-
Notifications
You must be signed in to change notification settings - Fork 2
/
M2L6d.txt
141 lines (135 loc) · 5.78 KB
/
M2L6d.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
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
#
# File: content-mit-8-421-2x-subtitles/M2L6d.txt
#
# Captions for 8.421x module
#
# This file has 131 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
OK then, let me spend five to 10 minutes
on spectroscopic notation.
So the next five or 10 minutes, how
we describe the configuration of an atom,
it's, well, I don't particularly like to teach it,
because it's more nomenclature about old-fashioned symbols.
On the other hand, if you're working with atoms,
you have to learn the language of how to describe atoms.
And I also know that in, an appreciable fraction
of oral exams, there will be one person in the committee,
and say, what is your favorite atom,
and what is the configuration of your atom?
So, it's something, if you're an atomic physicist,
you're supposed to know.
The spectroscopic notation, the term designation
focuses on the fact that if you have an isolated atom,
we have angular momentum conservation,
and so we have at least two quantum numbers, which I was
also two good quantum numbers.
We have some approximate quantum numbers,
where we have additional terms which break certain symmetries,
but an isolated atom lives in isotropic space.
The total angular momentum of this atom is conserved.
It's absolutely conserved.
It's an absolutely good quantum number,
and the good quantum numbers is the total angular momentum J,
and its projection M j.
So, in the language of atomic physics, we call J a level,
it's different from states, so one level
has now, 2 J plus 1 sublevels, or states.
So usually when we talk about a level,
we assume the level has degeneracy, degeneracies,
because there is the M j quantum number.
So J-- we're talking about electronic structure--
so J can have, when we have a isolated atom,
can have contributions from several electrons.
It can have contributions, and these electrons
can contribute through spin, S, and orbital angular momentum L.
In many situations, especially with alkali atoms,
the inner core is completely filled shell.
There is no S, no L from the inner electrons
which contribute, and all the contribution
of the angular momentum comes from the outer electron.
Especially for the lighter atoms,
the non-relativistic atoms, the different electrons
undergo LS coupling.
In other words, if you have multiple electrons,
their orbital angular momentum couple up
to the total orbital angular momentum L,
and all the spins couple up to the total spin, s.
So therefore, before we introduce spin orbit coupling,
L is a good quantum number, S is a good quantum number,
and then they couple to a good quantum number J.
Of course, once L and S couple to J, orbital and spin,
angular momentum precess around the total angular momentum.
But anyway, I just want to say when
I talk about J, what are possible ingredients?
So let's assume we have an atom which has total angular
momentum J, which is the sum of orbital angular momentum
and spin angular momentum.
And then, in this case, we use a term designation,
a level is designated by a term which
is written as L, the value of orbital angular momentum.
The spin multiplicity 2s plus 1, is an upper left index,
and a lower right index is J. And of course if L is zero,
we use the letter S P D. This is sort of the historic letter
designation for L equals zero, one, and two.
So, in other words, if you have an atom where the total angular
momentum is composed of orbital angular momentum and spin,
you can always write the symbol
and this symbol is to turn designation which characterizes
the state, the ground state, or an excited state of your atom.
If you have the hydrogenic atom, you
precede the term by the principal quantum number n.
So let me give you an example.
If you have the sodium atom, the outer electron
has n equals three, it has zero orbital angular momentum,
it has spin one-half, and 2 S plus 1 is 2,
and the total angular momentum is one-half.
If you go to the first excited state,
you are still in n equals three, but you
have promoted the electron from an S state to a P state.
So therefore, the orbital angular momentum
is now one, designated by P.
The spin is still spin one-half, but now,
orbital angular momentum of one, and spin one-half,
can form a total angular momentum, which
can either be one-half or 3/2.
So, if you ask what is the state you prepare your atom in,
and you would give it a symbol three doublet p one-half,
and I've explained to you what it means.
There is one addition, and sometimes you
want to not just mention what is the principal quantum
number of the outer electrons, sometimes
you want to specify the whole configuration.
So this would mean you want to sort of build up the electron
shell, and day that I have two electrons in 1 s,
tow electrons in 2 s, one electron in 2 p, and so on.
So you use-- I think this would now be, so 1 s is hydrogen,
1 s 2 is helium, OK, so this would
be boron-- so what you use is, here we use the products,
we use products of symbols n, l, m.
So to come back to the example of sodium,
so sodium is filled up, in the first two shells,
so it's 2 p 6.
And then we have one electron, the outer electron in 3 s.
However let me point out, that this way
to specify the configuration, strongly
depends on a hydrogenic model.
It assumes that the electrons are non-interacting, and is
therefore an approximation.
In contrast, the term designation,
with a total angular momentum, is always exact,
because-- well, it is the total angular
momentum is an exact quantum number--
whereas the configuration is based
on the independent electron approximation
and hydrogenic orbits.
Usually when you have a real atom,
and you calculate, with high precision,
what the electronic wavefunction is,
you find, actually, that's the total many body
wavefunction is a superposition of many such configurations.
But as long as one configuration is dominant,
this configuration designation make sense.