# Problem from Quantum Optics

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1. Jul 25, 2016

### Muthumanimaran

1. The problem statement, all variables and given/known data
What is the Expectation value of

$$\sum_{l=0}^{n} \bra{n} (a\dagger)^l(a)^l\ket{n}$$

where $$a\dagger$$ and $$a$$ are lowering operators
2. Relevant equations
$$a\dagger\ket{n}=\sqrt{n+1}\ket{n}$$
$$a\ket{n}=\sqrt{n}\ket{n}$$
3. The attempt at a solution
I get the solution as $$\sum_{l=0}^{n} (n)!(n+1)!$$
But the correct answer is $$\sum_{l=0}^{n} \frac{n!}{(n-l)!}$$

2. Jul 25, 2016