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41 lines
1.3 KiB
Plaintext
41 lines
1.3 KiB
Plaintext
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7.35 is nothing more than a definition of spherical coordinates.
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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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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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⎪ ∂y ∂y
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⎪
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⎨ L̂𝓎 = zp𝓍 - xp𝓏 = -ιħ (z ∂͟_ - x ∂͟_ )
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⎪ ∂y ∂y
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⎪
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⎪ L̂𝓏 = xp𝓎 - yp𝓍 = -ιħ (x ∂͟_ - y ∂͟_ )
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⎩ ∂y ∂y
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Substituing 7.35 into 7.47,
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⎧ L̂𝓍 = -ιħ (r sinθ sinϕ ∂͟_ - r cosθ ∂͟_ )
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⎪ ∂y ∂y
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⎪
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⎨ L̂𝓎 = -ιħ (r cosθ ∂͟_ - r sinθ cosϕ ∂͟_ )
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⎪ ∂y ∂y
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⎪
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⎪ L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟_ - r sinθ sinϕ ∂͟_ )
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⎩ ∂y ∂y
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⎧ L̂𝓍 = -ιħ (r sinθ sinϕ ∂͟_ - r cosθ ∂͟_ )
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⎪ ∂y ∂y
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⎪
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⎨ L̂𝓎 = ιħ (r cosθ ∂͟_ + r sinθ cosϕ ∂͟_ )
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⎪ ∂y ∂y
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⎪
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⎪ L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟_ - r sinθ sinϕ ∂͟_ )
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⎩ ∂y ∂y
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