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Chapter 11 Total Angular Momentum
With possibilities of both orbital L and spin S angular momentum in a system we must consider
that total angular momentum J is conserved J = L + S.
In a system in which the Hamiltonian commutes with J and J
2 , [ H , JI ]=0 and [ H , J
2 ]=0 ,
we need to establish how J is related to L and S.
1 ) Given
L x
L = i"
L and
S x
S = i"
S then
J x
J = i"
J ,
J follows the angular momentum algebra.
!
J x
J =
L +
S
x
L +
S
L x
L +
S x
S +
Sx
L +
Lx
S
cancel to zero
= i"
J
J x
J = i"
J QED quod erat demonstrantum
2 ) J
2 ,L
2 ! "
= 0 J
2 commutes with L
2
L
2
2
2 ! "
= L
2 ,L
2 ! "
= 0
+ S
2 ,S
2 ! "
= 0
+ 2 S
X
L
X
,L
2 ! "
[L
2 ,L X ]= 0
+ 2 S
Y
L
Y
,L
2 ! "
[L
2 ,L Y ]= 0
+ 2 S
Z
L
Z
,L
2 ! "
[L
2 ,L Z ]= 0
3) J
2 ,S
2 ! "
= 0 J
2 commutes with S
2
L
2
2
2 ! "
= L
2 ,L
2 ! "
= 0
+ S
2 ,S
2 ! "
= 0
+ 2 L
X
S
X
,S
2 ! "
[S 2 ,S X ]= 0
+ 2 L
Y
S
Y
,S
2 ! "
[S 2 ,L Y ]= 0
+ 2 L
Z
S
Z
,S
2 ! "
[S 2 ,L Z ]= 0
4) J
2 ,J Z
= 0 or J
2 ,J I
= 0 I = 1 , 2 , 3 J
2 commutes with J x
J
Y
J
Z
L
2
2
+ S
Z
= L
2 ,L Z
= 0
+ L
2 ,S Z
= 0
+ S
2 ,L Z
= 0
+ S
2 ,S Z
= 0
+ 2 S
X
L
X
,L
Z
=% i"L Y
+ 2 S
Y
L
Y
,L
Z
=+ i"L X
+ 2 S
Z
L
z
,L
Z
= 0
+ 2 L
X
S
X
,S
Z
=% i"S Y
+ 2 L
Y
S
Y
,S
Z
=+ i"S X
+ 2 L
Z
LS
z
,S
Z
= 0
= % 2 i"S X
L
Y
L
X
% 2 i"L X
S
Y
S
X
J
S
L
J
Z
S and
L have constant projections on
J
L i
J = L
2
L i
S = L
2
J
2 ! L
2 ! S
2
( ) =^
J
2
2 ! S
2
( j (^ j^ +^1 )^ +^ l ( l^ +^1 )^!^ s ( s^ +^1 )) "
2 = cons tan t
S i
J = S
2
L i
S = S
2
J
2 ! L
2 ! S
2
( ) =^
J
2
2 ! S
2
( j (^ j^ +^1 )^!^ l ( l^ +^1 )^ +^ s ( s^ +^1 )) "
2 = cons tan t
Set of Commuting Operators for the System- Symmetries of the Hamiltonian
The complete set of commuting operators is then H , J
2 , J Z
, L
2 , S
2
. Each operator represents
a constant of motion for the system, and each represented by a quantum number n , j , m j
, l , s
The most general wave function can be written
n , j , m j , l , s
= n , j , m j
, l , s = C m L , m S
L , S
Clesbch " Gordan Coef !
n
R ( r )
l , m l
Y lm
(^ # $)
n , m l , m s m j
= m l
% s , m S
& S
H n , j , m j
, l , s = E n
n , j , m j
, l , s
J
2 n , j , m j
, l , s = j ( j + 1 )&
2 n , j , m j
, l , s
J
Z
n , j , m j
, l , s = m j
& n , j , m j
, l , s
L
2 n , j , m j
, l , s = l ( l + 1 )&
2 n , j , m j
, l , s
S
2 n , j , m j
, l , s = s ( s + 1 )&
2 n , j , m j
, l , s
Addition of Angular Momentum
J = L + S
j = | L + S | ........ | L! S |
m j
= + j , + j! 1 ( ) ,+ j! 2 ( ) .......! j = 2 j + 1 states
Classification of J States for the Hydrogen Atom J=L+1/
J = L + 1 / 2 :
J = 0 + 1 / 2 j = 1 / 2 m j
2 s + 1 L J
2 S 1 / 2
J = 1 + 1 / 2 j = 3 / 2 m j
2 s + 1 L J
2 P 3 / 2 , 1 / 2
j = 1 / 2 m j
J = 2 + 1 / 2 j = 5 / 2 m j
2 s + 1 L J
2 D 5 / 2 , 3 / 2 , 1 / 2
j = 3 / 2 m j
j = 1 / 2 m j
e-
P+
r
ParaHelium and OrthoHelium
H =
p 1
2
2 m
!
2 e
2
4 "# 0
r 1
$
%
&
'
(
)
p 2
2
2 m
!
2 e
2
4 "# 0
r 2
$
%
&
'
(
)
e
2
4 "# 0
r 12
Hee Interaction term
!# " #$
E =
Z
2
n
2
E 0
Z
2
n
2
E 0
e
2
4 "# 0
r 12
! #" #$
J = 1 / 2 + 1 / 2
j = 1 m j
= ± 1 , 0
2 S + 1 L J
j = 0 m j
= 0
+( 1 , 2 ) = , 100
( 1 ) , 100 ( (^2 ))^ -^
A ( 1 , 2 ) ,
S ( 1 , 2 ) -
A ( 1 , 2 ) E = 4 E 0
+( 1 , 2 ) =
1
2
, 100
( 1 ) , nlm
( 2 ) + , nlm
( 1 ) , 100 ( (^2 ))
r 12 < r 12
! ######" ######$
A ( 1 , 2 ) ,
S ( 1 , 2 ) -
A ( 1 , 2 ) E =
4
1
E 0
4
n
2
E 0
para! He
+( 1 , 2 ) =
1
2
, 100
( 1 ) , nlm
( 2 )! , nlm
( 1 ) , 100 ( (^2 ))
r 12
r 12
! ######" ######$
S ( 1 , 2 ) ,
A ( 1 , 2 ) -
S ( 1 , 2 ) E =
4
1
E 0
4
n
2
E 0
! * E 12
ortho! He
HUND’s RULES (not covered)
Highest Spin States have the lowest energy. Consistent w Pauli Principle
For a given S state highest L lies lowest. Consistent w Pauli Principle
If Shell < half filled lowest J lies lowest, if >half filled highest J lies lowest.
Pauli Principle - The total wave function must me antisymmetric for electron states.
S states 0
A
L States (-1)
L 0
S 1
A 2
S 3
A …….
4 He 2
: 1 s
2 S = 1 / 2 + 1 / 2 S = 1
S , 0
A
L = 0 + 0 L = 0
S
J = 0 + 0 J = 0
1 S 0
ground state
12 C 6
: 1 s
2 2 s
2 2 p
2 S = 1 / 2 + 1 / 2 S = 1
S , 0
A
L = 1 + 1 L = 2
S , 1
A , 0
S
J = 1 + 1 J = 2 , 1 , 0
3 S 1
ground state
r
12
e-
e-
+2e
r 1
r
2
e-
e-
e- e-
