Quantum Mechanics: Concepts and Applications · Problem 5.17

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · angular-momentum-commutator-identities

Problem

(a) Using the commutation relations of angular momentum, verify the validity of the (Jacobi) identity: [J^x, [J^y, J^z]]+[J^y, [J^z, J^x]]+[J^z, [J^x, J^y]]=0[\hat{J}_x,\ [\hat{J}_y,\ \hat{J}_z]] + [\hat{J}_y,\ [\hat{J}_z,\ \hat{J}_x]] + [\hat{J}_z,\ [\hat{J}_x,\ \hat{J}_y]] = 0. (b) Prove the following identity: [J^x2, J^y2]=[J^y2, J^z2]=[J^z2, J^x2][\hat{J}_x^2,\ \hat{J}_y^2] = [\hat{J}_y^2,\ \hat{J}_z^2] = [\hat{J}_z^2,\ \hat{J}_x^2]. (c) Calculate the expressions of L^−L^+Ylm(θ, φ)\hat{L}_-\hat{L}_+Y_{lm}(\theta,\ \varphi) and L^+L^−Ylm(θ, φ)\hat{L}_+\hat{L}_-Y_{lm}(\theta,\ \varphi), and then infer the commutator [L^+L^−, L^−L^+]Ylm(θ, φ)[\hat{L}_+\hat{L}_-,\ \hat{L}_-\hat{L}_+]Y_{lm}(\theta,\ \varphi).

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