1
00:00:01,340 --> 00:00:04,340
PROFESSOR: So again,
much of the difficulty

2
00:00:04,340 --> 00:00:08,660
here is just making sure you're
doing the perturbation theory

3
00:00:08,660 --> 00:00:14,840
right and working with
the two basis correctly.

4
00:00:14,840 --> 00:00:17,380
But now we're in
the good position.

5
00:00:17,380 --> 00:00:19,410
We can combine our results.

6
00:00:24,220 --> 00:00:26,380
So here is what I
want to combine.

7
00:00:39,370 --> 00:00:46,930
I want, in principle, to combine
all the things, all the terms.

8
00:00:46,930 --> 00:00:52,190
And we've calculated all
the terms to some degree.

9
00:00:52,190 --> 00:00:56,650
So we saw the Darwin term
only affects l equals 0.

10
00:00:56,650 --> 00:01:04,440
On the other hand, spin orbit
really requires s dot l.

11
00:01:04,440 --> 00:01:12,970
And the state should be
acted nontrivially by l.

12
00:01:12,970 --> 00:01:18,880
If all your states are
singlets, l gives 0 on them.

13
00:01:18,880 --> 00:01:21,320
So actually, you
have a little bit

14
00:01:21,320 --> 00:01:25,480
of an uncomfortable situation
because l acting on l

15
00:01:25,480 --> 00:01:27,640
equals 0 states will give you 0.

16
00:01:27,640 --> 00:01:30,400
But you still have
the 1 over r cubed.

17
00:01:30,400 --> 00:01:33,860
And the 1 over r cubed
has l dependencies here.

18
00:01:33,860 --> 00:01:38,260
So it's 0 in the denominator,
0 in the numerator.

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00:01:38,260 --> 00:01:41,770
The whole spin orbit
coupling doesn't seem

20
00:01:41,770 --> 00:01:44,110
to make sense for l equals 0.

21
00:01:44,110 --> 00:01:49,030
So most people say, physically,
the spin orbit coupling

22
00:01:49,030 --> 00:01:51,050
vanishes for l equals 0.

23
00:01:51,050 --> 00:01:53,990
It should not be there.

24
00:01:53,990 --> 00:01:57,100
But then something
very funny happens.

25
00:01:57,100 --> 00:02:02,890
If you take a proper limit
of this as l goes to 0,

26
00:02:02,890 --> 00:02:08,550
it gives you, for the spin
orbit result, the same thing

27
00:02:08,550 --> 00:02:13,510
as the Darwin result.
So for l equals 0,

28
00:02:13,510 --> 00:02:19,750
the spin orbit limit is
actually the same as the Darwin.

29
00:02:19,750 --> 00:02:22,750
It's almost as if
you say, oh, the spin

30
00:02:22,750 --> 00:02:24,370
orbit has everything in it.

31
00:02:24,370 --> 00:02:27,280
But it's not legal
for l equals 0.

32
00:02:27,280 --> 00:02:28,800
But the Darwin does it.

33
00:02:28,800 --> 00:02:32,560
So if I put together the
spin orbit and relativistics

34
00:02:32,560 --> 00:02:37,570
for l different
from 0, I'm legal.

35
00:02:37,570 --> 00:02:43,090
For l equals 0, I should really
sum relativistic and Darwin.

36
00:02:43,090 --> 00:02:45,160
But actually, it
turns out that it's

37
00:02:45,160 --> 00:02:48,190
the same as summing
relativistic plus spin orbit.

38
00:02:48,190 --> 00:02:50,680
Because the limit of
spin orbit for l equals 0

39
00:02:50,680 --> 00:02:52,270
is equal to the Darwin.

40
00:02:52,270 --> 00:02:58,240
So we will just sum
relativistic plus spin orbit.

41
00:02:58,240 --> 00:03:01,120
And that gives the result
for everything, including

42
00:03:01,120 --> 00:03:03,340
Darwin for l equals 0.

43
00:03:03,340 --> 00:03:06,520
So this is minor subtlety.

44
00:03:06,520 --> 00:03:11,080
But the end result is
that we can combine it.

45
00:03:11,080 --> 00:03:17,210
So the happy thing, as well,
is that we now can do this.

46
00:03:17,210 --> 00:03:26,260
mj delta H relativistic plus
delta H spin orbit nljmj.

47
00:03:29,410 --> 00:03:34,300
Because whatever
we calculated here

48
00:03:34,300 --> 00:03:38,200
was actually the same
that H relativistic

49
00:03:38,200 --> 00:03:40,630
in the coupled basis.

50
00:03:40,630 --> 00:03:44,830
And anyway, the spin orbit,
we use the coupled basis

51
00:03:44,830 --> 00:03:46,100
to get this number.

52
00:03:46,100 --> 00:03:51,040
So this thing, you can
add the two results,

53
00:03:51,040 --> 00:04:01,360
and you get En0 squared
over 2mc squared 3 plus n

54
00:04:01,360 --> 00:04:12,670
J j plus 1 minus 3l l
plus 1 minus 3/4 over l l

55
00:04:12,670 --> 00:04:16,060
plus 1/2 l plus 1--

56
00:04:16,060 --> 00:04:16,825
a little messy.

57
00:04:23,860 --> 00:04:26,390
OK.

58
00:04:26,390 --> 00:04:31,510
Something very
unexpected happens now.

59
00:04:31,510 --> 00:04:34,880
It's something that when you do
the algebra yourself, you say,

60
00:04:34,880 --> 00:04:35,780
wow.

61
00:04:35,780 --> 00:04:36,625
How did that happen?

62
00:04:39,990 --> 00:04:40,815
Here is the issue.

63
00:04:43,820 --> 00:04:49,810
We are trying to compute the
splittings of the hydrogen

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00:04:49,810 --> 00:04:51,340
atom.

65
00:04:51,340 --> 00:04:56,300
And for this, as we said
in the coupled basis,

66
00:04:56,300 --> 00:05:01,480
the degeneracies happen
for a given value of n.

67
00:05:07,879 --> 00:05:10,775
I'm getting good at
doing this diagram now.

68
00:05:13,420 --> 00:05:15,390
So we're looking at the fixed n.

69
00:05:18,640 --> 00:05:22,390
And then, you have all kind of
degenerate states for various

70
00:05:22,390 --> 00:05:25,090
l's and j's.

71
00:05:25,090 --> 00:05:30,120
So is there a better way
to rewrite this formula?

72
00:05:30,120 --> 00:05:33,130
And you say the following--

73
00:05:33,130 --> 00:05:38,630
suppose you think of
some states of fixed j--

74
00:05:38,630 --> 00:05:43,620
fixed j states-- j.

75
00:05:47,920 --> 00:05:54,070
For example, this state-- we'll
use our l equals 1 over here--

76
00:05:54,070 --> 00:05:57,160
I have 1, 2, 3--

77
00:05:57,160 --> 00:06:04,960
3P 3/2 and 3P 1/2.

78
00:06:04,960 --> 00:06:11,020
Here, I have 3D 5/2
because this is l equals 2.

79
00:06:11,020 --> 00:06:14,850
So 2 plus 1/2 and 2 minus 1/2.

80
00:06:17,950 --> 00:06:24,280
So I have here
states of fixed j--

81
00:06:24,280 --> 00:06:27,340
two states with the same j.

82
00:06:27,340 --> 00:06:30,610
But they come from
different l's.

83
00:06:30,610 --> 00:06:34,480
Because when you get a
total j of some value,

84
00:06:34,480 --> 00:06:37,195
it can come from a lower l--

85
00:06:37,195 --> 00:06:39,640
l plus 1/2 gives you that j--

86
00:06:39,640 --> 00:06:45,850
or it can come from a top l with
l minus 1/2 giving you that j.

87
00:06:45,850 --> 00:06:53,230
So a given j value can arise
from an l that is 1/2 higher

88
00:06:53,230 --> 00:06:56,450
or an l that is 1/2 lower.

89
00:06:56,450 --> 00:07:04,300
So for a fixed j, it may be
that l is equal to j minus 1/2,

90
00:07:04,300 --> 00:07:08,250
or l is equal to j plus 1/2.

91
00:07:11,780 --> 00:07:14,260
Now, look at this quantity.

92
00:07:14,260 --> 00:07:20,165
We'll call this whole
quantity f of jl.

93
00:07:27,130 --> 00:07:40,660
Very astonishingly, f of jl when
l is j minus 1/2, or f of jl

94
00:07:40,660 --> 00:07:46,150
where l is j plus 1/2,
you can calculate it.

95
00:07:46,150 --> 00:07:51,250
Put l equals j minus
1/2 on that formula.

96
00:07:51,250 --> 00:07:56,185
And then put l equals j
plus 1/2 in that formula.

97
00:07:56,185 --> 00:07:59,870
You would say, it's
going to be a mess.

98
00:07:59,870 --> 00:08:03,460
In fact, both
cases are the same.

99
00:08:03,460 --> 00:08:10,410
And it give you minus
2 over a j plus 1/2.

100
00:08:10,410 --> 00:08:16,880
So the whole l dependence
here, amazingly, is fake.

101
00:08:16,880 --> 00:08:19,200
There's no l dependence
in this factor.

102
00:08:19,200 --> 00:08:21,950
It is a little strange.

103
00:08:21,950 --> 00:08:27,500
There's no l dependence because,
given j, l can be two values.

104
00:08:27,500 --> 00:08:30,470
And for those two values, that
function gives you the same.

105
00:08:30,470 --> 00:08:33,720
It's one of those, like x
squared gives the same for 1

106
00:08:33,720 --> 00:08:34,804
and minus 1.

107
00:08:34,804 --> 00:08:40,039
This function, once you
fix j, l can be two values.

108
00:08:40,039 --> 00:08:42,960
And it so happens that these
two values give the same.

109
00:08:42,960 --> 00:08:46,160
So at the end of the day,
this just depends on j.

110
00:08:46,160 --> 00:08:50,000
That's the most important
result of this lecture.

111
00:08:50,000 --> 00:08:54,760
The whole structure,
once you put relativity,

112
00:08:54,760 --> 00:08:59,830
Darwin, and spin orbit,
just depends on j.

113
00:08:59,830 --> 00:09:03,835
And what is the result
when you simplify this?

114
00:09:09,160 --> 00:09:20,620
Our result is that the
fine structure Enljmj--

115
00:09:24,700 --> 00:09:36,460
fine structure 1-- is equal to
minus alpha to the fourth mc

116
00:09:36,460 --> 00:09:45,050
squared 1 over 2n to the fourth
n over j plus 1/2 minus 3/4.

117
00:09:49,570 --> 00:09:53,900
Whole answer, all together--

118
00:09:53,900 --> 00:10:01,180
Darwin fine spin orbit
and relativistic.

119
00:10:01,180 --> 00:10:06,880
So a few comments about this.

120
00:10:06,880 --> 00:10:10,900
People that look at
the Dirac equation

121
00:10:10,900 --> 00:10:14,210
more seriously would have
expected this result.

122
00:10:14,210 --> 00:10:18,160
It turns out that in the
Dirac equation, the symmetry

123
00:10:18,160 --> 00:10:19,440
and the rotations--

124
00:10:19,440 --> 00:10:21,520
the generator of rotations--

125
00:10:21,520 --> 00:10:26,890
is exactly j, which is l plus s.

126
00:10:26,890 --> 00:10:28,420
That generates rotation.

127
00:10:28,420 --> 00:10:30,640
That commutes with
the Hamiltonian.

128
00:10:30,640 --> 00:10:33,820
So you should expect that
the energy eigenstates

129
00:10:33,820 --> 00:10:35,590
are j eigenstates.

130
00:10:35,590 --> 00:10:40,240
So here, we're seeing
that, yes, they can

131
00:10:40,240 --> 00:10:42,150
be simultaneously diagonalized.

132
00:10:42,150 --> 00:10:47,000
The eigenstates can also
be labeled by the j value.

133
00:10:47,000 --> 00:10:50,050
So the exact eigenstates
in the Dirac equation

134
00:10:50,050 --> 00:10:52,060
can be labeled by the
j value, and we're

135
00:10:52,060 --> 00:10:55,060
seeing a reflection of this.

136
00:10:55,060 --> 00:10:59,950
This means that the j multiplets
are not going to be split,

137
00:10:59,950 --> 00:11:02,770
and there's going to be,
moreover, some degeneracy.

138
00:11:02,770 --> 00:11:07,030
They're not split because
there's no mj dependence.

139
00:11:07,030 --> 00:11:09,820
And they are all
going to be the same.

140
00:11:09,820 --> 00:11:14,740
So let me finish by drawing
how the spectrum looks.

141
00:11:14,740 --> 00:11:19,060
It's pretty important
to see that.

142
00:11:19,060 --> 00:11:23,590
This quantity over here,
this numerical quantity,

143
00:11:23,590 --> 00:11:27,590
is always positive--

144
00:11:27,590 --> 00:11:31,640
positive-- for any state
in the hydrogen atom.

145
00:11:31,640 --> 00:11:34,940
You can see that because j max--

146
00:11:34,940 --> 00:11:38,250
the maximum value--
so this is negative.

147
00:11:38,250 --> 00:11:41,960
So you need to know what is the
minimum value of this quantity

148
00:11:41,960 --> 00:11:43,550
to see if it goes negative.

149
00:11:43,550 --> 00:11:48,530
The minimum value of this
ratio is when j is maximum.

150
00:11:48,530 --> 00:11:55,580
The maximum j is when j is l
plus 1/2 with the maximum l.

151
00:11:55,580 --> 00:11:59,090
So this is l plus 1 maximum.

152
00:11:59,090 --> 00:12:03,260
But l plus 1 maximum
is, in fact, n.

153
00:12:03,260 --> 00:12:06,320
So the minimum
value of this is 1.

154
00:12:06,320 --> 00:12:08,720
So this is always positive.

155
00:12:08,720 --> 00:12:12,090
As n increases, the
corrections become smaller.

156
00:12:12,090 --> 00:12:13,880
So what happens?

157
00:12:13,880 --> 00:12:16,170
All the corrections
are negative.

158
00:12:16,170 --> 00:12:24,630
So if you had the original
states, they all go down a bit.

159
00:12:24,630 --> 00:12:27,750
Even the ground state
goes down a bit.

160
00:12:27,750 --> 00:12:31,110
1S 1/2 is down.

161
00:12:34,150 --> 00:12:38,920
Because for n equal
1, you do get a state.

162
00:12:38,920 --> 00:12:40,300
Then what do you get?

163
00:12:40,300 --> 00:12:45,350
2S 1/2 is here.

164
00:12:45,350 --> 00:12:46,720
It's also down a bit.

165
00:12:46,720 --> 00:12:54,270
3S 1/2-- so this is l equals 0.

166
00:12:54,270 --> 00:12:58,035
Then you have l equals 1.

167
00:12:58,035 --> 00:13:01,130
Remember, what did we have here?

168
00:13:01,130 --> 00:13:11,330
We have 2P 1/2 and 2P 3/2.

169
00:13:11,330 --> 00:13:15,440
Because l equal 1 gives
you j equals 3/2 and 1/2.

170
00:13:15,440 --> 00:13:20,540
Look, these two remain
degenerate because they

171
00:13:20,540 --> 00:13:23,870
have the same n and the same j.

172
00:13:23,870 --> 00:13:25,890
And that's all that matters.

173
00:13:25,890 --> 00:13:31,060
So 1/2 and 1/2 have the same
j, so they remain degenerate.

174
00:13:31,060 --> 00:13:35,970
2P 3/2 has a higher
j, therefore,

175
00:13:35,970 --> 00:13:37,500
has a smaller number.

176
00:13:37,500 --> 00:13:42,050
So it's lowered less, and
it appears a little higher.

177
00:13:42,050 --> 00:13:50,530
So here, you would have
3P 1/2 and here, 3P 3/2.

178
00:13:53,620 --> 00:13:56,765
And the next state here is 3D.

179
00:14:02,990 --> 00:14:05,270
Now you're combining l equal 2.

180
00:14:05,270 --> 00:14:08,840
So you get 5/2 and 3/2.

181
00:14:08,840 --> 00:14:15,470
The 3/2 is degenerate with this
3/2 because it has the same j.

182
00:14:15,470 --> 00:14:19,630
And the 5/2 is a little higher--

183
00:14:19,630 --> 00:14:20,920
3D 5/2.

184
00:14:23,930 --> 00:14:28,780
So this is our final picture.

185
00:14:28,780 --> 00:14:32,470
The hydrogen atom, all the
states get pushed down.

186
00:14:32,470 --> 00:14:40,390
The various multiplets with
different l but the same j

187
00:14:40,390 --> 00:14:41,610
are still degenerate.

188
00:14:41,610 --> 00:14:43,660
This formula has
no l dependence.

189
00:14:43,660 --> 00:14:45,760
So these two are the same.

190
00:14:45,760 --> 00:14:47,050
These two are the same.

191
00:14:47,050 --> 00:14:49,930
These two are the same.

192
00:14:49,930 --> 00:14:55,000
Moreover, the j
multiplets are not split.

193
00:14:55,000 --> 00:14:57,700
Every j multiplet differs
[? because of ?] a collection

194
00:14:57,700 --> 00:15:00,370
of states with different mj.

195
00:15:00,370 --> 00:15:01,780
But mj doesn't appear.

196
00:15:01,780 --> 00:15:05,740
So this is your fine
structure of hydrogen atom.

197
00:15:05,740 --> 00:15:11,140
And this is where you study
in detail the Zeeman splitting

198
00:15:11,140 --> 00:15:13,030
and the Stark splitting.

199
00:15:13,030 --> 00:15:14,860
And we'll talk a
little bit about them

200
00:15:14,860 --> 00:15:17,710
next time, as we
will begin, also,

201
00:15:17,710 --> 00:15:20,680
our study of the
semiclassical approximation.

202
00:15:20,680 --> 00:15:22,560
See you then.