- #1
div curl F= 0
- 19
- 0
Homework Statement
"Write down the operator [tex] \hat{a}^2 [/tex] in the basis of the energy states [tex] |n> [/tex]. Determine the eigenvalues and eigenvectors of the operator [tex] \hat{a}^2 [/tex] working in the same basis.
You may use the relation: [tex] \sum_{k = 0}^{\infty} \frac{|x|^{2k}}{(2k)!} = cosh(|x|) [/tex]"
Homework Equations
The Attempt at a Solution
For the first part, I've got the abstract version of the operator to be:
[tex] \hat{a}^2 = \sum_{n=0}^{\infty} \sqrt{n(n-1)} |n-2><n| [/tex]
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: [tex] \hat{a} |n> = \lambda |n> [/tex], and I've tried a ket composed on the basis n: [tex] |\psi> = \sum_{n=0}^{\infty} C_n |n> [/tex].
I'd be grateful if somebody could show me the way with this question, I've just hit a brick wall with it.