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PROFESSOR: All right, it
is time to get started.
9
00:00:25,940 --> 00:00:30,750
Thanks for coming for
this cold and rainy
10
00:00:30,750 --> 00:00:32,540
Wednesday before Thanksgiving.
11
00:00:35,490 --> 00:00:40,150
Today we're supposed to talk
about the radial equation.
12
00:00:40,150 --> 00:00:43,710
That's our main subject today.
13
00:00:43,710 --> 00:00:47,640
We discussed last
time the states
14
00:00:47,640 --> 00:00:51,430
of angular momentum from
the abstract viewpoint,
15
00:00:51,430 --> 00:00:55,460
and now we make contact with
some important problems,
16
00:00:55,460 --> 00:01:00,390
and differential equations,
and things like that.
17
00:01:00,390 --> 00:01:05,730
And there's a few concepts
I want to emphasize today.
18
00:01:05,730 --> 00:01:09,570
And basically, the
main concept is
19
00:01:09,570 --> 00:01:13,480
that I want you to just become
familiar with what we would
20
00:01:13,480 --> 00:01:18,330
call the diagram, the key
diagram for the states
21
00:01:18,330 --> 00:01:23,250
of a theory, of a particle in
a three dimensional potential.
22
00:01:23,250 --> 00:01:27,860
I think you have to have a good
understanding of what it looks,
23
00:01:27,860 --> 00:01:30,700
and what is special
about it, and when
24
00:01:30,700 --> 00:01:32,790
it shows particular properties.
25
00:01:32,790 --> 00:01:37,730
So to begin with, I'll
have to do a little aside
26
00:01:37,730 --> 00:01:43,296
on a object that is
covered in many courses.
27
00:01:43,296 --> 00:01:46,050
I don't know to what
that it's covered,
28
00:01:46,050 --> 00:01:48,920
but it's the subject
of spherical harmonics.
29
00:01:48,920 --> 00:01:53,730
So we'll talk about spherical
harmonics for about 15 minutes.
30
00:01:53,730 --> 00:01:56,295
And then we'll do
the radial equation.
31
00:01:56,295 --> 00:01:59,360
And for the radial equation,
after we discuss it,
32
00:01:59,360 --> 00:02:01,550
we'll do three examples.
33
00:02:01,550 --> 00:02:05,610
And that will be the
end of today's lecture.
34
00:02:05,610 --> 00:02:10,830
Next time, as you come back
from the holiday next week,
35
00:02:10,830 --> 00:02:14,470
we are doing the addition of
angular momentum basically.
36
00:02:14,470 --> 00:02:19,070
And then the last week,
more examples and a few more
37
00:02:19,070 --> 00:02:23,420
things for emphasis to
understand it all well.
38
00:02:23,420 --> 00:02:26,610
All right, so in terms
of spherical harmonics,
39
00:02:26,610 --> 00:02:37,530
I wanted to emphasize that
our algebraic analysis led
40
00:02:37,530 --> 00:02:42,380
to states that we
called jm, but today I
41
00:02:42,380 --> 00:02:47,750
will call lm, because they
will refer to orbital angular
42
00:02:47,750 --> 00:02:49,180
momentum.
43
00:02:49,180 --> 00:02:52,860
And as you've seen in
one of your problems,
44
00:02:52,860 --> 00:02:55,360
orbital angular
momentum has to do
45
00:02:55,360 --> 00:03:01,620
with values of j,
which are integers.
46
00:03:01,620 --> 00:03:09,670
So half integers values of j
cannot be realized for orbital
47
00:03:09,670 --> 00:03:10,820
angular momentum.
48
00:03:10,820 --> 00:03:12,460
It's a very interesting thing.
49
00:03:12,460 --> 00:03:18,840
So spin states don't have wave
functions in the usual way.
50
00:03:18,840 --> 00:03:23,490
It's only states of
integer angular momentum
51
00:03:23,490 --> 00:03:26,260
that have wave functions.
52
00:03:26,260 --> 00:03:27,905
And those are the
spherical harmonics.
53
00:03:27,905 --> 00:03:32,330
So I will talk about
lm, and l, as usual,
54
00:03:32,330 --> 00:03:36,040
will go from 0 to infinity.
55
00:03:36,040 --> 00:03:40,910
And m goes from l to minus l.
56
00:03:43,420 --> 00:03:48,660
And you had these states, and
we said that algebraically you
57
00:03:48,660 --> 00:03:56,440
would have L squared equals h
squared l times l plus 1 lm.
58
00:03:56,440 --> 00:04:03,960
And Lz lm equal hm lm.
59
00:04:07,610 --> 00:04:10,440
Now basically, the
spherical harmonics
60
00:04:10,440 --> 00:04:17,339
are going to be wave
functions for these states.
61
00:04:17,339 --> 00:04:20,019
And the way we
can approach it is
62
00:04:20,019 --> 00:04:24,790
that we did a little bit of work
already with constructing the l
63
00:04:24,790 --> 00:04:27,120
squared operator.
64
00:04:27,120 --> 00:04:34,050
And in last lecture we derived,
starting from the fact that L
65
00:04:34,050 --> 00:04:39,720
is r cross p and using
x,y, and z, px, py, pz,
66
00:04:39,720 --> 00:04:44,270
and passing through spherical
coordinates that L squared is
67
00:04:44,270 --> 00:04:52,230
the operator minus h squared 1
over sine theta d d theta sine
68
00:04:52,230 --> 00:04:59,295
theta d d theta again plus
1 over sine squared theta d
69
00:04:59,295 --> 00:05:00,866
second d phi squared.
70
00:05:03,830 --> 00:05:10,220
And we didn't do it,
but Lz, which you know
71
00:05:10,220 --> 00:05:23,790
is h bar over i x d dy
minus y d dx can also
72
00:05:23,790 --> 00:05:28,320
be translated into
angular variables.
73
00:05:28,320 --> 00:05:31,820
And it has a very simple form.
74
00:05:31,820 --> 00:05:33,740
Also purely angular.
75
00:05:33,740 --> 00:05:38,550
And you can interpret it Lz is
rotations around the z-axis,
76
00:05:38,550 --> 00:05:40,420
so they change phi.
77
00:05:40,420 --> 00:05:42,530
So it will not
surprise you, if you
78
00:05:42,530 --> 00:05:48,840
do this exercise, that
this is h over i d d phi.
79
00:05:48,840 --> 00:05:52,870
And you should really check it.
80
00:05:52,870 --> 00:05:57,020
There's another one that
is a bit more laborious.
81
00:05:57,020 --> 00:06:03,630
L plus minus, remember,
is Lx plus minus i Lz.
82
00:06:03,630 --> 00:06:06,010
We have a big attendance today.
83
00:06:10,250 --> 00:06:13,900
Is equal to-- more people.
84
00:06:13,900 --> 00:06:30,630
h bar e to the plus minus i phi
i cosine theta over sine theta
85
00:06:30,630 --> 00:06:36,130
d d phi plus minus d d theta.
86
00:06:41,000 --> 00:06:44,010
And that takes a bit of algebra.
87
00:06:44,010 --> 00:06:45,300
You could do it.
88
00:06:45,300 --> 00:06:47,520
It's done in many books.
89
00:06:47,520 --> 00:06:50,040
It's probably there
in Griffith's.
90
00:06:50,040 --> 00:06:52,590
And these are the
representations
91
00:06:52,590 --> 00:06:56,840
of these operators as
differential operators that
92
00:06:56,840 --> 00:06:59,690
act and function
on theta and phi
93
00:06:59,690 --> 00:07:03,940
and don't care about radius.
94
00:07:03,940 --> 00:07:08,910
So in mathematical physics,
people study these things
95
00:07:08,910 --> 00:07:14,460
and invent these things
called spherical harmonics
96
00:07:14,460 --> 00:07:19,210
Ylm's of theta and phi.
97
00:07:19,210 --> 00:07:22,940
And the way you
could see their done
98
00:07:22,940 --> 00:07:28,230
is in fact, such that
this L squared viewed
99
00:07:28,230 --> 00:07:34,870
as this operator, differential
operator, acting on Ylm
100
00:07:34,870 --> 00:07:43,690
is indeed equal to h squared
l times l plus 1 Ylm.
101
00:07:43,690 --> 00:07:49,890
And Lz thought also as a
differential operator, the one
102
00:07:49,890 --> 00:07:51,930
that we've written there.
103
00:07:51,930 --> 00:07:58,960
On the Ylm is h bar m Ylm.
104
00:08:01,700 --> 00:08:06,430
So they are constructed
in this way,
105
00:08:06,430 --> 00:08:10,100
satisfying these equations.
106
00:08:10,100 --> 00:08:12,980
These are important equations
in mathematical physics,
107
00:08:12,980 --> 00:08:14,930
and these functions
were invented
108
00:08:14,930 --> 00:08:18,380
to satisfy those equations.
109
00:08:18,380 --> 00:08:24,130
Well, these are the properties
of those states over there.
110
00:08:24,130 --> 00:08:30,620
So we can think
of these functions
111
00:08:30,620 --> 00:08:34,429
as the wave functions
associated with those states.
112
00:08:34,429 --> 00:08:39,289
So that's interpretation that
is natural in quantum mechanics.
113
00:08:39,289 --> 00:08:42,210
And we want to think
of them like that.
114
00:08:42,210 --> 00:08:50,780
We want to think of the Ylm's
as the wave functions associated
115
00:08:50,780 --> 00:08:52,900
to the states lm.
116
00:08:52,900 --> 00:08:56,760
So lm.
117
00:08:56,760 --> 00:09:02,575
And here you would put a
position state theta phi.
118
00:09:08,610 --> 00:09:11,360
This is analogous to the
thing that we usually
119
00:09:11,360 --> 00:09:17,090
call the wave function being a
position state times the state
120
00:09:17,090 --> 00:09:18,590
side.
121
00:09:18,590 --> 00:09:25,440
So we want to think of
the Ylm's in this way
122
00:09:25,440 --> 00:09:29,850
as pretty much the wave
functions associated
123
00:09:29,850 --> 00:09:32,550
to those states.
124
00:09:32,550 --> 00:09:42,160
Now there is a little bit of
identities that come once you
125
00:09:42,160 --> 00:09:45,150
accept that this is what
you think of the Ylm's.
126
00:09:45,150 --> 00:09:50,480
And then the compatibility
of these equations.
127
00:09:50,480 --> 00:09:57,390
Top here with these ones makes
in this identification natural.
128
00:09:57,390 --> 00:10:01,920
Now in order to manipulate
and learn things
129
00:10:01,920 --> 00:10:03,950
about those spherical
harmonics the way
130
00:10:03,950 --> 00:10:06,220
we do things in
quantum mechanics,
131
00:10:06,220 --> 00:10:10,190
we think of the
completeness relation.
132
00:10:10,190 --> 00:10:17,450
If we have d cube x x x, this
is a completeness relation
133
00:10:17,450 --> 00:10:19,770
for position states.
134
00:10:24,500 --> 00:10:31,320
And I want to derive or
suggest a completeness relation
135
00:10:31,320 --> 00:10:33,760
for these theta phi states.
136
00:10:33,760 --> 00:10:38,900
For that, I would
pass this integral
137
00:10:38,900 --> 00:10:41,170
to do it in spherical
coordinates.
138
00:10:41,170 --> 00:10:52,290
So I would do dr rd
theta r sine theta d phi.
139
00:10:52,290 --> 00:10:59,100
And I would put r theta
phi position states
140
00:10:59,100 --> 00:11:01,050
for these things.
141
00:11:01,050 --> 00:11:06,180
And position states r theta phi.
142
00:11:06,180 --> 00:11:07,810
Still being equal to 1.
143
00:11:13,510 --> 00:11:21,920
And we can try to
split this thing.
144
00:11:21,920 --> 00:11:25,360
It's natural for us to
think of just theta phi,
145
00:11:25,360 --> 00:11:30,390
because these wave functions
have nothing to do with r,
146
00:11:30,390 --> 00:11:34,940
so I will simply do
the integrals this way.
147
00:11:34,940 --> 00:11:39,780
d theta sine theta d phi.
148
00:11:39,780 --> 00:11:45,360
And think just like a
position state in x, y, z.
149
00:11:45,360 --> 00:11:48,770
It's a position state in x,
in y, and in z multiplied.
150
00:11:48,770 --> 00:11:52,280
We'll just split these
things without trying
151
00:11:52,280 --> 00:11:55,790
to be too rigorous about it.
152
00:11:55,790 --> 00:12:00,700
Theta and phi like this.
153
00:12:00,700 --> 00:12:11,845
And you would have the integral
dr r squared r r equal 1.
154
00:12:15,800 --> 00:12:21,410
And at this point,
I want to think
155
00:12:21,410 --> 00:12:29,580
of this as the natural way of
setting a completeness relation
156
00:12:29,580 --> 00:12:32,060
for theta and phi.
157
00:12:32,060 --> 00:12:35,360
And this doesn't
talk to this one,
158
00:12:35,360 --> 00:12:40,260
so I will think of this that
in the space of theta and phi,
159
00:12:40,260 --> 00:12:42,740
objects that just
depend on theta and phi,
160
00:12:42,740 --> 00:12:45,570
this acts as a complete thing.
161
00:12:45,570 --> 00:12:48,220
And if objects depend
also in r, this
162
00:12:48,220 --> 00:12:50,580
will act as a complete thing.
163
00:12:50,580 --> 00:12:53,140
So I will-- I don't know.
164
00:12:53,140 --> 00:12:56,130
Maybe the right way
to say is postulate
165
00:12:56,130 --> 00:12:59,960
that we'll have a completeness
relation of this form.
166
00:12:59,960 --> 00:13:12,975
d theta sine theta d phi
theta phi theta phi equals 1.
167
00:13:23,260 --> 00:13:31,070
And then with this we can
do all kinds of things.
168
00:13:31,070 --> 00:13:34,230
First, this integral
is better written.
169
00:13:34,230 --> 00:13:42,520
This integral really represents
0 to pi d theta sine theta 0
170
00:13:42,520 --> 00:13:46,690
to 2 pi d phi.
171
00:13:46,690 --> 00:13:49,785
Now this is minus
d cosine theta.
172
00:13:57,530 --> 00:14:02,020
And when theta is equal
to 0, cosine theta
173
00:14:02,020 --> 00:14:10,280
is 1 to minus 1 integral
d phi 0 to 2 pi.
174
00:14:10,280 --> 00:14:18,550
So this integral, really
d theta sine theta d
175
00:14:18,550 --> 00:14:25,090
phi this is really the
integral from minus 1 to 1.
176
00:14:25,090 --> 00:14:32,955
Change that order of d cos theta
integral d phi from 0 to 2 pi.
177
00:14:37,650 --> 00:14:43,227
And this is called the
integral over solid angle.
178
00:14:43,227 --> 00:14:44,060
That's a definition.
179
00:14:47,710 --> 00:14:51,930
So we could write the
completeness relation
180
00:14:51,930 --> 00:14:57,350
in the space theta phi as
integral over solid angle theta
181
00:14:57,350 --> 00:15:02,800
phi theta phi equals 1.
182
00:15:08,280 --> 00:15:12,460
Then the key property of
the spherical harmonics,
183
00:15:12,460 --> 00:15:20,220
or the lm states, is
that they are orthogonal.
184
00:15:20,220 --> 00:15:25,870
So delta l, l prime,
delta m, m prime.
185
00:15:25,870 --> 00:15:28,180
So the orthogonality
are of this state
186
00:15:28,180 --> 00:15:31,860
is guaranteed because
Hermitian operators,
187
00:15:31,860 --> 00:15:36,790
different eigenvalues,
they have to be orthogonal.
188
00:15:36,790 --> 00:15:38,990
Eigenstates of Hermitian.
189
00:15:38,990 --> 00:15:41,400
Operators with
different eigenvalues.
190
00:15:41,400 --> 00:15:46,670
Here, you introduce a complete
set of states of theta phi.
191
00:15:46,670 --> 00:15:58,950
So you put l prime m prime
theta phi theta phi lm.
192
00:16:02,340 --> 00:16:12,850
And this is the integral over
solid angle of Yl prime m
193
00:16:12,850 --> 00:16:17,610
prime of theta phi star.
194
00:16:17,610 --> 00:16:21,400
This is in the wrong position.
195
00:16:21,400 --> 00:16:27,866
And here Ylm of theta
phi being equal delta l
196
00:16:27,866 --> 00:16:31,440
l prime delta m m prime.
197
00:16:37,650 --> 00:16:47,630
So this is orthogonality
of the spherical harmonics.
198
00:16:47,630 --> 00:16:50,570
And this is pretty
much all we need.
199
00:16:50,570 --> 00:16:58,690
Now there's the standard ways
of constructing these things
200
00:16:58,690 --> 00:17:02,860
from the quantum mechanical
sort of intuition.
201
00:17:02,860 --> 00:17:07,450
Basically, you can
try to first build
202
00:17:07,450 --> 00:17:15,425
Yll, which corresponds
to the state ll.
203
00:17:20,050 --> 00:17:23,390
Now the kind of
differential equations
204
00:17:23,390 --> 00:17:27,349
this Yll satisfies
are kind of simple.
205
00:17:27,349 --> 00:17:30,100
But in particular,
the most important one
206
00:17:30,100 --> 00:17:34,980
is that L plus kills this state.
207
00:17:34,980 --> 00:17:37,720
So basically you
use the condition
208
00:17:37,720 --> 00:17:40,920
that L plus kills this
state to find a differential
209
00:17:40,920 --> 00:17:46,050
equation for this, which
can be solved easily.
210
00:17:46,050 --> 00:17:47,850
Not a hard
differential equation.
211
00:17:47,850 --> 00:17:50,300
Then you find Yll.
212
00:17:50,300 --> 00:17:57,700
And then you can find Yll
minus 1 and all the other ones
213
00:17:57,700 --> 00:18:01,400
by applying the
operator L minus.
214
00:18:01,400 --> 00:18:03,960
The lowering operator of m.
215
00:18:03,960 --> 00:18:06,420
So in principle, if you
have enough patience,
216
00:18:06,420 --> 00:18:10,130
you can calculate all the
spherical harmonics that way.
217
00:18:10,130 --> 00:18:14,240
There's no obstruction.
218
00:18:14,240 --> 00:18:17,170
But the form is a
little messy, and if you
219
00:18:17,170 --> 00:18:21,640
want to find the normalizations
so that these things work out
220
00:18:21,640 --> 00:18:25,790
correctly, well, it takes some
work at the end of the day.
221
00:18:25,790 --> 00:18:30,680
So we're not going
to do that here.
222
00:18:30,680 --> 00:18:34,500
We'll just leave it at
that, and if we ever
223
00:18:34,500 --> 00:18:40,260
need some special harmonics,
we'll just hold the answers.
224
00:18:40,260 --> 00:18:44,250
And they are in most textbooks.
225
00:18:44,250 --> 00:18:46,940
So if you do need
them, well, you'll
226
00:18:46,940 --> 00:18:49,620
have to do with
complicated normalizations.
227
00:18:52,230 --> 00:18:56,190
So that's really all I wanted to
say about spherical harmonics,
228
00:18:56,190 --> 00:19:00,210
and we can turn then to
the real subject, which
229
00:19:00,210 --> 00:19:02,720
is the radial equation.
230
00:19:02,720 --> 00:19:03,870
So the radial equation.
231
00:19:12,440 --> 00:19:17,480
So we have a Hamiltonian
H equals p squared vector
232
00:19:17,480 --> 00:19:21,160
over 2m plus v of r.
233
00:19:21,160 --> 00:19:23,960
And we've seen that
this is equal to h over
234
00:19:23,960 --> 00:19:33,030
2m 1 over r d second dr
squared r plus 1 over 2mr
235
00:19:33,030 --> 00:19:37,755
squared L squared plus v of r.
236
00:19:40,500 --> 00:19:42,695
So this is what we're
trying to solve.
237
00:19:45,510 --> 00:19:48,070
And the way we
attempt to solve this
238
00:19:48,070 --> 00:19:49,920
is by separation of variables.
239
00:19:49,920 --> 00:19:54,600
So we'll try to write the wave
function, psi, characterized
240
00:19:54,600 --> 00:19:56,980
by three things.
241
00:19:56,980 --> 00:20:04,680
Its energy, the value of
l, and the value of m.
242
00:20:04,680 --> 00:20:08,350
And it's a function of
position, because we're
243
00:20:08,350 --> 00:20:12,380
trying to solve H
psi equal E psi.
244
00:20:12,380 --> 00:20:16,580
And that's the energy
that we want to consider.
245
00:20:16,580 --> 00:20:20,980
So I will write here to begin
with something that will not
246
00:20:20,980 --> 00:20:25,840
turn out to be exactly
right, but it's
247
00:20:25,840 --> 00:20:29,520
important to do
it first this way.
248
00:20:29,520 --> 00:20:36,700
A function of art r that
has labels E, l, and m.
249
00:20:36,700 --> 00:20:40,000
Because it certainly could
depend on E, could depend on l,
250
00:20:40,000 --> 00:20:43,100
and could depend on m,
that radial function.
251
00:20:43,100 --> 00:20:45,420
And then the angular
function will
252
00:20:45,420 --> 00:20:50,530
be the Ylm's of theta and phi.
253
00:20:50,530 --> 00:20:53,060
So this is the [INAUDIBLE]
sets for the equation.
254
00:20:57,870 --> 00:21:03,390
If we have that, we can plug
into the Schrodinger equation,
255
00:21:03,390 --> 00:21:04,790
and see what we get.
256
00:21:04,790 --> 00:21:10,010
Well, this operator
will act on this f.
257
00:21:10,010 --> 00:21:15,520
This will have the
operator L squared,
258
00:21:15,520 --> 00:21:18,900
but L squared over Ylm,
you know what it is.
259
00:21:18,900 --> 00:21:23,040
And v of r is multiplicative,
so it's no big problem.
260
00:21:23,040 --> 00:21:24,230
So what do we have?
261
00:21:24,230 --> 00:21:30,070
We have minus h squared
over 2m 1 over r.
262
00:21:30,070 --> 00:21:32,720
Now I can talk
normal derivatives.
263
00:21:32,720 --> 00:21:46,900
d r squared r times fElm
plus 1 over 2mr squared.
264
00:21:46,900 --> 00:21:49,690
And now have L squared
acting on this,
265
00:21:49,690 --> 00:21:54,680
but L squared acting on the
Ylm is just this factor.
266
00:21:54,680 --> 00:22:03,640
So we have h squared l times
l plus 1 times the fElm.
267
00:22:08,850 --> 00:22:12,910
Now I didn't put the
Ylm in the first term
268
00:22:12,910 --> 00:22:15,200
because I'm going to
cancel it throughout.
269
00:22:15,200 --> 00:22:28,700
So we have this term here plus
v of r fElm equals E fElm.
270
00:22:34,140 --> 00:22:39,670
That is substituting into the
equation h psi equal E psi.
271
00:22:39,670 --> 00:22:42,490
So first term here.
272
00:22:42,490 --> 00:22:46,300
Second term, it acted on
the spherical harmonic.
273
00:22:46,300 --> 00:22:47,840
v of r is multiplicative.
274
00:22:47,840 --> 00:22:50,030
E on that.
275
00:22:50,030 --> 00:22:52,500
But then what you
see immediately
276
00:22:52,500 --> 00:22:55,960
is that this differential
equation doesn't depend on m.
277
00:22:59,550 --> 00:23:04,200
It was L squared, but no
Lz in the Hamiltonian.
278
00:23:04,200 --> 00:23:06,610
So no m dependent.
279
00:23:06,610 --> 00:23:12,290
So actually we
were overly proven
280
00:23:12,290 --> 00:23:17,370
in thinking that f
was a function of m.
281
00:23:17,370 --> 00:23:21,390
What we really have
is that psi Elm
282
00:23:21,390 --> 00:23:28,530
is equal to a function of E
and l or r Ylm of theta phi.
283
00:23:31,350 --> 00:23:33,760
And then the
differential equation
284
00:23:33,760 --> 00:23:37,230
is minus h squared over 2m.
285
00:23:37,230 --> 00:23:40,220
Let's multiply all by r.
286
00:23:40,220 --> 00:23:44,942
d second dr squared of r fEl.
287
00:23:50,410 --> 00:23:51,540
Plus look here.
288
00:23:55,760 --> 00:24:00,070
The r that I'm multiplying
is going to go into the f.
289
00:24:00,070 --> 00:24:01,700
Here it's going
to go into the f.
290
00:24:01,700 --> 00:24:03,260
Here it's going
to go into the f.
291
00:24:03,260 --> 00:24:04,680
It's an overall thing.
292
00:24:04,680 --> 00:24:09,100
But here we keep h
squared l times l
293
00:24:09,100 --> 00:24:26,090
plus 1 over 2mr squared rfEl
plus v of r fEl rfEl equal
294
00:24:26,090 --> 00:24:27,555
e times rfEl.
295
00:24:36,470 --> 00:24:42,825
So what you see here is that
this function is quite natural.
296
00:24:46,390 --> 00:24:53,432
So it suggests the
definition of uEl to be rfEl.
297
00:24:59,530 --> 00:25:03,060
So that the differential
equation now finally becomes
298
00:25:03,060 --> 00:25:13,930
minus h squared over 2m d
second dr squared of uEl plus
299
00:25:13,930 --> 00:25:18,035
there's the u here, the u
here, and this potential
300
00:25:18,035 --> 00:25:20,310
that has two terms.
301
00:25:20,310 --> 00:25:26,920
So this will be v of
r plus h squared l
302
00:25:26,920 --> 00:25:36,036
times l plus 1 over 2mr
squared uEl equals E times eEl.
303
00:25:40,540 --> 00:25:42,920
And this is the famous
radial equation.
304
00:25:47,100 --> 00:25:50,870
It's an equation for you.
305
00:25:50,870 --> 00:25:57,250
And here, this whole
thing is sometimes
306
00:25:57,250 --> 00:25:58,940
called the effective potential.
307
00:26:04,670 --> 00:26:07,280
So look what we've got.
308
00:26:07,280 --> 00:26:16,056
This f, if you wish here,
is now of the form uEl
309
00:26:16,056 --> 00:26:24,040
of r Ylm over r theta phi.
310
00:26:24,040 --> 00:26:25,960
f is u over r.
311
00:26:25,960 --> 00:26:30,520
So this is the way we've
written the solution,
312
00:26:30,520 --> 00:26:35,010
and u satisfies this equation,
which is a one dimensional
313
00:26:35,010 --> 00:26:38,856
Schrodinger equation
for the radius r.
314
00:26:38,856 --> 00:26:42,990
One dimensional equation
with an effective potential
315
00:26:42,990 --> 00:26:44,990
that depends on L.
316
00:26:44,990 --> 00:26:49,580
So actually the first
thing you have to notice
317
00:26:49,580 --> 00:26:52,890
is that the central
potential problem
318
00:26:52,890 --> 00:27:00,080
has turned into an
infinite collection of one
319
00:27:00,080 --> 00:27:01,510
dimensional problems.
320
00:27:01,510 --> 00:27:04,956
One for each value of l.
321
00:27:04,956 --> 00:27:09,940
For different values of l, you
have a different potential.
322
00:27:09,940 --> 00:27:12,310
Now they're not
all that different.
323
00:27:12,310 --> 00:27:15,530
They have different
intensity of this term.
324
00:27:15,530 --> 00:27:20,200
For l equals 0, well
you have some solutions.
325
00:27:20,200 --> 00:27:23,780
And for l equal 1, the answer
could be quite different.
326
00:27:23,780 --> 00:27:26,410
For l equal 2, still different.
327
00:27:26,410 --> 00:27:30,210
And you have to solve
an infinite number
328
00:27:30,210 --> 00:27:35,330
of one dimensional problems.
329
00:27:35,330 --> 00:27:39,380
That's what the Schrodinger
equation has turned into.
330
00:27:39,380 --> 00:27:45,210
So we filled all
these blackboards.
331
00:27:45,210 --> 00:27:48,420
Let's see, are there questions?
332
00:27:48,420 --> 00:27:50,470
Anything so far?
333
00:28:01,068 --> 00:28:01,567
Yes?
334
00:28:07,536 --> 00:28:09,326
AUDIENCE: You might
get to this later,
335
00:28:09,326 --> 00:28:13,303
but what does it mean
in our wave equations,
336
00:28:13,303 --> 00:28:18,780
in our wave function there,
psi of Elm is equal to fEl,
337
00:28:18,780 --> 00:28:22,516
and the spherical harmonic
of that one mean that one has
338
00:28:22,516 --> 00:28:24,320
an independence and
the other doesn't.
339
00:28:24,320 --> 00:28:27,550
Can they be separated
on the basis of m?
340
00:28:27,550 --> 00:28:33,520
PROFESSOR: So it is just a fact
that the radial solution is
341
00:28:33,520 --> 00:28:37,010
independent of n, so it's
an important property.
342
00:28:37,010 --> 00:28:39,900
n is fairly simple.
343
00:28:39,900 --> 00:28:43,500
The various state, the
states with angular momentum
344
00:28:43,500 --> 00:28:48,180
l, but different m's just differ
in their angular dependence,
345
00:28:48,180 --> 00:28:52,010
not in the radial dependence.
346
00:28:52,010 --> 00:28:54,230
And practically,
it means that you
347
00:28:54,230 --> 00:28:59,960
have an infinite set of one
dimensional problems labeled
348
00:28:59,960 --> 00:29:04,790
by l, and not labeled by m,
which conceivably could have
349
00:29:04,790 --> 00:29:07,240
happened, but it doesn't happen.
350
00:29:07,240 --> 00:29:10,380
So just a major simplicity.
351
00:29:10,380 --> 00:29:11,614
Yes?
352
00:29:11,614 --> 00:29:13,030
AUDIENCE: Does the
radial equation
353
00:29:13,030 --> 00:29:15,784
have all the same properties as
a one dimensional Schrodinger
354
00:29:15,784 --> 00:29:16,284
equation?
355
00:29:16,284 --> 00:29:18,720
Or does the divergence in
the effect [INAUDIBLE] 0
356
00:29:18,720 --> 00:29:19,220
change that?
357
00:29:19,220 --> 00:29:21,470
PROFESSOR: Well,
it changes things,
358
00:29:21,470 --> 00:29:24,640
but the most serious
change is the fact
359
00:29:24,640 --> 00:29:28,650
that, in one
dimensional problems,
360
00:29:28,650 --> 00:29:32,640
x goes from minus
infinity to infinity.
361
00:29:32,640 --> 00:29:35,100
And here it goes
from 0 to infinity,
362
00:29:35,100 --> 00:29:38,510
so we need to worry
about what happens at 0.
363
00:29:38,510 --> 00:29:41,590
Basically that's the
main complication.
364
00:29:41,590 --> 00:29:46,670
One dimensional potential,
but it really just
365
00:29:46,670 --> 00:29:48,370
can't go below 0.
366
00:29:48,370 --> 00:29:53,470
r is a radial variable,
and we can't forget that.
367
00:29:53,470 --> 00:29:54,290
Yes?
368
00:29:54,290 --> 00:29:58,224
AUDIENCE: The potential v of r
will depend on whatever problem
369
00:29:58,224 --> 00:29:59,140
you're solving, right?
370
00:29:59,140 --> 00:30:00,139
PROFESSOR: That's right.
371
00:30:00,139 --> 00:30:04,670
AUDIENCE: Could you find
the v of r [INAUDIBLE]?
372
00:30:04,670 --> 00:30:07,090
PROFESSOR: Well
that doesn't quite
373
00:30:07,090 --> 00:30:09,620
make sense as a Hamiltonian.
374
00:30:09,620 --> 00:30:13,270
You see, if you
have a v of r, it's
375
00:30:13,270 --> 00:30:17,180
something that is supposed to
be v of r for any wave function.
376
00:30:17,180 --> 00:30:18,380
That's the definition.
377
00:30:18,380 --> 00:30:20,900
So it can depend
on some parameter,
378
00:30:20,900 --> 00:30:25,230
but that parameter cannot be
the l of the particular wave
379
00:30:25,230 --> 00:30:28,092
function.
380
00:30:28,092 --> 00:30:32,370
AUDIENCE: [INAUDIBLE] or
something that would interact
381
00:30:32,370 --> 00:30:32,870
with the--
382
00:30:32,870 --> 00:30:34,790
PROFESSOR: If you
have magnetic fields,
383
00:30:34,790 --> 00:30:38,860
things change, because
then you can split levels
384
00:30:38,860 --> 00:30:40,900
with respect to m.
385
00:30:40,900 --> 00:30:43,640
Break degeneracies and
things change indeed.
386
00:30:46,780 --> 00:30:49,960
We'll take care of those by
using perturbation theory
387
00:30:49,960 --> 00:30:50,760
mostly.
388
00:30:50,760 --> 00:30:53,840
Use this solution and
then perturbation theory.
389
00:30:53,840 --> 00:31:01,420
OK, so let's proceed
a little more on this.
390
00:31:01,420 --> 00:31:07,710
So the first thing that we
want to talk a little about
391
00:31:07,710 --> 00:31:11,440
is the normalization and
some boundary conditions,
392
00:31:11,440 --> 00:31:14,950
because otherwise
we can't really
393
00:31:14,950 --> 00:31:17,610
understand what's going on.
394
00:31:17,610 --> 00:31:22,630
And happily the discussion
is not that complicated.
395
00:31:22,630 --> 00:31:24,450
So we want to normalize.
396
00:31:24,450 --> 00:31:25,600
So what do we want?
397
00:31:25,600 --> 00:31:39,230
Integral d cube x psi Elm
of x squared equals 1.
398
00:31:39,230 --> 00:31:44,200
So clearly we want to go
into angular variables.
399
00:31:44,200 --> 00:31:53,290
So again, this is r squared
dr integral d solid angle,
400
00:31:53,290 --> 00:31:54,530
r squared Er.
401
00:31:54,530 --> 00:32:10,800
And this thing is now uEl
squared absolute value
402
00:32:10,800 --> 00:32:12,250
over r squared.
403
00:32:12,250 --> 00:32:14,230
Look at the right
most blackboard.
404
00:32:14,230 --> 00:32:19,570
uEl of r, I must square it
because the wave function
405
00:32:19,570 --> 00:32:20,770
is squared.
406
00:32:20,770 --> 00:32:22,720
Over r squared.
407
00:32:22,720 --> 00:32:31,565
And then I have Ylm star of
theta phi Ylm of theta phi.
408
00:32:34,560 --> 00:32:37,330
And if this is supposed
to be normalized,
409
00:32:37,330 --> 00:32:39,245
this is supposed
to be the number 1.
410
00:32:42,280 --> 00:32:46,220
Well happily, this
part, this is why
411
00:32:46,220 --> 00:32:50,570
we needed to talk a little
about spherical harmonics.
412
00:32:50,570 --> 00:32:55,630
This integral is 1, because
it corresponds precisely
413
00:32:55,630 --> 00:32:59,650
to l equal l prime
m equal m prime.
414
00:32:59,650 --> 00:33:04,610
And look how lucky
or nice this is.
415
00:33:04,610 --> 00:33:07,020
r squared cancels
with r squared,
416
00:33:07,020 --> 00:33:10,120
so the final condition
is the integral from 0
417
00:33:10,120 --> 00:33:23,000
to infinity dr uEl of r
squared is equal to 1,
418
00:33:23,000 --> 00:33:26,770
which shows that
kind of the u really
419
00:33:26,770 --> 00:33:30,120
plays a role for wave
function and a line.
420
00:33:30,120 --> 00:33:32,760
And even though it was
a little complicated,
421
00:33:32,760 --> 00:33:35,520
there was the r here,
and angular dependence,
422
00:33:35,520 --> 00:33:38,530
and everything, a
good wave function
423
00:33:38,530 --> 00:33:43,240
is one that is just
think of psi as being u.
424
00:33:43,240 --> 00:33:46,630
A one dimensional wave
function psi being u,
425
00:33:46,630 --> 00:33:49,784
and if you can integrate
it square, you've got it.
426
00:33:49,784 --> 00:33:50,700
AUDIENCE: [INAUDIBLE].
427
00:33:53,890 --> 00:33:57,900
PROFESSOR: Because I
had to square this,
428
00:33:57,900 --> 00:33:59,662
so there was u over r.
429
00:33:59,662 --> 00:34:03,280
AUDIENCE: But
that's [INAUDIBLE].
430
00:34:03,280 --> 00:34:05,000
PROFESSOR: Oh, I'm sorry.
431
00:34:05,000 --> 00:34:09,449
That parenthesis is a remnant.
432
00:34:09,449 --> 00:34:11,284
I tried to erase it a little.
433
00:34:14,600 --> 00:34:16,830
It's not squared anymore.
434
00:34:16,830 --> 00:34:20,500
The square is on the
absolute value is r squared.
435
00:34:24,980 --> 00:34:27,276
So this is good news
for our interpretation.
436
00:34:34,230 --> 00:34:39,050
So now before I discuss the
peculiarities of the boundary
437
00:34:39,050 --> 00:34:45,352
conditions, I want to
introduce really the main point
438
00:34:45,352 --> 00:34:47,310
that we're going to
illustrate in this lecture.
439
00:34:47,310 --> 00:34:51,029
This is the thing that
should remain in your heads.
440
00:34:54,900 --> 00:35:00,390
It's a picture, but
it's an important one.
441
00:35:03,470 --> 00:35:06,330
When you want to
organize the spectrum,
442
00:35:06,330 --> 00:35:08,465
you'll draw the
following diagram.
443
00:35:12,740 --> 00:35:17,820
Energy is here and l here.
444
00:35:17,820 --> 00:35:19,590
And it's a funny
kind of diagram.
445
00:35:19,590 --> 00:35:21,660
It's not like a curve or a plot.
446
00:35:21,660 --> 00:35:26,760
It's like a histogram or
kind of thing like that.
447
00:35:26,760 --> 00:35:33,656
So what will happen is that you
have a one dimensional problem.
448
00:35:37,480 --> 00:35:44,200
If these potentials are normal,
there will be bound states.
449
00:35:44,200 --> 00:35:46,940
And let's consider the
case of bound states
450
00:35:46,940 --> 00:35:50,840
for the purposes of this
graph, just bound states.
451
00:35:50,840 --> 00:35:53,670
Now you look at
this, and you say OK,
452
00:35:53,670 --> 00:35:55,110
what am I supposed to do?
453
00:35:55,110 --> 00:36:00,335
I'm going to have states
for all values of l,
454
00:36:00,335 --> 00:36:03,560
and m, and probably
some energies.
455
00:36:03,560 --> 00:36:06,780
So m doesn't affect
the radial equation.
456
00:36:06,780 --> 00:36:08,160
That's very important.
457
00:36:08,160 --> 00:36:11,310
But l does, so I have
a different problem
458
00:36:11,310 --> 00:36:12,890
to solve for different l.
459
00:36:12,890 --> 00:36:18,790
So I will make my histogram
here and put here l
460
00:36:18,790 --> 00:36:22,570
equals 0 at this region.
461
00:36:22,570 --> 00:36:27,890
l equals 1, l equals 2,
l equals 3, and go on.
462
00:36:30,490 --> 00:36:35,200
Now suppose I fix an l.
463
00:36:35,200 --> 00:36:36,520
l is fixed.
464
00:36:36,520 --> 00:36:41,010
Now it's a Schrodinger equation
for a one dimensional problem.
465
00:36:41,010 --> 00:36:47,390
You would expect that if the
potential suitably grows, which
466
00:36:47,390 --> 00:36:51,790
is a typical case,
E will be quantized.
467
00:36:51,790 --> 00:36:54,830
And there will not
be degeneracies,
468
00:36:54,830 --> 00:36:57,740
because the bound state
spectrum in one dimension
469
00:36:57,740 --> 00:36:59,530
is not degenerate.
470
00:36:59,530 --> 00:37:03,730
So I should expect
that for each l there
471
00:37:03,730 --> 00:37:08,500
are going to be energy values
that are going to appear.
472
00:37:08,500 --> 00:37:14,780
So for l equals 0, I expect that
there will be some energy here
473
00:37:14,780 --> 00:37:16,600
for which I've got a state.
474
00:37:16,600 --> 00:37:19,490
And that line means
I got a state.
475
00:37:19,490 --> 00:37:23,385
And there's some energy here
that could be called E1,
476
00:37:23,385 --> 00:37:35,720
0 is the first energy that
is allowed with l equals 0.
477
00:37:35,720 --> 00:37:39,570
Then there will be
another one here maybe.
478
00:37:39,570 --> 00:37:44,480
E-- I'll write it down-- 2,0.
479
00:37:44,480 --> 00:37:53,620
So basically I'm labeling the
energies with En,l which means
480
00:37:53,620 --> 00:37:56,970
the first solution
with l equals 0,
481
00:37:56,970 --> 00:37:58,980
the second solution
with l equals 0,
482
00:37:58,980 --> 00:38:04,430
the third solution E 3,0.
483
00:38:04,430 --> 00:38:08,460
Then you come to l
equals 1, and you
484
00:38:08,460 --> 00:38:11,640
must solve the equation again.
485
00:38:11,640 --> 00:38:15,530
And then for l equal 1, there
will be the lowest energy,
486
00:38:15,530 --> 00:38:19,290
the ground state energy of
the l equal 1 potential,
487
00:38:19,290 --> 00:38:21,240
and then higher and higher.
488
00:38:21,240 --> 00:38:25,130
Since the l equal 1
potential is higher
489
00:38:25,130 --> 00:38:30,270
than the l equals 0
potential, it's higher up.
490
00:38:30,270 --> 00:38:33,010
The energies should
be higher up,
491
00:38:33,010 --> 00:38:35,600
at least the first
one should be.
492
00:38:35,600 --> 00:38:38,980
And therefore the
first one could
493
00:38:38,980 --> 00:38:44,190
be a little higher than this,
or maybe by some accident
494
00:38:44,190 --> 00:38:49,180
it just fits here, or
maybe it should fit here.
495
00:38:49,180 --> 00:38:51,540
Well, we don't know
but know, but there's
496
00:38:51,540 --> 00:38:55,760
no obvious reason why it
should, so I'll put it here.
497
00:38:55,760 --> 00:38:56,750
l equals 1.
498
00:38:56,750 --> 00:39:04,150
And this would be E1,1.
499
00:39:04,150 --> 00:39:07,460
The first state with l equals 1.
500
00:39:07,460 --> 00:39:11,150
Then here it could be E2,1.
501
00:39:11,150 --> 00:39:16,040
The second state with l
equal 1 and higher up.
502
00:39:16,040 --> 00:39:21,570
And then for l equal-- my
diagram is a little too big.
503
00:39:21,570 --> 00:39:25,200
E1,1.
504
00:39:25,200 --> 00:39:25,930
E2,1.
505
00:39:25,930 --> 00:39:30,230
And then you have states here,
so maybe this one, l equals 2,
506
00:39:30,230 --> 00:39:31,590
I don't know where it goes.
507
00:39:31,590 --> 00:39:36,350
It just has to be higher than
this one, so I'll put it here.
508
00:39:36,350 --> 00:39:39,760
And this will be E1,2.
509
00:39:39,760 --> 00:39:43,230
Maybe there's an E2,2.
510
00:39:43,230 --> 00:39:49,980
And here an E1,3.
511
00:39:49,980 --> 00:39:54,210
But this is the answer
to your problem.
512
00:39:54,210 --> 00:39:59,520
That's the energy levels
of a central potential.
513
00:39:59,520 --> 00:40:02,510
So it's a good,
nice little diagram
514
00:40:02,510 --> 00:40:07,210
in which you put the states,
you put the little line wherever
515
00:40:07,210 --> 00:40:08,590
you find the state.
516
00:40:08,590 --> 00:40:12,140
And for l equals 0,
you have those states.
517
00:40:12,140 --> 00:40:17,760
Now because there's no
degeneracies in the bound
518
00:40:17,760 --> 00:40:21,800
states of a one
dimensional potential,
519
00:40:21,800 --> 00:40:27,000
I don't have two lines here
that coincide, because there's
520
00:40:27,000 --> 00:40:30,180
no two states with
the same energy here.
521
00:40:30,180 --> 00:40:34,030
It's just one state.
522
00:40:34,030 --> 00:40:35,200
And this one here.
523
00:40:35,200 --> 00:40:38,460
I cannot have two things there.
524
00:40:38,460 --> 00:40:41,500
That's pretty important to.
525
00:40:41,500 --> 00:40:46,230
So you have a list
of states here.
526
00:40:46,230 --> 00:40:50,260
And just one state here, one
state, but as you can see,
527
00:40:50,260 --> 00:40:54,500
you're probably are catching
me in a little wrong play
528
00:40:54,500 --> 00:40:58,420
of words, because I say
there's one state here.
529
00:40:58,420 --> 00:41:01,130
Yes, it's one state,
because it's l equals 0.
530
00:41:01,130 --> 00:41:02,570
One state, one state.
531
00:41:02,570 --> 00:41:05,890
But this state,
which is one single--
532
00:41:05,890 --> 00:41:11,410
this should be called one
single l equal 1 multiplet.
533
00:41:11,410 --> 00:41:15,660
So this is not really one
state at the end of the day.
534
00:41:15,660 --> 00:41:19,940
It's one state of the one
dimensional radial equation,
535
00:41:19,940 --> 00:41:24,600
but you know that l
equals 1 comes accompanied
536
00:41:24,600 --> 00:41:28,250
with three values of m.
537
00:41:28,250 --> 00:41:31,750
So there's three states
that are degenerate,
538
00:41:31,750 --> 00:41:33,560
because they have
the same energy.
539
00:41:33,560 --> 00:41:37,430
The energy doesn't depend on l.
540
00:41:37,430 --> 00:41:43,440
So this thing is an
l equal 1 multiplet,
541
00:41:43,440 --> 00:41:47,730
which means really three states.
542
00:41:47,730 --> 00:41:50,090
And this is three states.
543
00:41:50,090 --> 00:41:52,420
And this is three states.
544
00:41:52,420 --> 00:41:57,860
And this is 1 l equal
2 multiplet, which
545
00:41:57,860 --> 00:42:03,160
has possibility of m equals
2, 1, 0 minus 1 and minus 2.
546
00:42:03,160 --> 00:42:06,960
So in this state is just
one l equal 2 multiplet,
547
00:42:06,960 --> 00:42:12,180
but it really means five states
of the central potential.
548
00:42:12,180 --> 00:42:15,860
Five degenerate
states, because the m
549
00:42:15,860 --> 00:42:17,810
doesn't change the energy.
550
00:42:17,810 --> 00:42:19,480
And this is five states.
551
00:42:19,480 --> 00:42:21,850
And this is seven states.
552
00:42:21,850 --> 00:42:28,250
One l equal 3 multiplet,
which contains seven states.
553
00:42:28,250 --> 00:42:30,870
OK, so questions?
554
00:42:30,870 --> 00:42:35,240
This is the most
important graph.
555
00:42:35,240 --> 00:42:37,330
If you have that
picture in your head,
556
00:42:37,330 --> 00:42:39,900
then you can
understand really where
557
00:42:39,900 --> 00:42:41,850
you're going with any potential.
558
00:42:41,850 --> 00:42:44,334
Any confusion here
above the notation?
559
00:42:44,334 --> 00:42:44,834
Yes?
560
00:42:44,834 --> 00:42:46,929
AUDIENCE: So normally
when we think about a one
561
00:42:46,929 --> 00:42:49,220
dimensional problem, we say
that there's no degeneracy.
562
00:42:49,220 --> 00:42:51,770
Not really.
563
00:42:51,770 --> 00:42:53,287
No multiple degeneracy,
so should we
564
00:42:53,287 --> 00:42:56,816
think of the radial equation as
having copies for each m value
565
00:42:56,816 --> 00:42:59,760
and each having the
same eigenvalue?
566
00:42:59,760 --> 00:43:02,450
PROFESSOR: I don't
think it's necessary.
567
00:43:02,450 --> 00:43:05,930
You see, you've got your uEl.
568
00:43:05,930 --> 00:43:07,660
And you have here you solutions.
569
00:43:07,660 --> 00:43:10,700
Once the uEl is
good, you're supposed
570
00:43:10,700 --> 00:43:13,340
to be able to put any Ylm.
571
00:43:13,340 --> 00:43:18,060
So put l, and now the m's that
are allowed are solutions.
572
00:43:18,060 --> 00:43:19,560
You're solving the problem.
573
00:43:19,560 --> 00:43:26,100
So think of a master radial
function as good for a fixed l,
574
00:43:26,100 --> 00:43:30,030
and therefore it works
for all values of m.
575
00:43:30,030 --> 00:43:33,310
But don't try to think of
many copies of this equation.
576
00:43:33,310 --> 00:43:37,010
I don't think it would help you.
577
00:43:37,010 --> 00:43:38,060
Any other questions?
578
00:43:42,956 --> 00:43:44,932
Yes?
579
00:43:44,932 --> 00:43:47,155
AUDIENCE: Sorry to ask,
but if you could just
580
00:43:47,155 --> 00:43:49,790
review how is degeneracy
built one more time?
581
00:43:49,790 --> 00:43:50,670
PROFESSOR: Yeah.
582
00:43:50,670 --> 00:43:53,390
Remember last time we
were talking about,
583
00:43:53,390 --> 00:44:02,020
for example, what is a
j equal to multiplet.
584
00:44:02,020 --> 00:44:05,860
Well, these were a collection
of states jm with j
585
00:44:05,860 --> 00:44:09,310
equals 2 an m sum value.
586
00:44:09,310 --> 00:44:14,320
And they are all obtained by
acting with angular momentum
587
00:44:14,320 --> 00:44:15,990
operators in each other.
588
00:44:15,990 --> 00:44:17,260
And there are five states.
589
00:44:17,260 --> 00:44:28,310
The 2,2, the 2,1, the 2,0, the
2, minus 1, and the 2, minus 2.
590
00:44:28,310 --> 00:44:30,240
And all these
states are obtained
591
00:44:30,240 --> 00:44:35,630
by acting with, say, lowering
operators l minus and this.
592
00:44:35,630 --> 00:44:38,930
Now all these angular
momentum operators,
593
00:44:38,930 --> 00:44:43,560
all of the Li's commute
with the Hamiltonian.
594
00:44:43,560 --> 00:44:46,200
Therefore all of
these states are
595
00:44:46,200 --> 00:44:49,720
obtained by acting with Li
must have the same energy.
596
00:44:49,720 --> 00:44:52,390
That's why we say that
this comes in a multiplet.
597
00:44:52,390 --> 00:44:59,100
So when you get j-- in this case
we'll call it l-- l equals 2.
598
00:44:59,100 --> 00:45:01,880
You get five states.
599
00:45:01,880 --> 00:45:04,980
They correspond to the
various values of m.
600
00:45:04,980 --> 00:45:07,320
So when you did that
radial equation that
601
00:45:07,320 --> 00:45:10,820
has a solution for
l equals 2, you're
602
00:45:10,820 --> 00:45:12,450
getting the full multiplet.
603
00:45:12,450 --> 00:45:14,460
You're getting five states.
604
00:45:14,460 --> 00:45:16,970
1 l equal 2 multiplet.
605
00:45:16,970 --> 00:45:19,280
That's why one line here.
606
00:45:19,280 --> 00:45:21,555
That is equivalent
to five states.
607
00:45:24,740 --> 00:45:31,770
OK, so that diagram, of course,
is really quite important.
608
00:45:31,770 --> 00:45:41,430
So now we want to understand
the boundary conditions.
609
00:45:41,430 --> 00:45:43,050
So we have here this.
610
00:45:43,050 --> 00:45:46,890
So this probably
shouldn't erase yet.
611
00:45:46,890 --> 00:45:48,480
Let's do the
boundary conditions.
612
00:45:58,390 --> 00:46:03,315
So behavior here
at r equals to 0.
613
00:46:08,720 --> 00:46:11,300
At r going to 0.
614
00:46:17,360 --> 00:46:21,780
The first claim is that
surprisingly, you would think,
615
00:46:21,780 --> 00:46:24,560
well, normalization is king.
616
00:46:24,560 --> 00:46:26,760
If it's normalized, it's good.
617
00:46:26,760 --> 00:46:29,470
So just any number.
618
00:46:29,470 --> 00:46:33,910
Just don't let it diverge
near 0, and that will be OK.
619
00:46:33,910 --> 00:46:36,940
But it turns out
that that's not true.
620
00:46:36,940 --> 00:46:38,290
It's not right.
621
00:46:38,290 --> 00:46:56,100
And you need the limit as r goes
to 0 of uEl of r be equal to 0.
622
00:46:56,100 --> 00:47:03,610
And we'll take this and
explore the simplest case.
623
00:47:03,610 --> 00:47:08,860
That is corresponds to
saying what if the limit of r
624
00:47:08,860 --> 00:47:15,270
goes to 0 or uEl of
r was a constant?
625
00:47:15,270 --> 00:47:17,870
What goes wrong?
626
00:47:17,870 --> 00:47:21,250
Certainly normalization
doesn't go wrong.
627
00:47:21,250 --> 00:47:23,880
It can be a constant.
628
00:47:23,880 --> 00:47:26,830
u could be like that, and
it would be normalized,
629
00:47:26,830 --> 00:47:29,270
and that doesn't go wrong.
630
00:47:29,270 --> 00:47:31,670
So let's look at
the wave function.
631
00:47:31,670 --> 00:47:35,440
What happens with this?
632
00:47:35,440 --> 00:47:38,160
I actually will
take for simplicity,
633
00:47:38,160 --> 00:47:45,840
because we'll analyze it later,
the example of l equals 0.
634
00:47:45,840 --> 00:47:48,720
So let's put even 0.
635
00:47:48,720 --> 00:47:49,750
l equals 0.
636
00:47:53,830 --> 00:48:04,006
Well, suppose you look
at the wave function now,
637
00:48:04,006 --> 00:48:06,700
and how does it look?
638
00:48:06,700 --> 00:48:16,270
Psi of E0-- if l is equal
to 0, m must be equal to 0--
639
00:48:16,270 --> 00:48:21,990
would be this u over
r times a constant.
640
00:48:21,990 --> 00:48:25,520
So a constant, because
y 0, 0 is a constant.
641
00:48:25,520 --> 00:48:30,640
And then you uE0 of r over r.
642
00:48:30,640 --> 00:48:41,050
So when r approaches 0, psi
goes like c prime over r,
643
00:48:41,050 --> 00:48:44,140
some other constant over r.
644
00:48:44,140 --> 00:48:46,240
So I'm doing
something very simple.
645
00:48:46,240 --> 00:48:53,360
I'm saying if uE0 is approaching
the constant at the origin,
646
00:48:53,360 --> 00:48:57,930
if it's uE0, well, this is
a constant because it's 0,0.
647
00:48:57,930 --> 00:48:59,800
So this is going to constant.
648
00:48:59,800 --> 00:49:01,890
So at the end of the
day, the wave function
649
00:49:01,890 --> 00:49:03,360
looks like 1 over r.
650
00:49:08,280 --> 00:49:16,820
But this is impossible, because
the Schrodinger equation H
651
00:49:16,820 --> 00:49:26,070
psi has minus h squared over 2m
Laplacian on psi plus dot dot
652
00:49:26,070 --> 00:49:26,570
dot.
653
00:49:30,150 --> 00:49:39,330
And the up Laplacian of 1 over
r is minus 4 pi times a delta
654
00:49:39,330 --> 00:49:43,355
function at x equals 0.
655
00:49:43,355 --> 00:49:49,220
So this means that the
Schrodinger equation,
656
00:49:49,220 --> 00:49:52,680
you think oh I put
psi equals c over r.
657
00:49:52,680 --> 00:49:56,300
Well, if you calculate the
Laplacian, it seems to be 0.
658
00:49:56,300 --> 00:49:59,680
But if you're more careful,
as you know for [? emm ?]
659
00:49:59,680 --> 00:50:03,750
the Laplacian of 1 over r is
minus 4 pi times the delta
660
00:50:03,750 --> 00:50:05,240
function.
661
00:50:05,240 --> 00:50:09,990
So in the Schrodinger
equation, the kinetic term
662
00:50:09,990 --> 00:50:12,330
produces a delta function.
663
00:50:12,330 --> 00:50:14,720
There's no reason
to believe there's
664
00:50:14,720 --> 00:50:17,600
a delta function
in the potential.
665
00:50:17,600 --> 00:50:21,310
We'll not try such
crazy potentials.
666
00:50:21,310 --> 00:50:25,460
A delta function in a one
dimensional potential,
667
00:50:25,460 --> 00:50:27,800
you've got the solution.
668
00:50:27,800 --> 00:50:31,480
A delta function in a
three dimensional potential
669
00:50:31,480 --> 00:50:35,235
is absolutely crazy.
670
00:50:35,235 --> 00:50:38,760
It has infinite number
of bound states,
671
00:50:38,760 --> 00:50:40,840
and they just go
all the way down
672
00:50:40,840 --> 00:50:43,030
to energies of minus infinity.
673
00:50:43,030 --> 00:50:46,190
It's a very horrendous
thing, a delta function
674
00:50:46,190 --> 00:50:49,660
in three dimensions,
for quantum mechanics.
675
00:50:49,660 --> 00:50:54,070
So this thing, there's no delta
function in the potential.
676
00:50:54,070 --> 00:50:56,630
And you've got a delta
function from the kinetic term.
677
00:50:56,630 --> 00:50:58,800
You're not going to
be able to cancel it.
678
00:50:58,800 --> 00:51:01,840
This is not a solution.
679
00:51:07,570 --> 00:51:14,060
So you really cannot
approach a constant there.
680
00:51:14,060 --> 00:51:16,190
It's quite bad.
681
00:51:16,190 --> 00:51:20,130
So the wave functions
will have to vanish,
682
00:51:20,130 --> 00:51:25,990
and we can prove that, or at
least under some circumstances
683
00:51:25,990 --> 00:51:26,920
prove it.
684
00:51:26,920 --> 00:51:30,310
And as all these
things are, they all
685
00:51:30,310 --> 00:51:33,500
depend on how crazy
potentials you want to accept.
686
00:51:33,500 --> 00:51:37,110
So we should say something.
687
00:51:37,110 --> 00:51:41,990
So I'll say something
about these potentials,
688
00:51:41,990 --> 00:51:46,606
and we'll prove a result.
689
00:51:50,830 --> 00:52:09,460
So my statement will be the
centrifugal barrier, which
690
00:52:09,460 --> 00:52:14,860
is a name for this
part of the potential,
691
00:52:14,860 --> 00:52:23,370
dominates as r goes to 0.
692
00:52:23,370 --> 00:52:27,050
If this doesn't happen,
all bets are off.
693
00:52:27,050 --> 00:52:33,720
So let's assume that v of
r, maybe it's 1 over r,
694
00:52:33,720 --> 00:52:36,210
but it's not worse
than 1 over r squared.
695
00:52:36,210 --> 00:52:39,370
It's 1 over r
cubed, for example,
696
00:52:39,370 --> 00:52:41,430
or something like that.
697
00:52:41,430 --> 00:52:43,790
You would have to
analyze it from scratch
698
00:52:43,790 --> 00:52:45,080
if it would be that bad.
699
00:52:45,080 --> 00:52:50,260
But I will assume that the
centrifugal barrier dominates.
700
00:52:50,260 --> 00:52:54,270
And then look at the
differential equation.
701
00:52:54,270 --> 00:52:56,430
Well, what differential
equation do I have?
702
00:52:56,430 --> 00:53:04,475
Well, I have this and this.
703
00:53:04,475 --> 00:53:07,210
This thing is less
important than that,
704
00:53:07,210 --> 00:53:10,970
and this is also less
important, because this is u
705
00:53:10,970 --> 00:53:12,530
divided by r squared.
706
00:53:12,530 --> 00:53:14,250
And here is just u.
707
00:53:14,250 --> 00:53:17,380
So this is certainly
less important than that,
708
00:53:17,380 --> 00:53:19,610
and this is less
important than that,
709
00:53:19,610 --> 00:53:22,740
and if I want to have
some variation of u,
710
00:53:22,740 --> 00:53:26,170
or understand how it
varies, I must keep this.
711
00:53:26,170 --> 00:53:31,880
So at this order, I should
keep just the kinetic term
712
00:53:31,880 --> 00:53:38,490
h squared over 2m d
second dr squared u of El.
713
00:53:42,400 --> 00:53:49,840
And h squared l times l
plus 1 over 2 mr squared.
714
00:53:49,840 --> 00:53:55,770
And I will try to cancel these
two to explore how the wave
715
00:53:55,770 --> 00:53:59,360
function looks near or equal 0.
716
00:53:59,360 --> 00:54:02,300
These are the two most important
terms of the differential
717
00:54:02,300 --> 00:54:06,050
equation, so I have the
right to keep those, and try
718
00:54:06,050 --> 00:54:12,450
to balance them out to leading
order, and see what I get.
719
00:54:12,450 --> 00:54:16,960
So all the h squared
over 2m's go away.
720
00:54:16,960 --> 00:54:26,300
So this is equivalent to
d second uEl dr squared is
721
00:54:26,300 --> 00:54:32,165
equal to l times l plus
1 uEl over r squared.
722
00:54:35,810 --> 00:54:39,400
And this is solved
by a power uEl.
723
00:54:43,080 --> 00:54:50,900
You can try r to the
s, some number s.
724
00:54:50,900 --> 00:55:01,390
And then this thing gives
you s times s minus 1.
725
00:55:01,390 --> 00:55:06,195
Taking two derivatives is
equal to l times l plus 1.
726
00:55:11,580 --> 00:55:14,830
As you take two derivatives,
you lose two powers of r,
727
00:55:14,830 --> 00:55:17,960
so it will work out.
728
00:55:17,960 --> 00:55:21,190
And from here, you see
that the possible solutions
729
00:55:21,190 --> 00:55:25,725
are s equals l plus 1.
730
00:55:25,725 --> 00:55:29,100
And s equals 2 minus l.
731
00:55:33,660 --> 00:55:39,310
So this corresponds
to a uEl that
732
00:55:39,310 --> 00:55:44,890
goes like r to the
l plus 1, or a uEl
733
00:55:44,890 --> 00:55:48,958
that goes like 1
over r to the l.
734
00:55:52,782 --> 00:55:56,660
This Is far too singular.
735
00:55:56,660 --> 00:56:01,020
For l equals 0, we argued
that the wave function
736
00:56:01,020 --> 00:56:02,586
should go like a constant.
737
00:56:06,900 --> 00:56:09,920
I'm sorry, cannot
go like a constant.
738
00:56:09,920 --> 00:56:11,860
Must vanish.
739
00:56:11,860 --> 00:56:13,480
This is not possible.
740
00:56:13,480 --> 00:56:14,510
It's not a solution.
741
00:56:14,510 --> 00:56:16,020
It must vanish.
742
00:56:16,020 --> 00:56:22,200
For l equals 0, uE0 goes
like r and vanishes.
743
00:56:22,200 --> 00:56:25,650
So that's consistent,
and this is good.
744
00:56:25,650 --> 00:56:29,650
For l equals 0, this would
be like a constant as well
745
00:56:29,650 --> 00:56:30,400
and would be fine.
746
00:56:30,400 --> 00:56:34,790
But for l equals 1
already, this is 1 over r,
747
00:56:34,790 --> 00:56:36,035
and this is not normalizable.
748
00:56:38,900 --> 00:56:50,210
So this time this is not
normalizable for l greater
749
00:56:50,210 --> 00:56:52,320
or equal than one.
750
00:56:52,320 --> 00:56:59,170
So this is the answer
[INAUDIBLE] this assumption,
751
00:56:59,170 --> 00:57:01,880
which is a very
reasonable assumption.
752
00:57:01,880 --> 00:57:05,480
But if you don't have
that you have to beware.
753
00:57:08,010 --> 00:57:16,490
OK, this is our
condition for u there.
754
00:57:16,490 --> 00:57:24,090
And so uEl goes like
this as r goes to 0.
755
00:57:24,090 --> 00:57:27,840
It would be the whole answer.
756
00:57:27,840 --> 00:57:37,910
So f, if you care about f still,
which is what appears here,
757
00:57:37,910 --> 00:57:40,970
goes like u divided by r.
758
00:57:40,970 --> 00:57:50,562
So fEl goes like cr to the l.
759
00:57:54,258 --> 00:58:02,460
And when l is equal to 0,
f behaves like a constant.
760
00:58:02,460 --> 00:58:06,400
u vanishes for l
equal to 0, but f
761
00:58:06,400 --> 00:58:08,760
goes like a
constant, which means
762
00:58:08,760 --> 00:58:13,540
that if you take 0
orbital angular momentum,
763
00:58:13,540 --> 00:58:16,670
you may have some
probability of finding
764
00:58:16,670 --> 00:58:21,660
the particle at the origin,
because this f behaves
765
00:58:21,660 --> 00:58:25,520
like a constant for l equals 0.
766
00:58:25,520 --> 00:58:28,720
On the other hand,
for any higher l,
767
00:58:28,720 --> 00:58:31,920
f will also vanish
at the origin.
768
00:58:31,920 --> 00:58:36,840
And that is intuitively said
that the centrifugal barrier
769
00:58:36,840 --> 00:58:39,990
prevents the particle
from reaching the origin.
770
00:58:39,990 --> 00:58:42,840
There's a barrier,
a potential barrier.
771
00:58:42,840 --> 00:58:47,020
This potential is
1 over r squared.
772
00:58:47,020 --> 00:58:49,420
Doesn't let you go to
close to the origin.
773
00:58:49,420 --> 00:58:53,480
But that potential
disappears for l equals 0,
774
00:58:53,480 --> 00:58:56,840
and therefore the particle
can reach the origin.
775
00:58:56,840 --> 00:59:00,500
But only for l equals 0
it can reach the origin.
776
00:59:03,120 --> 00:59:08,710
OK, one more thing.
777
00:59:08,710 --> 00:59:13,320
Behavior near infinity
is of interest as well.
778
00:59:20,130 --> 00:59:25,260
So what happens for
r goes to infinity?
779
00:59:32,440 --> 00:59:34,850
Well, for r goes to
infinity, you also
780
00:59:34,850 --> 00:59:39,770
have to be a little
careful what you assume.
781
00:59:39,770 --> 00:59:44,000
I wish I could tell you it's
always like this, but it's not.
782
00:59:44,000 --> 00:59:46,455
It's rich in all
kinds of problems.
783
00:59:49,040 --> 00:59:51,050
So there's two
cases where there's
784
00:59:51,050 --> 00:59:53,500
an analysis that is simple.
785
00:59:53,500 --> 00:59:59,890
Suppose v of r is equal to 0
for r greater than some r0.
786
01:00:02,520 --> 01:00:12,896
Or r times v of f goes to
0 as r goes to infinity.
787
01:00:12,896 --> 01:00:13,645
Two possibilities.
788
01:00:16,660 --> 01:00:22,440
The potential is plane
0 after some distance.
789
01:00:22,440 --> 01:00:30,140
Or the potential
multiplied by r goes to 0
790
01:00:30,140 --> 01:00:31,310
as r goes to infinity.
791
01:00:31,310 --> 01:00:36,660
And you would say, look, you've
missed the most important case.
792
01:00:36,660 --> 01:00:40,390
The hydrogen atom, the
potential is 1 over r.
793
01:00:40,390 --> 01:00:43,640
r times v of r doesn't go to 0.
794
01:00:43,640 --> 01:00:45,850
And indeed, what I'm
going to write here
795
01:00:45,850 --> 01:00:49,380
doesn't quite apply to the wave
functions of the hydrogen atom.
796
01:00:49,380 --> 01:00:51,440
They're a little unusual.
797
01:00:51,440 --> 01:00:56,805
The potential of the hydrogen
atom is felt quite far away.
798
01:00:59,480 --> 01:01:04,520
So never the less, if you
have those conditions,
799
01:01:04,520 --> 01:01:13,000
we can ignore the potential
as we go far away.
800
01:01:13,000 --> 01:01:17,595
And we'll consider the
following situation.
801
01:01:33,100 --> 01:01:37,930
Look that the centrifugal
barrier satisfies this as well.
802
01:01:37,930 --> 01:01:41,330
So the full effective
potential satisfies.
803
01:01:41,330 --> 01:01:44,870
If v of r satisfies
that, r times 1
804
01:01:44,870 --> 01:01:48,780
over r squared of effective
potential also satisfies that.
805
01:01:48,780 --> 01:01:52,250
So we can ignore
all the potential,
806
01:01:52,250 --> 01:01:58,170
and we're left
ignore the effective.
807
01:01:58,170 --> 01:02:01,440
And therefore we're left
with minus h squared
808
01:02:01,440 --> 01:02:09,793
over 2m d second uEl dr
squared is equal to EuEl.
809
01:02:13,900 --> 01:02:17,920
And that's a very
trivial equation.
810
01:02:17,920 --> 01:02:19,481
Yes, Matt?
811
01:02:19,481 --> 01:02:20,954
AUDIENCE: When you
say v of r goes
812
01:02:20,954 --> 01:02:23,409
to 0 for r greater
than [INAUDIBLE] 0.
813
01:02:23,409 --> 01:02:25,870
Are you effectively
[INAUDIBLE] the potential?
814
01:02:25,870 --> 01:02:30,870
PROFESSOR: Right, there may
be some potentials like this.
815
01:02:30,870 --> 01:02:34,340
A potential that is like that.
816
01:02:34,340 --> 01:02:38,180
An attractive potential, and it
vanishes after some distance.
817
01:02:38,180 --> 01:02:41,940
Or a repulsive potential that
vanishes after some distance.
818
01:02:41,940 --> 01:02:44,440
AUDIENCE: But say the potential
was a [INAUDIBLE] potential.
819
01:02:44,440 --> 01:02:47,072
Are you just approximating it
to 0 after it's [INAUDIBLE]?
820
01:02:47,072 --> 01:02:49,280
PROFESSOR: Well, if I'm in
the [INAUDIBLE] potential,
821
01:02:49,280 --> 01:02:53,460
unfortunately I'm
neither here nor here,
822
01:02:53,460 --> 01:02:55,830
so this doesn't apply.
823
01:02:55,830 --> 01:02:58,390
So the [INAUDIBLE]
potential is an exception.
824
01:02:58,390 --> 01:02:59,994
The solutions are
a little more--
825
01:02:59,994 --> 01:03:02,160
AUDIENCE: The conditions
you're saying. [INAUDIBLE].
826
01:03:02,160 --> 01:03:05,230
PROFESSOR: So these
are conditions
827
01:03:05,230 --> 01:03:08,490
that allow me to say something.
828
01:03:08,490 --> 01:03:10,900
If they're not
satisfied, I sort of
829
01:03:10,900 --> 01:03:14,810
have to analyze
them case by case.
830
01:03:14,810 --> 01:03:18,110
That's the price we have to pay.
831
01:03:18,110 --> 01:03:22,280
It's a little more complicated
than you would think naively.
832
01:03:22,280 --> 01:03:27,000
Now here, it's interesting to
consider two possibilities.
833
01:03:27,000 --> 01:03:30,800
The case when E is less
than 0, or the case when
834
01:03:30,800 --> 01:03:33,660
E is greater than 0.
835
01:03:33,660 --> 01:03:37,910
So scattering solutions
or bound state solutions.
836
01:03:37,910 --> 01:03:42,800
For these ones, if the
energy is less than 0
837
01:03:42,800 --> 01:03:47,480
and there's no potential, you're
in the forbidden zone far away,
838
01:03:47,480 --> 01:03:51,080
so you must have a
decaying exponential.
839
01:03:51,080 --> 01:03:57,200
El goes like exponential
of minus square root
840
01:03:57,200 --> 01:04:03,280
of 2m E over h squared r.
841
01:04:03,280 --> 01:04:04,845
That solves that equation.
842
01:04:07,962 --> 01:04:10,040
You see, the solution
of these things
843
01:04:10,040 --> 01:04:15,810
are either exponential
decays or exponential growths
844
01:04:15,810 --> 01:04:19,040
and oscillatory solutions,
sines and cosines,
845
01:04:19,040 --> 01:04:21,790
or E to the i things.
846
01:04:21,790 --> 01:04:27,840
So here we have a decay,
because with energy less than 0,
847
01:04:27,840 --> 01:04:29,080
the potential is 0.
848
01:04:29,080 --> 01:04:32,850
So you're in a forbidden region,
so you must decay like that.
849
01:04:32,850 --> 01:04:35,520
In this hydrogen
atom what happens
850
01:04:35,520 --> 01:04:39,090
is that there's a power
of r multiplying here.
851
01:04:39,090 --> 01:04:43,710
Like r to the n, or r to the
k or something like that.
852
01:04:43,710 --> 01:04:51,216
If E is less than 0, you
have uE equal exponential
853
01:04:51,216 --> 01:05:04,561
of plus minus ikr, where k
is square root of 2m E over h
854
01:05:04,561 --> 01:05:05,060
squared.
855
01:05:08,420 --> 01:05:11,670
And those, again,
solve that equation.
856
01:05:11,670 --> 01:05:16,280
And they are sort of
wave solutions far away.
857
01:05:22,350 --> 01:05:25,150
Now with this
information, the behavior
858
01:05:25,150 --> 01:05:31,460
of the u's near the origin, the
behavior of the u's far away,
859
01:05:31,460 --> 01:05:35,040
you can then make
qualitative plots
860
01:05:35,040 --> 01:05:37,820
of how solutions would
look at the origin.
861
01:05:37,820 --> 01:05:39,610
They grow up like r to the l.
862
01:05:39,610 --> 01:05:42,150
Then it's a one
dimensional potential,
863
01:05:42,150 --> 01:05:46,310
so they oscillate maybe, but
then decay exponentially.
864
01:05:46,310 --> 01:05:47,940
And the kind of
thing you used to do
865
01:05:47,940 --> 01:05:51,170
in 804 of plotting
how things look,
866
01:05:51,170 --> 01:05:55,550
it's feasible at this stage.
867
01:05:55,550 --> 01:05:59,580
So it's about time
to do examples.
868
01:05:59,580 --> 01:06:01,260
I have three examples.
869
01:06:01,260 --> 01:06:04,900
Given time, maybe
I'll get to two.
870
01:06:04,900 --> 01:06:06,030
That's OK.
871
01:06:06,030 --> 01:06:08,320
The last example is
kind of the cutest,
872
01:06:08,320 --> 01:06:13,550
but maybe it's OK to
leave it for Monday.
873
01:06:13,550 --> 01:06:17,100
So are there
questions about this
874
01:06:17,100 --> 01:06:19,015
before we begin our examples?
875
01:06:24,840 --> 01:06:26,262
Andrew?
876
01:06:26,262 --> 01:06:30,472
AUDIENCE: What is consumption
of [INAUDIBLE] [? barrier ?]
877
01:06:30,472 --> 01:06:30,972
dominates.
878
01:06:30,972 --> 01:06:33,340
But why is that a
reasonable assumptions?
879
01:06:33,340 --> 01:06:35,720
PROFESSOR: Well,
potentials that are just
880
01:06:35,720 --> 01:06:41,450
too singular at the
origin are not common.
881
01:06:41,450 --> 01:06:45,820
Just doesn't happen.
882
01:06:45,820 --> 01:06:50,770
So mathematically
you could try them,
883
01:06:50,770 --> 01:06:54,730
but I actually don't
know of useful examples
884
01:06:54,730 --> 01:06:57,315
if a potential is very
singular at the origin.
885
01:07:00,595 --> 01:07:03,976
AUDIENCE: [INAUDIBLE] in
the potential [INAUDIBLE]
886
01:07:03,976 --> 01:07:05,425
the centrifugal barrier.
887
01:07:05,425 --> 01:07:08,323
That [INAUDIBLE].
888
01:07:08,323 --> 01:07:09,310
PROFESSOR: Right.
889
01:07:09,310 --> 01:07:12,900
An effective potential, the
potential doesn't blow up--
890
01:07:12,900 --> 01:07:17,880
your potential doesn't blow
up more than 1 over r squared
891
01:07:17,880 --> 01:07:19,090
or something like that.
892
01:07:19,090 --> 01:07:24,640
So we'll just take it like that.
893
01:07:24,640 --> 01:07:31,720
OK, our first example
is the free particle.
894
01:07:31,720 --> 01:07:33,260
You would say come on.
895
01:07:33,260 --> 01:07:34,890
That's ridiculous.
896
01:07:34,890 --> 01:07:35,525
Too simple.
897
01:07:38,510 --> 01:07:42,980
But it's fairly non-trivial
in spherical coordinates.
898
01:07:42,980 --> 01:07:45,720
And you say, well, so what.
899
01:07:45,720 --> 01:07:49,550
Free particles, you say
what the momentum is.
900
01:07:49,550 --> 01:07:51,070
You know the energy.
901
01:07:51,070 --> 01:07:52,780
How do you label the states?
902
01:07:52,780 --> 01:07:55,250
You label them by three momenta.
903
01:07:55,250 --> 01:07:58,540
Or energy and direction.
904
01:07:58,540 --> 01:08:01,830
So momentum
eigenstates, for example
905
01:08:01,830 --> 01:08:04,000
But in spherical
coordinates, these will not
906
01:08:04,000 --> 01:08:06,080
be momentum
eigenstates, and these
907
01:08:06,080 --> 01:08:08,220
are interesting
because they allow
908
01:08:08,220 --> 01:08:12,450
us to solve for more
complicated problems, in fact.
909
01:08:12,450 --> 01:08:15,390
And they allow you to
understand scattering out
910
01:08:15,390 --> 01:08:16,810
of central potential.
911
01:08:16,810 --> 01:08:18,725
So these are actually
pretty important.
912
01:08:22,410 --> 01:08:25,010
You can label these
things by three numbers.
913
01:08:25,010 --> 01:08:27,729
p1, p2, p3.
914
01:08:27,729 --> 01:08:33,840
Or energy and theta and phi,
the directions of the momenta.
915
01:08:33,840 --> 01:08:39,700
What we're going to label
them are by energy l and m.
916
01:08:39,700 --> 01:08:45,399
So you might say how do we
compare all these infinities,
917
01:08:45,399 --> 01:08:48,779
but it somehow works out.
918
01:08:48,779 --> 01:08:53,600
There's the same number of
states really in either way.
919
01:08:53,600 --> 01:08:57,279
So what do we have?
920
01:08:57,279 --> 01:09:01,300
It's a potential
that v is equal to 0.
921
01:09:01,300 --> 01:09:03,790
So let's write the
differential equation.
922
01:09:03,790 --> 01:09:06,390
v is equal to 0.
923
01:09:06,390 --> 01:09:07,920
But not v effective.
924
01:09:07,920 --> 01:09:16,760
So you have minus h squared
over 2m d second uEl
925
01:09:16,760 --> 01:09:23,950
dr squared plus h
squared over 2m l times
926
01:09:23,950 --> 01:09:32,279
l plus 1 over r
squared uEl equal EuEl.
927
01:09:32,279 --> 01:09:35,494
This is actually
quite interesting.
928
01:09:35,494 --> 01:09:39,590
As you will see, it's a bit
puzzling the first time.
929
01:09:39,590 --> 01:09:43,649
Well, let's cancel
this h squared over 2m,
930
01:09:43,649 --> 01:09:47,270
because they're
kind of annoying.
931
01:09:47,270 --> 01:09:54,090
So we'll put d second uEl over
dr squared with a minus-- I'll
932
01:09:54,090 --> 01:10:03,350
keep that minus-- plus l times
l plus 1 over r squared uEl.
933
01:10:03,350 --> 01:10:08,610
And here I'll put k
squared times uEl.
934
01:10:08,610 --> 01:10:13,070
And k squared is the
same k as before.
935
01:10:13,070 --> 01:10:16,880
And E is positive because
you have a free particle.
936
01:10:16,880 --> 01:10:19,330
E is positive.
937
01:10:19,330 --> 01:10:25,940
And k squared is given by
this, 2m E over h squared.
938
01:10:25,940 --> 01:10:28,145
So this is the equation
we have to solve.
939
01:10:33,330 --> 01:10:40,990
And it's kind of interesting,
because on the one hand,
940
01:10:40,990 --> 01:10:45,590
there is an energy on
the right hand side.
941
01:10:45,590 --> 01:10:48,700
And then you would say, look,
it looks like this just typical
942
01:10:48,700 --> 01:10:51,290
one dimensional
Schrodinger equation.
943
01:10:51,290 --> 01:10:53,610
Therefore that
energy probably is
944
01:10:53,610 --> 01:10:57,250
quantized because it shows
in the right hand side.
945
01:10:57,250 --> 01:11:03,340
Why wouldn't it be quantized
if it just shows this way?
946
01:11:03,340 --> 01:11:06,990
On the other hand, it
shouldn't be quantized.
947
01:11:06,990 --> 01:11:12,550
So what is it about this
differential equation that
948
01:11:12,550 --> 01:11:16,780
shows that the energy
never gets quantized?
949
01:11:16,780 --> 01:11:20,330
Well, the fact is that
the energy in some sense
950
01:11:20,330 --> 01:11:22,850
doesn't show up in this
differential equation.
951
01:11:22,850 --> 01:11:27,630
You think it's here, but
it's not really there.
952
01:11:27,630 --> 01:11:29,710
What does that mean?
953
01:11:29,710 --> 01:11:31,760
It actually means
that you can define
954
01:11:31,760 --> 01:11:41,240
a new variable rho
equal kr, scale r.
955
01:11:41,240 --> 01:11:47,870
And basically chain rule
or your intuition, this k
956
01:11:47,870 --> 01:11:49,500
goes down here.
957
01:11:49,500 --> 01:11:53,810
k squared r squared k squared
r squared, it's all rho.
958
01:11:53,810 --> 01:11:56,730
So chain rule or
changing variables
959
01:11:56,730 --> 01:12:03,650
will turn this equation into
a minus d second uEl d rho
960
01:12:03,650 --> 01:12:08,850
squared plus l times
l plus 1 rho squared
961
01:12:08,850 --> 01:12:14,900
is equal to-- times uEl--
is equal to uEl here.
962
01:12:20,450 --> 01:12:23,270
And the energy has
disappeared from the equation
963
01:12:23,270 --> 01:12:27,400
by rescaling, a trivial
rescaling of coordinates.
964
01:12:27,400 --> 01:12:30,860
That doesn't mean that
the energy is not there.
965
01:12:30,860 --> 01:12:35,510
It is there, because
you will find solutions
966
01:12:35,510 --> 01:12:37,720
that depend on rho,
and then you will
967
01:12:37,720 --> 01:12:40,830
put rho equal kr and
the energies there.
968
01:12:40,830 --> 01:12:44,070
But there's no
quantization of energy,
969
01:12:44,070 --> 01:12:49,490
because the energy doesn't
show in this equation anymore.
970
01:12:49,490 --> 01:12:55,310
It's kind of a neat thing, or
rather conceptually interesting
971
01:12:55,310 --> 01:12:59,350
thing that energy is
not there anymore.
972
01:12:59,350 --> 01:13:04,560
And then you look at this
differential equation,
973
01:13:04,560 --> 01:13:09,720
and you realize that
it's a nasty one.
974
01:13:09,720 --> 01:13:15,280
So this equation is
quite easy without this.
975
01:13:15,280 --> 01:13:18,130
It's a power solution.
976
01:13:18,130 --> 01:13:23,700
It's quite easy without this,
it's exponentials are this.
977
01:13:23,700 --> 01:13:26,580
But whenever you have a
differential equation that
978
01:13:26,580 --> 01:13:31,930
has two derivatives,
a term with 1
979
01:13:31,930 --> 01:13:35,440
over x squared
times the function,
980
01:13:35,440 --> 01:13:40,600
and a term with 1
times the function,
981
01:13:40,600 --> 01:13:42,255
you're in Bessel territory.
982
01:13:44,960 --> 01:13:47,300
All these functions
have Bessel things.
983
01:13:49,820 --> 01:13:53,940
And then you have another
term like 1 over x d dx.
984
01:13:53,940 --> 01:13:57,540
That is not a problem, but the
presence of these two things,
985
01:13:57,540 --> 01:14:01,170
one with 1 over x squared
and one with this,
986
01:14:01,170 --> 01:14:02,640
complicates this equation.
987
01:14:02,640 --> 01:14:05,270
So Bessel, without
this, would be
988
01:14:05,270 --> 01:14:08,880
exponential solution without
this would be powers.
989
01:14:08,880 --> 01:14:12,610
In the end, the fact is that
this is spherical Bessel,
990
01:14:12,610 --> 01:14:15,200
and it's a little complicated.
991
01:14:15,200 --> 01:14:16,860
Not terribly complicated.
992
01:14:16,860 --> 01:14:20,280
The solutions are
spherical Bessel functions,
993
01:14:20,280 --> 01:14:22,970
which are not all that bad.
994
01:14:22,970 --> 01:14:25,190
And let me say what they are.
995
01:14:38,550 --> 01:14:41,960
So what are the
solutions to this thing?
996
01:14:41,960 --> 01:14:45,390
In fact, the solutions
that are easier to find
997
01:14:45,390 --> 01:14:54,960
is that the uEl's are r
times the Bessel function
998
01:14:54,960 --> 01:14:58,020
jl is called spherical
Bessel functions.
999
01:14:58,020 --> 01:15:01,860
So it's not capital
j that people
1000
01:15:01,860 --> 01:15:06,320
use for the normal
Bessel, but lower case l.
1001
01:15:06,320 --> 01:15:08,640
Of kr.
1002
01:15:08,640 --> 01:15:13,430
As you know, you solve this,
and the solutions for this
1003
01:15:13,430 --> 01:15:18,050
would be of the
form rho jl for rho.
1004
01:15:18,050 --> 01:15:22,530
But rho is kr, so we don't
care about the constant,
1005
01:15:22,530 --> 01:15:25,590
because this is a
homogeneous linear equation.
1006
01:15:25,590 --> 01:15:26,890
So some number here.
1007
01:15:26,890 --> 01:15:29,320
You could put a
constant if you wish.
1008
01:15:29,320 --> 01:15:32,130
But that's the solution.
1009
01:15:32,130 --> 01:15:34,820
Therefore your
complete solutions
1010
01:15:34,820 --> 01:15:42,590
is like the psi's of
Elm would be u divided
1011
01:15:42,590 --> 01:15:50,406
by r, which is jl of kr
times Ylm's of theta phi.
1012
01:15:50,406 --> 01:15:51,780
These are the
complete solutions.
1013
01:15:56,280 --> 01:15:59,400
This is a second order
differential equation.
1014
01:15:59,400 --> 01:16:03,870
Therefore it has to
have two solutions.
1015
01:16:03,870 --> 01:16:07,960
And this is what is called a
regular solution at the origin.
1016
01:16:07,960 --> 01:16:12,250
The Bessel functions
come in j and n type.
1017
01:16:12,250 --> 01:16:15,670
And the n type is
singular at the origins,
1018
01:16:15,670 --> 01:16:17,340
so we won't care about it.
1019
01:16:20,810 --> 01:16:23,870
So what do we get from here?
1020
01:16:23,870 --> 01:16:27,780
Well, some behavior
that is well known.
1021
01:16:27,780 --> 01:16:36,110
Rho jl of rho behaves like
rho to the l plus 2 over 2l
1022
01:16:36,110 --> 01:16:42,690
plus 1 double factorial
as rho goes to 0.
1023
01:16:42,690 --> 01:16:46,540
So that's a fact about
these Bessel functions.
1024
01:16:46,540 --> 01:16:50,570
They behave that
way, which is good,
1025
01:16:50,570 --> 01:16:56,640
because rho jl
behaves like that,
1026
01:16:56,640 --> 01:17:00,770
so u behaves like r
to the l plus 1, which
1027
01:17:00,770 --> 01:17:03,700
is what we derived
a little time ago.
1028
01:17:03,700 --> 01:17:05,950
So this behavior of
the Bessel function
1029
01:17:05,950 --> 01:17:09,370
is indeed consistent
with our solution.
1030
01:17:09,370 --> 01:17:13,890
Moreover, there's another
behavior that is interesting.
1031
01:17:13,890 --> 01:17:15,910
This Bessel
function, by the time
1032
01:17:15,910 --> 01:17:19,580
it's written like
that, when you go
1033
01:17:19,580 --> 01:17:25,390
far off to infinity
jl of rho, it
1034
01:17:25,390 --> 01:17:31,950
behaves like sine of
rho minus l pi over 2.
1035
01:17:35,900 --> 01:17:40,120
This is as rho goes to infinity.
1036
01:17:40,120 --> 01:17:45,080
So as rho goes to
infinity, this is
1037
01:17:45,080 --> 01:17:47,550
behaving like a
trigonometric function.
1038
01:17:47,550 --> 01:17:54,680
It's consistent with this,
because rho-- this is rho jl
1039
01:17:54,680 --> 01:17:58,120
is what we call u essentially.
1040
01:17:58,120 --> 01:18:02,470
So u behaves like this
with rho equal kr.
1041
01:18:02,470 --> 01:18:03,730
And that's consistent.
1042
01:18:03,730 --> 01:18:07,350
This superposition of
a sine and a cosine.
1043
01:18:07,350 --> 01:18:10,570
But it's kind of interesting
though that this l pi over 2
1044
01:18:10,570 --> 01:18:12,940
shows up here.
1045
01:18:12,940 --> 01:18:15,890
You see the fact
that this function
1046
01:18:15,890 --> 01:18:17,940
has to vanish at the origin.
1047
01:18:17,940 --> 01:18:20,800
It vanishes at the origin
and begins to vary.
1048
01:18:20,800 --> 01:18:23,990
And by the time you go
far away, you contract.
1049
01:18:23,990 --> 01:18:28,970
And the way it
behaves is this way.
1050
01:18:28,970 --> 01:18:32,580
The face is determined.
1051
01:18:32,580 --> 01:18:36,060
So that actually gives
a lot of opportunity
1052
01:18:36,060 --> 01:18:43,190
to physicists because
the free particle--
1053
01:18:43,190 --> 01:18:51,090
so for the free
particle, uEl behaves
1054
01:18:51,090 --> 01:18:59,285
like sine of kr minus l pi
over 2 as r goes to infinity.
1055
01:19:05,560 --> 01:19:20,590
So from that people have
asked the following question.
1056
01:19:20,590 --> 01:19:25,720
What if you have a
potential that, for example
1057
01:19:25,720 --> 01:19:30,140
for simplicity, a potential
that is localized.
1058
01:19:30,140 --> 01:19:32,650
Well, if this
potential is localized,
1059
01:19:32,650 --> 01:19:35,850
the solution far
away is supposed
1060
01:19:35,850 --> 01:19:39,280
to be a superposition
of sines and cosines.
1061
01:19:39,280 --> 01:19:42,750
So if there is no
potential, the solution
1062
01:19:42,750 --> 01:19:44,780
is supposed to be this.
1063
01:19:44,780 --> 01:19:48,270
Now another superposition
of sines and cosines,
1064
01:19:48,270 --> 01:19:50,330
at the end of the
day, can always
1065
01:19:50,330 --> 01:19:55,040
be written as some sine of
this thing plus a change
1066
01:19:55,040 --> 01:20:01,230
in this phase So in
general, uEl will
1067
01:20:01,230 --> 01:20:11,100
go like sine of kr minus l pi
over 2 plus a shift, a phase
1068
01:20:11,100 --> 01:20:17,140
shift, delta l that can
depend on the energy.
1069
01:20:21,490 --> 01:20:26,170
So if you haven't tried to
find the radial solutions
1070
01:20:26,170 --> 01:20:28,600
of a problem with
some potential,
1071
01:20:28,600 --> 01:20:32,400
if the potential is 0,
there's no such term.
1072
01:20:32,400 --> 01:20:37,780
But if the potential is
here, it will have an effect
1073
01:20:37,780 --> 01:20:41,330
and will give you a phase shift.
1074
01:20:41,330 --> 01:20:44,550
So if you're doing particle
scattering experiments,
1075
01:20:44,550 --> 01:20:47,300
you're sending
waves from far away
1076
01:20:47,300 --> 01:20:50,560
and you just see how the
wave behaves far away,
1077
01:20:50,560 --> 01:20:54,190
you do have measurement
information on this phase
1078
01:20:54,190 --> 01:20:55,140
shift.
1079
01:20:55,140 --> 01:20:58,310
And from this phase shift,
you can learn something
1080
01:20:58,310 --> 01:21:00,900
about the potential.
1081
01:21:00,900 --> 01:21:05,850
So this is how this
problem of free particle
1082
01:21:05,850 --> 01:21:10,130
suddenly becomes very
important and very interesting.
1083
01:21:10,130 --> 01:21:13,370
For example, as a way
through the behavior at
1084
01:21:13,370 --> 01:21:17,410
infinity learning something
about the potential.
1085
01:21:17,410 --> 01:21:20,800
For example, if the
potential is attractive,
1086
01:21:20,800 --> 01:21:26,020
it pulls the wave function
in and produces some sign
1087
01:21:26,020 --> 01:21:30,520
of delta that the corresponds
to a positive delta.
1088
01:21:30,520 --> 01:21:33,120
If the potential
is repulsive, it
1089
01:21:33,120 --> 01:21:36,580
pushes the wave
function out, repels it
1090
01:21:36,580 --> 01:21:39,120
and produces a delta
that is negative.
1091
01:21:39,120 --> 01:21:42,740
You can track those
signs thinking carefully.
1092
01:21:42,740 --> 01:21:48,100
But the potentials will teach
you something about delta.
1093
01:21:48,100 --> 01:21:53,460
The other case that this is
interesting-- I will just
1094
01:21:53,460 --> 01:21:58,400
introduce it and stop, because
we might as well stop--
1095
01:21:58,400 --> 01:22:03,880
is a very important case.
1096
01:22:03,880 --> 01:22:05,890
The square well.
1097
01:22:05,890 --> 01:22:09,250
Well, we've studied
in one dimension
1098
01:22:09,250 --> 01:22:11,640
the infinite square well.
1099
01:22:11,640 --> 01:22:14,580
That's one potential that
you now how to solve,
1100
01:22:14,580 --> 01:22:18,440
and sines and
cosines is very easy.
1101
01:22:18,440 --> 01:22:21,740
Now imagine a
spherical square well,
1102
01:22:21,740 --> 01:22:25,770
which is some sort of cavity
in which a particle is
1103
01:22:25,770 --> 01:22:28,500
free to move here, but
the potential becomes
1104
01:22:28,500 --> 01:22:30,900
infinite at the boundary.
1105
01:22:33,740 --> 01:22:38,940
It's a hollow sphere,
so the potential v of r
1106
01:22:38,940 --> 01:22:42,180
is equal to 0 for r less than a.
1107
01:22:42,180 --> 01:22:46,090
And it's infinity
for r greater than a.
1108
01:22:46,090 --> 01:22:51,590
So it's like a bag, a
balloon with solid walls
1109
01:22:51,590 --> 01:22:53,460
impossible to penetrate.
1110
01:22:53,460 --> 01:22:58,590
So this is the most
symmetric simple potential
1111
01:22:58,590 --> 01:23:00,200
you could imagine in the world.
1112
01:23:02,990 --> 01:23:05,520
And we're going to solve it.
1113
01:23:05,520 --> 01:23:07,190
How can we solve this?
1114
01:23:07,190 --> 01:23:12,900
Well, we did 2/3 of the
work already in solving it.
1115
01:23:12,900 --> 01:23:14,050
Why?
1116
01:23:14,050 --> 01:23:18,890
Because inside here
the potential is 0,
1117
01:23:18,890 --> 01:23:22,060
so the particle is free.
1118
01:23:22,060 --> 01:23:29,480
So inside here the solutions
are of the form uEl
1119
01:23:29,480 --> 01:23:33,180
go like rjl of kr.
1120
01:23:36,130 --> 01:23:40,750
And the only thing you will need
is that they vanish at the end.
1121
01:23:40,750 --> 01:23:44,410
So you will fix
this by demanding
1122
01:23:44,410 --> 01:23:57,050
that ka is a number z such--
well, the jl of ka will be 0.
1123
01:23:57,050 --> 01:23:59,460
So that the wave
function vanishes
1124
01:23:59,460 --> 01:24:03,290
at this point where the
potential becomes infinite.
1125
01:24:03,290 --> 01:24:05,530
So you've solved
most of the problem.
1126
01:24:05,530 --> 01:24:10,700
And we'll discuss it in detail,
because it's an important one.
1127
01:24:10,700 --> 01:24:16,940
But this is the most symmetric
potential, you may think.
1128
01:24:16,940 --> 01:24:23,380
This potential is very
symmetric, very pretty,
1129
01:24:23,380 --> 01:24:27,520
but nothing to write home about.
1130
01:24:27,520 --> 01:24:30,250
If you tried to
look-- and we're going
1131
01:24:30,250 --> 01:24:34,060
to calculate this diagram.
1132
01:24:34,060 --> 01:24:35,820
You would say well
it's so symmetric
1133
01:24:35,820 --> 01:24:39,950
that something pretty
is going to happen here.
1134
01:24:39,950 --> 01:24:41,720
Nothing happens.
1135
01:24:41,720 --> 01:24:44,290
These states will show up.
1136
01:24:44,290 --> 01:24:49,040
And these ones will show
up, and no state ever
1137
01:24:49,040 --> 01:24:50,460
will match another one.
1138
01:24:50,460 --> 01:24:55,130
There's no pattern, or
rhyme, or reason for it.
1139
01:24:55,130 --> 01:24:57,630
On the other hand,
if you would have
1140
01:24:57,630 --> 01:25:07,550
taken a potential v of r
of the form beta r squared,
1141
01:25:07,550 --> 01:25:10,960
that potential will
exhibit enormous amounts
1142
01:25:10,960 --> 01:25:14,130
of degeneracies all over.
1143
01:25:14,130 --> 01:25:17,480
And we will have to
understand why that happens.
1144
01:25:17,480 --> 01:25:19,500
So we'll see you next Monday.
1145
01:25:19,500 --> 01:25:21,390
Enjoy your break.
1146
01:25:21,390 --> 01:25:26,540
Homework will only happen
late after Thanksgiving.
1147
01:25:26,540 --> 01:25:28,940
And just have a great time.
1148
01:25:28,940 --> 01:25:32,190
Thank you for coming today,
and will see you soon.
1149
01:25:32,190 --> 01:25:33,740
[APPLAUSE]