I How to pick an operator onto which the wavefunction collapses

[fluidistic]

Imagine a system of 1 particle in a superposition of eigenstates of some operator(s). If one were to make a measurement of a property of that particle, how is the operator (or observable) "picked" so that the wavefunction collapses into an eigenstate of said operator? In other words, how do one make the wavefunction collapse into say an energy eigenstate as opposed to a position eigenstate? I already know that it's by "measuring the energy of the particle", this is not what I am asking. I am asking what makes the measurement measuring the energy instead of other properties, in that case.

Another concrete example would be a Bloch electron in a solid. Atoms in the lattice are constantly measuring its energy, I think, and never its momentum nor position. I'd like to derive this fact from first principle. And then move on to Cooper pairs, etc. (where I am not sure, but I think the lattice still constantly measure the energies of Cooper pairs, but I'd like to derive it from first principles).

 

[PeterDonis]

I am asking what makes the measurement measuring the energy instead of other properties

The physical configuration of the measuring device.

 

[fluidistic]

I'm still expecting a more elaborate answer. One that deals with entanglement between the measurement apparatus and the studied particle, for instance.

 

[atyy]

I already know that it's by "measuring the energy of the particle", this is not what I am asking. I am asking what makes the measurement measuring the energy instead of other properties, in that case.

The answer involves some sort of circular reasoning. We know an instrument measures energy, if on an energy eigenstate, it produces the definite energy of the state. More generally, we know an instrument measures energy, if it produces results distributed according to the Born rule for the energy measurement.

Another way to answer the question that may seem less circular is to say that we only know need to define how to make a position measurement. For all other things, we know how to prepare the initial quantum state of the instrument, and to write the Hamiltonian for the combined system and instrument. If we make a position measurement on the instrument (eg. the position of the pointer on the instrument), then it is equivalent to making a measurement on the quantum system being measured. To make an energy measurement, we can arrange the initial state and Hamiltonian of the instrument such that the position readings are distributed according to the Born rule for energy measurements on the quantum system being measured.

 

[kith]

It might be worthy to aks yourself how to determine which observable / property of a classical system a given measurement apparatus measues.

 

[A. Neumaier]

To make an energy measurement, we can arrange [...]

? I'd like to know a reference discussing how to arrange an energy measurement of a bound electron, say.