1
P313 Notes 10
Electron Configurations
Addition, or “coupling”, of Angular Momenta
Electron Configurations
The energy of an electron depends mostly upon the
principal quantum number n
, and to
a lesser extent on the other quantum numbers.
All electrons with
n
= 1 are said to be in the
K
shell.
(The notation comes from Xray
spectroscopy).
These are the electrons with the lowestlying energy.
To remove a K
shell electron from an atom requires a lot of energy.
All electrons with
n =
2 are said to be in the
L
shell.
All electrons with
n =
3 are said to be in the
M
shell.
All electrons with
n =
4 are said to be in the
N
shell.
etc.
A electron is specified by four quantum numbers:
n
,
l
,
m
l
,
m
x
.
(
m
j
l
n
,
,
,
will also do,
but for the present discussion we are going to use
n
,
l
,
m
l
,
m
x
.)
No two electrons can
have the same set of these four quantum numbers. This is
Pauli’s Exclusion Principle
.
If you think of an eigenfunction (or rather of
ψψ
*) as representing a charge distribution,
a charge distribution is uniquely determined by the four quantum numbers, and there
cannot be two different eigenfunctions with the same set of quantum numbers.
In a more
sophisticated alternative interpretation, which you will meet in courses on Statistical
Mechanics and Quantum Mechanics, you will understand that the Exclusion Principle is a
consequence of the indistinguishability of electrons.
Furthermore, the principle applies
only to particles with integralplusonehalf spin.
Such particles are known collectively
as
fermions
, of which electrons, protons and neutrons are examples.
Photons and pions
have integral spin, and are known collectively as
bosons
.
For a given
n
, there are
n
possible values of
l
, namely
.
1
,...
3
,
2
,
1
,
0

=
n
l
Such electrons are referred to as
s, p. d, f
... electrons.
For a given
l
, there are
1
2
+
l
values of
m
l
, namely
l
l
l
l
l
m
l
,
1
,
....
,
0
....
,
2
,
1
,

+

+


=
For a given
n
that’s
2
1
0
)
1
2
(
n
l
n
l
=
+
∑

=
combinations so far.
But for each of these
n
2
combinations, there are two possible values of
s
m
, namely
.