1
00:00:00,500 --> 00:00:05,240
PROFESSOR: We spoke
about the hydrogen atom.

2
00:00:05,240 --> 00:00:10,240
And in the hydrogen
atom, we drew

3
00:00:10,240 --> 00:00:16,059
the spectrum, so the table, the
data of spectrum of a quantum

4
00:00:16,059 --> 00:00:16,700
system.

5
00:00:16,700 --> 00:00:22,750
So this is a question that I
you in general to be aware of.

6
00:00:22,750 --> 00:00:26,560
What do we mean by a
diagram of the energy

7
00:00:26,560 --> 00:00:30,440
levels in a central potential?

8
00:00:30,440 --> 00:00:33,020
So this is something we
did for the hydrogen atom.

9
00:00:33,020 --> 00:00:38,230
But in general, the diagram
of energy eigenstates

10
00:00:38,230 --> 00:00:41,740
in a central potential
looks like this.

11
00:00:41,740 --> 00:00:43,480
You put the energies here.

12
00:00:43,480 --> 00:00:47,140
And they could be bound states
that have negative energy.

13
00:00:47,140 --> 00:00:50,410
They could be even bound
states with positive energy,

14
00:00:50,410 --> 00:00:53,600
depending on the system
you're discussing.

15
00:00:53,600 --> 00:00:56,920
You remember the
harmonic oscillator,

16
00:00:56,920 --> 00:01:00,670
the potential is naturally
defined to be positive.

17
00:01:00,670 --> 00:01:04,780
And all these energy states
that the harmonic oscillator

18
00:01:04,780 --> 00:01:07,560
has represent bound states.

19
00:01:07,560 --> 00:01:09,600
They are normalizeable
wave functions.

20
00:01:09,600 --> 00:01:12,080
In fact, you don't
have scattering states

21
00:01:12,080 --> 00:01:15,350
because the potential
just reaches forever.

22
00:01:15,350 --> 00:01:18,320
So in general, for
a central potential,

23
00:01:18,320 --> 00:01:21,850
however, the system
is shown like that.

24
00:01:21,850 --> 00:01:25,210
And we plot here l.

25
00:01:25,210 --> 00:01:27,770
But l is not a
continuous variable,

26
00:01:27,770 --> 00:01:35,410
so we'll put l equals 0
here, l equals 1, l equals 2,

27
00:01:35,410 --> 00:01:37,210
l equals 3.

28
00:01:37,210 --> 00:01:41,260
And then you start
plotting the energy levels.

29
00:01:41,260 --> 00:01:43,780
You solve the radial equation.

30
00:01:43,780 --> 00:01:47,470
Remember, the radial equation
is a Schrodinger equation,

31
00:01:47,470 --> 00:01:56,660
minus h squared over 2m, d
second u d r squared, plus V

32
00:01:56,660 --> 00:02:02,660
effective of r u equals Eu.

33
00:02:02,660 --> 00:02:08,479
And V effective
of r is the V of r

34
00:02:08,479 --> 00:02:13,060
that your system had, plus
a contribution from angular

35
00:02:13,060 --> 00:02:15,360
momentum, 2m.

36
00:02:19,350 --> 00:02:24,820
So this radial equation is a
collection of radial equations.

37
00:02:24,820 --> 00:02:28,270
We've tried to emphasize
that many times already.

38
00:02:28,270 --> 00:02:31,090
And you solve it
first for l equal 0,

39
00:02:31,090 --> 00:02:34,720
then for l equal 1, for l
equal 2, you go on and on.

40
00:02:34,720 --> 00:02:38,260
So you solve it for l equal 0.

41
00:02:38,260 --> 00:02:41,920
And just like as in any
one-dimensional problem,

42
00:02:41,920 --> 00:02:44,720
for l equals 0, you
solve this equation

43
00:02:44,720 --> 00:02:46,810
and you find energy levels.

44
00:02:46,810 --> 00:02:50,620
So you sketch them like that.

45
00:02:50,620 --> 00:02:54,430
Those are the energy
levels for l equals 0.

46
00:02:54,430 --> 00:02:59,470
This is the ground state of
the l equal 0 radial equation.

47
00:02:59,470 --> 00:03:04,600
Now just to remind you,
this is the hard part

48
00:03:04,600 --> 00:03:06,400
of solving the
Schrodinger equation.

49
00:03:06,400 --> 00:03:08,800
Because at the end
of the day, the psi

50
00:03:08,800 --> 00:03:12,550
is u of r over r times some Ylm.

51
00:03:16,960 --> 00:03:21,630
So the l that you have
here, the term is a Y

52
00:03:21,630 --> 00:03:23,440
and the m is arbitrary.

53
00:03:23,440 --> 00:03:28,080
In fact, the u doesn't
know about the m value.

54
00:03:28,080 --> 00:03:31,210
So here you have
these energy levels.

55
00:03:31,210 --> 00:03:37,630
And then what happened with
this hydrogen atom is you keep

56
00:03:37,630 --> 00:03:39,790
solving, of course,
for all the l's.

57
00:03:39,790 --> 00:03:43,360
And in general, when you
solve for l equals 1,

58
00:03:43,360 --> 00:03:45,940
you may find some
levels like this.

59
00:03:45,940 --> 00:03:49,450
It's a miracle when
the levels coincide.

60
00:03:49,450 --> 00:03:51,970
There's no reason why
they should coincide.

61
00:03:51,970 --> 00:03:54,970
They happen to coincide
for the hydrogen atom.

62
00:03:54,970 --> 00:03:59,290
And that's because of a very
special symmetry of the 1 over

63
00:03:59,290 --> 00:04:01,660
r potential orbit.

64
00:04:01,660 --> 00:04:04,355
Then for l equal
2, you solve it.

65
00:04:04,355 --> 00:04:06,580
And for l equal 3, you solve.

66
00:04:06,580 --> 00:04:08,530
And you find these
states, and that's

67
00:04:08,530 --> 00:04:14,370
the diagram of states
of a central potential.

68
00:04:14,370 --> 00:04:16,709
For the hydrogen
atom, of course,

69
00:04:16,709 --> 00:04:18,680
first, all the
energies were negative

70
00:04:18,680 --> 00:04:23,010
and the energy levels coincide.

71
00:04:23,010 --> 00:04:27,920
So this is the ground state of
the l equal 0 radial equation.

72
00:04:27,920 --> 00:04:31,110
This is the ground state of
the l equal 1 radial equation.

73
00:04:31,110 --> 00:04:34,170
This is the ground state of
the l equal 2 radial equation.

74
00:04:34,170 --> 00:04:36,495
This is the ground state
of the whole system.

75
00:04:40,280 --> 00:04:43,240
So this is what we call
plotting the spectrum

76
00:04:43,240 --> 00:04:46,360
in a radial potential problem.

77
00:04:46,360 --> 00:04:48,400
And it's a generic form.

78
00:04:48,400 --> 00:04:54,470
So we were doing this
for the hydrogen atom

79
00:04:54,470 --> 00:04:58,730
last time to try to
understand the various orbits.

80
00:04:58,730 --> 00:05:07,400
And we had for the
hydrogen atom, V of r

81
00:05:07,400 --> 00:05:14,280
was a potential like this,
minus e squared over r.

82
00:05:14,280 --> 00:05:18,870
And then you have sometimes
the l contribution

83
00:05:18,870 --> 00:05:24,350
that this function diverges
towards the origin, 1

84
00:05:24,350 --> 00:05:26,190
over r squared.

85
00:05:26,190 --> 00:05:29,530
And by the time you
add them together,

86
00:05:29,530 --> 00:05:32,490
This is the original
potential, so this could

87
00:05:32,490 --> 00:05:35,070
be thought as l equals 0 case.

88
00:05:35,070 --> 00:05:42,600
Then if you have some l
over here and some other l,

89
00:05:42,600 --> 00:05:45,310
maybe like that, these
are the various potentials

90
00:05:45,310 --> 00:05:46,790
that you get.

91
00:05:46,790 --> 00:05:50,260
And in general, you
may want to figure out,

92
00:05:50,260 --> 00:05:52,060
for example, if
you have an energy

93
00:05:52,060 --> 00:05:59,560
level, some particular energy,
what are the turning points.

94
00:05:59,560 --> 00:06:04,030
So let's consider for
that case that's just one

95
00:06:04,030 --> 00:06:07,310
curve that we care about.

96
00:06:11,840 --> 00:06:18,110
And the electron will be going
from some value of the radius,

97
00:06:18,110 --> 00:06:20,105
so this is the plot of
the effective potential

98
00:06:20,105 --> 00:06:22,330
of the function of radius.

99
00:06:22,330 --> 00:06:25,020
And we'll go from
one to another.

100
00:06:25,020 --> 00:06:29,840
They could be called
r minus to r plus.

101
00:06:29,840 --> 00:06:33,110
And our semi-classical
interpretation,

102
00:06:33,110 --> 00:06:37,760
which is roughly good if you're
talking about high quantum

103
00:06:37,760 --> 00:06:41,660
numbers, high principle
quantum numbers, high l

104
00:06:41,660 --> 00:06:49,600
quantum numbers, is
that you have an ellipse

105
00:06:49,600 --> 00:06:53,630
and the radial distance
to the center where

106
00:06:53,630 --> 00:06:59,555
the proton is located goes
from r plus to r minus.

107
00:07:04,450 --> 00:07:06,880
The electron is
bouncing back and forth.

108
00:07:06,880 --> 00:07:08,800
That is the classical picture.

109
00:07:08,800 --> 00:07:10,720
In the quantum
mechanical picture,

110
00:07:10,720 --> 00:07:13,490
you expect something
somewhat similar.

111
00:07:13,490 --> 00:07:16,720
There's going to be a wave
function, maybe a wave

112
00:07:16,720 --> 00:07:19,690
function here, psi squared.

113
00:07:19,690 --> 00:07:25,480
And it's going to be vanishingly
small before this point.

114
00:07:25,480 --> 00:07:27,670
And then by the
time you get here,

115
00:07:27,670 --> 00:07:33,620
it's going to be very
fast, and then decay again.

116
00:07:33,620 --> 00:07:36,890
So the probability
distribution will

117
00:07:36,890 --> 00:07:40,590
mimic the time spent
by the particles,

118
00:07:40,590 --> 00:07:43,640
as we used to argue before.

119
00:07:43,640 --> 00:07:50,730
So let's do a little exercise of
calculating the turning points.

120
00:07:50,730 --> 00:07:53,940
So how do we do that?

121
00:07:53,940 --> 00:08:03,890
Well, we set h squared l times
l plus 1 over 2mr squared,

122
00:08:03,890 --> 00:08:06,980
minus e squared over r--

123
00:08:06,980 --> 00:08:10,580
that's the effective potential--

124
00:08:10,580 --> 00:08:17,390
equal to the energy of some
level n, principle quantum

125
00:08:17,390 --> 00:08:19,700
number n.

126
00:08:19,700 --> 00:08:26,840
So it would be minus e squared
over 2a0, 1 over n squared.

127
00:08:26,840 --> 00:08:31,970
That's the value
of the energy En.

128
00:08:31,970 --> 00:08:35,570
And the solutions of
this quadratic equation

129
00:08:35,570 --> 00:08:39,725
are going to give us the r plus
and the r minus of the orbit.

130
00:08:43,100 --> 00:08:51,560
So it's probably worthwhile
to do a little transformation

131
00:08:51,560 --> 00:09:00,920
and to say r equal a0 times
x, where x is unit free.

132
00:09:07,740 --> 00:09:13,590
And then the equation becomes
h squared, l times l plus 1,

133
00:09:13,590 --> 00:09:21,900
over 2ma0 squared,
times l times l plus 1,

134
00:09:21,900 --> 00:09:36,090
over x squared, minus e squared
over a0x, is equal to minus

135
00:09:36,090 --> 00:09:41,180
e squared over 2a0,
1 over n squared.

136
00:09:44,450 --> 00:09:49,280
So the unit should work out.

137
00:09:49,280 --> 00:09:52,680
We should get a nice
equation without units.

138
00:09:52,680 --> 00:09:57,860
So what must be happening
is that the coefficient

139
00:09:57,860 --> 00:10:05,070
in front of here, h
squared 2ma0 squared,

140
00:10:05,070 --> 00:10:11,250
let's take the other a0 and
separate it out, and transform

141
00:10:11,250 --> 00:10:11,750
this.

142
00:10:11,750 --> 00:10:16,670
Remember, a0 was h
squared over m e squared.

143
00:10:21,600 --> 00:10:27,430
So here we get h
squared over 2ma0.

144
00:10:27,430 --> 00:10:30,340
And that a0 now
has an h squared.

145
00:10:30,340 --> 00:10:32,670
And there's m e squared.

146
00:10:32,670 --> 00:10:37,110
So the h squared
cancels, the m cancels.

147
00:10:37,110 --> 00:10:41,280
And this is e squared over 2a0.

148
00:10:41,280 --> 00:10:49,190
So this whole
coefficient is e squared

149
00:10:49,190 --> 00:10:55,100
over 2a0, which is nice because
now the e squared over a0,

150
00:10:55,100 --> 00:10:59,680
the e squared over a0, a
squared over a0, cancel.

151
00:10:59,680 --> 00:11:08,890
And we get l times l
plus 1 over x squared--

152
00:11:08,890 --> 00:11:13,520
I've canceled all this factor--

153
00:11:13,520 --> 00:11:20,885
minus 2 over x is equal
to minus 1 over n squared.

154
00:11:25,600 --> 00:11:30,700
So I multiplied by the
inverse of this quantity.

155
00:11:30,700 --> 00:11:34,360
It clears up the factor
in the first term.

156
00:11:34,360 --> 00:11:37,720
It produces an extra factor
of 2 in the second term.

157
00:11:37,720 --> 00:11:42,640
And you've got a nice
simple quadratic equation.

158
00:11:42,640 --> 00:11:43,902
AUDIENCE: Question?

159
00:11:43,902 --> 00:11:45,026
PROFESSOR: Yes?

160
00:11:45,026 --> 00:11:48,232
AUDIENCE: Where do you get
the extra [INAUDIBLE] l,

161
00:11:48,232 --> 00:11:50,140
l plus 1 [? come from? ?]

162
00:11:50,140 --> 00:11:52,890
PROFESSOR: Oh,
there's no such thing.

163
00:11:52,890 --> 00:11:54,880
There's too many of them.

164
00:11:54,880 --> 00:11:57,490
Thank you very much.

165
00:11:57,490 --> 00:12:10,350
Yes, too many, 1 over x squared.

166
00:12:10,350 --> 00:12:10,990
Thank you.

167
00:12:14,430 --> 00:12:21,470
So let's move this
to the other side--

168
00:12:21,470 --> 00:12:25,280
plus 1 over n squared equals 0.

169
00:12:25,280 --> 00:12:28,015
So this is the main equation.

170
00:12:31,120 --> 00:12:35,720
And we can write this--

171
00:12:35,720 --> 00:12:40,600
well, the solution for 1 over
x is a quadratic equation in 1

172
00:12:40,600 --> 00:12:42,190
over x.

173
00:12:42,190 --> 00:12:47,215
So I'll write it here.

174
00:12:49,940 --> 00:12:56,600
1 plus/minus square root of
1 minus l times l plus 1,

175
00:12:56,600 --> 00:13:01,660
over n squared, divided
by l times l plus 1.

176
00:13:08,120 --> 00:13:10,920
That's just from the
quadratic formula.

177
00:13:10,920 --> 00:13:13,680
So then you invert it.

178
00:13:13,680 --> 00:13:21,500
So x is now l times l plus 1,
over 1 plus/minus square root

179
00:13:21,500 --> 00:13:25,755
of 1 minus l, l plus
1 over n squared.

180
00:13:30,400 --> 00:13:38,330
And we multiply by
the opposite factor

181
00:13:38,330 --> 00:13:39,710
to clear the square root.

182
00:13:39,710 --> 00:13:48,035
So 1 minus/plus square root of
1 minus l times l plus 1 over n

183
00:13:48,035 --> 00:13:48,535
squared.

184
00:13:51,380 --> 00:13:52,970
And the same factor here--

185
00:13:52,970 --> 00:13:57,980
1 minus/plus square
root of 1 minus l,

186
00:13:57,980 --> 00:14:04,870
l plus 1 over n squared,
all of these things.

187
00:14:04,870 --> 00:14:07,780
But that's not so bad.

188
00:14:07,780 --> 00:14:15,880
You get l times l plus
1, times this factor

189
00:14:15,880 --> 00:14:18,100
that you had in the numerator--

190
00:14:18,100 --> 00:14:19,450
that still is the same--

191
00:14:19,450 --> 00:14:24,550
1 minus l, l plus
1 over n squared.

192
00:14:24,550 --> 00:14:31,180
Now we're after an interesting
piece of information, the two

193
00:14:31,180 --> 00:14:36,200
sizes of the ellipse, so it's
worth simplifying what you got.

194
00:14:36,200 --> 00:14:40,450
This equation is not nice
enough, so we're simplifying.

195
00:14:40,450 --> 00:14:46,840
And in the denominator, you
have a plus b times a minus b.

196
00:14:46,840 --> 00:14:56,370
So it's 1 minus 1 minus l,
l plus 1, over n squared.

197
00:14:56,370 --> 00:15:01,470
And the good thing is that the
denominator, the 1 cancels.

198
00:15:01,470 --> 00:15:04,620
And you get the plus,
l times l plus 1 over n

199
00:15:04,620 --> 00:15:07,350
squared, that cancels, that one.

200
00:15:07,350 --> 00:15:12,780
So that, at the end of the day,
we get a pretty nice formula.

201
00:15:16,160 --> 00:15:20,620
The formula is x is
equal to n squared,

202
00:15:20,620 --> 00:15:31,770
1 minus/plus, 1 minus l times
l plus 1, over n squared,

203
00:15:31,770 --> 00:15:34,100
like that.

204
00:15:34,100 --> 00:15:42,620
So if we wish, it's r plus/minus
is x multiplied by a0,

205
00:15:42,620 --> 00:15:49,730
so it's n squared a0, 1
minus/plus square root

206
00:15:49,730 --> 00:15:54,750
of 1 minus l times l
plus 1, over n squared.

207
00:16:03,930 --> 00:16:14,430
OK, so the ellipse is
defined by those two values

208
00:16:14,430 --> 00:16:16,260
that we have here.

209
00:16:16,260 --> 00:16:20,660
But the surprising
thing is that the sum

210
00:16:20,660 --> 00:16:23,640
over the ellipse,
and its eccentricity,

211
00:16:23,640 --> 00:16:29,130
is dramatically affected
by the values of l.

212
00:16:29,130 --> 00:16:37,320
In fact, for l equals largest,
l will be comparable to n.

213
00:16:37,320 --> 00:16:40,700
Remember, l can go
up to n minus 1.

214
00:16:40,700 --> 00:16:44,070
So at that point,
this is essentially 1.

215
00:16:44,070 --> 00:16:47,670
You cancel this and
there's nothing here.

216
00:16:47,670 --> 00:16:55,830
So r plus and r minus become
about the same for l equals n

217
00:16:55,830 --> 00:17:00,990
minus 1. r plus is almost
the same as r minus

218
00:17:00,990 --> 00:17:05,730
And the orbit is circular,
completely circular.

219
00:17:05,730 --> 00:17:13,260
On the other hand,
for l equals 0,

220
00:17:13,260 --> 00:17:19,170
the orbit is completely
elliptical in that the radius

221
00:17:19,170 --> 00:17:23,319
for l equal 0, this
is 0, 1, plus/minus 1.

222
00:17:23,319 --> 00:17:27,210
So sometimes it's twice this
value, sometimes it's 0.

223
00:17:27,210 --> 00:17:33,120
So you have the case
where r minus can be 0,

224
00:17:33,120 --> 00:17:38,650
and you have just an orbit
that is like that, r minus 0,

225
00:17:38,650 --> 00:17:40,930
so extremely elliptical--

226
00:17:44,325 --> 00:17:45,295
elliptical.

227
00:17:48,210 --> 00:17:51,990
Of course, that is the
semi-classical approximation.

228
00:17:51,990 --> 00:17:56,175
So it's more reliable when
you have a reasonable l.

229
00:17:59,850 --> 00:18:05,370
And finally we can
say here, for example,

230
00:18:05,370 --> 00:18:14,310
one interesting thing, that
r plus plus r minus, over 2,

231
00:18:14,310 --> 00:18:17,290
which is this--

232
00:18:17,290 --> 00:18:21,460
r plus plus r minus
is the total longest

233
00:18:21,460 --> 00:18:25,450
axis of the ellipse, divided by
2, the center of the ellipse,

234
00:18:25,450 --> 00:18:31,330
not the focus, that distance
is independent of l,

235
00:18:31,330 --> 00:18:34,610
so you have n squared a0.

236
00:18:38,790 --> 00:18:46,890
So a typical Rydberg atom
will have n equal 100.

237
00:18:46,890 --> 00:18:49,810
So this is an example--

238
00:18:49,810 --> 00:19:02,420
n equal 100, l equals 60, in
which case, for n equal 100,

239
00:19:02,420 --> 00:19:15,370
r plus/minus is equal to
10,000a0, n squared a0,

240
00:19:15,370 --> 00:19:21,100
times this factor, which in one
case is 1.8 and in the other

241
00:19:21,100 --> 00:19:24,290
case is 0.2.

242
00:19:24,290 --> 00:19:28,880
That's what you get for l
equal 60 and n equal 100.

243
00:19:28,880 --> 00:19:39,220
So this orbit, you have
r plus about 18,000a0,

244
00:19:39,220 --> 00:19:52,090
and r minus about 2,000a0.

245
00:19:52,090 --> 00:19:56,350
And all of our orbits
satisfy this property

246
00:19:56,350 --> 00:20:06,040
that if you have this,
r minus and r plus,

247
00:20:06,040 --> 00:20:09,340
all of the orbits
with different l

248
00:20:09,340 --> 00:20:13,130
have the same r
plus plus r minus.

249
00:20:13,130 --> 00:20:20,190
So if this is the total
length of the major axis when

250
00:20:20,190 --> 00:20:27,330
the orbit becomes
circular, it's the same.

251
00:20:27,330 --> 00:20:34,260
And this distance, when the
orbit becomes very elliptical,

252
00:20:34,260 --> 00:20:35,340
is the same as well.

253
00:20:38,040 --> 00:20:46,040
I mentioned last time that this
nice property is degeneracy.

254
00:20:46,040 --> 00:20:52,310
We're here, if you're keeping
n fixed but changing l,

255
00:20:52,310 --> 00:20:55,250
you're going from
all these ellipses.

256
00:20:55,250 --> 00:20:59,150
This is for l equals n minus 1.

257
00:20:59,150 --> 00:21:01,730
And this one is for l equals 0.

258
00:21:01,730 --> 00:21:04,550
And all this ellipses arc here.

259
00:21:04,550 --> 00:21:09,350
And those are all the
semi-classical picture

260
00:21:09,350 --> 00:21:13,950
of those degenerate states in
the diagram of the hydrogen

261
00:21:13,950 --> 00:21:14,450
atom.

262
00:21:14,450 --> 00:21:18,020
The diagram of the hydrogen
atom was something like this.

263
00:21:22,650 --> 00:21:25,460
And you're looking at
all the degenerate states

264
00:21:25,460 --> 00:21:28,650
that you have there.

265
00:21:28,650 --> 00:21:32,330
And they are degenerate,
and you would say, well,

266
00:21:32,330 --> 00:21:37,210
why are ellipses that
look like that degenerate.

267
00:21:37,210 --> 00:21:42,370
Well, even Kepler apparently
knew that in Kepler's laws.

268
00:21:42,370 --> 00:21:47,410
That he observed that the
period of motion of an orbit

269
00:21:47,410 --> 00:21:50,200
just depended on
the semi-major axis.

270
00:21:50,200 --> 00:21:53,670
So periods are
related to energies.

271
00:21:53,670 --> 00:21:57,430
And it's reasonable
that we have this thing

272
00:21:57,430 --> 00:21:59,150
in quantum mechanics.

273
00:21:59,150 --> 00:22:02,740
Now this degeneracy, I
want to just finish up

274
00:22:02,740 --> 00:22:06,760
by emphasizing
what you have here.

275
00:22:06,760 --> 00:22:13,660
When somebody asks what is the
number of states you have here,

276
00:22:13,660 --> 00:22:16,780
well, you have to be precise
in what you're counting.

277
00:22:16,780 --> 00:22:22,990
The number of full physical
states of the quantum system

278
00:22:22,990 --> 00:22:26,850
is one here, one here, one here.

279
00:22:26,850 --> 00:22:30,960
But each one of this
corresponds to l equals 1,

280
00:22:30,960 --> 00:22:34,900
so each one of this
is triply degenerate,

281
00:22:34,900 --> 00:22:38,770
because m can be
minus 1, 0, and 1.

282
00:22:38,770 --> 00:22:43,700
So here-- three states,
three states, three states.

283
00:22:43,700 --> 00:22:48,690
Here-- this is five states,
five states, five states,

284
00:22:48,690 --> 00:22:50,830
because they all have l equal 2.

285
00:22:50,830 --> 00:22:54,100
And l equal 2 goes m
from minus 2 to plus 2.

286
00:22:54,100 --> 00:22:59,030
So we don't actually
put three things here.

287
00:22:59,030 --> 00:23:01,030
I think that would be confusing.

288
00:23:01,030 --> 00:23:04,900
We could not put five
and we cannot see.

289
00:23:04,900 --> 00:23:08,620
But it should be
remembered that there's

290
00:23:08,620 --> 00:23:11,830
the implicit extra
degeneracy here

291
00:23:11,830 --> 00:23:17,680
associated with the
azimuthal quantum number

292
00:23:17,680 --> 00:23:22,070
that we sometimes just don't
represent it in a figure.