The most general and fundamental definition of angular momentum is as the generator of rotations. More specifically, let R ( n ^ ^ , ϕ ϕ ) be a rotation operator, which rotates any quantum state about axis n ^ ^ by angle ϕ ϕ . As ϕ ϕ → → 0 , the operator R ( n ^ ^ , ϕ ϕ ) approaches the identity operator, because a rotation of 0° maps all states to themselves. Then the angular momentum operator J n ^ ^ about axis n ^ ^ is defined as: J n ^ ^ ≡ ≡ i ℏ ℏ lim ϕ ϕ → → 0 R ( n ^ ^ , ϕ ϕ ) − − 1 ϕ ϕ = i ℏ ℏ ∂ ∂ R ( n ^ ^ , ϕ ϕ ) ∂ ∂ ϕ ϕ | ϕ ϕ = 0