Relation between actual measurement and Mathematical observables

[MHD93]

I'm having a gap in understanding the relation between them, and resolving my confusion is really appreciated.

For example, the Hamiltonian operator, why do we call its eigenvalues energies? how do we actually measure it in the laboratory, quantum mechanically?

And maybe I need a better sense in which I understand energy? What is it, in quantum mechanics?

More generally, can I write down any hermitian operator and claim that it can be measured? How do I know how it is measured?

 

[MHD93]

Well, I know I asked many questions, but I am not asking for answers for all question. Any clues, partial answers or links for any question will also help..

 

[Jolb]

For example, the Hamiltonian operator, why do we call its eigenvalues energies? how do we actually measure it in the laboratory, quantum mechanically?

To the zeroth order approximation, you just measure the energy! This basic correspondence actually does hold, albeit very roughly.

To the first order approximation, remember that energy is not an absolute number: we only measure energy differences. This corresponds to the fact that we can always add a uniform potential to the Schrodinger equation without changing the form of the eigenstates--only a global shift in energies results. So an energy measurement for a system basically looks at what possible energy transitions the system can undergo. For example, to measure the "Hamiltonian operator" of a Harmonic oscillator would correspond (again--first order approximation) to doing emission/absorption experiments to see what the differences in energies between eigenstates are. If you had a harmonic oscillator in an unknown energy eigenstate and you want to figure out which eigenstate it's in, you'd prepare a bunch of identical copies and do an emission/absorption experiment on each one, and you'd eventually narrow down which eigenstate it is.

To second order, you might argue that all observable operators, including the Hamiltonian, are made out of the position and momentum operators, plus the operators for spin degrees of freedom. [There are no other physical degrees of freedom!] So if you believe you can get position and momentum measurements, then you can build all the other ones.

To third order, you might argue that none of these measurements are actually possible. To do any sort of measurement, there will always be some effect on the system due to the measurement device, and thus the Hamiltonian gets perturbed. For example, measuring the position of a particle might be accomplished by shining a photon onto it... but the photon actually perturbs the Hamiltonian a little bit so the eigenstates get perturbed. In fact measurement has a lot to do with perturbation theory.

Really this is a deep question that you should worry about even in experiments. Making sure your measurement doesn't perturb the system too much is very important.

And maybe I need a better sense in which I understand energy? What is it, in quantum mechanics?

Energy is basically the same as what it is in classical mechanics. First remember it's not absolute and only energy differences are relevant. Second, it's made of kinetic and potential energies, and the total energy is conserved for an isolated system. In the same way a conserved total energy dictates a surface in Phase space on which the classical state of the system lives, in quantum mechanics, conservation of energy dictates that the quantum state be trapped inside the subset of Hilbert space corresponding to eigenstates with that energy.

More generally, can I write down any hermitian operator and claim that it can be measured? How do I know how it is measured?

This is a good question I've asked to many professors. First, all observables MUST correspond to hermitian operators. Second, all the Hermitian operators I know of can be observed, even some very weird ones like the parity operator (because you can represent it in terms of position and momentum operators). I have not seen a proof though that all Hermitian operators are observable. Is the identity operator observable? Maybe someone else knows the answer to that.