The transformation from polar coordinates to cartesian is the set of equations ⎧ x = r sinθ cosϕ ⎪ ⎨ y = r sinθ sinϕ ⎪ ⎩ z = r cosθ The transformation from cartesian to polar coordinates is the set ⎧ r = √(x²+y²+z²) ⎪ ⎪ cos(θ) = z͟ = _͟z͟ ⎨ r √(x²+y²+z²) ⎪ ⎪ tan(ϕ) = y͟ ⎩ x Some differential forms may come in handy. ⎧ dcos(θ) = -sin(θ) dθ ⎨ ⎪ dtan(ϕ) = _͟1͟ dϕ ⎩ cos²(ϕ) ∂/∂θ: ⎧ ∂͟x = r cosϕ cosθ ⎪ ∂θ ⎪ ⎨ ∂͟y = r sinϕ cosθ ⎪ ∂θ ⎪ ⎪ ∂͟z = -r sinθ ⎩ ∂θ ∂/∂ϕ: ⎧ ∂͟x = -r sinθ sinϕ ⎪ ∂ϕ ⎪ ⎨ ∂͟y = r sinθ cosϕ ⎪ ∂ϕ ⎪ ⎪ ∂͟z = 0 ⎩ ∂ϕ ∂/∂x: ⎧ ∂͟r = √(x²+y²+z²) = x͟ ⎪ ∂x r ⎪ ⎪ ∂͟cos(θ) = -z͟x͟ ⎨ ∂x r³ ⎪ ⎪ ∂͟tan(ϕ) = -y͟ ⎩ ∂x x² ∂/∂y: ⎧ ∂͟r = √(x²+y²+z²) = y͟ ⎪ ∂y r ⎪ ⎪ ∂͟cos(θ) = -z͟y͟ ⎨ ∂y r³ ⎪ ⎪ ∂͟tan(ϕ) = 1͟ ⎩ ∂y x ∂/∂z: ⎧ ∂͟r = √(x²+y²+z²) = z͟ ⎪ ∂z r ⎪ ⎪ ∂͟cos(θ) = z͟ = 1͟ ⎛r ∂͟z͟ - z ∂͟r͟⎞ = 1͟ - z͟² ⎨ ∂z r r² ⎝ ∂z ∂z⎠ r r³ ⎪ ⎪ ∂͟tan(ϕ) = 0 ⎩ ∂z These differential forms can be used to transform each cartesian differentiation operator. ∂͟ = ∂͟r͟ ∂͟ + ∂͟c͟o͟s͟θ͟ ∂͟ + ∂͟t͟a͟n͟ϕ͟ ∂͟ ; ∂x ∂x ∂r ∂x ∂cosθ ∂x ∂tanϕ ∂͟ = x͟ ∂͟ - z͟x͟ _͟1͟ ∂͟ - y͟ cos²(ϕ) ∂͟ ; ∂x r ∂r r³ -sinθ ∂θ x² ∂ϕ ∂͟ = sin(θ) cos(ϕ) ∂͟ ∂x ∂r + 1͟ sin(θ) cos(ϕ) cos(θ) _͟1͟ ∂͟ r sin(θ) ∂θ - r͟ s͟i͟n͟(ϕ͟) s͟i͟n͟(θ͟) cos²(ϕ͟) ∂͟ ; r² cos²(ϕ) sin²(θ) ∂ϕ ∂͟ = sin(θ) cos(ϕ) ∂͟ + 1͟ cos(ϕ) cos(θ) ∂͟ - 1͟ s͟i͟n͟(ϕ͟) ∂͟ . ∂x ∂r r ∂θ r sin(θ) ∂ϕ ─────────────────────────────────────────────── ∂͟ = ∂͟r͟ ∂͟ + ∂͟c͟o͟s͟θ͟ ∂͟ + ∂͟t͟a͟n͟ϕ͟ ∂͟ ; ∂y ∂y ∂r ∂y ∂cosθ ∂y ∂tanϕ ∂͟ = y͟ ∂͟ - z͟y͟ ∂͟ + 1͟ ∂͟ ; ∂y r ∂r r³ ∂cosθ x ∂tanϕ ∂͟ = sin(θ) sin(ϕ) ∂͟ - 1͟ sin(θ) cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ ; ∂y ∂r r -sin(θ) ∂θ r sin(θ) ∂ϕ ∂͟ = sin(θ) sin(ϕ) ∂͟ + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ . ∂y ∂r r ∂θ r sin(θ) ∂ϕ ─────────────────────────────────────────────── ∂͟ = ∂͟r͟ ∂͟ + ∂͟c͟o͟s͟θ͟ ∂͟ + ∂͟t͟a͟n͟ϕ͟ ∂͟ ; ∂z ∂z ∂r ∂z ∂cosθ ∂z ∂tanϕ ∂͟ = z͟ ∂͟ + ⎛1͟ - z͟²⎞∂͟ + 0 ; ∂z r ∂r ⎝r r³⎠∂cosθ ∂͟ = cos(θ) ∂͟ + 1͟⎛1 - cos²(θ)⎞ _͟1͟ ∂͟ ; ∂z ∂r r⎝ ⎠ -sin(θ) ∂θ ∂͟ = cos(θ) ∂͟ - _͟1͟_ ⎛1 - cos²(θ)⎞ ∂͟ ; ∂z ∂r rsin(θ)⎝ ⎠ ∂θ ∂͟ = cos(θ) ∂͟ - _͟1͟_ sin²(θ) ∂͟ ; ∂z ∂r rsin(θ) ∂θ ∂͟ = cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ . ∂z ∂r r ∂θ The set of differential operator transformations is ⎧ ∂͟ = sin(θ) cos(ϕ) ∂͟ + 1͟ cos(ϕ) cos(θ) ∂͟ - 1͟ s͟i͟n͟(ϕ͟) ∂͟ ⎪ ∂x ∂r r ∂θ r sin(θ) ∂ϕ ⎪ ⎨ ∂͟ = sin(θ) sin(ϕ) ∂͟ + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ ⎪ ∂y ∂r r ∂θ r sin(θ) ∂ϕ ⎪ ⎪ ∂͟ = cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ ⎩ ∂z ∂r r ∂θ 7.47 is the set of algebraic conditions expressed by the vector definition 𝐋 = 𝐫 × 𝐩. ⎧ L̂𝓍 = yp𝓏 - zp𝓎 = -ιħ (y ∂͟_ - z ∂͟_ ) ⎪ ∂z ∂y ⎪ ⎨ L̂𝓎 = zp𝓍 - xp𝓏 = -ιħ (z ∂͟_ - x ∂͟_ ) ⎪ ∂x ∂z ⎪ ⎪ L̂𝓏 = xp𝓎 - yp𝓍 = -ιħ (x ∂͟_ - y ∂͟_ ) ⎩ ∂y ∂x Substituting 7.35 into 7.47, ⎧ L̂𝓍 = -ιħ (r sinθ sinϕ ∂͟_ - r cosθ ∂͟_ ) ⎪ ∂z ∂y ⎪ ⎨ L̂𝓎 = -ιħ (r cosθ ∂͟_ - r sinθ cosϕ ∂͟_ ) ⎪ ∂x ∂z ⎪ ⎪ L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟_ - r sinθ sinϕ ∂͟_ ) ⎩ ∂y ∂x Substituting the transformed differentiation operators, ⎧ L̂𝓍 = -ιħ⎛r sinθ sinϕ ⎛cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ ⎞ ⎪ ⎝ ⎝ ∂r r ∂θ⎠ ⎪ ⎪ - r cosθ ⎛sin(θ) sin(ϕ) ∂͟ + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ ⎞⎞ ⎪ ⎝ ∂r r ∂θ r sin(θ) ∂ϕ⎠⎠ ⎪ ⎪ ⎪ L̂𝓎 = -ιħ⎛r cosθ⎛sin(θ) cos(ϕ) ∂͟ + 1͟ cos(ϕ) cos(θ) ∂͟ - 1͟ s͟i͟n͟(ϕ͟) ∂͟ ⎞ ⎪ ⎝ ⎝ ∂r r ∂θ r sin(θ) ∂ϕ⎠ ⎨ ⎪ - r sinθ cosϕ ⎛cos(θ) ∂͟ - s͟i͟n͟(θ͟) ∂͟ ⎞⎞ ⎪ ⎝ ∂r r ∂θ⎠⎠ ⎪ ⎪ ⎪ L̂𝓏 = -ιħ⎛r sinθ cosϕ⎛sin(θ) sin(ϕ) ∂͟ + 1͟ cos(θ) s͟i͟n͟(ϕ͟) ∂͟ + c͟o͟s͟(ϕ͟) ∂͟ ⎞ ⎪ ⎝ ⎝ ∂r r ∂θ r sin(θ) ∂ϕ⎠ ⎪ ⎪ - r sinθ sinϕ ⎛sin(θ) cos(ϕ) ∂͟ + 1͟ cos(ϕ) cos(θ) ∂͟ - 1͟ s͟i͟n͟(ϕ͟) ∂͟ ⎞⎞ ⎩ ⎝ ∂r r ∂θ r sin(θ) ∂ϕ⎠⎠ Simplifying... ⎧ L̂𝓍 = -ιħ⎛r sinθ sinϕ cosθ ∂͟ - sinϕ sin²θ ∂͟ ⎪ ⎝ ∂r ∂θ ⎪ ⎪ - r cosθ sinθ sinϕ ∂͟ - cos²θ sinϕ ∂͟ - c͟o͟s͟θ͟ cosϕ ∂͟ ⎞ ⎪ ∂r ∂θ sinθ ∂ϕ⎠ ⎪ ⎪ ⎪ L̂𝓎 = -ιħ⎛r cosθ sinθ cosϕ ∂͟ + cosθ cosϕ cosθ ∂͟ - c͟o͟s͟θ͟ sinϕ ∂͟ ⎪ ⎝ ∂r ∂θ sinθ ∂ϕ ⎨ ⎪ - r sinθ cosϕ cosθ ∂͟ + sinθ cosϕ sinθ ∂͟ ⎞ ⎪ ∂r ∂θ⎠ ⎪ ⎪ ⎪ L̂𝓏 = -ιħ⎛r sin²θ cosϕ sinϕ ∂͟ + sinθ cosθ sinϕ cosϕ ∂͟ + cosϕ c͟o͟s͟ϕ͟ ∂͟ ⎪ ⎝ ∂r ∂θ ∂ϕ ⎪ ⎪ - r sin²θ sinϕ cosϕ ∂͟ - sinθ cosθ sinϕ cosϕ ∂͟ + sinθ sinϕ s͟i͟n͟ϕ͟ ∂͟ ⎞ ⎩ ∂r ∂θ sinθ ∂ϕ⎠ ⎧ L̂𝓍 = -ιħ⎛- sinϕ sin²θ ∂͟ - cos²θ sinϕ ∂͟ - c͟o͟s͟θ͟ cosϕ ∂͟ ⎞ ⎪ ⎝ ∂θ ∂θ sinθ ∂ϕ⎠ ⎪ ⎨ L̂𝓎 = -ιħ⎛cosθ cosϕ cosθ ∂͟ - c͟o͟s͟θ͟ sinϕ ∂͟ + sinθ cosϕ sinθ ∂͟ ⎞ ⎪ ⎝ ∂θ sinθ ∂ϕ ∂θ⎠ ⎪ ⎪ L̂𝓏 = -ιħ⎛cosϕ c͟o͟s͟ϕ͟ ∂͟ + sinθ sinϕ s͟i͟n͟ϕ͟ ∂͟ ⎞ ⎩ ⎝ ∂ϕ sinθ ∂ϕ⎠ ⎧ L̂𝓍 = ιħ⎛ sinϕ sin²θ ∂͟ + cos²θ sinϕ ∂͟ + c͟o͟s͟θ͟ cosϕ ∂͟ ⎞ ⎪ ⎝ ∂θ ∂θ sinθ ∂ϕ⎠ ⎪ ⎨ L̂𝓎 = ιħ⎛-cos²θ cosϕ ∂͟ - sin²θ cosϕ ∂͟ + c͟o͟s͟θ͟ sinϕ ∂͟ ⎞ ⎪ ⎝ ∂θ ∂θ sinθ ∂ϕ ⎠ ⎪ ⎪ L̂𝓏 = -ιħ⎛cos²ϕ ∂͟ + sin²ϕ ∂͟ ⎞ ⎩ ⎝ ∂ϕ ∂ϕ⎠ ⎧ L̂𝓍 = ιħ⎛ sinϕ ∂͟ + cotθ cosϕ ∂͟ ⎞ ⎪ ⎝ ∂θ ∂ϕ⎠ ⎪ ⎨ L̂𝓎 = ιħ⎛-cosϕ ∂͟ + cotθ sinϕ ∂͟ ⎞ ⎪ ⎝ ∂θ ∂ϕ ⎠ ⎪ ⎪ L̂𝓏 = -ιħ∂͟ ⎩ ∂ϕ The spherical representation is the following set, which perfectly matches the obtained result. ⎧ L̂𝓍 = ιħ (sinϕ ∂͟_ + cosϕ cotθ ∂͟_ ) ⎪ ∂θ ∂ϕ ⎪ ⎨ L̂𝓎 = ιħ (-cosϕ ∂͟_ + sinϕ cotθ ∂͟_ ) ⎪ ∂θ ∂ϕ ⎪ ⎪ L̂𝓏 = -ιħ ∂͟_ ⎩ ∂ϕ