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QM probability and normal distribution

  1. Oct 4, 2015 #1
    Is there a relationship between QM probability and normal distribution ?

    I'm thinking about drawing probability densities as functions of phase.

    Thanks
     
  2. jcsd
  3. Oct 4, 2015 #2

    mfb

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    You can find things that follow a normal distribution in QM, but I would not call this "relationship between QM and normal distribution" in the same way I don't see a "relationship between the normal distribution and the number 3".
     
  4. Oct 4, 2015 #3
    You dropped out an essential word which was "probability". I'm looking for a mathematical relationship to calculate QM probabilities from probability densities.
     
  5. Oct 4, 2015 #4

    mfb

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    Probability densitites are used in quantum mechanics, but quantum mechanics is more than pure probability theory.
     
  6. Oct 4, 2015 #5
    That's why I said QM probability.
     
  7. Oct 4, 2015 #6

    Simon Phoenix

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    Hi Forcefield, I'm really not at all sure what you're trying to get at here.

    The probability function you get depends on what observable you're trying to measure - and the state of the system. Suppose we prepared a single qubit (a spin-1/2 particle say) in an eigenstate of the spin-z operator, and let's suppose we prepared the spin up state. Then, assuming ideal measurements, the probability of getting spin up in a measurement of spin-z is 1, and the probability of getting spin down is 0. Not really very Gaussian :-)

    Now suppose we measure spin-x, then the probability of getting a spin up result is now 1/2 and the probability of getting a spin down result is 1/2. A uniformly random distribution.

    Let's take some more examples - if we have a coherent state of light and make a measurement of photon number - then we'll get a Poisson distribution (in many runs of the same experiments, of course). Measurement of the field quadrature operator of the same coherent state will give us a Gaussian (if I recall correctly). Take a photon number state of the EM field and measure it's phase - you'll get a uniformly random distribution.

    So the probabilities depend crucially on what property we're choosing to measure and what state the system is in.

    Could you perhaps be a bit more specific because your question doesn't make a lot of sense to me.
     
  8. Oct 4, 2015 #7
    Hi Simon,

    Actually I think it is Gaussian but the variance is zero.

    I think that makes it Gaussian too just like throwing coins. It is just difficult to see because there are only two options.

    I have only a vague idea what you are talking about here. Remember I marked this thread high school level.

    You know there is this so called phase that you can throw away when you calculate probabilities in QM. I am thinking about whether it is possible to calculate QM probabilities by assuming that the measurement probability changes like normal distribution as time goes by.
     
  9. Oct 4, 2015 #8

    blue_leaf77

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    Given ##\psi(x,t)##, you want ##|\psi(x,t)|^2## at fixed position ##x=x_0## to follow Gaussian profile with time, is that what you mean?
     
  10. Oct 4, 2015 #9

    Simon Phoenix

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    You're talking here about a global phase factor - you certainly can't ignore the relative phases between terms in a superposition when calculating probabilities. Recall that to calculate a probability in QM we're taking the square modulus of sums of complex numbers.

    Well only in the sense that for sufficiently large numbers of trials then the Gaussian is a good approximation to the binomial (I think the rule of thumb is that np ≥ 5 where n is the number of trials and p is the probability)
     
  11. Oct 4, 2015 #10

    mfb

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    Only phases in the basis where you want to calculate the probabilities, at the time when you want to calculate them.
    No.
     
  12. Oct 4, 2015 #11
    No, I'm not considering wave functions. Actually I haven't studied them seriously yet.
     
  13. Oct 4, 2015 #12
    Yes, I thought about that and it looks like a relative phase of pi/2 is required.
     
  14. Oct 4, 2015 #13

    mfb

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    The relative phase in a superposition can be everything (well, as it is a phase, it is usually restricted to the range from 0 to 2pi).
     
  15. Oct 4, 2015 #14

    blue_leaf77

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    The only entity which connects QM with probabilistic interpretation is the wavefunction.
     
  16. Oct 4, 2015 #15
    I think one should consider a period of π here.
     
  17. Oct 4, 2015 #16

    mfb

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    That does not make sense.
     
  18. Oct 5, 2015 #17
    Yes, I was sloppy there. The length of the period doesn't really matter but if you look at the shape of normal distribution it is more like half-circle than full-circle.
     
  19. Oct 5, 2015 #18
    Sorry, but that made even less sense.
     
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