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I understand that operators act on a quantum state to produce a new state. But I don't know whether they are merely abstract mathematical operations, or they do exist in real life, as in a quantum mechanics lab for example.

What are examples of hamiltonian operator in the lab? Is the action of measuring the state of an electron in a hydrogen atom considered as an operator? If yes, then what is this operator called?

Is the action of transition of the electron from one state to another considered as an operator?

Is the position operator applied to a quantum state considered as the action of measuring the position of the electron in that quantum state? So for the momentum operator? Please try to clarify this point for me as well as you can.

Thank you!

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What are examples of hamiltonian operator in the lab?

Anything that you can measure about a system is an observable. Each observable corresponds to an operator, and the eigenvalues of that operator are the possible values of the observable that you can measure.

The Hamiltonian operator usually* corresponds to the energy of the system. Whenever you measure a system's energy, you are measuring an eigenvalue of the Hamiltonian operator.

Is the action of measuring the state of an electron in a hydrogen atom considered as an operator?

You can't "measure the state of an electron". "The state" isn't an observable; at best, it's a bundle of other observables, which may or may not be compatible with each other. In all but the most trivial possible cases, simultaneously measuring all of the variables you would associate with the "state" of an object is impossible. For example, any electron has an associated position and momentum that you can measure. Position and momentum are linked by the Heisenberg Uncertanity Principle in such a way that a more precise measurement of one leads to a less precise measurement of the other. This is because measurement changes the state of the object, so in general measuring one quantity and then the other is not the same as measuring them in the reverse order.

Is the action of transition of the electron from one state to another considered as an operator?

Transition of the electron from one state to another is, overall, governed by the time evolution operator. When I said earlier that each observable corresponds to an operator, I meant exactly what I said, and not the reverse. There are many operators that don't correspond to observables$^{\dagger}$, and the time evolution operator is one of them.

This is a bit of a cop-out, though, since the time evolution operator itself is entirely determined by the Hamiltonian (which is regarded as the "generator of time translations"), so if you know the Hamiltonian and the initial state, you can predict the time evolution of the wavefunction.

In the particular case of the transition of an electron in an atom, the Hamiltonian consists of a few different parts:

  • The kinetic energy of the electron (represented by the squared momentum operator),
  • The potential energy of the electron (represented by the atomic potential operator), and
  • Terms corresponding to the energy contained in the electromagnetic field (which, if quantized, will consist of creation and annihilation operators for photons).

*Technically, it corresponds to the classical Hamiltonian of the system, which is a number that may or may not represent the total energy depending on the system's construction. In general, for time-independent Hamiltonians in most practical situations, the total-energy definition is fairly safe.

$^{\dagger}$The formal condition an operator must meet if it corresponds to an observable is self-adjointness.

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