# Observables - what are they?

I often read in books that an observable is represented by an Hermitean operator. But it is deceiving as operator isn't the observable.

As far as I've read the observable is denoted like $\langle \psi|\hat{x}|\psi\rangle$ which is equivalent of $\langle \psi |\hat{x} \psi\rangle$ or $\langle \hat{x}^\dagger\psi| \psi \rangle$. So I would say that an observable is represented by an inner product of a (1st) wavefunction with (2nd) an operator acted on a wavefunction. (If I look the second equation).

The inner product isn't the observable itself; rather, it's the expectation value of that observable. If I repeat the experiment I did many times over, the average value of the measurement will be $\langle\psi|\hat{x}|\psi\rangle$. This is why we say that, mathematically, the operator itself is the observable: it's the mathematical object that we use to make predictions regarding the actual measured numbers we get from experiments.