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