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21 lines
746 B
Plaintext
21 lines
746 B
Plaintext
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This problem associated with chapter 7 was assigned during lecture.
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Does L̂𝓏 commute with 𝐫̂²?
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[L̂𝓏,𝐫̂²] = L̂𝓏 𝐫̂² - 𝐫̂² L̂𝓏.
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L̂𝓏 𝐫̂² - 𝐫̂² L̂𝓏.
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Using the position representations, in spherical coordinates,
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L̂𝓏 ≐ -ιħ∂/∂θ and 𝐫̂² ≐ 𝐫²,
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L̂𝓏 𝐫̂² - 𝐫̂² L̂𝓏 = 𝐫² ιħ∂/∂θ - ιħ∂/∂θ 𝐫².
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𝐫² has no θ dependence, so it can be separated from any quantity differentiated with respect to theta, I.E.,
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∂/∂θ 𝐫² = 𝐫² ∂/∂θ.
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𝐫² ιħ∂/∂θ - ιħ∂/∂θ 𝐫² = 𝐫² ιħ∂/∂θ - 𝐫² ιħ∂/∂θ = 0 = [L̂𝓏,𝐫̂²] = 0.
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[L̂𝓏,𝐫̂²] = 0, so these quantities commute.
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