
gives you h bar square
1023
01:20:00,350 --> 01:20:05,520
root of 1 times 2
minus 1 times 0.
1024
01:20:05,520 --> 01:20:10,720
1, 0-- it lowers
it-- times 1/2, 1/2.
1025
01:20:15,080 --> 01:20:19,270
Let me go here-- plus 1, 1.
1026
01:20:19,270 --> 01:20:21,980
And what is S minus on this?
1027
01:20:21,980 --> 01:20:26,390
Use the formula
with J equals 1/2.
1028
01:20:26,390 --> 01:20:37,460
So this is h bar square root of
1/2 times 3/2 minus 1/2 times
1029
01:20:37,460 --> 01:20:43,522
minus 1/2 times 1/2 minus 1/2.
1030
01:20:43,522 --> 01:20:48,620
Whew-- well not too difficult.
1031
01:20:48,620 --> 01:20:57,630
But this gives you h over square
root of 2, 1, 0 tensor 1/2,
1032
01:20:57,630 --> 01:21:01,440
1/2 plus just h bar.
1033
01:21:01,440 --> 01:21:11,030
This whole thing is 1-- 1,
1 tensor 1/2, minus 1/2.
1034
01:21:11,030 --> 01:21:16,800
OK, stop a second to
see what's happened.
1035
01:21:16,800 --> 01:21:18,730
We had this equality.
1036
01:21:18,730 --> 01:21:20,260
And we acted with J minus.
1037
01:21:20,260 --> 01:21:23,390
Acting on the left,
it gives us a number
1038
01:21:23,390 --> 01:21:25,980
times the state we want.
1039
01:21:25,980 --> 01:21:29,950
Acting on the
right, we got this.
1040
01:21:29,950 --> 01:21:36,220
So actually, equating this
to that, or left hand side
1041
01:21:36,220 --> 01:21:41,270
to right hand side, we finally
found the state 3/2, 1/2.
1042
01:21:41,270 --> 01:21:49,630
So the state 3/2,
1/2 is as follows.
1043
01:21:54,410 --> 01:22:06,930
3/2, 1/2 is-- you must
divide by that square root.
1044
01:22:06,930 --> 01:22:09,435
So you get the square
root of 3 down.
1045
01:22:09,435 --> 01:22:12,280
The h bars cancel.
1046
01:22:12,280 --> 01:22:18,290
So here it is, a very nice
little formula-- 2 over 3,
1047
01:22:18,290 --> 01:22:26,450
1, 0 tensor 1/2, 1/2 plus
1 over square root of 3,
1048
01:22:26,450 --> 01:22:34,010
1, 1 tensor 1/2, minus 1/2.
1049
01:22:34,010 --> 01:22:36,730
So we have the top
state of the multiplet.
1050
01:22:40,070 --> 01:22:43,910
We have the next state
of the multiplet.
1051
01:22:43,910 --> 01:22:47,240
We have-- I'm sorry, the
top state of the multiplet
1052
01:22:47,240 --> 01:22:49,250
was this.
1053
01:22:49,250 --> 01:22:51,830
You have the bottom
state of the multiplet,
1054
01:22:51,830 --> 01:22:53,890
the middle state
of the multiplet.
1055
01:22:53,890 --> 01:22:58,920
What you're missing is the
bottom and the middle term.
1056
01:22:58,920 --> 01:23:02,935
And this one can be
obtained in many ways.
1057
01:23:05,660 --> 01:23:08,990
One way would be to
raise this state.
1058
01:23:08,990 --> 01:23:12,330
The minus 3/2 could
be raised by one unit
1059
01:23:12,330 --> 01:23:14,710
and do exactly the same thing.
1060
01:23:14,710 --> 01:23:24,760
Well, the result is square root
of 2 over 3, 1, 0 tensor 1/2,
1061
01:23:24,760 --> 01:23:30,550
minus 1/2 plus 1 over
square root of 3.
1062
01:23:30,550 --> 01:23:35,240
That square root of 2
doesn't look right to me now.
1063
01:23:35,240 --> 01:23:37,830
I must have copied it wrong.
1064
01:23:37,830 --> 01:23:43,940
It's 1 over square root of
3-- 1 over square root of 3,
1065
01:23:43,940 --> 01:23:50,660
1, minus 1 tensor 1/2, 1/2.
1066
01:23:50,660 --> 01:23:52,365
So you've built that
whole multiplet.
1067
01:23:55,420 --> 01:24:00,480
And this state, as we said,
was a linear combination
1068
01:24:00,480 --> 01:24:01,996
of the two possible states.
1069
01:24:05,800 --> 01:24:09,430
This 3 minus 1/2 was
a linear combination
1070
01:24:09,430 --> 01:24:11,530
of these two possible states.
1071
01:24:11,530 --> 01:24:14,690
So the other states
that are left over,
1072
01:24:14,690 --> 01:24:17,785
the other linear
combinations, form
1073
01:24:17,785 --> 01:24:21,090
the J equals 1/2 multiplet.
1074
01:24:21,090 --> 01:24:26,050
So basically, every state must
be orthogonal to each other.
1075
01:24:26,050 --> 01:24:35,200
So the other state, the 1/2, 1/2
and the 1/2, minus 1/2 of the J
1076
01:24:35,200 --> 01:24:41,430
equals 1/2 multiplet must
be this orthogonal to this.
1077
01:24:41,430 --> 01:24:44,810
And this must be
orthogonal to that.
1078
01:24:44,810 --> 01:24:50,760
So those formulas are easily
found by orthogonality.
1079
01:24:50,760 --> 01:24:56,760
So I'll conclude by
writing them-- minus 1
1080
01:24:56,760 --> 01:25:04,010
over square root of
3, 1, 0, 1/2, 1/2
1081
01:25:04,010 --> 01:25:13,170
plus the square root of 2
over 3, 1, 1, 1/2, minus 1/2.
1082
01:25:13,170 --> 01:25:19,770
And here, you get 1 over
square root of 3, 1, 0, 1/2,
1083
01:25:19,770 --> 01:25:31,340
minus 1/2 minus 2 over
square root of 3, 1, minus 1
1084
01:25:31,340 --> 01:25:33,380
tensor 1/2, 1/2.
1085
01:25:36,350 --> 01:25:45,870
So lots of terms, a little
hard to read-- I apologize.
1086
01:25:45,870 --> 01:25:53,120
Now, the punchline here is
that you've found these states.
1087
01:25:53,120 --> 01:25:57,050
And the claim is
that these are states
1088
01:25:57,050 --> 01:26:00,610
in which L dot S is diagonal.
1089
01:26:00,610 --> 01:26:04,310
And it's kind of obvious
that that should be the case.
1090
01:26:04,310 --> 01:26:07,830
Because what was L dot S?
1091
01:26:07,830 --> 01:26:20,150
So one last formula--
L dot S equals
1092
01:26:20,150 --> 01:26:26,220
1/2 of J squared minus L
squared minus S squared.
1093
01:26:26,220 --> 01:26:29,770
Now, in terms of
eigenvalues, this
1094
01:26:29,770 --> 01:26:36,930
is 1/2 h squared J times
J plus 1 minus L times L
1095
01:26:36,930 --> 01:26:42,440
plus 1 minus S times S plus 1.
1096
01:26:42,440 --> 01:26:45,160
Now, all the states
that we built
1097
01:26:45,160 --> 01:26:48,070
have definite
values of J squared,
1098
01:26:48,070 --> 01:26:50,450
definite values of S squared.
1099
01:26:50,450 --> 01:26:52,740
Because L was 1.
1100
01:26:52,740 --> 01:26:55,270
And S is 1/2.
1101
01:26:55,270 --> 01:27:03,400
So here you go h squared over 2
J times J plus 1 minus 1 times
1102
01:27:03,400 --> 01:27:09,450
2 is 2 minus 1/2
times 3/2 is 3/4.
1103
01:27:12,050 --> 01:27:13,630
And that's the whole story.
1104
01:27:13,630 --> 01:27:17,170
The whole story in a sense
has been summarized by this.
1105
01:27:17,170 --> 01:27:22,210
We have four states
with J equals 3/2
1106
01:27:22,210 --> 01:27:25,850
and two states
with J equals 1/2.
1107
01:27:25,850 --> 01:27:31,190
So these six states
that you have here--
1108
01:27:31,190 --> 01:27:33,690
split because of
this interaction
1109
01:27:33,690 --> 01:27:42,700
into four states that have
J equal to 3/2 and two
1110
01:27:42,700 --> 01:27:47,370
states that have J equal to 1/2.
1111
01:27:47,370 --> 01:27:49,640
And you plug the numbers here.
1112
01:27:49,640 --> 01:27:51,456
And that gives you the
amount of splitting.
1113
01:27:54,790 --> 01:28:02,810
So actually, this height
that this goes up here
1114
01:28:02,810 --> 01:28:05,230
is h squared over 2.
1115
01:28:05,230 --> 01:28:08,030
And this is minus h
squared by the time you
1116
01:28:08,030 --> 01:28:11,860
put the numbers J, 3/2, and 1/2.
1117
01:28:11,860 --> 01:28:17,220
So all our work was because
the Hamiltonian at the end
1118
01:28:17,220 --> 01:28:20,360
was simple in J squared.
1119
01:28:20,360 --> 01:28:23,240
And therefore, we
needed J multiplets.
1120
01:28:23,240 --> 01:28:27,900
J multiplets are the addition
of angular momentum multiplets.
1121
01:28:27,900 --> 01:28:31,270
In a sense, we don't have
to construct these things
1122
01:28:31,270 --> 01:28:34,910
if you don't want to calculate
very explicit details.
1123
01:28:34,910 --> 01:28:37,960
Once you have that,
you have everything.
1124
01:28:37,960 --> 01:28:42,620
This product of angular
momentum 1, angular momentum 1/2
1125
01:28:42,620 --> 01:28:45,800
gave you total
angular momentum 3/2
1126
01:28:45,800 --> 01:28:48,990
and 1/2-- four
states, two states.
1127
01:28:48,990 --> 01:28:53,380
So four states split one way,
two states split the other way,
1128
01:28:53,380 --> 01:28:55,210
and that's the end of the story.
1129
01:28:55,210 --> 01:28:58,340
So more of this in recitation
and more of this all
1130
01:28:58,340 --> 01:29:00,580
of next week.
1131
01:29:00,580 --> 01:29:02,790
We'll see you then.