Operator Equation Msg Deleted: Solving Info Issues

[JohanL]

message deleted

 

[HallsofIvy]

Proof by induction (on m) should work.

If m= 1, then
$$a^{m}(a\dagger)^{m}= aa\dagger= n+1$$
which is the correct formula.

Now, assume that, for some k,
$$a^{k}(a\dagger)^{k}=(n+1)(n+2)...(n+k)$$
and look at
$$a^{k+1}(a\dagger)^{k+1}= a(a^{k}(a\dagger)^k) a\dagger$$
$$= a((n+1)(n+2)...(n+k))a\dagger$$
and apply the commutativity relation to that.