Squared orbital angular momentum in position and momentum operators

Operator ordering gives

L^{2} = X^{2} P^{2} - (X \cdot P)^{2} + i ℏ X \cdot P .

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- Spherical Laplacian from orbital angular momentum Squared orbital angular momentum in position and momentum operators

Spherical Laplacian from orbital angular momentum

In position representation,

L^{2} = - ℏ^{2} \nabla_{S^{2}}^{2} .

The radial derivatives cancel after substituting

P = - i ℏ\nabla

into the Cartesian identity for

L^{2}

.

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 33C / b / Solution

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