Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. The angular momentum in the spatial representation is L = i ℏ ℏ ( θ θ ^ ^ sin ⁡ ⁡ ( θ θ ) ∂ ∂ ∂ ∂ ϕ ϕ − − ϕ ϕ ^ ^ ∂ ∂ ∂ ∂ θ θ ) = i ℏ ℏ ( x ^ ^ ( sin ⁡ ⁡ ( ϕ ϕ ) ∂ ∂ ∂ ∂ θ θ + cot ⁡ ⁡ ( θ θ ) cos ⁡ ⁡ ( ϕ ϕ ) ∂ ∂ ∂ ∂ ϕ ϕ ) + y ^ ^ ( − − cos ⁡ ⁡ ( ϕ ϕ ) ∂ ∂ ∂ ∂ θ θ + cot ⁡ ⁡ ( θ θ ) sin ⁡ ⁡ ( ϕ ϕ ) ∂ ∂ ∂ ∂ ϕ ϕ ) − − z ^ ^ ∂ ∂ ∂ ∂ ϕ ϕ ) L + = ℏ ℏ e i ϕ ϕ ( ∂ ∂ ∂ ∂ θ θ + i cot ⁡ ⁡ ( θ θ ) ∂ ∂ ∂ ∂ ϕ ϕ ) , L − − = ℏ ℏ e − − i ϕ ϕ ( − − ∂ ∂ ∂ ∂ θ θ + i cot ⁡ ⁡ ( θ θ ) ∂ ∂ ∂ ∂ ϕ ϕ ) , L 2 = − − ℏ ℏ 2 ( 1 sin ⁡ ⁡ ( θ θ ) ∂ ∂ ∂ ∂ θ θ ( sin ⁡ ⁡ ( θ θ ) ∂ ∂ ∂ ∂ θ θ ) + 1 sin 2 ⁡ ⁡ ( θ θ ) ∂ ∂ 2 ∂ ∂ ϕ ϕ 2 ) , L z = − − i ℏ ℏ ∂ ∂ ∂ ∂ ϕ ϕ .