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ATOMIC PHYSICS REVISION NOTES:
1 Electron Spin
An electron has spin s =
, wehich means that in units of h the electron can have a comp onent
of spin
in any given direction (or more generally the electron can b e in a quantum
sup erp osition of the two states).
An atom with two electrons (helium) can have a total electron spin S = 1 or S = 0.
For the S = 1 state the p ossible values of the comp onent of spin in any one direction (the
z -direction) are 1, 0 or -1 (in units of h).
The states with S z
= 1 and S z
= 1 are the states in wich the two spin
electrons
both have s z
or b oth have s z
, resp ectively. However, the states with S z = 0 are
sup erp ositions of states in which one electron has s z
and the other has s z
, and
vice versa, i.e.
p
("# + #") S = 1 ; S
z
p
("# #") S = 0 ; S
z
For an atom with n electrons the total spin can take values from S =
(if n is o dd)
or S = 0 (if n is even) up to
n
, in integer steps. For a given value of total spin, S , the
comp onent of total spin in any given direction will b e from S to S in integer steps.
2 Pauli Exclusion Principle
No two identical fermions (half o dd-integer spin particles) can o ccupy the same state (in-
cluding the spin state). A maximum of two fermions can have the same spatial wavefunction
provided their combined spin is zero (i.e. the spin part of the wavefunction is antisymmetric
under interchange).
The complete wavefunction of a system of identical fermions is antisymmetric under the
interchange of any two of the fermions.
3 Spin-Orbit Interaction (j-j Coupling Scheme)
The spin-orbit coupling b etween the magnetic moment due to the spin angular momentum
and the magnetic eld due to the orbital angular momentum, splits the energy levels of the
electron with a given orbital angular momentum quantum numb er, l , into states dep ending
on the total angular momentum quantum numb er, j.
j = l
The splitting is prop ortional to l s and using j = l + s, we have
l s =
j
l
s
The eigenvalues of j
, l
and s
are j (j + 1)h
, l (l + 1)h
and s(s + 1)h
; (s = 1 =2), resp ectively
so that the splitting is prop ortional to
(j (j + 1) l (l + 1) s(s + 1))
The states of the electrons are denoted as
nfl g j
where n is the principle quantum numb er, fl g is a co de which indicates the orbital angular
momentum as follows
s l = 0
p l = 1
d l = 2
f l = 3
g l = 4 ; (etc:)
For example, the outer shell of Bi has three electrons with n = 6 and l = 1 - the rst two
have j = 1 = 2 and since this is then full, the third must have j = 3 =2, This outer sub-shell is
denoted by
(6p)
(6p) 3
Below this sub-shell Bi has a closed sub-shell with n = 4 ; 3, which has a total of 14 electrons
(j can take the values 7 = 2 and 5 = 2 p ermitting 8 and 6 electrons resp ectively), a closed sub-
shell with n = 5 ; l = 2 with a total of 10 electrons, and a closed subshell with n = 6 ; l = 0
containing two electrons,. Since these are closed sub-shells we do not usually indicate the
values of j and we would write the con guration for the entire outer shell as
(4f )
(5d)
(6s)
(6p)
(6p) 3
7 Weak Field Zeeman E ect
Apply an external magnetic eld in the z -direction.
The e ective magnetic moment in the z -direction is
z
= (g s
s
z
z
B
where
B
e
2 m e
is the Bohr magneton and g s
is the electron gyro-magnetic ratio (almost equal to 2 in the
case of the electron).
The exp ectation value of
z
in a state which is an eigenstate of s
; l
; j
and j
z
may b e
written
h
z
i = B
h( g s s j + l j)i
hj
i
j
z
Using j = l + s ( or rather l = j s and squaring b oth sides) leads to
s j =
j
s
so that the exp ectation value b ecomes
hs ji =
h
(j (j + 1) + l (l + 1) s(s + 1)) ;
and likewise using s = j l and squaring we get
hl ji =
h
(j (j + 1) + s(s + 1) l (l + 1)) ;
and we also have
hj
i = j (j + 1)h
so that the e ective magnetic moment is
h i = B
(g s (j (j + 1) + l (l + 1) s(s + 1)) + j (j + 1) + s(s + 1) l (l 1))
j (j + 1)
j
8 Natural Linewidths
The natural linewidth, , of a sp ectral line is related to the lifetime of the excited state by
h
