The angular-momentum commutators are and its cyclic permutations

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

The Structure of Quantum Mechanics, Ch Position, momentum and the origin of the canonical commutator

Problem

The angular-momentum commutators are [L^x,L^y]=iℏL^z[\hat L_x,\hat L_y] = i\hbar\hat L_z and its cyclic permutations. Consider the quadratic combination L^x2\hat L_x^2. (a) Using the commutator product rule [A^2,B^]=A^[A^,B^]+[A^,B^]A^[\hat A^2,\hat B] = \hat A[\hat A,\hat B] + [\hat A,\hat B]\hat A, show that [L^x2,L^y]=iℏ(L^xL^z+L^zL^x).[\hat L_x^2,\hat L_y] = i\hbar(\hat L_x\hat L_z + \hat L_z\hat L_x). (b) Using the Leibniz rule for Poisson brackets, {fg,h}PB=f{g,h}PB+g{f,h}PB\{fg,h\}_{\rm PB} = f\{g,h\}_{\rm PB} + g\{f,h\}_{\rm PB}, and the classical bracket {Lx,Ly}PB=Lz\{L_x,L_y\}_{\rm PB}=L_z, compute {Lx2,Ly}PB\{L_x^2,L_y\}_{\rm PB}. (c) Show that [L^x2,L^y]/iℏ[\hat L_x^2,\hat L_y]/i\hbar equals 2L^xL^z+iℏL^y2\hat L_x\hat L_z + i\hbar\hat L_y, and state the extra term by which this differs from the naive quantisation 2L^xL^z2\hat L_x\hat L_z of the classical result of part (b).

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