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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 f(x,y,z)f(x,y,z) (so that partial derivatives may be taken in either order). You may quote the canonical commutators [x^,p^x]=[y^,p^y]=[z^,p^z]=iℏ[\hat x,\hat p_x] = [\hat y,\hat p_y] = [\hat z,\hat p_z] = i\hbar and that any position and momentum operator referring to different axes commute. (a) Show, starting from those commutators and the commutator 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, that [L^z,x^]=iℏy^[\hat L_z,\hat x] = i\hbar\hat y. (b) Confirm the same result in spherical coordinates: use L^z=−iℏ ∂/∂ϕ\hat L_z = -i\hbar\,\partial/\partial\phi together with x=rsin⁡θcos⁡ϕx = r\sin\theta\cos\phi, y=rsin⁡θsin⁡ϕy = r\sin\theta\sin\phi, z=rcos⁡θz = r\cos\theta, by letting [L^z,x^][\hat L_z,\hat x] act on a smooth test function f(r,θ,ϕ)f(r,\theta,\phi).

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