Question 5.P.11: Using [ X , P] = i h, calculate the various commutation rela...

The operators \hat{T}_{1},\hat{T}_{2}, and \hat{T}_{3} can be viewed as describing some sort of collective vibrations; \hat{T}_{3} has the structure of a harmonic oscillator Hamiltonian. The first commutator can be calculated as follows:

[\hat{T}_{1},\hat{T}_{2}]=\frac{1}{4}[\hat{P}^{2}-\hat {X}^{2},\hat{T}_{2}]=\frac{1}{4}[\hat{P}^{2},\hat{T}_{2}]-\frac{1}{4}[\hat{X}^{2},\hat{T}_{2}],              (5.289)

where, using the commutation relation [\hat{X},\hat{P} ]=i\hbar , we have

=-\frac{i\hbar }{4} \hat{P}^{2} -\frac{i\hbar }{4} \hat {P}^{2}-\frac{i\hbar }{4} \hat{P}^{2}-\frac{i\hbar }{4} \hat{P}^{2} =i\hbar \hat{P}^{2},                               (5.290)

=\frac{i\hbar }{4} \hat{X}^{2} +\frac{i\hbar }{4} \hat{X}^{2} +\frac{i\hbar }{4} \hat{X}^{2}+\frac{i\hbar }{4} \hat{X}^{2}=i\hbar \hat{X}^{2};                          (5.291)

hence

[\hat{T}_{1},\hat{T}_{2}]=\frac{1}{4}[\hat{P}^{2}-\hat {X}^{2},\hat{T}_{2}]=-\frac{1}{4}[i\hbar \hat{P}^{2}+i\hbar \hat {X}^{2}]=-i\hbar \hat{T}_{3}.                  (5.292)

The second commutator is calculated as follows:

[\hat{T}_{2},\hat{T}_{3}]=\frac{1}{4}[\hat{T}_{2},\hat {P}^{2}+\hat{X}^{2}]=\frac{1}{4}[\hat{T}_{2},\hat{P}^{2}]+\frac{1}{4}[\hat{T}_{2},\hat{X}^{2}],           (5.293)

where [\hat{T}_{2},\hat{P}^{2}] and [\hat {T}_{2},\hat{X}^{2}] were calculated in (5.290) and (5.291):

[\hat{T}_{2},\hat{P}^{2}]=i\hbar \hat{P}^{2},      [\hat{T}_{2},\hat{X}^{2}]=-i\hbar \hat{X}^{2}.           (5.294)

Thus, we have

[\hat{T}_{2},\hat{T}_{3}]=\frac{1}{4}(i\hbar \hat{P}^{2}-i\hbar \hat{X}^{2})=i\hbar \hat{T}_{1}.                     (5.295)

The third commutator is

[\hat{T}_{3},\hat{T}_{1}]=\frac{1}{4}[\hat{T}_{3},\hat {P}^{2}-\hat{X}^{2}]=\frac{1}{4}[\hat{T}_{3},\hat{P}^{2}]-\frac{1}{4}[\hat{T}_{3},\hat{X}^{2}],              (5.296)

where

=\frac{i\hbar }{4}(2\hat{X}\hat{P}+2\hat{P}\hat{X}) =\frac{i\hbar }{2}(\hat{X}\hat{P}+\hat{P}\hat{X}),               (5.297)

[\hat{T}_{3},\hat{X}^{2}]=\frac{1}{4}[\hat{P}^{2},\hat {X}^{2}]+\frac{1}{4}[\hat{X}^{2},\hat{X}^{2}]=\frac{1}{4}[\hat {P}^{2},\hat{X}^{2}]=-\frac{i\hbar }{2}(\hat{X}\hat{P}+\hat {P}\hat{X});           (5.298)

hence

=\frac{i\hbar }{4}(\hat{X}\hat{P}+\hat{P}\hat{X})=i\hbar \hat{T}_{2}.                                      (5.299)

In sum, the commutation relations between \hat{T}_{1},\hat {T}_{2}, and \hat{T}_{3} are

[\hat{T}_{1},\hat{T}_{2}]=-i\hbar \hat{T}_{3},     [\hat{T}_{2} ,\hat{T}_{3}]=i\hbar \hat{T}_{1}, [\hat{T}_{3},    \hat{T}_{1}] =i\hbar \hat{T}_{2}.                    (5.300)

These relations are similar to those of ordinary angular momentum, save for the minus sign in [\hat{T}_{1},\hat{T}_{2}]=-i\hbar \hat{T}_{3}.