4 ms·

I think that GP is trying to say the following. In quantum mechanics, probabilities are given by the square of the absolute value of more fundamental quantitie

I think that GP is trying to say the following. In quantum mechanics, probabilities are given by the square of the absolute value of more fundamental quantities called amplitudes. When something happens in two ways that can be distinguished, you must add the probabilities. When something happens in two indistinguishable ways, you must add the amplitudes, which yields a different probability after squaring. For example, .3^2+.4^2 != (.3+.4)^2. Thus, you can verify experimentally whether particles are or are not distinguishable.

JumpCrisscross 5y ago
> you can verify experimentally whether particles are or are not distinguishable Thank you for a terrific explanation. Could you please go one layer deeper? Why does whether probabilities or amplitudes are summed imply fungibility (or its absence)?
- wizzwizz4 5y ago
  Because you can construct experiments where the proton from source A ends up at location B, and the proton from source C ends up at location D, or A ends up at D and B ends up at C. (Or some other possibilities.) You find that the A→B, C→D possibility's amplitude sums with the A→D, B→C possibility: i.e., that they're the same indistinguishable final state. If swapping two things around gives a result indistinguishable from not swapping them, that's fungibility.
  - whatshisface 5y ago
    Photons of different polarizations are nonidentical, but if this argument is true, it would also prove that horizontally and vertically polarized photons were identical... in an experiment insensitive to polarization. I do not believe this answers the original question.
    - wizzwizz4 5y ago
      Two coins aren't identical, but they're fungible. Is "identical" even well-defined on fundamental-ish particles?
      - whatshisface 5y ago
        Identical means that either bosonic or fermionic conditions are applied to the joint tensor-product'd wavefunctions of multi-particle systems. It's well defined, and although I was hoping this thread would offer a more grounded definition I am not sure if it succeeded.
- MatteoFrigo 5y ago
  Well, I am not a physicist, but I can fake it well enough for HN :-) Before I say anything, if you have never heard of amplitudes before, you should read the Feynman lectures on physics vol. III, which you can find here: https://www.feynmanlectures.caltech.edu/III_toc.html https://www.feynmanlectures.caltech.edu/III_toc.html Specifically read chapter 1 and chapter 3. The specific situation about distinguishable/indistinguishable particles is described in section 3-4. Your question is kind of phrased backwards. Probabilities and amplitudes are human inventions to describe Nature's behavior, so by themselves they don't imply anything about Nature. The implication is the other way around: Nature has decided that some particle pairs are indistinguishable (proton vs. proton) and some are distinguishable (proton vs. neutron). Different rules apply to the two cases. This indistinguishability is a pure quantum phenomenon. In a classical world, all objects are distinguishable---you can in principle label all protons and know which is which. But Nature does not work like that, and there exists this peculiar notion that you cannot tell two protons apart not even in principle. There is no deeper explanation of this phenomenon AFAIK---it is what it is. You can tell experimentally whether two particles are or are not distinguishable by running the experiment as in Feynman 3-4. If you observe a distribution consistent with the add-amplitude rule, then the particles are indistinguishable. Any attempt to distinguish them leads to the contradictions explained in Chapter 1. There is an even more peculiar phenomenon. Despite being indistinguishable, swapping two indistinguishable particles is not a no-op because it changes the amplitudes. You must still add amplitudes, but the amplitudes are different, yet different in such a subtle way that you still cannot tell the particles apart. Feynman 3-4 tells you how the amplitudes change during the swap, with a deeper explanation in chapter 4.
whatshisface 5y ago
I am sorry but I am lost. If they both have probabilities or amplitudes of 1 would that not lead to a joint probability of 200% in the nonidentical case and 400% in the identical one?
- MatteoFrigo 5y ago
  Sorry, I didn't mean to imply that my comment should apply literally to all cases. I am just pointing out to parent that particles can be indistinguishable in principle, and not just as a technological limitation of our measurements. Moreover, there is an experimental way to tell the difference between distinguishable and indistinguishable, roughly based on the difference between probabilities and amplitudes. To dig deeper one must look at the details, e.g. in Feynman's lectures vol. III. The statement that protons are indistinguishable is not strictly correct either, because protons have a spin. Protons with the same spin are indistinguishable, but you can tell apart protons with different spin. The spin of protons can only assume two values, so effectively there are two classes of protons, indistinguishable within the class. In your specific case, it is clearly false that the probability of having one particle in one place is 200%. However, my statement still holds for expectations, and you end up with an expected two particles in one place. In the indistinguishable case, you must compute expectations based on amplitudes, not probabilities.
  - db48x 5y ago
    Yea, spin adds a new level of complications. You can distinguish between two otherwise–indistinguishable protons if they have opposite spin, but the spin of a proton can also change over time (usually due to interactions with other particles, such as stray radio waves passing through your experiment). Going back to the experiment that I described, you can imagine that the particles are released at A and B with opposite spins, and then the detector at A’ only detects the spin that corresponds to the particle at A. This causes you to measure yet another probability, distinct from the other two, because there are now more possibilities and there are still multiple ways to cause the detector to find something. It could detect the proton from A, but the proton from A could also have its spin flipped and thus not be detected. The particle from B could arrive at A’ with the wrong spin and not be counted, or it could have its spin flipped along the way and be counted. You still cannot tell which proton you detected! Similar complications occur with polarization of photons, which someone else mentioned in one of the comments. It’s worse though because polarization is a continuous quantity, and there are more ways to change it.
db48x 5y ago
Thank you, I should have mentioned amplitudes.

JumpCrisscross 5y ago

> you can verify experimentally whether particles are or are not distinguishable Thank you for a terrific explanation. Could you please go one layer deeper? Why does whether probabilities or amplitudes are summed imply fungibility (or its absence)?

wizzwizz4 5y ago
Because you can construct experiments where the proton from source A ends up at location B, and the proton from source C ends up at location D, or A ends up at D and B ends up at C. (Or some other possibilities.) You find that the A→B, C→D possibility's amplitude sums with the A→D, B→C possibility: i.e., that they're the same indistinguishable final state. If swapping two things around gives a result indistinguishable from not swapping them, that's fungibility.
- whatshisface 5y ago
  Photons of different polarizations are nonidentical, but if this argument is true, it would also prove that horizontally and vertically polarized photons were identical... in an experiment insensitive to polarization. I do not believe this answers the original question.
  - wizzwizz4 5y ago
    Two coins aren't identical, but they're fungible. Is "identical" even well-defined on fundamental-ish particles?
    - whatshisface 5y ago
      Identical means that either bosonic or fermionic conditions are applied to the joint tensor-product'd wavefunctions of multi-particle systems. It's well defined, and although I was hoping this thread would offer a more grounded definition I am not sure if it succeeded.
MatteoFrigo 5y ago
Well, I am not a physicist, but I can fake it well enough for HN :-) Before I say anything, if you have never heard of amplitudes before, you should read the Feynman lectures on physics vol. III, which you can find here: https://www.feynmanlectures.caltech.edu/III_toc.html https://www.feynmanlectures.caltech.edu/III_toc.html Specifically read chapter 1 and chapter 3. The specific situation about distinguishable/indistinguishable particles is described in section 3-4. Your question is kind of phrased backwards. Probabilities and amplitudes are human inventions to describe Nature's behavior, so by themselves they don't imply anything about Nature. The implication is the other way around: Nature has decided that some particle pairs are indistinguishable (proton vs. proton) and some are distinguishable (proton vs. neutron). Different rules apply to the two cases. This indistinguishability is a pure quantum phenomenon. In a classical world, all objects are distinguishable---you can in principle label all protons and know which is which. But Nature does not work like that, and there exists this peculiar notion that you cannot tell two protons apart not even in principle. There is no deeper explanation of this phenomenon AFAIK---it is what it is. You can tell experimentally whether two particles are or are not distinguishable by running the experiment as in Feynman 3-4. If you observe a distribution consistent with the add-amplitude rule, then the particles are indistinguishable. Any attempt to distinguish them leads to the contradictions explained in Chapter 1. There is an even more peculiar phenomenon. Despite being indistinguishable, swapping two indistinguishable particles is not a no-op because it changes the amplitudes. You must still add amplitudes, but the amplitudes are different, yet different in such a subtle way that you still cannot tell the particles apart. Feynman 3-4 tells you how the amplitudes change during the swap, with a deeper explanation in chapter 4.

whatshisface 5y ago

I am sorry but I am lost. If they both have probabilities or amplitudes of 1 would that not lead to a joint probability of 200% in the nonidentical case and 400% in the identical one?

MatteoFrigo 5y ago
Sorry, I didn't mean to imply that my comment should apply literally to all cases. I am just pointing out to parent that particles can be indistinguishable in principle, and not just as a technological limitation of our measurements. Moreover, there is an experimental way to tell the difference between distinguishable and indistinguishable, roughly based on the difference between probabilities and amplitudes. To dig deeper one must look at the details, e.g. in Feynman's lectures vol. III. The statement that protons are indistinguishable is not strictly correct either, because protons have a spin. Protons with the same spin are indistinguishable, but you can tell apart protons with different spin. The spin of protons can only assume two values, so effectively there are two classes of protons, indistinguishable within the class. In your specific case, it is clearly false that the probability of having one particle in one place is 200%. However, my statement still holds for expectations, and you end up with an expected two particles in one place. In the indistinguishable case, you must compute expectations based on amplitudes, not probabilities.
- db48x 5y ago
  Yea, spin adds a new level of complications. You can distinguish between two otherwise–indistinguishable protons if they have opposite spin, but the spin of a proton can also change over time (usually due to interactions with other particles, such as stray radio waves passing through your experiment). Going back to the experiment that I described, you can imagine that the particles are released at A and B with opposite spins, and then the detector at A’ only detects the spin that corresponds to the particle at A. This causes you to measure yet another probability, distinct from the other two, because there are now more possibilities and there are still multiple ways to cause the detector to find something. It could detect the proton from A, but the proton from A could also have its spin flipped and thus not be detected. The particle from B could arrive at A’ with the wrong spin and not be counted, or it could have its spin flipped along the way and be counted. You still cannot tell which proton you detected! Similar complications occur with polarization of photons, which someone else mentioned in one of the comments. It’s worse though because polarization is a continuous quantity, and there are more ways to change it.

db48x 5y ago

Thank you, I should have mentioned amplitudes.