Apuntes de Matemáticas de Mecánica Cuántica

8 -Angular momentum operators

1- Definition of the angular momentum operators

2- Commutation relations

3- Step-up and step-down operators

4- Angular momentum eigenvalues

5- Angular momentum eigenfunctions

6- Spin angular momentum

7. Conservation of angular momenta: monoelectronic atoms

8- Composition of angular momenta

, 1- Definition of the angular momentum operators

In classical mechanics, angular momentum is defined as:
   
L = L x i + L y j + Lz k =
  
  
( ) ( )
ypz − zpy i + ( zpx − xpz ) j + xpy − ypx k
i j k Lx = ypz − zpy

L= x y z
Ly = zpx − xpz
p x p y pz
Lz = xpy − ypx
2
L = Lx2 + Ly2 + Lz2

In QM the components, using the substitution rule are:
∂
⎛ ∂ ∂⎞ q
No need to symmetrize i ∂q
L̂x = −i ⎜ y − z ⎟
⎝ ∂z ∂y ⎠ j


⎛ ∂ ∂⎞
L̂y = −i ⎜ z − x ⎟ L̂ = L̂x + L̂y + L̂z
⎝ ∂x ∂z ⎠
⎛ ∂ ∂⎞ L̂2 = L̂x2 + L̂y2 + L̂z2
L̂z = −i ⎜ x − y ⎟
⎝ ∂y ∂x ⎠

, Definition of the angular momentum operators

Operators in Spherical coordinates

⎛ ∂ ∂⎞ ⎛ ∂ ∂⎞
Z Lx = −i⎜ y − z ⎟ = −i⎜ −senϕ ⋅ − cot θ ⋅ cosϕ ⋅ ⎟
⎝ ∂z ∂y ⎠ ⎝ ∂θ ∂ϕ ⎠
θ
⎛ ∂ ∂⎞ ⎛ ∂ ∂⎞
Ly = −i⎜ z − x ⎟ = −i⎜ cosϕ ⋅ − cot θ ⋅ senϕ ⋅ ⎟
riA
⎝ ∂y ∂z ⎠ ⎝ ∂θ ∂ϕ ⎠

⎛ ∂ ∂⎞ ⎛ ∂⎞
A
y Lz = −i⎜ x − y ⎟ = −i⎜ ⎟
⎝ ∂y ∂x ⎠ ⎝ ∂ϕ ⎠
φ
⎛ 1 ∂ ∂ 1 ∂2 ⎞
L = − ⎜
2 2
senθ + ⎟
x
⎝ senθ ∂θ ∂θ sen 2θ ∂ϕ 2 ⎠

x = r ⋅ senθ ⋅ cos ϕ
y = r ⋅ senθ ⋅ senϕ 2 ⎛ 1 ∂ ⎛ 2 ∂ ⎞ 1 ∂ ∂ 1 ∂2 ⎞
Ĥ = − ⎜ ⎜⎝ r ⎟⎠ + 2 senθ + 2 2 2⎟
+ V (r)
z = r ⋅ cosθ 2m ⎝ r ∂r
2
∂r r senθ ∂θ ∂θ r sen θ ∂ϕ ⎠
2 ∂ ⎛ 2 ∂ ⎞ L2
=− ⎜⎝ r ⎟⎠ + + V (r)
2mr ∂r
2
∂r 2mr 2