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Is this an accurate explainantion of QM?

  1. Mar 12, 2010 #1
    A buddy of mine asked me to explain QM. I don't by any means tout myself as having any sort of worthwhile knowledge about it so I was a little surprised he asked me. But nevertheless, I explained it to the best of my ability. I'm hoping you guys can tell me if I've royally messed anything up here.

    In the short of it, envision one of these (cubic lattice):


    In philosophical terms, consider that the above is the matter (conf. materialism) that makes up the entire universe. Everything. The lattice is going to be used to be used to represent what the Copenhagen interpretation (the standard interpretation) calls the "wavefunction." Now imagine that fundamental particles (e.g., photons, electrons) can only be located at one of the points on the lattice. At a macroscopic level, things smooth out (e.g., a beam of light, which appears to travel in a straight line through spacetime at the speed of light), but on a microscopic level, particles are jumping from point to point, never existing in between. There is measurable time between the particle's jump to each point, but it's literally impossible to observe it during that time. The unique quality here is that, for all intents and purposes, their movement is "random." When we try to catch a particle "between" (again, it has fallen out of existence, so it's not "between" anything, in terms of the space part of spacetime) points, we fail. It necessarily jumps to a point. Now, it's most likely to jump to a point nearest to where it last existed (in fact, it's very predictable where the particle will go, but it cannot be deduced), but it can jump to any point on the lattice.

    Now, the above understanding (Copenhagen's interpretation) relies on Bell's theorem. A well known alternative, and at this point probably considered fringe, is Bohm's (not to be confused with Bohr) interpretation, which is what purports the so-called "hidden variable" theories that would bring the existence of randomness to a grinding halt. To better picture this, there is the famous thought experiment popularly known as "Schrodinger's Cat." In short, there is a chance that a cat that was put into a box is dead (that's a given in the experiment; Schrodinger explains how to yield an increasing probability of a dead cat). The only way to know for sure if the cat is actually alive or dead is to open the box and check it out, but we can't open the box because of the observer effect; when we do open the box, the cat is either alive or dead. Copenhagen interpretation (Bell's theorem) says that we must conclude that the cat is both alive and dead. Bohm interpretation (hidden variable) says that there is a way to figure it out for sure without opening the box, which if we're remaining analogous to the simplicity of the experiment, poses a problem for us. Rather, Bohm is wrong.​

    He went on to ask how the observer effect works, mechanically. I explained that Bell's theorem only actually applies to a closed system, so when we say that a particle falls out of existence, what we really mean is that it has wandered off somewhere, possibly another galaxy, and the term for all of this is "entanglement," and the theory that explains this is known as "quantum information."

    I'd appreciate any feedback if you have any. I don't want to give this guy wrong information, y'know?


    I should add that I've never taken a class or anything on it. That's just based on what I've read on my own. I'd imagine that this is not exactly an area well suited for "self-teaching." Anyway, please be gentle. I don't doubt that I've muddled some concepts somewhere in there.
    Last edited by a moderator: Apr 24, 2017
  2. jcsd
  3. Mar 12, 2010 #2


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    What you describe in your first paragraph sounds a lot more like energy-time uncertainty relation than anything to do with position. As far as we know, space-time isn't quantized, and therefore, the electron may take any value on a continuous space, it need not move on a lattice. In other words, in 1-D the electron may be at position x, or x+dx for any dx no matter how small.

    Energy is quantized, the electron may NOT take just any energy value E, or E+dE. It can only take quantized energy levels. Caveat: this is not quite true for a free particle which may assume a continuous level of energies.

    Lastly, I don't believe the Bohm interpretation says that in principle we could find out whether the cat is alive or dead without opening the box. The hidden variable is "hidden" from us! I don't believe the theory makes any definitive claims on whether we can ever know this hidden variable. It only says that this hidden variable is there, not that we can know it.

    So, according to the Bohm theory then, the cat is either alive or dead before we open the box. We may never know until we open the box, but the cat definitely is not in the superposition of states alive and dead.
  4. Mar 12, 2010 #3
    It is also about Zeno's paradox and there is only one possible solution as far as I see, which is that space and/or time is indeed quantized, or so to say "digital" rather than "analog". Just like Pac-man in its digital world defined by its "Planck scale" called PIXEL size. It can move only in those discrete and quantized steps, though it can still have various velocities and accelerations, its movement can still appear smooth, like you see in computer games and the smoother will appear the smaller is pixel size or "Planck size".

    Basically, if Achilles is to ever overrun the tortoise we must be living in a world made of "pixels" like a computer screen or that lattice in OP. Which does not mean that pixels themselves are not infinitely divisible after all, that still is most likely to be. Thought, we might have a chances to learn about what is beyond this Planck scale of ours, as much as Pac-man has a chance to get out of the computer screen.

    When looking at QM, it is really a question whether trajectories are continuous and the world seem quantized because of underlying grid, or is space-time indeed analog and continuous and that what is quantized is matter itself. I think we would not really be able to see the difference as long as this "pixel size" is too small to "see" and take apart.
  5. Mar 12, 2010 #4


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    Zeno's paradox hasn't really been an issue since integral calculus came about. We can handle adding infinitesimal intervals pretty well now...
  6. Mar 12, 2010 #5
    I'm sorry, but this description has very little to do with quantum mechanics. It's a very confused explanation. At this level, you can begin with wikipedia
  7. Mar 13, 2010 #6
    It is not an issue, never was. Zeno too knew what would actually happen, with the concept of 'velocity', even without integral calculus, it was self-evident, hence "paradox". Integral calculus does not address the problem at all, it actually makes it more clear if you try to associate those mathematical variables with the real world properties.

    Just think about it as Zeno describes it, in the name of logic I say there is no other possible solution as to how Achilles can overrun the tortoise but that the space is quantized. [EDIT: time may stay continuous independently.] I'm not talking related to QM, but surely this paradox must have some connection with it, though it is really hard say for me if QM can provide any insights about it at all.
    Last edited: Mar 13, 2010
  8. Mar 13, 2010 #7


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    Zeno presented his paradox to argue that motion was and illusion. He argued that because of his paradoxes, motion is absurd. He therefore argued that the correct world view was in oneness; that plurality and change were illusions.

    Zeno's paradox is quite philosophical. We CAN, in fact, think of space-time as continuous without dealing with his paradox. Even though there's infinite halves within some distance, each time you halve the distance you halve the time required and in the end you end up with infinite halves that require infinitesimal time. You can do this kind of addition in Calculus, that's why I bring it up.

    Zeno's paradox, in no way, assures us that space-time is discrete.
  9. Mar 13, 2010 #8
    Yes, that's funny bit, however it has no direct implication. Some think QM predicts parallel universes and I accept all theories, but we need to decide which one makes most sense.

    More philosophical than the implications of Quantum Mechanics? It is philosophical because it questions the deepest and most essential aspects of the reality itself, just like QM. There is a solution for Zeno's "arrow paradox" explained by relativistic effect of length contraction, so I do hope QM might be able to say something about this one as it concern "micro-cosmos" aka quantum mechanics.

    What end? What "bottom" did we hit, "infinitesimal time"?

    There you did it yourself, you quantized the time! But, you have not solved the paradox as you are still left with the "infinite halves" of space. Are you suggesting Achilles should catch up with the tortoise "between" the "real" time segments?

    Define "infinitesimal time" and you will see you are talking about Planck scale and "grains of time", and again quantization of the time alone does not solve the paradox, you have to have the space to be made of "pixels" or "voxels", "grid"... whatever you call it, the point is humans exist and move in the world where there is FINITE number of "distance segments" between you and me or anyone else, and the size of these segments might very well be what we already defined by 'Planck scale', so in a way humanity already accepted the space (and time) is quantized and resolved Zeno's paradox by actually defining the size of these "pixels".

    - "The Planck length is related to Planck energy by the uncertainty principle. At this scale, the concepts of size and distance break down, as quantum indeterminacy becomes virtually absolute."
  10. Mar 13, 2010 #9
    To varga:

    Zeno's paradox has been solved for centuries. You clearly don't know/understand
    the concept of a convergent series. There's little anybody can do to educate you
    in a forum; that you have to do by yourself.

    I'll ask you a question: take the sum 1 + 1/2 + 1/4 + 1/8 + ... + 1/2^n, with
    n some nonnegative integer; what do you think happens to the sum when n grows
    arbitrarily large?
    Last edited: Mar 13, 2010
  11. Mar 13, 2010 #10


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    The quantization of space-time may or may not be true. I do not pretend to know. The article you showed regarding the Planck scale shows us, in fact, how little we know of matters.

    However, my point is only that Zeno's paradox is not sufficient to prompt us to assume quantized space-time.
  12. Mar 15, 2010 #11
    That I can agree with, though I'd prefer to continue this argument. I think QM actually does have solution to this paradox too, in a way to suggest that matter itself does not travel continuous trajectories but appears-disappears, makes *jumps*, that solves the paradox as well.


    Everything is fine when n grows arbitrarily large, as long as it is not INFINITELY large.

    I'll ask you a question too: how do you explain anything can move from point A to B if there is infinite number of "half-distances", if n=infinity?
    Last edited: Mar 15, 2010
  13. Mar 15, 2010 #12
    @Varga: You might find the book "Prime Obsession" useful for some of the mathematical background on series and sets, all the way through matrices.
  14. Mar 15, 2010 #13


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    Why? What happens when it grows infinitely large?

    The distance to move is finite, equal to the length d, of segment AB in meters. Set the particle moving with velocity v=1 m/s. It will move through the distance AB in a time interval equal to d seconds. See, everything hinges on the distance to move being finite ... that is why the convergent series is important. You can also break it up into the time intervals required to travel through each half-interval ... that series is also convergent, showing that the time taken to traverse the entire interval is finite.
  15. Mar 15, 2010 #14
    First i think its correct to say if spacetime is quantized Zeno's Paradox (Dichotomy,Achilles) dissapears.
    For me thats quite a compelling solution.

    And what if you change the paradox a bit ?
    Achilles does not approach in 1/2,1/4...1/2^n
    but in 1/(rnd*10)+1/(rnd*100) ... + 1/(rnd*10^n) rnd = random number 1-9

    anyway this explains much better what i mean:

    "there's a tradition among some high school calculus teachers to present them as "Zeno's Paradox", and then "resolve the paradox" by pointing out that an infinite series can have a finite sum. This may be a useful pedagogical device for beginning calculus students, but it misses an interesting and important philosophical point implied by Zeno's arguments."
    Last edited: Mar 15, 2010
  16. Mar 15, 2010 #15
    That's not how QM works.

    As n grows infinitely large, the sum converges to a finite number, 2.
    IOW: 1 + 1/2 + 1/4 + ... = 2.

    Now, your question:

    We have an arrow moving in a straight line with speed v going from A to B. Let's
    denote by A1 the point right in the middle:

    A ..... A1 ..... B

    Let d = (distance between A and B) = d(A, B); then, d(A, A1) = d/2;
    Let t1 = time for the arrow to arrive at A1: t1 = d(A, A1) / v = 1/2 d/v;

    Now, let A2 be the point halfway between A1 and B:

    A1 ..... A2 ..... B

    The time for the arrow to cover the distance between A1 and A2 will be

    t2 = d(A1, A2) / v;

    Now, d(A1, A2) = 1/2 d(A, A1) = 1/2 d/2 = d/4; therefore:

    t2 = 1/4 d/v;

    After n steps we have:

    t1 = 1/2 d/v
    t2 = 1/4 d/v = t1/2
    t3 = 1/8 d/v = t1/2^2
    tn = 1/2^n d/v = t1/2^(n-1)

    So, after n steps, the total accumulated flight time will amount to

    t1 + t2 + ... + tn = t1 + t1/2 + ... + t1/2^(n-1) = t1[1 + 1/2 + ... + 1/2^(n-1)]

    After an *infinite* number of steps, the total flight time will be:

    t = t1 (1 + 1/2 + 1/4 + ...).

    The quantity between the parentheses is the same infinite sum I wrote above, which
    is equal to 2. So,

    t = 2 t1 = 2 1/2 d/v = d/v,

    as it should be.
    Last edited: Mar 15, 2010
  17. Mar 15, 2010 #16


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    If it's a philosophical point you are making, then fine. But what I'm saying is that in the context of current physics and mathematics, Zeno's paradox does not prompt us to dig any further than integral calculus and infinite series.
  18. Mar 15, 2010 #17
    It is rather not a "philosophical point".
    I think infinite series do not solve Zeno's Paradox in all aspects.
  19. Mar 16, 2010 #18
    Who are you, then?
  20. Jul 21, 2010 #19
    I agee. Zeno's point was about logic not being sufficient by itself. Reason must temper logical thought.
  21. Jul 21, 2010 #20
    No. Hidden variables theories suggest that we don't understand qm fully right now. If we did, we would know why the cats survival or demise was determined (by something), and not as random and inexplicable as the laid back Copenhagen supporters are, or as plain crazy as the many worlds supporters clearly are.

    There is no one less true to science than one who insists on many universes when we only have evidence for one, apart from those who believe that the entire universe splits on each quantum interaction. Who let these crazies in ?

    The scientific fundamentalists seem even less grounded in reason than the religious fundamentalists. That is a rediculous state of affairs, perhaps exagerated, but a serious and sad reflection on the nature of modern science.

    Einstein predicted this travesty
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