-
Notifications
You must be signed in to change notification settings - Fork 2
/
M2L7h.txt
114 lines (110 loc) · 4.35 KB
/
M2L7h.txt
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
#
# File: content-mit-8-421-2x-subtitles/M2L7h.txt
#
# Captions for 8.421x module
#
# This file has 104 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
The last conclusion to the fine structure is the Darwin term.
The Darwin term is sort of peculiar,
but there is a sort of an intuitive picture
in which I can derive it.
It starts by saying that an electron-- well, we know that an electron
is, as far as we know, a point object.
But in order to derive the Darwin term,
we assume that the electron is not arbitrarily, accurately
localized, but it is smeared out.
And you may remember when I discussed
fundamental units and the fine structure constant alpha
and I told you that alpha-- the fact that alpha is small
leads to the fact that people say
electromagnetic interactions are weak.
I discussed with you that if you want to do similar particle physics
and use single particle pictures and single particle equations
you can never assume that a particle is localized better
than its Compton wavelength.
If you would localize a particle better than the Compton
wavelength, it would have a momentum uncertainty, which
would lead to an energy uncertainty, which
is larger than its rest mass and you get into quantum field theory
and pair production.
So at some point you have to be aware
that single particle pictures break down
when you assume a localization of particles
better than the Compton wavelength.
So therefore-- and I'm waving my hands
because it's a hand-waving argument--
but we can derive the Darwin term
by assuming that the electron is sort of smeared out
over a dimension.
And this dimension can only be the Compton wavelength.
Which is h bar over mc.
And there is a nice German word for it,
how you can imagine that the electron is smeared out.
It goes by the word "Zitterbewegung".
And I think the German word is much nicer
than its translation, which is trembling motion.
So at least in this picture you assume that the electron
is sort of smeared out.
It's not because the electron has a size.
It just trembles.
It just trembles.
And it leads to a smear out over the Compton wavelength.
But that implies now that if this trembling motion is
sufficiently fast we should not use in our Schroedinger
equation the Coulomb potential.
You should use the Coulomb potential
which is spatially averaged over the Compton wavelength.
So therefore, the phenomenological derivation
of the Darwin term goes by replacing the Coulomb potential
in the Schroedinger equation by the spatially averaged Coulomb
potential.
So let me assume-- let me now derive what that means.
So if you assume there is a very small displacement s,
now we can take the Coulomb potential
and expand it into-- it's sy squ potential
plus derivative times s plus the second order Taylor expansion
the xi, the xj component.
And then we need the second order derivative xi
xj of the Coulomb potential.
So the correction is now-- well, if we average it
in an isotopic way, the gradient term,
the linear term doesn't contribute.
It just averages to zero.
So the leading correction comes from the curvature term.
So therefore we have to take the curvature or the Laplacian
of the Coulomb potential.
This scale of sx square sy square sz square is
the Compton wavelength squared.
And, well, maybe I'm taking the argument off the Compton
wavelength now too literally, but at least
if I want to carry it through, sx square, sy square,
sz square are each of them 1/3 of s absolute value squared.
So if I now use the Coulomb potential
I obtain the result where I have the Laplacian of 1 over r.
And as of course you know, the 1
over r function has a non-vanishing Laplacian
only at the origin.
So therefore the correction is only valid for s electrons,
which have a non-vanishing probably
to feel the origin of the Coulomb potential.
I did a lot of hand waving here.
And, yes, those factors of 1/3 are
the fact that the electron is smeared out
over the Compton radius there may be prefactors.
But, OK.
This is the physical picture.
You can get an exact result by simply
taking the Dirac equation, doing the
non-relativistic approximation.
And the term which appears in the Pauli equation
is actually the same.
The only thing which is different
is that the prefactor is not 1/6.
the exact prefactor is 1/8.
So this, the exact result, actually
validates that the physical picture we have used
is at least a reasonable approximation.