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I always feel that we are inclined to ask this question because we want to treat the wavefunction as if it were a probability distribution. While they share som

by stared 2y ago

I always feel that we are inclined to ask this question because we want to treat the wavefunction as if it were a probability distribution. While they share some properties, fundamentally they are not the same thing. In typical probability, we deal with an ensemble of fixed states, or at least phenomena that can be simulated as such. In quantum physics, the wavefunction is fundamental. The question "what was the exact path?" is meaningless. In particular, if we take the approach of Feynman path integrals, we find that particles take many paths - including circular paths through each slit - before arriving somewhere else where they interact (i.e., become entangled) with, say, an electron in the screen. Sure, we may consider different experiments (e.g., quantum erasers, see https://lab.quantumflytrap.com/lab/quantum-eraser https://lab.quantumflytrap.com/lab/quantum-eraser), but analogies with deterministic particles are whimsical - sometimes they work, sometimes not.

bop24 2y ago
So I only have a B.S. in physics but my impression is that the weird parts of quantum mechanics are fundamentally a measurement problem. At the quantum level, we are very very limited in what we can use to measure properties of a quantum system - which is why we resort to probabilities. Wave functions are just a mathematical representation of a physical property that are (only?) ever operated on using quantum operators which result in a statistical distribution. Because they are so closely tied to probabilities I struggle with interpretations that try to say that perhaps these wave functions are something physical and based in reality (i.e. they are in superposition so particles must take on every possible state at once). An analogy I use is its like when we talk about sample sizes of a population of people, what is an ‘average person’? An average person is not something physical we can pick out, it exists in abstract. I’m curious if anyone with more experience in QM can shed light on how sound my thinking is here.
- stared 2y ago
  > Wave functions are just a mathematical representation of a physical property that are (only?) ever operated on using quantum operators which result in a statistical distribution. It is not correct— at least not unless you subscribe to the Copenhagen interpretation. Yet, while this interpretation is a simple heuristic for interaction with big systems (e.g., a photon hits a CCD array), none of the quantum physicists I know treat it seriously (for that matter, I have a PhD in quantum optics theory). I mean, at some certain level, everything is "just a mathematical representation" - in the spirit of "all models are wrong but some are useful". But the wavefunction is more fundamental than measurement. The other can be thought of as a particle entangling with a system so large that, for statistical reasons, it becomes irreversible - because of chaos, not fundamental rules. For some materials, I recommend materials on decoherence by WH Zurek, e.g. https://arxiv.org/pdf/quant-ph/0105127 https://arxiv.org/pdf/quant-ph/0105127. Some other references (here a shameless self ad) in https://www.spiedigitallibrary.org/journals/optical-engineering/volume-61/issue-08/081808/Visualizing-quantum-mechanics-in-an-interactive-simulation--Virtual-Lab/10.1117/1.OE.61.8.081808.full https://www.spiedigitallibrary.org/journals/optical-engineer... - mostly in the introduction and, speaking about interpretations, section 3.7. EDIT: or actually even simpler toy model of measurement, look at the Schrodinger cat in this one: https://arxiv.org/abs/2312.07840 https://arxiv.org/abs/2312.07840
  - bop24 2y ago
    Thank you, I will look into these resources
  - tsimionescu 2y ago
    The measurement, i.e. the Born rule, is just as fundamental as the wavefunction. The wavefunction doesn't mean anything on its own, it's not a measurable quantity that can be used to make any observable prediction whatsoever. If I claim that the wavefunction amplitude for some electron being at some location at some point in time is 1/2(1+i), how would you verify this prediction without invoking the Born rule?
    - stared 2y ago
      You may say that nuclear fusion in stars does not mean anything; only what matters is that we see light. At a certain level it is true, but to simulate a system we need to simulate its inner workings, not only - the end effect. The exact phase of a wavefunction does not matter - but it is an important phenomenon, giving raise to gauge invariance. The Born rule can be derived. In short, since we use unitary operators, length is preserved. For a derivation, see https://journals.aps.org/pra/abstract/10.1103/PhysRevA.71.052105 https://journals.aps.org/pra/abstract/10.1103/PhysRevA.71.05.... Also, to be nitpicky, we also never measure probabilities. Something (macroscopic) happened or not. It gives rise to quite fundamental and philosophical questions, including "what is (classical) probability" (I don't know an answer that fully satisfies me), many world interpretations (maybe all possible things just are), and in general what on indeterminism and free will.
      - tsimionescu 2y ago
        I'm not a quantum physicist and so can't really comment on why, but it's clear that the paper you linked is not widely accepted, as the Born rule is still taught as a postulate of quantum mechanics, not a derived property of the wavefunction. I'd wager a guess that the paper ends up inventing some other postulate that is itself not derivable from the wavefunction, so it becomes at best a philosophical matter which postulate you actually prefer. I also don't agree with your comparison of what I said to the nuclear reactions happening inside a star. The problem with the wavefunction without the Born rule is not that it's difficult to observe, it's that it's literally meaningless: knowing the value of the wavefunction for some state of a system doesn't tell you anything at all unless you apply the Born rule to this value. And as for probabilities, certain kinds of probabilities at least have a very clear and simple definition (though they are rather narrow cases): if you repeat an experiment in exactly the same conditions N times, and an outcome O happens in p/N times and doesn't happen (1-p/N) times, then we define P(O), the probability of outcome O, as the value p/N. For systems where this applies, it is very much a measurable quantity (with some noise, of course, related to the fidelity with which you can reproduce the same experiment). I do agree that this well-defined, measurable, concept of probability is rarely what we mean by "the probability of O", since (a) it's often hard or impossible to repeat (or even perform) the experiment, and (b) we often care about what will happen the next M times we repeat this experiment, and the measure P(O) I defined above does not tell us anything about future events.
- GoblinSlayer 2y ago
  Sounds like Einstein's hidden variables theory: below wave function picture there's more fundamental newtonian reality that produces the higher level wave function behavior, but is itself inaccessible due to insufficiently fine instruments, aka hidden variables. "God doesn't throw dice" is about that.
- l33tman 2y ago
  You're partially correct, but describing it in that way makes it sound like if you could "just look a little bit closer" the statistics would disappear, which doesn't happen. So it's more subtle than this. Fundamentally it's because QM doesn't use additive probabilities, but rather additive amplitudes which are complex numbers, and the probability is the square of the sum of these, so you can get interference between amplitudes. You can never get interference by adding probabilities. In the dual slit experiment this is visible as you can't get the interference effects by summing the probabilities for "particle through slit 1" and "particle through slit 2" but rather you need to sum the amplitudes of the processes. Working physicists (since 100 years) just do this, there is no practical need to interpret it further, but it would be cool if someone could figure out some prediction/experiment mismatch that does indeed require tweaking this!

by stared 2y ago

bop24 2y ago

So I only have a B.S. in physics but my impression is that the weird parts of quantum mechanics are fundamentally a measurement problem. At the quantum level, we are very very limited in what we can use to measure properties of a quantum system - which is why we resort to probabilities. Wave functions are just a mathematical representation of a physical property that are (only?) ever operated on using quantum operators which result in a statistical distribution. Because they are so closely tied to probabilities I struggle with interpretations that try to say that perhaps these wave functions are something physical and based in reality (i.e. they are in superposition so particles must take on every possible state at once). An analogy I use is its like when we talk about sample sizes of a population of people, what is an ‘average person’? An average person is not something physical we can pick out, it exists in abstract. I’m curious if anyone with more experience in QM can shed light on how sound my thinking is here.

stared 2y ago
> Wave functions are just a mathematical representation of a physical property that are (only?) ever operated on using quantum operators which result in a statistical distribution. It is not correct— at least not unless you subscribe to the Copenhagen interpretation. Yet, while this interpretation is a simple heuristic for interaction with big systems (e.g., a photon hits a CCD array), none of the quantum physicists I know treat it seriously (for that matter, I have a PhD in quantum optics theory). I mean, at some certain level, everything is "just a mathematical representation" - in the spirit of "all models are wrong but some are useful". But the wavefunction is more fundamental than measurement. The other can be thought of as a particle entangling with a system so large that, for statistical reasons, it becomes irreversible - because of chaos, not fundamental rules. For some materials, I recommend materials on decoherence by WH Zurek, e.g. https://arxiv.org/pdf/quant-ph/0105127 https://arxiv.org/pdf/quant-ph/0105127. Some other references (here a shameless self ad) in https://www.spiedigitallibrary.org/journals/optical-engineering/volume-61/issue-08/081808/Visualizing-quantum-mechanics-in-an-interactive-simulation--Virtual-Lab/10.1117/1.OE.61.8.081808.full https://www.spiedigitallibrary.org/journals/optical-engineer... - mostly in the introduction and, speaking about interpretations, section 3.7. EDIT: or actually even simpler toy model of measurement, look at the Schrodinger cat in this one: https://arxiv.org/abs/2312.07840 https://arxiv.org/abs/2312.07840
- bop24 2y ago
  Thank you, I will look into these resources
- tsimionescu 2y ago
  The measurement, i.e. the Born rule, is just as fundamental as the wavefunction. The wavefunction doesn't mean anything on its own, it's not a measurable quantity that can be used to make any observable prediction whatsoever. If I claim that the wavefunction amplitude for some electron being at some location at some point in time is 1/2(1+i), how would you verify this prediction without invoking the Born rule?
  - stared 2y ago
    You may say that nuclear fusion in stars does not mean anything; only what matters is that we see light. At a certain level it is true, but to simulate a system we need to simulate its inner workings, not only - the end effect. The exact phase of a wavefunction does not matter - but it is an important phenomenon, giving raise to gauge invariance. The Born rule can be derived. In short, since we use unitary operators, length is preserved. For a derivation, see https://journals.aps.org/pra/abstract/10.1103/PhysRevA.71.052105 https://journals.aps.org/pra/abstract/10.1103/PhysRevA.71.05.... Also, to be nitpicky, we also never measure probabilities. Something (macroscopic) happened or not. It gives rise to quite fundamental and philosophical questions, including "what is (classical) probability" (I don't know an answer that fully satisfies me), many world interpretations (maybe all possible things just are), and in general what on indeterminism and free will.
    - tsimionescu 2y ago
      I'm not a quantum physicist and so can't really comment on why, but it's clear that the paper you linked is not widely accepted, as the Born rule is still taught as a postulate of quantum mechanics, not a derived property of the wavefunction. I'd wager a guess that the paper ends up inventing some other postulate that is itself not derivable from the wavefunction, so it becomes at best a philosophical matter which postulate you actually prefer. I also don't agree with your comparison of what I said to the nuclear reactions happening inside a star. The problem with the wavefunction without the Born rule is not that it's difficult to observe, it's that it's literally meaningless: knowing the value of the wavefunction for some state of a system doesn't tell you anything at all unless you apply the Born rule to this value. And as for probabilities, certain kinds of probabilities at least have a very clear and simple definition (though they are rather narrow cases): if you repeat an experiment in exactly the same conditions N times, and an outcome O happens in p/N times and doesn't happen (1-p/N) times, then we define P(O), the probability of outcome O, as the value p/N. For systems where this applies, it is very much a measurable quantity (with some noise, of course, related to the fidelity with which you can reproduce the same experiment). I do agree that this well-defined, measurable, concept of probability is rarely what we mean by "the probability of O", since (a) it's often hard or impossible to repeat (or even perform) the experiment, and (b) we often care about what will happen the next M times we repeat this experiment, and the measure P(O) I defined above does not tell us anything about future events.
GoblinSlayer 2y ago
Sounds like Einstein's hidden variables theory: below wave function picture there's more fundamental newtonian reality that produces the higher level wave function behavior, but is itself inaccessible due to insufficiently fine instruments, aka hidden variables. "God doesn't throw dice" is about that.
l33tman 2y ago
You're partially correct, but describing it in that way makes it sound like if you could "just look a little bit closer" the statistics would disappear, which doesn't happen. So it's more subtle than this. Fundamentally it's because QM doesn't use additive probabilities, but rather additive amplitudes which are complex numbers, and the probability is the square of the sum of these, so you can get interference between amplitudes. You can never get interference by adding probabilities. In the dual slit experiment this is visible as you can't get the interference effects by summing the probabilities for "particle through slit 1" and "particle through slit 2" but rather you need to sum the amplitudes of the processes. Working physicists (since 100 years) just do this, there is no practical need to interpret it further, but it would be cool if someone could figure out some prediction/experiment mismatch that does indeed require tweaking this!