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00:00:01,350 --> 00:00:02,800
PROFESSOR: So far, so good.

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00:00:02,800 --> 00:00:05,030
We've really calculated.

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00:00:05,030 --> 00:00:06,310
And we could stop here.

4
00:00:06,310 --> 00:00:09,000
But there is a
nice interpretation

5
00:00:09,000 --> 00:00:14,460
of this Darwin term
that gives you intuition

6
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as to what's really happening.

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See, when you look at
these terms overall--

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00:00:22,110 --> 00:00:25,740
these interactions
with this class--

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00:00:25,740 --> 00:00:29,610
you say, well, the kinetic
energy was not relativistic.

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00:00:29,610 --> 00:00:31,860
That gives rise to this term.

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00:00:31,860 --> 00:00:35,280
There is a story for
the spin orbit as well.

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00:00:35,280 --> 00:00:38,490
We think of the proton--

13
00:00:38,490 --> 00:00:40,020
creates an electric field.

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You, the electron,
are moving around.

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As you move inside the
static-electric field,

16
00:00:46,680 --> 00:00:48,370
you see a magnetic field.

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00:00:48,370 --> 00:00:52,080
The magnetic field interacts
with your dipole moment

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00:00:52,080 --> 00:00:52,950
of the electron.

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That's the story here.

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00:00:55,060 --> 00:00:57,140
So what's the story
for this term?

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00:00:57,140 --> 00:00:59,910
Where does it come from?

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What is the physics behind it?

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That's what I want
to discuss now.

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So the physics interpretation
of the Darwin term

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is that the electron
behaves not as

26
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if it would be a point
particle but as if it would be,

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kind of, a little ball where
the charge is spread out.

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We know that any mechanical
model of the electron, where

29
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you think of it as
a ball, the spinning

30
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doesn't work for the spin.

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But somehow, here, we can see
that this extra correction

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00:01:48,060 --> 00:01:50,220
to the energy is
the correction that

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00:01:50,220 --> 00:01:55,710
would appear if, somehow, the
electron, instead of feeling

34
00:01:55,710 --> 00:01:59,790
the potential of the
nucleus at one point,

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as if it would be
a point particle,

36
00:02:01,350 --> 00:02:03,960
it's as if it would be spread.

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So let me make a drawing here.

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Here is the proton.

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And it creates a potential.

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And we will put an
electron at the point r.

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00:02:23,425 --> 00:02:24,300
There's the electron.

42
00:02:24,300 --> 00:02:26,940
But now, we're going to think
that this electron is really

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a cloud, like this.

44
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And the charge is
distributed over there.

45
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And now, I'm going
to try to estimate

46
00:02:46,340 --> 00:02:49,460
what happens to the
potential energy

47
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if it's really
behaving like that.

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So for that, I'm going to
make another coordinate system

49
00:02:59,600 --> 00:03:01,710
sort of starting here.

50
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I'm going to call
the vector that

51
00:03:03,330 --> 00:03:08,760
goes to an arbitrary
point here vector u.

52
00:03:08,760 --> 00:03:12,600
So from the origin or center
of this charge distribution,

53
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we put the vector u.

54
00:03:14,460 --> 00:03:18,870
And then I have this vector
over here, which is r plus u.

55
00:03:23,150 --> 00:03:29,840
And this u points to a little
bit of charge, a little cubit

56
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here, for example.

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So this is our setup.

58
00:03:38,250 --> 00:03:43,275
Now, the potential--
due to the proton--

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00:03:45,840 --> 00:03:51,360
proton-- we have a
potential V of r,

60
00:03:51,360 --> 00:03:58,740
which is equal to minus the
charge of the electron times

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00:03:58,740 --> 00:04:02,340
the scalar potential, phi of r--

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it's minus e times e over r,
which is our familiar minus e

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00:04:10,060 --> 00:04:12,070
squared over r.

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00:04:12,070 --> 00:04:14,110
So the potential energy--

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this is the potential energy.

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00:04:17,911 --> 00:04:27,320
And so if you had
a point particle--

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00:04:27,320 --> 00:04:28,610
for a point particle--

68
00:04:34,710 --> 00:04:41,040
you have that the proton creates
a potential at a distance r.

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00:04:41,040 --> 00:04:42,870
You multiply by the charge.

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And you get the
potential energy.

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00:04:47,670 --> 00:04:53,500
So how do we do it a
little more generally?

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00:04:53,500 --> 00:04:58,555
I'm going to call
this V tilde of r.

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The true potential energy--

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potential energy-- when the
center of the electron is

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00:05:19,010 --> 00:05:19,640
at r--

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00:05:26,040 --> 00:05:32,090
electron at r.

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00:05:32,090 --> 00:05:34,765
So what is the true
potential energy

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00:05:34,765 --> 00:05:36,980
when the center of
the electron is at r?

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00:05:36,980 --> 00:05:40,490
Well, I would have
to do an integral

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00:05:40,490 --> 00:05:48,240
over the electron of the
little amount of charges

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00:05:48,240 --> 00:05:52,205
times the potential
at those points.

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00:05:58,240 --> 00:06:01,990
For every piece
of charge, I must

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00:06:01,990 --> 00:06:06,430
multiply the charge times
the potential generated

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00:06:06,430 --> 00:06:07,990
by the proton.

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00:06:07,990 --> 00:06:14,770
And that would give me the
total energy of this electron

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00:06:14,770 --> 00:06:18,500
in this potential, V of r.

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00:06:18,500 --> 00:06:23,900
So let me use that
terminology here.

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00:06:23,900 --> 00:06:29,100
I will describe the
charge density--

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00:06:29,100 --> 00:06:36,310
density-- rho of u--

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00:06:36,310 --> 00:06:42,040
the density of charge at a
position given by a u vector.

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00:06:42,040 --> 00:06:49,160
I'll write it as minus
e times rho 0 of u--

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00:06:49,160 --> 00:06:50,690
a little bit of notation.

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00:06:50,690 --> 00:06:55,610
I apologize, but it's
necessary to do things cleanly.

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00:06:55,610 --> 00:07:00,820
The charge density
of the electron--

95
00:07:00,820 --> 00:07:05,750
electron-- is given by
rho of u minus e rho 0.

96
00:07:05,750 --> 00:07:10,340
And then it should be
true that the integral

97
00:07:10,340 --> 00:07:21,740
over the electron of d cube
u rho 0 of u is equal to 1.

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00:07:21,740 --> 00:07:23,240
Look at that integral.

99
00:07:23,240 --> 00:07:26,870
Why should it be equal to 1?

100
00:07:26,870 --> 00:07:31,820
It should be equal to 1, because
then the integral of the charge

101
00:07:31,820 --> 00:07:32,960
density--

102
00:07:32,960 --> 00:07:39,630
the integral of this over all of
space would be minus e times 1.

103
00:07:39,630 --> 00:07:42,830
And that's exactly
what you expect.

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00:07:42,830 --> 00:07:48,380
So rho 0 is a unit free--

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00:07:48,380 --> 00:07:53,390
well, 1 over length cubed
is a charge-free quantity.

106
00:07:53,390 --> 00:07:58,600
The charge is carried by
this constant over here.

107
00:07:58,600 --> 00:08:03,340
So what happens then
with this V of r?

108
00:08:03,340 --> 00:08:06,910
This V of r is the charge--

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00:08:06,910 --> 00:08:19,090
little charge q is d cubed
u times rho of u times phi

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00:08:19,090 --> 00:08:20,900
at r plus u.

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00:08:23,960 --> 00:08:29,700
Look, the charge is this
at some little element.

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00:08:29,700 --> 00:08:33,919
And then the potential there
is the potential at r plus u.

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00:08:36,809 --> 00:08:39,510
So one more step--

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00:08:39,510 --> 00:08:51,560
V of r-- if you take from
rho this extra minus e,

115
00:08:51,560 --> 00:08:57,290
minus e times phi
of r is what we call

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00:08:57,290 --> 00:09:01,400
the potential energy due to r.

117
00:09:01,400 --> 00:09:05,780
So take the minus
e out and append it

118
00:09:05,780 --> 00:09:13,900
to the capital Phi here, so that
we have integral d cube u rho

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00:09:13,900 --> 00:09:19,240
0 of u V of r plus u.

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OK, this was our goal.

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00:09:29,730 --> 00:09:33,530
It gives you the new energy--

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00:09:33,530 --> 00:09:39,320
the true potential
energy of this electron

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00:09:39,320 --> 00:09:44,270
when this center is
at r as the smearing

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00:09:44,270 --> 00:09:48,920
of the potential energy
of a point particle

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00:09:48,920 --> 00:09:51,650
smeared over the electron.

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00:09:56,930 --> 00:10:02,230
So this is a formula that
represents our intuition.

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00:10:02,230 --> 00:10:04,780
And moreover, it
has the rho 0 here

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00:10:04,780 --> 00:10:08,770
that tells you the weight that
you should apply at any point,

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00:10:08,770 --> 00:10:13,370
because this rho 0 is
proportional to the charge--

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very good.

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00:10:14,420 --> 00:10:15,680
We have a formula.

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But I was supposed
to explain that term.

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00:10:20,030 --> 00:10:22,260
And we have not
explained it yet.

134
00:10:22,260 --> 00:10:24,470
But now, it's time
to explain it.

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00:10:24,470 --> 00:10:27,350
We're going to try to
do a computation that

136
00:10:27,350 --> 00:10:29,330
helps us do that.

137
00:10:29,330 --> 00:10:33,890
The idea is that we're going
to expand the potential

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00:10:33,890 --> 00:10:39,830
around the point r and treat
this as a small deviation

139
00:10:39,830 --> 00:10:44,330
because, after all, we expect
this little ball that I drew

140
00:10:44,330 --> 00:10:46,970
big here for the
purposes of illustration

141
00:10:46,970 --> 00:10:48,900
to be rather small.

142
00:10:48,900 --> 00:11:02,430
So let's write V of r plus u
as V of r in Taylor series--

143
00:11:02,430 --> 00:11:04,710
plus the next term
is the derivative

144
00:11:04,710 --> 00:11:08,390
of V with respect to
position evaluated

145
00:11:08,390 --> 00:11:13,030
at r multiplied by the
deviation that you moved.

146
00:11:13,030 --> 00:11:27,160
So sum over i dv dx i
evaluated at r times u i.

147
00:11:27,160 --> 00:11:31,610
This is the component
of this vector u.

148
00:11:31,610 --> 00:11:35,030
That's the first term
in the Taylor series.

149
00:11:35,030 --> 00:11:37,130
And we need one more--

150
00:11:37,130 --> 00:11:49,625
plus 1/2 sum over i and j,
d 2nd V vx i vx j evaluated

151
00:11:49,625 --> 00:11:52,820
at r ui uj.

152
00:11:55,940 --> 00:11:58,640
And then we have
to integrate here.

153
00:11:58,640 --> 00:12:02,190
We substituted in
here and integrate.

154
00:12:02,190 --> 00:12:09,560
So V tilde of r--
let's see what it is.

155
00:12:09,560 --> 00:12:12,210
Well, I can put
this whole thing--

156
00:12:12,210 --> 00:12:14,180
we don't want to write so much.

157
00:12:14,180 --> 00:12:18,280
So let's try to do
it a little quick.

158
00:12:18,280 --> 00:12:20,470
We're integrating over u.

159
00:12:20,470 --> 00:12:24,800
And here, let's think
of the first term here.

160
00:12:24,800 --> 00:12:29,300
When you plug in the first
term, this function of r

161
00:12:29,300 --> 00:12:30,830
has nothing to do with u.

162
00:12:30,830 --> 00:12:38,380
So it goes out and you get V of
r times the integral d cube u

163
00:12:38,380 --> 00:12:39,800
rho 0 of u.

164
00:12:42,790 --> 00:12:49,160
For the next term in
here, these derivatives

165
00:12:49,160 --> 00:12:52,490
are evaluated at r-- have
nothing to do with u.

166
00:12:52,490 --> 00:12:53,660
They go out.

167
00:12:53,660 --> 00:13:05,770
So plus sum over I dv dx
i of r integral d cube u

168
00:13:05,770 --> 00:13:09,210
rho 0 of u, ui.

169
00:13:12,530 --> 00:13:16,720
Last term-- these
derivatives go out.

170
00:13:16,720 --> 00:13:18,910
They're evaluated at r.

171
00:13:18,910 --> 00:13:20,450
They have nothing
to do with you.

172
00:13:20,450 --> 00:13:29,140
So plus 1/2 sum over i
and j d second V dx i dx

173
00:13:29,140 --> 00:13:40,430
j evaluated at r integral
d cube u rho of u ui uj.

174
00:13:40,430 --> 00:13:42,985
OK-- all these terms!

175
00:13:47,620 --> 00:13:52,480
Happily we can interpret
much of what we have here.

176
00:13:52,480 --> 00:13:57,445
So what is this first integral?

177
00:14:03,170 --> 00:14:10,070
V of r equals--

178
00:14:10,070 --> 00:14:13,050
this integral is our
normalization integral.

179
00:14:16,290 --> 00:14:18,520
Over here, that's equal to 1.

180
00:14:18,520 --> 00:14:22,050
So this is very nice.

181
00:14:22,050 --> 00:14:25,110
That's what you
would expect that

182
00:14:25,110 --> 00:14:29,450
to first approximation, the
total energy of the electron,

183
00:14:29,450 --> 00:14:31,200
when it is at r--

184
00:14:31,200 --> 00:14:34,680
as if it would be a
point particle at r.

185
00:14:34,680 --> 00:14:38,670
Now, let's look
at the next terms.

186
00:14:38,670 --> 00:14:44,970
Now, we will assume that
the distribution of charge

187
00:14:44,970 --> 00:14:46,720
is spherically symmetric.

188
00:14:46,720 --> 00:14:50,850
So that means that
rho of u vector

189
00:14:50,850 --> 00:14:54,690
is actually a
function rho 0 of u,

190
00:14:54,690 --> 00:15:00,560
where u is the length
of the u vector.

191
00:15:00,560 --> 00:15:04,837
So it just depends on the
distance from the point

192
00:15:04,837 --> 00:15:05,670
that you're looking.

193
00:15:05,670 --> 00:15:07,280
That's the charge.

194
00:15:07,280 --> 00:15:11,390
Why would it be
more complicated?

195
00:15:11,390 --> 00:15:17,450
If that is the case, if this
is spherically symmetric,

196
00:15:17,450 --> 00:15:25,880
you can already see that
these integrals would vanish,

197
00:15:25,880 --> 00:15:30,350
because you're integrating the
spherically symmetric quantity

198
00:15:30,350 --> 00:15:32,450
times a power of a coordinate.

199
00:15:32,450 --> 00:15:34,700
That's something you
did in the homework

200
00:15:34,700 --> 00:15:37,760
as well for this Stark effect.

201
00:15:37,760 --> 00:15:41,150
You realized that integrals of
spherically symmetric functions

202
00:15:41,150 --> 00:15:44,610
then powers of x, y,
and z-- well, they get

203
00:15:44,610 --> 00:15:47,030
killed, unless those
are even powers.

204
00:15:47,030 --> 00:15:49,340
So this is an odd power.

205
00:15:49,340 --> 00:15:50,810
And that's 0.

206
00:15:50,810 --> 00:15:54,500
So this term is gone.

207
00:15:54,500 --> 00:16:02,420
And this integral is
interesting as well.

208
00:16:02,420 --> 00:16:05,060
When you have a spherically
symmetric quantity

209
00:16:05,060 --> 00:16:09,440
and you have i and j different,
the integral would be 0.

210
00:16:09,440 --> 00:16:12,320
It's like having a power
of x and a power of y

211
00:16:12,320 --> 00:16:13,860
in your homework.

212
00:16:13,860 --> 00:16:17,710
So if that integral would be 0,
it would only be non-zero if i

213
00:16:17,710 --> 00:16:21,030
is equal to j-- the first
component, for example--

214
00:16:21,030 --> 00:16:23,600
1 u1 u1.

215
00:16:23,600 --> 00:16:28,940
But that integral would be
the same as u2 u2 or u3 u3.

216
00:16:28,940 --> 00:16:32,630
So in fact, each
of these integrals

217
00:16:32,630 --> 00:16:41,630
is proportional to delta
i j, but equal to 1/3

218
00:16:41,630 --> 00:16:47,390
of the integral of
d cube u rho of u--

219
00:16:47,390 --> 00:16:51,600
I'll put the delta ij here--

220
00:16:51,600 --> 00:16:54,840
times u squared.

221
00:16:54,840 --> 00:17:00,510
u squared is-- u1 squared plus
u2 squared plus u3 squared--

222
00:17:00,510 --> 00:17:03,270
each one gives
the same integral.

223
00:17:03,270 --> 00:17:08,380
Therefore, the result
has a 1/3 in front.

224
00:17:08,380 --> 00:17:09,940
So what do we get?

225
00:17:09,940 --> 00:17:20,430
Plus 1/6-- and here,
we get a big smile.

226
00:17:20,430 --> 00:17:20,970
Why?

227
00:17:20,970 --> 00:17:25,079
Because delta i j,
with this thing,

228
00:17:25,079 --> 00:17:30,195
is the Laplacian of
V evaluated at r.

229
00:17:30,195 --> 00:17:33,390
They have the same
derivative summed.

230
00:17:33,390 --> 00:17:34,830
So what do we get here?

231
00:17:34,830 --> 00:17:41,920
1/6 of the Laplacian
of V evaluated

232
00:17:41,920 --> 00:17:51,100
at r times the integral d
cube u rho 0 of u u squared.

233
00:17:57,810 --> 00:18:02,700
Very nice-- already starting
to look like what we wanted,

234
00:18:02,700 --> 00:18:05,040
a Laplacian.

235
00:18:05,040 --> 00:18:06,720
Can we do a little better?

236
00:18:06,720 --> 00:18:09,120
Yes, we can.

237
00:18:09,120 --> 00:18:10,260
It's possible.

238
00:18:10,260 --> 00:18:17,070
Let's assume the charged
particle has a size.

239
00:18:17,070 --> 00:18:30,330
So let's assume the
electron is a ball of radius

240
00:18:30,330 --> 00:18:35,290
the Compton wavelength,
which is h bar over mc.

241
00:18:38,840 --> 00:18:50,020
So that means rho 0
of u is non-zero when

242
00:18:50,020 --> 00:18:53,050
u is less than this lambda.

243
00:18:53,050 --> 00:18:57,940
And it's 0 when u is
greater than this lambda.

244
00:18:57,940 --> 00:19:01,860
That's a ball-- some
density up to there.

245
00:19:01,860 --> 00:19:04,280
And this must integrate to 1.

246
00:19:04,280 --> 00:19:11,710
So it's 1 over the volume of
that ball, 4 pi lambda cubed

247
00:19:11,710 --> 00:19:12,415
over 3.

248
00:19:22,370 --> 00:19:23,385
Do this integral.

249
00:19:26,920 --> 00:19:34,750
This integral gives you,
actually, 3/5 of lambda

250
00:19:34,750 --> 00:19:40,320
squared with that function.

251
00:19:40,320 --> 00:19:47,540
So the end result is that V
tilde of r is equal to V of r.

252
00:19:51,100 --> 00:19:59,270
Plus 3/5 of this thing
times that is 1/10 h

253
00:19:59,270 --> 00:20:08,080
squared m squared c
squared Laplacian of V.

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00:20:08,080 --> 00:20:11,410
The new correction to the
potential, when this mole came

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00:20:11,410 --> 00:20:13,930
out to this 1/10--

256
00:20:13,930 --> 00:20:16,600
that's pretty close, I must say.

257
00:20:16,600 --> 00:20:19,720
There's an 1/8 there.

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00:20:19,720 --> 00:20:24,670
We assume the electron is
a ball of size, the Compton

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00:20:24,670 --> 00:20:27,970
wavelength of the electron.

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00:20:27,970 --> 00:20:29,830
It's a very rough assumption.

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00:20:29,830 --> 00:20:35,170
So even to see the number not
coming off by a factor of 100

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00:20:35,170 --> 00:20:38,300
is quite nice.

263
00:20:38,300 --> 00:20:40,510
So that's the interpretation.

264
00:20:40,510 --> 00:20:42,910
The known locality--
the spread out

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00:20:42,910 --> 00:20:47,070
of the electron into a little
ball of its Compton size

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00:20:47,070 --> 00:20:51,330
is quite significant.

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00:20:51,330 --> 00:20:53,290
The Compton wavelength
of the electron

268
00:20:53,290 --> 00:20:57,010
is fundamental in
quantum field theory.

269
00:20:57,010 --> 00:20:59,020
The Compton wavelength
of an electron

270
00:20:59,020 --> 00:21:05,980
is the size of a photon whose
energy is equal to the rest

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00:21:05,980 --> 00:21:07,430
mass of the electron.

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00:21:07,430 --> 00:21:11,050
So if you have a photon
of this size equal

273
00:21:11,050 --> 00:21:15,010
of wavelength equal to Compton
length of the electron,

274
00:21:15,010 --> 00:21:17,680
that photon packs
enough energy to create

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00:21:17,680 --> 00:21:21,520
electron-positron pairs.

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00:21:21,520 --> 00:21:24,790
That's why, in quantum field
theory, you care about this.

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00:21:24,790 --> 00:21:27,670
And the quantum field
theory is really

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00:21:27,670 --> 00:21:30,290
the way you do relativistic
quantum mechanics.

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00:21:30,290 --> 00:21:35,500
So it's not too surprising
that, in relativistic analysis

280
00:21:35,500 --> 00:21:37,990
such as that of
the Dirac equation,

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00:21:37,990 --> 00:21:42,910
we find a role for the Compton
wavelength of the electron

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00:21:42,910 --> 00:21:46,364
and a correction
associated with it.