Answer to Question #86639 in Atomic and Nuclear Physics for Anand

Question #86639
For a motion of a particle of mass μ in a spherically symmetric potential show that
L²and Lzcommute with the Hamiltonian
1
Expert's answer
2019-03-22T10:37:20-0400

The Hamiltonianof a particle of mass μ in a spherically symmetric potential:


H=p22μ+V(r)H = \frac{p^2}{2\mu}+V(r)

Angular momentum operators:

Lz=i(xyyx),L2=Lx2+Ly2+Lz2L_z = -i\hbar \left(x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}\right), L^2 = L^2_x+L^2_y+L^2_z

Angular momentum in spherical coordinates:


Lz=iϕ,L2=2(1sinθθ(sinθθ)+1sin2θ2ϕ2)L_z = -i\hbar \frac{\partial}{\partial \phi}, L^2 = - \hbar^2 \left(\frac{1}{\sin{\theta}}\frac{\partial}{\partial \theta}\left( \sin{\theta}\frac{\partial}{\partial \theta} \right)+\frac{1}{\sin^2{\theta}}\frac{\partial^2}{\partial \phi^2} \right)

As the hamiltonian is independent of θ,ϕ\theta, \phi coordinates,


[Lz,H]=0,[L2,H]=0[L_z, H] =0, \, [L^2, H] =0


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