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Fascinating. May I ask, how does one predict the behavior of an electron without using probability amplitudes? I'm extremely interested, and the question isn'

Fascinating. May I ask, how does one predict the behavior of an electron without using probability amplitudes? I'm extremely interested, and the question isn't meant as anything but an inquiry. Finding a gap in one's knowledge is one of the more exciting aspects of life.

loup-vaillant 11y ago
First, you need to stop calling them "probability amplitudes". Amplitudes are complex numbers, out there in the real world (as far as we understand it). This is quite different from probabilities, who are real numbers between 0 and 1 who exist only as a mental constructs. Simply put, while amplitudes share some mathematical properties with probabilities, they are not probabilities. Now that we're done turning colloquial words into misleading jargon, we can talk about the theory itself. Namely Everett's many-worlds and decoherence. We could play the "where's the electron" game, but I'd rather play "where's the photon" instead —it's simpler. So you throw a photon through a half-sieved mirror, to be detected by one of to judiciously placed detectors. Oh, and have one detector linked to a kitten murdering system for good measure. If you repeat the experiment often enough, you will witness a kitten death half the time, with absolutely no way to predict the outcome in advance. The results are the same if you put the kitten in a box, and open that box after the fact. So it certainly looks like the universe is not deterministic. The equations on the other hand are definitely deterministic. Future amplitude distributions are perfectly predicted by past amplitude distributions —which by the way you can't fully observe, but that's another issue entirely. So, if you look at the amplitude distribution, you'll see that once the photon hit the mirror, there will be a blob of amplitude for both cases: passing through and being reflected. Going further, there will be a blob of amplitude for each of the detectors being hit. Finally, there will be a blob of amplitude for the living kitten, and another one for the dead kitten. Oh, and the equations also says that the blobs quickly cease to interact —that's decoherence. Basically, what the equations say is that the universe splits itself in two, generating one version with the dead kitten, and one version with the live kitten. The equations also say that the inhabitants of either version don't get to see the other one (they've ceased to interact). Now the only question left is why we experimentally find ourselves to be in one version and not in another. But never forget that every time you run that Schrödinger experiment, a cat will die. If not in your universe, then the other. --- The Copenhagen interpretation, which would have the blob of amplitude corresponding to the other universe just collapse into nothingness (that is, set to zero), has no basis in the equations which by now are backed up by mountains of evidence. It is an additional hypothesis layered on top of the equations, conveniently formulated in a way that wouldn't falsify any experiment. On top of that, it violates a number of long standing principles, such as locality. You could also call those amplitudes "probabilities", but that's just a word trick. It doesn't explain anything.
- wpietri 11y ago
  I so appreciate this explanation, as I have wondered about this for years. Thanks! My understanding of (real) probabilities is that they're really about our ignorance of a system. E.g., if I'm flipping a coin but put very little spin on it, it's easy for a human to predict what side will show. As we put more spin on it, the probability of guessing correctly goes down, but that's about perceptual and cognitive limitations. My (layman's) understanding of quantum mechanics, though, is that there are no hidden variables. In which case, I couldn't understand why anybody would call the quantum stuff "probability". It's a relief to know that they're fundamentally different.
  - ccvannorman 11y ago
    I read once that you can look at QM without randomness by comparing it to interference waves, similar to what happens if you sprinkle water on the surface of a pond -- the result looks chaotic but it is 100% deterministic, and QM can be viewed in this way too.
- jessaustin 11y ago
  It's fun that wave-function reduction is "an additional hypothesis layered on top of the equations, conveniently formulated in a way that wouldn't falsify any experiment", while infinities of universes that have "ceased to interact" is not.
  - loup-vaillant 11y ago
    Indeed. And it doesn't help that Many-Worlds, despite being much more reasonable than any collapse hypothesis, came decades after the Copenhagen interpretation. Science tends to reject theories that don't make additional predictions. http://lesswrong.com/lw/qa/the_dilemma_science_or_bayes/ http://lesswrong.com/lw/qa/the_dilemma_science_or_bayes/ But the fact remains that blindly following the equations leads you to Many-Words. You have to modify the results to get to any sort of collapse hypotheses. Also, simply postulating that what you don't see doesn't exist doesn't help. http://lesswrong.com/lw/pb/belief_in_the_implied_invisible/ http://lesswrong.com/lw/pb/belief_in_the_implied_invisible/

loup-vaillant 11y ago

First, you need to stop calling them "probability amplitudes". Amplitudes are complex numbers, out there in the real world (as far as we understand it). This is quite different from probabilities, who are real numbers between 0 and 1 who exist only as a mental constructs. Simply put, while amplitudes share some mathematical properties with probabilities, they are not probabilities. Now that we're done turning colloquial words into misleading jargon, we can talk about the theory itself. Namely Everett's many-worlds and decoherence. We could play the "where's the electron" game, but I'd rather play "where's the photon" instead —it's simpler. So you throw a photon through a half-sieved mirror, to be detected by one of to judiciously placed detectors. Oh, and have one detector linked to a kitten murdering system for good measure. If you repeat the experiment often enough, you will witness a kitten death half the time, with absolutely no way to predict the outcome in advance. The results are the same if you put the kitten in a box, and open that box after the fact. So it certainly looks like the universe is not deterministic. The equations on the other hand are definitely deterministic. Future amplitude distributions are perfectly predicted by past amplitude distributions —which by the way you can't fully observe, but that's another issue entirely. So, if you look at the amplitude distribution, you'll see that once the photon hit the mirror, there will be a blob of amplitude for both cases: passing through and being reflected. Going further, there will be a blob of amplitude for each of the detectors being hit. Finally, there will be a blob of amplitude for the living kitten, and another one for the dead kitten. Oh, and the equations also says that the blobs quickly cease to interact —that's decoherence. Basically, what the equations say is that the universe splits itself in two, generating one version with the dead kitten, and one version with the live kitten. The equations also say that the inhabitants of either version don't get to see the other one (they've ceased to interact). Now the only question left is why we experimentally find ourselves to be in one version and not in another. But never forget that every time you run that Schrödinger experiment, a cat will die. If not in your universe, then the other. --- The Copenhagen interpretation, which would have the blob of amplitude corresponding to the other universe just collapse into nothingness (that is, set to zero), has no basis in the equations which by now are backed up by mountains of evidence. It is an additional hypothesis layered on top of the equations, conveniently formulated in a way that wouldn't falsify any experiment. On top of that, it violates a number of long standing principles, such as locality. You could also call those amplitudes "probabilities", but that's just a word trick. It doesn't explain anything.

wpietri 11y ago
I so appreciate this explanation, as I have wondered about this for years. Thanks! My understanding of (real) probabilities is that they're really about our ignorance of a system. E.g., if I'm flipping a coin but put very little spin on it, it's easy for a human to predict what side will show. As we put more spin on it, the probability of guessing correctly goes down, but that's about perceptual and cognitive limitations. My (layman's) understanding of quantum mechanics, though, is that there are no hidden variables. In which case, I couldn't understand why anybody would call the quantum stuff "probability". It's a relief to know that they're fundamentally different.
- ccvannorman 11y ago
  I read once that you can look at QM without randomness by comparing it to interference waves, similar to what happens if you sprinkle water on the surface of a pond -- the result looks chaotic but it is 100% deterministic, and QM can be viewed in this way too.
jessaustin 11y ago
It's fun that wave-function reduction is "an additional hypothesis layered on top of the equations, conveniently formulated in a way that wouldn't falsify any experiment", while infinities of universes that have "ceased to interact" is not.
- loup-vaillant 11y ago
  Indeed. And it doesn't help that Many-Worlds, despite being much more reasonable than any collapse hypothesis, came decades after the Copenhagen interpretation. Science tends to reject theories that don't make additional predictions. http://lesswrong.com/lw/qa/the_dilemma_science_or_bayes/ http://lesswrong.com/lw/qa/the_dilemma_science_or_bayes/ But the fact remains that blindly following the equations leads you to Many-Words. You have to modify the results to get to any sort of collapse hypotheses. Also, simply postulating that what you don't see doesn't exist doesn't help. http://lesswrong.com/lw/pb/belief_in_the_implied_invisible/ http://lesswrong.com/lw/pb/belief_in_the_implied_invisible/