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98 Quantum physics<br />
4.5 Angular momentum<br />
Orbital angular momentum<br />
Angular<br />
momentum<br />
operators<br />
Ladder<br />
operators<br />
Eigenfunctions<br />
and<br />
eigenvalues<br />
ˆL = r× ˆp (4.101)<br />
Lˆ<br />
z = ¯h (<br />
x ∂<br />
i ∂y −y ∂ )<br />
(4.102)<br />
∂x<br />
= ¯h ∂<br />
i ∂φ<br />
(4.103)<br />
Lˆ<br />
2 2 2 2<br />
= Lˆ<br />
x + Lˆ<br />
y + Lˆ<br />
z (4.104)<br />
= −¯h 2 [ 1<br />
sinθ<br />
∂<br />
∂θ<br />
(<br />
sinθ ∂<br />
∂θ<br />
)<br />
+ 1<br />
sin 2 θ<br />
∂ 2 ]<br />
∂φ 2<br />
(4.105)<br />
L ˆ ± = Lˆ<br />
x ±iLˆ<br />
y (4.106)<br />
(<br />
=¯he ±iφ icotθ ∂<br />
∂φ ± ∂ )<br />
(4.107)<br />
∂θ<br />
L ˆ ± Y m l<br />
l<br />
=¯h[l(l +1)−m l (m l ±1)] 1/2 Y m l±1<br />
l<br />
(4.108)<br />
Lˆ<br />
2 Y m l<br />
l<br />
= l(l +1)¯h 2 Y m l<br />
l<br />
(l ≥ 0) (4.109)<br />
Lˆ<br />
z Y m l<br />
l<br />
= m l¯hY m l<br />
l<br />
(|m l |≤l) (4.110)<br />
Lˆ<br />
z [ L ˆ ± Y m l<br />
l<br />
(θ,φ)]=(m l ±1)¯h L ˆ ± Y m l<br />
l<br />
(θ,φ) (4.111)<br />
l-multiplicity = (2l +1) (4.112)<br />
L<br />
angular<br />
momentum<br />
p linear momentum<br />
r position vector<br />
xyz Cartesian<br />
coordinates<br />
rθφ spherical polar<br />
coordinates<br />
¯h (Planck<br />
constant)/(2π)<br />
L ˆ ± ladder operators<br />
Y m l<br />
l spherical<br />
harmonics<br />
l,m l integers<br />
Angular momentum commutation relations a<br />
Conservation <strong>of</strong> angular<br />
momentum b<br />
[ Lˆ<br />
z ,x]=i¯hy (4.114)<br />
[ Lˆ<br />
z ,y]=−i¯hx (4.115)<br />
[ Lˆ<br />
z ,z] = 0 (4.116)<br />
[ Lˆ<br />
z , pˆ<br />
x ]=i¯h pˆ<br />
y (4.117)<br />
[ Lˆ<br />
z , pˆ<br />
y ]=−i¯h pˆ<br />
x (4.118)<br />
[ Lˆ<br />
z , pˆ<br />
z ] = 0 (4.119)<br />
[Ĥ, ˆ L z ] = 0 (4.113)<br />
[ L ˆ2<br />
, Lˆ<br />
x ]=[ L ˆ2<br />
, Lˆ<br />
y ]=[ L ˆ2<br />
, Lˆ<br />
z ] = 0 (4.127)<br />
L angular momentum<br />
p momentum<br />
H Hamiltonian<br />
L ˆ ± ladder operators<br />
[ Lˆ<br />
x , Lˆ<br />
y ]=i¯h Lˆ<br />
z (4.120)<br />
[ Lˆ<br />
z , Lˆ<br />
x ]=i¯h Lˆ<br />
y (4.121)<br />
[ Lˆ<br />
y , Lˆ<br />
z ]=i¯h Lˆ<br />
x (4.122)<br />
[ L ˆ + , Lˆ<br />
z ]=−¯h L ˆ + (4.123)<br />
[ Lˆ<br />
− , Lˆ<br />
z ]=¯h Lˆ<br />
− (4.124)<br />
[ L ˆ + , Lˆ<br />
− ]=2¯h Lˆ<br />
z (4.125)<br />
[ L ˆ2<br />
, L ˆ ± ] = 0 (4.126)<br />
a <strong>The</strong> commutation <strong>of</strong> a and b is defined as [a,b]=ab−ba (see page 26). Similar expressions hold for S and J.<br />
b For motion under a central force.
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