I need to find the eigenvalue of the operator of the form $\int dx k \vert x \rangle \langle x \vert$ ( x goes from -a to +a). It seems that the eigenvector is $\int dx c(x) \vert x \ rangle$. But the problem is we can't use eigenvalue of 'k' here as it is in position space.
Apply the operator to the vector $\langle x_0 \vert$ and see what happens.
Dear Claude Pierre Massé,
I have tried it earlier, it gives -i \frac{\partial}{\partial x_0} \langle x_0 \vert. But that still don't solve the problem because it don't gives the eigenvalue but the operation of -i \frac{\partial}{\partial x_0} on the bra .
As far as I understand k in your question is an operator, and you are required to find its eigenvalues. I understand that the expression
(1) ∫dx K |x> a, and V=0 for |x| ≤ a. You can immediately calculate the eigenfunctions of the Hamiltonian :
(2) φn(x) = (2a)-1/2 sin(nπx/a) for |x| < a, and φn(x) = 0 for |x| ≥ a.
n is integer, though note that negative n gives the same eigenfunctions as positive n, up to a leading sign. The completion relations are immediate,
(3) (2a)-1 ∫dx sin(nπx/a) sin(n'πx/a) = δn,n',
where δ is the Kronecker symbol,
(4) ∑n = -∞∞ (2a)-1 sin(nπx/a) sin(nπx'/a) = δ(x - x')
where δ is Dirac's symbol.
So, try to apply the operator (1) to the eigenfunctions (2).
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