Is energy $E$ in Schrödinger equation an observable/ Can $E$ be measured? Take this quantum approach to estimate mean energy of a molecule:
$$\langle\psi|H|\psi\rangle=\overline E$$
Question:
Is $E$ an observable? How we can compare it to an experimental value? i.e how to experimentally measure it and what are the states involved (as energy is all about differences there must be two states)
Edit
It is not a question about how is theoretically defined an observable.
Any help?
 A: What you wrote is an expectation value, which means an average on some state $|\psi\rangle$ over all possible eigenvalues of the operator under analysis, weighted with the probability of that eigenvalue occurring on that state $\psi$.
So, yes, $\hat H$ is an observable (we reserve this term for operators and the quantities they represent) and this means that its eigenvalues (the energy levels) can actually be measured by means of suitable experiments.
But $\bar E$ is not an observable, and in general each single result of your measurements might have nothing to do with that value.
How to reconcile the two things?
Actually, repeating the same measurement an incredibly large number $N$ of times (or on an incredibly large number of systems) in the same initial conditions, should provide you with a set of results, whose average, weighted with the frequency each value occurs, should provide a value which bill be closer to $\bar E$ the larger $N$ is (they will coincide in the limit $N\rightarrow\infty$).
A: That is the expectation value of the energy. $\hat{H}$ is the Hamiltonian operator which corresponds to the energy of the system. So by evaluating $\langle\psi|\hat{H}|\psi\rangle$, you get the expectation value of the energy. The expectation value is the average of measurements performed on particles that are all in the state $\psi$. So either you make sure that after you measure the particle, you return it in the original state, or else you prepare an ensemble of particles in the same state $\psi$, and measure all of them. 
An observable is a special kind of operator. Its eigenvalues are always real and it describes a physical quantity. That is why all observables are described by Hermitian Operators, because Hermitian Operators are self-adjoint ($A=A^{\dagger}$).
A: Absolute energy is observable since it is the source of gravity. In practice this is hard to do but its observable status is indisputable. The Schrödinger energy is not absolute but relative to the rest energy of the particles that make up the system. Relative energy is observable by observing the products of a transition between states. For example binding energy can be observed by calorimetry, or if the binding reaction is optical such as in atomic deionisation, by observing emitted light. 
A: You can measure the energy of a molecule in a number of ways.
If what you want is to measure the energy difference between an excited state and the ground state, then you can drive the transition using e.m. waves of suitable frequency. You need a way to determine that the transition has happened, and a way to measure the frequency of the waves. To determine that the transition has happened, you could for example use further transitions in the molecule and ultimately look for fluorescence. This is the way really precise measurements are done in atomic and molecular physics labs. There are also clever techniques involving the motion in molecular beams.
If you want to measure the total energy including kinetic energy and internal energy, then you could drop the molecule into something cold, such as liquid helium. The amount of energy released (from kinetic energy and internal energy of the molecule) can be determined from the amount of helium that boils off. This method is not normally used, but it is the principle that counts. Instruments called bolometers do this kind of total energy measurement using a variety of ingenious strategies.
