Take with and its partners

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Orbital angular momentum, the spherical harmonics and rotations

Problem

Take L^z=x^p^y−y^p^x\hat L_z = \hat x\hat p_y - \hat y\hat p_x with p^x=−iℏ ∂/∂x\hat p_x = -i\hbar\,\partial/\partial x and its partners, acting on smooth functions. You may quote [x^,p^x]=[y^,p^y]=iℏ[\hat x,\hat p_x] = [\hat y,\hat p_y] = i\hbar, that operators referring to different axes commute, the product rule [A^,B^C^]=B^[A^,C^]+[A^,B^]C^[\hat A,\hat B\hat C] = \hat B[\hat A,\hat C] + [\hat A,\hat B]\hat C, and the result [L^z,x^]=iℏy^[\hat L_z,\hat x] = i\hbar\hat y. (a) Derive [L^z,y^]=−iℏx^[\hat L_z,\hat y] = -i\hbar\hat x from the canonical commutators. (b) Hence show that the combination x^+iy^\hat x + i\hat y obeys [L^z,x^+iy^]=ℏ(x^+iy^)[\hat L_z,\hat x + i\hat y] = \hbar(\hat x + i\hat y). (c) Confirm (b) in spherical coordinates, using L^z=−iℏ ∂/∂ϕ\hat L_z = -i\hbar\,\partial/\partial\phi and x+iy=rsin⁡θ eiϕx + iy = r\sin\theta\,e^{i\phi}, by letting the commutator act on a smooth test function.

Related theory

Browse the problem library