Yes, I know all of this, I’m a mathematician, just not one researching QM. The arxiv link looks interesting, but I have no time to read it right now. The question isn’t “why are eigenvectors of Hermitian operators interesting”, it is “why would we expect a system doing something as reasonable as evolving via the Schrödinger equation to do something as unreasonable as to suddenly collapse to one of its eigenfunctions”.
I guess I don’t understand the question. If we accept that mutually exclusive states are represented by orthogonal vectors, and we want to distinguish mutually exclusive states of some interesting subsystem, then what’s unreasonable with defining a “measurement” as something that correlates our apparatus with the orthogonal states of the interesting subsystem, or at least as an ideal form of a measurement?
I think my question isn’t really well-defined. I guess it’s more along the lines of “is there some ‘natural seeming’ reasoning procedure that gets me QM ”.
And it’s even less well-defined as I have no clear understanding of what QM is, as all my attempts to learn it eventually run into problems where something just doesn’t make sense—not because I can’t follow the math, but because I can’t follow the interpretation.
If we accept that mutually exclusive states are represented by orthogonal vectors, and we want to distinguish mutually exclusive states of some interesting subsystem, then what’s unreasonable with defining a “measurement” as something that correlates our apparatus with the orthogonal states of the interesting subsystem, or at least as an ideal form of a measurement?
Yes, this makes sense, though “mutually exclusive state are represented by orthogonal vectors” is still really weird. I kind of get why Hermitian operators here makes sense, but then we apply the measurement and the system collapses to one of its eigenfunctions. Why?
I kind of get why Hermitian operators here makes sense, but then we apply the measurement and the system collapses to one of its eigenfunctions. Why?
If I understand what you mean, this is a consequence of what we defined as a measurement (or what’s sometimes called a pre-measurement). Taking the tensor product structure and density matrix formalism as a given, if the interesting subsystem starts in a pure state, the unitary measurement structure implies that the reduced state of the interesting subsystem will generally be a mixed state after measurement. You might find parts of this review informative; it covers pre-measurements and also weak measurements, and in particular talks about how to actually implement measurements with an interaction Hamiltonian.
You could also turn around this question. If you find it somewhat plausible that that self-adjoint operators represent physical quantities, eigenvalues represent measurement outcomes and eigenvectors represent states associated with these outcomes (per the arguments I have given in my other post) one could picture a situation where systems hop from eigenvector to eigenvector through time. From this point of view, continuous evolution between states is the strange thing.
The paper by Hardy I cited in another answer to you tries to make QM as similar to a classical probabilistic framework as possible and the sole difference between his two frameworks is that there are continuous transformations between states in the quantum case. (But notice that he works in a finite-dimensional setting which doesn’t easily permit important features of QM like the canonical commutation relations).
Yes, I know all of this, I’m a mathematician, just not one researching QM. The arxiv link looks interesting, but I have no time to read it right now. The question isn’t “why are eigenvectors of Hermitian operators interesting”, it is “why would we expect a system doing something as reasonable as evolving via the Schrödinger equation to do something as unreasonable as to suddenly collapse to one of its eigenfunctions”.
I guess I don’t understand the question. If we accept that mutually exclusive states are represented by orthogonal vectors, and we want to distinguish mutually exclusive states of some interesting subsystem, then what’s unreasonable with defining a “measurement” as something that correlates our apparatus with the orthogonal states of the interesting subsystem, or at least as an ideal form of a measurement?
I think my question isn’t really well-defined. I guess it’s more along the lines of “is there some ‘natural seeming’ reasoning procedure that gets me QM ”.
And it’s even less well-defined as I have no clear understanding of what QM is, as all my attempts to learn it eventually run into problems where something just doesn’t make sense—not because I can’t follow the math, but because I can’t follow the interpretation.
Yes, this makes sense, though “mutually exclusive state are represented by orthogonal vectors” is still really weird. I kind of get why Hermitian operators here makes sense, but then we apply the measurement and the system collapses to one of its eigenfunctions. Why?
If I understand what you mean, this is a consequence of what we defined as a measurement (or what’s sometimes called a pre-measurement). Taking the tensor product structure and density matrix formalism as a given, if the interesting subsystem starts in a pure state, the unitary measurement structure implies that the reduced state of the interesting subsystem will generally be a mixed state after measurement. You might find parts of this review informative; it covers pre-measurements and also weak measurements, and in particular talks about how to actually implement measurements with an interaction Hamiltonian.
You could also turn around this question. If you find it somewhat plausible that that self-adjoint operators represent physical quantities, eigenvalues represent measurement outcomes and eigenvectors represent states associated with these outcomes (per the arguments I have given in my other post) one could picture a situation where systems hop from eigenvector to eigenvector through time. From this point of view, continuous evolution between states is the strange thing.
The paper by Hardy I cited in another answer to you tries to make QM as similar to a classical probabilistic framework as possible and the sole difference between his two frameworks is that there are continuous transformations between states in the quantum case. (But notice that he works in a finite-dimensional setting which doesn’t easily permit important features of QM like the canonical commutation relations).
Well yeah sure. But continuity is a much easier pill to swallow than “continuity only when you aren’t looking”.