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Angular momentum ladder operator ( $L_{\pm}$ )

The operators

L_{\pm} = L_{x} \pm i L_{y}

satisfy

[L_{z}, L_{\pm}] = \pm ℏ L_{\pm}, [L^{2}, L_{\pm}] = 0.

They therefore change the magnetic quantum number

m

by

\pm 1

while preserving the total angular-momentum quantum number

ℓ

.

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Solution