During a lesson on Rabi's oscillations, my professor computed the following series: $$H = \sum_n \left( (a |e_1 \rangle \langle e_0| + a^\dagger |e_0 \rangle \langle e_1|\right)^n$$ separating in odd and even cases. Where $e_0$, $e_1$ are a basis and $a$, $a^\dagger$ are the creation and annihilation operators acting on the fock states.
The even series is: $$H^{2n} = (a^{\dagger} a)^n |e_0\rangle \langle e_0| + (a a^\dagger )^n |e_1\rangle \langle e_1|$$ While to compute the odd one, he multiplied to the left for one term of the series. But, I noticed that the result changes if you multiply to the right or to the left, in fact: $$H^{1+2n} = a |e_1\rangle \langle e_0| (a^\dagger a)^n + a^\dagger |e_0 \rangle \langle e_1 |(a a^\dagger)^n$$ $$H^{2n+1} = |e_1\rangle \langle e_0| (a a^\dagger )^n a + |e_0 \rangle \langle e_1 |(a^\dagger a)^n a^\dagger$$ Why do I get two different results? They should commute, but I can't get the same result using the commutator between $a$ and $a^\dagger$.
You find the same answer, of course, by virtue of the basic maneuver of such operators, $$ a a^\dagger = a^\dagger a+1\equiv N+1, $$ and $$ f(N) a =a f(N-1), \\ f(N) a^\dagger = f(N+1) a^\dagger, $$ ∴ $$ a N^n= (N+1)^n a ,\\ a^\dagger (N+1)^n= N^n a^\dagger. $$
The creation and annihilation operator functions commute with the extraneous basis states, since the Fock space does.