Do the angular momentum operators Lx and Ly commute Evaluate

Do the angular momentum operators L_x and L_y commute? Evaluate the communator (L_x, L_Y) = L_xL_y-L_yL_x.

Solution

They do not commute.

We start from Lx = Y Pz ZPy, Ly = ZPx XPz, and Lz = XPy Y Px

[Lx,Ly] = [Y Pz ZPy,ZPx XPz]

= [Y Pz,ZPx] [Y Pz,XPz] [ZPy,ZPx] + [ZPy,XPz]

= Y[Pz,ZPx]+[Y,ZPx]Pz-Y[Pz,XPz]-[Y,XPz]Pz- Z[Py,ZPx]- [Z,ZPx]Py+Z[Py,XPz] +[Z,XPz]Py

= YZ[Pz,Px]+Y[Pz,Z]Px+Z[Y,Px]Pz+[Y,Z]PzPx-YX[Pz,Pz]-Y[Pz,X]Pz-X[Y,Pz]Pz-[Y,X]PzPz-Z[Z,Px]Py- [Z,Z]PyPx+ZX[Py,Pz] +Z[Py,X]Pz+ X[Z,Pz]Py+[Z,X]PyPz

= Y Px[Pz,Z] 0 0 + XPy[Z,Pz ]

= i(hbar)Y Px + i(hbar)XPy

= i(hbar)Lz