How do you calculate the commutator $$[b_i^n, (b_j^\dagger)^m]$$ where $b_i$ are annihilators and $b_j^\dagger$ are creators in second quantization. All annihalators should be swapped to the right side.
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Simply use the definition of the commutator $$[A B, C] = A B C - C A B = ABC - ACB + ACB - CAB \\= A (BC - CB) + (AC - CA)B = A[B, C] + [A,C]B$$ which generalizes to $$ [A^n, B] = \sum_{m=0}^{n-1} A^m[A,B]A^{n-m-1}$$ by repeated application. You should be able to do the rest yourself from here.
noah
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