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My current understanding is that a quantum system can be in a superposition of possible outcome states that are eigenstates of a particular operator on the wavefunction. However, it is also my understanding that these operators are essentially chosen based on brute experimental fact. If we always observed the particle appear in three locations, then we would use a different operator to account for this fact. My question is about the philosophical implications around which operators work...and what the particle "is" before measurement. It seems that interpretations of quantum mechanics are more concerned with the measurement problem, and less concerned with the nature of the particles themselves.

Can anyone point me toward investigations of this kind? I suppose you've got to just say "that's how things are" at some point, but I am wondering whether people have thought about this.

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  • $\begingroup$ Side remark: a particle in a state that is a superposition of three position eigenstates would not appear to be in three locations at once, it would be precisely in a superposition of three locations. "A particle" appearing in three locations at once would rather be three particles, each in a particular position eigenstate. $\endgroup$
    – user87745
    Commented Apr 1, 2021 at 18:09
  • $\begingroup$ @DvijD.C. Yes, but I was not saying that the particle was in a superposition of three eigenstates. I was inventing a new universe in which the particle being in three locations was a single eigenstate. $\endgroup$
    – Jeff Bass
    Commented Apr 1, 2021 at 18:12
  • $\begingroup$ A particle being in three locations cannot be a single eigenstate because I don't know what it means. What could be a single eigenstate (of some operator) is the state which is a superposition of three position eigenstate. I can write down what it would look like, for example, it could read $\vert x_1\rangle+\vert x_2\rangle +\vert x_3\rangle$ (forgetting normalization, etc. because, of course). Can you write down what you are proposing would look like? $\endgroup$
    – user87745
    Commented Apr 1, 2021 at 18:15
  • $\begingroup$ Again, three particles being in three locations could be an eigenstate. It would read $\vert x_1\rangle \otimes \vert x_2\rangle \otimes \vert x_3\rangle$. In fact, it is an eigenstate of the position operator $X_1\otimes X_2\otimes X_3$ on the Hilbert space obtained by taking outer product of three distinguishable particles. $\endgroup$
    – user87745
    Commented Apr 1, 2021 at 18:18

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The physically useful/interesting/relevant operators are the ones who are generators of certain physical symmetries. For example, the momentum operator is the generator of translations in space, the Hamiltonian is the generator of translations in time, the angular momenta/spin operators are the generators of rotations in space, etc. Such operators are useful/interesting/relevant because if the system possesses the relevant symmetry then the quantities represented by the respective operators are conserved.

There is something extra (and quite hand-wavy) to be said about the position operator in particular. If there were to be a candidate position operator such that the usual objects would appear to be in a superposition of its distant eigenstates, I agree with you that we would not call it a position operator. In some basic sense, the reasoning is that we require the position operator to be such that the Hamiltonian is local in its eigenspectrum, i.e., interactions are local in its eigenspectrum. This, via decoherence, leads to classical objects being well-localized in the position basis.

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  • $\begingroup$ Ahhhh so there is a more fundamental reason that these are the chosen operators. $\endgroup$
    – Jeff Bass
    Commented Apr 1, 2021 at 18:33
  • $\begingroup$ I assume this is connected to Noether's Theorem, but do you know of a way to get an intuition for how the operators are connected to symmetries? $\endgroup$
    – Jeff Bass
    Commented Apr 1, 2021 at 18:38
  • $\begingroup$ @JeffBass That the operators are generators of certain symmetries doesn't have much to do with Noether's theorem, it is just a matter of definition. It is more like whatever the generator of translations in space is is called momentum. But, yes, that they are conserved has to do with Noether's theorem and no, I don't have a good pre-mathematical intuition for it. Also, we call certain generators of symmetry by special names and remember them, etc. is because they are conserved -- so in that sense, all of it has to do with Noether's theorem. $\endgroup$
    – user87745
    Commented Apr 1, 2021 at 18:44
  • $\begingroup$ So maybe just to back up one step. I'm not sure I understand your point about the position operator. Can you elaborate on why the particle being in two distant locations at once is not a valid eigenstate? $\endgroup$
    – Jeff Bass
    Commented Apr 1, 2021 at 18:48
  • $\begingroup$ @JeffBass Because, as I said, I don't know what it means. How would you write it down? If you write down $\vert x_1 \rangle + \vert x_2\rangle$ then it is just a superposition of two position eigenstates -- which can, of course, be an eigenstate of some other operator but it is not "a particle being in two positions at once". $\endgroup$
    – user87745
    Commented Apr 1, 2021 at 18:50

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