5 ms·

For a single particle they are easy to confuse. A wave function ψ(t,x) for a single particle gives a probability amplitude to find the particle at coordinate x

by aap_ 2y ago

For a single particle they are easy to confuse. A wave function ψ(t,x) for a single particle gives a probability amplitude to find the particle at coordinate x at time t. In this case one can imagine an amplitude at each point in space and time, like a field. This interpretation however completely breaks down once you introduce a second particle: the wave function ψ(t,x1,x2) gives a probability amplitude to find particle 1 at x1 and particle 2 at x2 at time t. This no longer admits an interpretation of assigning some value to locations in space. Intuitively one might think you get one amplitude for each particle at some location but that's not how QM works, so we shouldn't think of the wave function as living in physical space.

whatshisface 2y ago
That's true, but it's also true of the classical probability distribution p(t,x1,x2).
- tsimionescu 2y ago
  Yes, which is exactly the point. The main difference is that the wave function has a complex value with norm <= 1, while a probability distribution function has a real value <= 1.
SiempreViernes 2y ago
But if you aren't trying to map the wave function to physical space somehow you are essentially saying that the central construct of your theory has no direct relation to the actual physical processes happening "underneath". This reduces to a kind of "shut up and calculate" attitude, so it seems poor starting point from which to write an interpretation text.
- tsimionescu 2y ago
  Space is a part of the wavefunction, as the article explains clearly. The wave function describes where the particles can be in physical space. And, the wave function has the same shape as the wave equations for traditional mechanical waves, like a sound wave or a sea wave. However, if a classical three-dimensional wave equation describes how matter osciallates in three-dimensional physical space, a quantum wavefunction doesn't do that. Quantum particles don't oscillate in physical space like that. A three-dimensional wavefunction might describe three particles' positions along a one-dimensional line, and it's oscillations are oscillations of probability, not position. The particles don't move, say, up and down. Their probability to be here or there on that 1-d line waxes and wanes. This is what the article is trying to explain: the basic mathematics of quantum mechanics, the definition of the wavefunction. The value of a wavefunction for the position of three particles is not a position in space at a moment in time. It is a (complex) probability for the position of every particle at that moment. This only seems confusing when looking at wavefunctions that describe positions. But wavefunctions often have many more observables, such as spin or polarization. A wavefunctions for two electrons moving around on a plane will not be a two-dimensional wave. It will be a wave in a six-dimensional space, whose axis may be "particle 1 has spin up/down, particle 2 has spin up/down, particle 1 position along x axis, particle 2 position along x axis, particle 1 position along y axis, particle two position along y axis".
  - SiempreViernes 2y ago
    I'm honestly confused; it's fine to say the wave function lives in some high dimensional phase space and that it's not actually describing some vibration of spacetime. But I don't recall ever imagining the wave function being a vibration of spacetime, is that really something people think? If I were to express some sort of wave-function-in-spacetime theory, I'd invoke lots of classical fields filling space and have those wiggle. In any case, the whole bit about the proper two-particle wave function living in a higher dimensional space is somewhat spoiled by the fact that you can factorise it into normal 3-space pieces (so long as you don't have your particles interacting), it doesn't seem such an alien space to me.
    - tsimionescu 2y ago
      Before the wavefunction, we used to explain the double slit experiment (the version without detectors at a slit) as light being an EM wave in physical space, essentially equivalent to a sound wave propagating through the EM field, which breaks on the wall and essentially transforms into two separate waves, each originating from one slit, which are then in phase and so they constructively interfere, forming the final pattern on the screen. Lots of people think that this is the same picture that the wavefunction gives, but this is wrong. In the QM picture, the emitter emits one photon, which is a quantum of energy described by a four-dimensional wavefunction which assigns some probability of a detection event at the slits, at the screen, etc. In this picture, there is no physical EM wave, any interaction with the light will happen at a single localized point in space. Of course, if you add more particles, especially those carrying charges, the picture changes, and you'll see probabilities that roughly correspond to a picture of an oscillating EM field. But the wavefunction, which is the "bedrock" physical theory, is separate from those waves in the EM field, which are just an approximate picture of the probabilities dictated by the wavefunciton.
- aap_ 2y ago
  I morally agree, but not quite: think of the wave function as not more than a bookkeeping device. It does get the job done but be careful to ascribe it too high an ontological status! The path integral formulation seems a lot more natural to me and it does not need a wave function, instead you can derive it and treat it as a bookkeeping device. The way I think about it is that it's an attempt to deterministically model non-deterministic behavior: you "pretend" that the system is deterministic by keeping track of all the possible ways it could have evolved in time. sure enough, once you make a measurement this probability distribution "collapses" and you find out what is actually the case.
  - deleted 2y ago
    [deleted]
  - GoblinSlayer 2y ago
    It's not just pretending, deterministic model works, unlike nondeterministic.
  - SiempreViernes 2y ago
    I think you are agreeing with my point that declaring the wave function to be mere bookkeeping is a poor foundation for writing about the interpretation of quantum mechanics? Can't really get any other sense out of your reply, but I'm not entirely sure.
    - aap_ 2y ago
      Also not sure I'm understanding you right :) My view is this: the wave function is mere bookkeeping and not anything ontologically fundamental. However the fact that such a seemingly bizarre concept lets you do quantum physics (even if it's not the only way) points to some fundamental questions about the nature of...well, nature. Of course this is not the only valid view...just one that makes sense to me. Thinking about these sorts of questions is a very fun endeavour.
      - SiempreViernes 2y ago

by aap_ 2y ago

whatshisface 2y ago

That's true, but it's also true of the classical probability distribution p(t,x1,x2).

tsimionescu 2y ago
Yes, which is exactly the point. The main difference is that the wave function has a complex value with norm <= 1, while a probability distribution function has a real value <= 1.

SiempreViernes 2y ago

But if you aren't trying to map the wave function to physical space somehow you are essentially saying that the central construct of your theory has no direct relation to the actual physical processes happening "underneath". This reduces to a kind of "shut up and calculate" attitude, so it seems poor starting point from which to write an interpretation text.

tsimionescu 2y ago
Space is a part of the wavefunction, as the article explains clearly. The wave function describes where the particles can be in physical space. And, the wave function has the same shape as the wave equations for traditional mechanical waves, like a sound wave or a sea wave. However, if a classical three-dimensional wave equation describes how matter osciallates in three-dimensional physical space, a quantum wavefunction doesn't do that. Quantum particles don't oscillate in physical space like that. A three-dimensional wavefunction might describe three particles' positions along a one-dimensional line, and it's oscillations are oscillations of probability, not position. The particles don't move, say, up and down. Their probability to be here or there on that 1-d line waxes and wanes. This is what the article is trying to explain: the basic mathematics of quantum mechanics, the definition of the wavefunction. The value of a wavefunction for the position of three particles is not a position in space at a moment in time. It is a (complex) probability for the position of every particle at that moment. This only seems confusing when looking at wavefunctions that describe positions. But wavefunctions often have many more observables, such as spin or polarization. A wavefunctions for two electrons moving around on a plane will not be a two-dimensional wave. It will be a wave in a six-dimensional space, whose axis may be "particle 1 has spin up/down, particle 2 has spin up/down, particle 1 position along x axis, particle 2 position along x axis, particle 1 position along y axis, particle two position along y axis".
- SiempreViernes 2y ago
  I'm honestly confused; it's fine to say the wave function lives in some high dimensional phase space and that it's not actually describing some vibration of spacetime. But I don't recall ever imagining the wave function being a vibration of spacetime, is that really something people think? If I were to express some sort of wave-function-in-spacetime theory, I'd invoke lots of classical fields filling space and have those wiggle. In any case, the whole bit about the proper two-particle wave function living in a higher dimensional space is somewhat spoiled by the fact that you can factorise it into normal 3-space pieces (so long as you don't have your particles interacting), it doesn't seem such an alien space to me.
  - tsimionescu 2y ago
    Before the wavefunction, we used to explain the double slit experiment (the version without detectors at a slit) as light being an EM wave in physical space, essentially equivalent to a sound wave propagating through the EM field, which breaks on the wall and essentially transforms into two separate waves, each originating from one slit, which are then in phase and so they constructively interfere, forming the final pattern on the screen. Lots of people think that this is the same picture that the wavefunction gives, but this is wrong. In the QM picture, the emitter emits one photon, which is a quantum of energy described by a four-dimensional wavefunction which assigns some probability of a detection event at the slits, at the screen, etc. In this picture, there is no physical EM wave, any interaction with the light will happen at a single localized point in space. Of course, if you add more particles, especially those carrying charges, the picture changes, and you'll see probabilities that roughly correspond to a picture of an oscillating EM field. But the wavefunction, which is the "bedrock" physical theory, is separate from those waves in the EM field, which are just an approximate picture of the probabilities dictated by the wavefunciton.
aap_ 2y ago
I morally agree, but not quite: think of the wave function as not more than a bookkeeping device. It does get the job done but be careful to ascribe it too high an ontological status! The path integral formulation seems a lot more natural to me and it does not need a wave function, instead you can derive it and treat it as a bookkeeping device. The way I think about it is that it's an attempt to deterministically model non-deterministic behavior: you "pretend" that the system is deterministic by keeping track of all the possible ways it could have evolved in time. sure enough, once you make a measurement this probability distribution "collapses" and you find out what is actually the case.
- deleted 2y ago
  [deleted]
- GoblinSlayer 2y ago
  It's not just pretending, deterministic model works, unlike nondeterministic.
- SiempreViernes 2y ago
  I think you are agreeing with my point that declaring the wave function to be mere bookkeeping is a poor foundation for writing about the interpretation of quantum mechanics? Can't really get any other sense out of your reply, but I'm not entirely sure.
  - aap_ 2y ago
    Also not sure I'm understanding you right :) My view is this: the wave function is mere bookkeeping and not anything ontologically fundamental. However the fact that such a seemingly bizarre concept lets you do quantum physics (even if it's not the only way) points to some fundamental questions about the nature of...well, nature. Of course this is not the only valid view...just one that makes sense to me. Thinking about these sorts of questions is a very fun endeavour.
    - SiempreViernes 2y ago