Help with Operators written as components.

[Inertia]

I would appreciate if someone could set me straight here. I understand if I have an arbitrary operator, I can express it in matrix component notation as follows:

Oi,j = <vi|O|vj>

Is it possible to get a representation of the operator O back from this component form. I'm more interested in what to do in the case of continuous i and j so I assume I will have to do a 2 dimensional integral. Also how could I go from this form to calculate an expectation value <O>. Again I would have thought this would be a 2D integral as-well but something a bit more as I would want this <X|O|X> where X is a wave-function.

 

[MisterX]

The answer to your questions can be found by inserting identity operators on either side of O. We express the identity operators in the v basis as $\sum_i \mid v_i \rangle \langle v_i \mid\; = \; \sum_j \mid v_j \rangle \langle v_j \mid \;= I$

\begin{align*} O &= \sum_{i,j} \mid v_i \rangle \langle v_i \mid O \mid v_j \rangle \langle v_j \mid \\ &= \sum_{i,j} \mid v_i \rangle O_{ij}\langle v_j \mid
\end{align*}

For the continuous case, we just replace sums by integrals.

Supposing $$\mid X \rangle = \sum_{i} \psi_v(i) \mid v_i \rangle$$

$$\langle X \mid O \mid X \rangle = \sum_{i,j}\psi^*_v(i) \langle v_i \mid O \mid v_j \rangle \psi_v(j) =\sum_{i,j}\psi^*_v(i) O_{ij} \psi_v(j)$$