1
00:00:00,770 --> 00:00:04,130
PROFESSOR: If you tell
someone solve this equation

2
00:00:04,130 --> 00:00:06,260
for l equals zero--

3
00:00:06,260 --> 00:00:08,150
so you have this
effective potential

4
00:00:08,150 --> 00:00:11,210
for the real equation--
solve it for l equals zero

5
00:00:11,210 --> 00:00:14,300
and you find some energies.

6
00:00:14,300 --> 00:00:19,040
You solve it for l equals 1,
and you've found some energies.

7
00:00:19,040 --> 00:00:21,230
You solve it now
for l equals two

8
00:00:21,230 --> 00:00:22,790
and you found some energies.

9
00:00:22,790 --> 00:00:25,670
Well, we found
them all together,

10
00:00:25,670 --> 00:00:30,090
but something
extraordinary happened

11
00:00:30,090 --> 00:00:34,290
there was no a priori
reason the system should

12
00:00:34,290 --> 00:00:35,610
have been so simple.

13
00:00:35,610 --> 00:00:39,830
It might have happened that
these states would have not

14
00:00:39,830 --> 00:00:41,810
been aligned with
the previous states.

15
00:00:44,620 --> 00:00:48,400
Nothing we've explained
in this course

16
00:00:48,400 --> 00:00:50,840
predicts that this
would have happened,

17
00:00:50,840 --> 00:00:57,620
this perfect alignment with
extreme amount of degeneracy.

18
00:00:57,620 --> 00:01:01,190
Because you have an l
equals 2 multiplet here.

19
00:01:01,190 --> 00:01:03,020
That means five states.

20
00:01:03,020 --> 00:01:06,140
So there is
degeneracy, but that's

21
00:01:06,140 --> 00:01:08,570
implicit in angular momentum.

22
00:01:08,570 --> 00:01:10,250
It has an explanation.

23
00:01:10,250 --> 00:01:13,430
But why would there
be a degeneracy

24
00:01:13,430 --> 00:01:18,846
between l equals 1 solutions
and l equals 2 solutions?

25
00:01:22,860 --> 00:01:24,520
Total mystery, actually.

26
00:01:27,410 --> 00:01:31,670
And this led to all kinds
of interesting discoveries

27
00:01:31,670 --> 00:01:36,380
that have to do with the
Runge-Lenz vector, which

28
00:01:36,380 --> 00:01:43,010
is some conserved vector
in planetary orbits.

29
00:01:43,010 --> 00:01:47,540
In planetary orbits,
in Newton's theory,

30
00:01:47,540 --> 00:01:51,960
an elliptical orbit is
the general solution.

31
00:01:51,960 --> 00:01:55,070
Nevertheless, elliptical
orbits do not precess.

32
00:01:55,070 --> 00:01:57,620
So you have an ellipse,
it goes like that.

33
00:01:57,620 --> 00:02:01,360
It's not going around
and rotating the ellipse

34
00:02:01,360 --> 00:02:02,080
at the same time.

35
00:02:02,080 --> 00:02:06,660
The precession of an ellipse is
not allowed by Newton's theory.

36
00:02:06,660 --> 00:02:10,880
It's allowed by
Einstein's gravity theory,

37
00:02:10,880 --> 00:02:15,250
and in fact, the precession
of mercury was measured.

38
00:02:15,250 --> 00:02:19,000
But there is no precession
in Newton's theory,

39
00:02:19,000 --> 00:02:23,620
and there's no precession
in a hydrogen atom,

40
00:02:23,620 --> 00:02:25,570
in a sense, as you will see.

41
00:02:25,570 --> 00:02:32,980
And that explains, actually
in a rather interesting way--

42
00:02:32,980 --> 00:02:35,830
but I'm not saying how yet--

43
00:02:35,830 --> 00:02:38,410
why there is this
extra degeneracy.

44
00:02:38,410 --> 00:02:42,430
In fact, if you had
solved the problem

45
00:02:42,430 --> 00:02:49,180
of an infinite spherical, well,
you will solve that in 805.

46
00:02:49,180 --> 00:02:51,410
Infinite spherical well.

47
00:02:51,410 --> 00:02:55,450
Suppose there's infinite square
well-- infinite spherical well.

48
00:02:55,450 --> 00:03:01,570
Inside a sphere of radius
A, the potential is 0.

49
00:03:01,570 --> 00:03:06,700
Outside the sphere of radius
A, the potential is infinite.

50
00:03:06,700 --> 00:03:09,440
That potential that
looks so symmetric--

51
00:03:09,440 --> 00:03:12,610
the l equals 0
states are like this.

52
00:03:12,610 --> 00:03:15,670
The l equal 1 states
are like that.

53
00:03:15,670 --> 00:03:18,330
The l equal 2 states
are like that,

54
00:03:18,330 --> 00:03:21,070
and there's never
any coincidence.

55
00:03:21,070 --> 00:03:23,585
So this coincidence
between the l

56
00:03:23,585 --> 00:03:28,780
equals 0, l equals 2, l
equals 2 is very special.

57
00:03:28,780 --> 00:03:31,150
It just doesn't happen often.

58
00:03:31,150 --> 00:03:35,170
It's a sign of an
extra symmetry.

59
00:03:35,170 --> 00:03:37,900
This could only be explained
because the hydrogen

60
00:03:37,900 --> 00:03:42,950
atom has an extra symmetry
you're not aware of.

61
00:03:42,950 --> 00:03:47,080
So that's why this
Runge-Lenz vector

62
00:03:47,080 --> 00:03:49,600
has to do with
that extra symmetry

63
00:03:49,600 --> 00:03:52,330
and explains this effect.

64
00:03:52,330 --> 00:03:56,470
And we'll get some
intuition about it today.

65
00:03:56,470 --> 00:04:00,490
A few more remarks
to get your intuition

66
00:04:00,490 --> 00:04:02,610
working on the hydrogen atoms.

67
00:04:02,610 --> 00:04:07,070
Z equals 1, we write
the wave function.

68
00:04:07,070 --> 00:04:09,690
This is the most
famous wave function.

69
00:04:09,690 --> 00:04:16,021
Pi a0 cubed e to
the minus r over a0.

70
00:04:16,021 --> 00:04:19,900
And this is for z equals 1.

71
00:04:22,520 --> 00:04:26,510
n equals 1, l equals
0, m equals zero--

72
00:04:26,510 --> 00:04:28,160
the complete ground state.

73
00:04:30,860 --> 00:04:38,510
It's interesting to note and
try to think, OK, suppose they

74
00:04:38,510 --> 00:04:42,410
gave you the z equals 1 answer.

75
00:04:42,410 --> 00:04:44,900
How do you get z
different from 1?

76
00:04:48,660 --> 00:04:51,960
Can I just do something
with this solution?

77
00:04:51,960 --> 00:04:56,130
Well, somehow it's written
here, but I don't give here

78
00:04:56,130 --> 00:04:59,220
the normalization because
it's impossibly complicated

79
00:04:59,220 --> 00:05:03,110
to write the general form
for the normalization.

80
00:05:03,110 --> 00:05:09,340
So how should I
think of changing

81
00:05:09,340 --> 00:05:10,930
if somebody would
have told me this

82
00:05:10,930 --> 00:05:13,810
is the answer for Z equals 1?

83
00:05:13,810 --> 00:05:17,380
How do I get to Z
different from 1?

84
00:05:17,380 --> 00:05:19,930
And then I think
of the potential,

85
00:05:19,930 --> 00:05:25,390
and the potential
was e squared over r.

86
00:05:25,390 --> 00:05:31,130
That was the potential
before, V of r.

87
00:05:31,130 --> 00:05:35,320
And it will pass
to a V of r that

88
00:05:35,320 --> 00:05:39,610
has minus Ze squared over r.

89
00:05:39,610 --> 00:05:45,130
Because now you have a nucleus
with Z protons interacting

90
00:05:45,130 --> 00:05:48,520
with one electron, so
that's how it changed.

91
00:05:48,520 --> 00:05:52,840
So naturally, what seems
to be the change here,

92
00:05:52,840 --> 00:05:57,400
and you could imagine just
solving it without the Z

93
00:05:57,400 --> 00:06:00,990
and then adding the Z, is that
everywhere that you have e

94
00:06:00,990 --> 00:06:05,385
squared you should
put Z times e squared.

95
00:06:08,470 --> 00:06:14,140
And then you think of a0.

96
00:06:14,140 --> 00:06:18,905
a0 was h squared
over me squared.

97
00:06:21,500 --> 00:06:24,290
We calculated that
some time ago.

98
00:06:24,290 --> 00:06:29,460
And then if e squared
goes to Ze squared,

99
00:06:29,460 --> 00:06:36,350
this will go to 1 over Z
squared over me squared.

100
00:06:36,350 --> 00:06:44,300
So it will go to a0 over Z.

101
00:06:44,300 --> 00:06:51,000
So you change a0 to a0 over Z.
And these are the two changes.

102
00:06:53,580 --> 00:06:56,570
One is implicit on the
other, but many times you

103
00:06:56,570 --> 00:06:58,970
write the formula
in a mixed way.

104
00:06:58,970 --> 00:07:00,200
Look at that energy.

105
00:07:04,390 --> 00:07:08,190
If you would have looked
at this without the Z

106
00:07:08,190 --> 00:07:11,280
and you would have
said, oh, e squared

107
00:07:11,280 --> 00:07:15,510
is replaced by Ze squared,
you would have put a single Z.

108
00:07:15,510 --> 00:07:21,930
But there is a Z squared here,
and it comes because 1z is here

109
00:07:21,930 --> 00:07:26,220
and the other z is in
the a0 because a0 also

110
00:07:26,220 --> 00:07:28,530
has the e squared.

111
00:07:28,530 --> 00:07:32,490
So you have to be aware
that we write these things.

112
00:07:32,490 --> 00:07:37,380
And this is intuitively a very
nice way to write the energy,

113
00:07:37,380 --> 00:07:40,040
because it has the right units--

114
00:07:40,040 --> 00:07:43,369
electric charge squared
divided by distance.

115
00:07:43,369 --> 00:07:44,910
But you could have
written everything

116
00:07:44,910 --> 00:07:48,270
with h bar, something
like that, in which case

117
00:07:48,270 --> 00:07:53,700
the z squared might have
been less surprising.

118
00:07:53,700 --> 00:07:58,020
We see here, however, the z is
appearing in the right place

119
00:07:58,020 --> 00:08:00,000
because of the a0.

120
00:08:00,000 --> 00:08:05,530
So here, I was right.

121
00:08:05,530 --> 00:08:10,096
e to the minus zr over a0.

122
00:08:13,824 --> 00:08:17,500
And can I get the
normalization even right?

123
00:08:17,500 --> 00:08:20,140
At this moment, yes.

124
00:08:20,140 --> 00:08:22,270
Let's do the same change here.

125
00:08:22,270 --> 00:08:27,870
Pi a0 cubed, z cubed.

126
00:08:27,870 --> 00:08:32,299
And this must be right,
because, in fact,

127
00:08:32,299 --> 00:08:36,940
if this wave function
was normalizable--

128
00:08:36,940 --> 00:08:40,706
not normalizable;
was normalized--

129
00:08:40,706 --> 00:08:45,720
when you do the integral,
somehow the a0 did not matter,

130
00:08:45,720 --> 00:08:46,625
must not matter.

131
00:08:49,200 --> 00:08:52,250
You checked psi squared
integrated over volume

132
00:08:52,250 --> 00:08:53,205
is equal to 1.

133
00:08:53,205 --> 00:08:56,720
The a0 must be canceling here.

134
00:08:56,720 --> 00:09:02,600
And therefore, if it works for
a0, it must work for a0 over z,

135
00:09:02,600 --> 00:09:04,050
and that must be
a wave function.

136
00:09:07,230 --> 00:09:09,390
So that's fine.

137
00:09:09,390 --> 00:09:10,965
That's one thing you could ask.

138
00:09:15,410 --> 00:09:21,630
Another thing that you could
ask is, at least intuitively,

139
00:09:21,630 --> 00:09:25,950
why did we get this factor here?

140
00:09:25,950 --> 00:09:29,860
Why did we get this exponential?

141
00:09:29,860 --> 00:09:36,970
And that's also not
mysterious at all.

142
00:09:36,970 --> 00:09:42,190
This comes from the
differential equation.

143
00:09:42,190 --> 00:09:47,950
It comes up rather immediately
from the differential equation.

144
00:09:47,950 --> 00:09:57,100
You have minus h squared over
2m d second u dr squared,

145
00:09:57,100 --> 00:10:00,310
plus some sort of number--

146
00:10:00,310 --> 00:10:03,720
you don't care how
much-- u over r squared

147
00:10:03,720 --> 00:10:08,500
in the effective potential,
minus some number over r

148
00:10:08,500 --> 00:10:12,340
times u is equal to Eu.

149
00:10:12,340 --> 00:10:15,880
This was your radial
differential equation.

150
00:10:15,880 --> 00:10:20,608
And this r goes to infinity.

151
00:10:20,608 --> 00:10:24,910
As r goes to infinity,
you get minus h

152
00:10:24,910 --> 00:10:35,110
squared over 2m d second u
dr squared is equal to Eu,

153
00:10:35,110 --> 00:10:36,430
roughly.

154
00:10:36,430 --> 00:10:38,820
That's the key terms.

155
00:10:38,820 --> 00:10:43,990
And from here, d
second u dr squared

156
00:10:43,990 --> 00:10:49,970
is equal to minus 2
mE over h squared u.

157
00:10:55,520 --> 00:10:58,100
And that gives you
an exponential,

158
00:10:58,100 --> 00:11:02,990
and the exponential must be
of the right value, which

159
00:11:02,990 --> 00:11:12,300
we can calculate easily from
the expression for the energy.

160
00:11:12,300 --> 00:11:16,460
So the expression for
the energy gives you

161
00:11:16,460 --> 00:11:28,930
em equals minus z squared
over 2a0 e squared, times 1

162
00:11:28,930 --> 00:11:30,000
over n squared.

163
00:11:34,720 --> 00:11:47,725
And I can change this
thing into minus z squared.

164
00:11:51,870 --> 00:11:58,660
Recall what's the value of
the fine structure constant.

165
00:11:58,660 --> 00:12:05,580
I can replace e squared from
a0 to get the following thing

166
00:12:05,580 --> 00:12:10,840
over 2a0 times h
squared over ma0.

167
00:12:14,190 --> 00:12:15,960
That is 1 over m squared.

168
00:12:20,340 --> 00:12:21,920
A little bit of manipulation.

169
00:12:21,920 --> 00:12:29,180
So at the end, minus
2mEn over h squared,

170
00:12:29,180 --> 00:12:31,735
which is what I need from
the differential equation.

171
00:12:36,010 --> 00:12:40,420
I must multiply by
minus 2m over h squared.

172
00:12:40,420 --> 00:12:45,470
You see that minus the 2m over
h squared, they will disappear.

173
00:12:45,470 --> 00:12:53,650
So you get here z squared
over n squared a0 squared.

174
00:12:53,650 --> 00:12:55,330
A little bit of manipulation.

175
00:12:55,330 --> 00:12:58,730
So what are we trying to get?

176
00:12:58,730 --> 00:13:00,610
We want to understand
immediately

177
00:13:00,610 --> 00:13:03,050
where this came from.

178
00:13:03,050 --> 00:13:04,640
And we see it.

179
00:13:04,640 --> 00:13:07,630
It comes from the asymptotic
form of the differential

180
00:13:07,630 --> 00:13:09,690
equation for a solution.

181
00:13:09,690 --> 00:13:14,440
So I calculate the value of
the right-hand side is this,

182
00:13:14,440 --> 00:13:16,540
and therefore this
differential equation

183
00:13:16,540 --> 00:13:24,850
now looks like the du dr
squared equals z squared over n

184
00:13:24,850 --> 00:13:27,820
squared a0 squared u.

185
00:13:27,820 --> 00:13:31,000
Are indeed, from
here, the solutions

186
00:13:31,000 --> 00:13:34,720
are exponentials
of the square root

187
00:13:34,720 --> 00:13:39,370
of this, which is z over na0r.

188
00:13:39,370 --> 00:13:46,490
And that's a quicker derivation
of a feature of the wave

189
00:13:46,490 --> 00:13:47,190
function.

190
00:13:47,190 --> 00:13:50,420
It's almost like you want to
look at this wave function,

191
00:13:50,420 --> 00:13:52,280
and you want to
say, I understand

192
00:13:52,280 --> 00:13:54,890
where everything comes from.

193
00:13:54,890 --> 00:13:58,340
And I don't have to solve
pages and pages of differential

194
00:13:58,340 --> 00:14:04,160
equations to see why I need
this, why I need that much.

195
00:14:04,160 --> 00:14:07,220
I know I need this
from r equals zero.

196
00:14:07,220 --> 00:14:08,450
I know this degree.

197
00:14:08,450 --> 00:14:11,120
I need it from the node theorem.

198
00:14:11,120 --> 00:14:16,500
Everything sort of has a
reason for being there,

199
00:14:16,500 --> 00:14:19,720
and we should understand.