(a) Calculate the commutators [\hat{X} ,\hat{L} _x],[\hat{X} ,\hat{L} _y], and [\hat{X} ,\hat{L} _z].
(b) Calculate the commutators: [\hat{P}_x ,\hat{L} _x],[\hat{P}_x ,\hat{L} _y], and [\hat{P}_x ,\hat{L} _z].
(c) Use the results of (a) and (b) to calculate [\hat{X} ,\hat{\vec{L^2} } ] and [\hat{P} _x,\hat{\vec{L^2}} ].
(a) The only nonzero commutator which involves \hat{X} and the various components of \hat{L}_x , \hat{L}_y , \hat{L}_z is [\hat{X} ,\hat{P}_x ]=i\hbar . Having stated this result, we can easily evaluate the needed commutators. First, since \hat{L}_x =\hat{Y} \hat{P}_z-\hat{Z} \hat{P} _y involves no \hat{P} _x , the operator \hat{X} commutes separately with \hat{Y} ,\hat{P}_z, \hat{Z} and \hat{P}_y; hence
[\hat{X} ,\hat{L}_x]= [\hat{X}, \hat{Y}\hat{P}_z-\hat{Z}\hat{P}_y]=0. (5.9)
The evaluation of the other two commutators is straightforward:
[\hat{X} ,\hat{L}_y]= [\hat{X}, \hat{Z}\hat{P}_x-\hat{X}\hat{P}_z]=[\hat{X} ,\hat{Z} \hat{P}_x]=\hat{Z}[\hat{X} ,\hat{P}_x] =i\hbar \hat{Z}, (5.10)
[\hat{X} ,\hat{L}_z]= [\hat{X}, \hat{X}\hat{P}_y-\hat{Y}\hat{P}_x]=-[\hat{X} ,\hat{Y} \hat{P}_x]=-\hat{Y}[\hat{X} ,\hat{P}_x] =-i\hbar \hat{Y}. (5.11)
(b) The only commutator between \hat{P}_x and the components of \hat{L}_x,\hat{L}_y,\hat{L}_z that survives is again [\hat{P}_x ,\hat{X}] =-i\hbar . We may thus infer
[\hat{P}_x ,\hat{L}_x] =[\hat{P} _x,\hat{Y} \hat{P} _z-\hat{Z} \hat{P} _y] =0, (5.12)
[\hat{P}_x ,\hat{L}_y] =[\hat{P} _x,\hat{Z} \hat{P} _x-\hat{X} \hat{P} _z]=-[\hat{P} _x,\hat{X} \hat{P} _z]=-[\hat{P} _x,\hat{X} ]\hat{P} _z=i\hbar \hat{P} _z, (5.13)
[\hat{P}_x ,\hat{L}_z] =[\hat{P} _x,\hat{X} \hat{P} _y-\hat{Y} \hat{P} _X]=[\hat{P} _x,\hat{X} \hat{P} _y]=[\hat{P} _x,\hat{X} ]\hat{P} _y=i\hbar \hat{P} _y. (5.14)
(c) Using the commutators derived in (a) and (b), we infer
[\hat{X} ,\hat{\vec{L^2} } ] =[\hat{X},\hat{L}^2_x ]+[\hat{X} ,\hat{L}^2_y]+[\hat{X} ,\hat{L}^2_z]
=0+\hat{L} _y[\hat{X} ,\hat{L} _y ] +[\hat{X},\hat{L}_y ]\hat{L} _y+\hat{L} _z[\hat{X} ,\hat{L}_z]+[\hat{X} ,\hat{L}_z ]\hat{L}_z
=i\hbar (\hat{L} _y\hat{Z}+\hat{Z}\hat{L}_y-\hat{L}_z\hat{Y}-\hat{Y}\hat{L}_y), (5.15)
[\hat{P}_x ,\hat{\vec{L^2} } ] =[\hat{P}_x,\hat{L}^2_x ]+[\hat{P}_x ,\hat{L}^2_y]+[\hat{P}_x ,\hat{L}^2_z]
=0+\hat{L} _y[\hat{P}_x ,\hat{L} _y ] +[\hat{P}_x,\hat{L}_y ]\hat{L} _y+\hat{L} _z[\hat{P}_x ,\hat{L}_z]+[\hat{P}_x ,\hat{L}_z ]\hat{L}_z
=i\hbar (\hat{L} _y\hat{P}_z+\hat{P}_z\hat{L}_y-\hat{L}_z\hat{P}_y-\hat{P}_y\hat{L}_y). (5.16)