How to represent operator in matrix form

[phyin]

I'm given some arbitrary operator call it O, how do I represent it in general matrix form while it still preserves the properties of the operator.

ex. if operator is hermitian how to i represent a most general matrix representation so it preserves properties of a hermitian matrix.

 

[vanhees71]

You need a basis of Hilbert space, $\{|n \rangle \}_{n \in \mathbb{N}}$, e.g., the harmonic-oscillator-energy eigen states. Then the matrix elements of an arbitrary operator, $\hat{O}$ are given by

$$O_{jk}=\langle j|\hat{O} k \rangle.$$

It's easy to verify that this is an Hermitean matrix, if $\hat{O}$, is selfadjoined.