div curl F= 0
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- 0
Homework Statement
"Write down the operator \hat{a}^2 in the basis of the energy states |n>. Determine the eigenvalues and eigenvectors of the operator \hat{a}^2 working in the same basis.
You may use the relation: \sum_{k = 0}^{\infty} \frac{|x|^{2k}}{(2k)!} = cosh(|x|)"
Homework Equations
The Attempt at a Solution
For the first part, I've got the abstract version of the operator to be:
\hat{a}^2 = \sum_{n=0}^{\infty} \sqrt{n(n-1)} |n-2><n|
but the second part is giving me some trouble. I'm not too sure how to set about it, I've tried a few different approaches but nothing ends up using the above relation. I've tried a coherent state: \hat{a} |n> = \lambda |n>, and I've tried a ket composed on the basis n: |\psi> = \sum_{n=0}^{\infty} C_n |n>.
I'd be grateful if somebody could show me the way with this question, I've just hit a brick wall with it.