Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Uncertainty principle and double-slit interference?

  1. Mar 19, 2008 #1
    Both uncertainty principle and double-slit interference are the most important contents in Quantum mechanics. Those days I am thinking about this question: Is there a hidden relation between them? Unfortunately, I couldn't give me a satisfied answer.
    How do you think about it?
    Thank you!
     
    Last edited: Mar 19, 2008
  2. jcsd
  3. Mar 19, 2008 #2
    What do you mean by a hidden relation? Double slit interference, like most quantum phenomina, is a consequence of the uncertainty principle.
     
  4. Mar 20, 2008 #3
    xylai, I'm replying to your private message here in the hopes that someone with more experience might chime in with a better answer, but to answer your request for more explanation:

    The uncertainty principle tells us that you cannot know the value of two non-commuting observables (such as position and momentum) with perfect accuracy; the more you know one, the less certain the other becomes. In the double slit experiment, when a photon passes through the double slit we know two possible, very precise, positions for that photon: the left slit or the right slit. Those two very precise position possibilities lead to great uncertainty in momentum.

    In a single-slit case, this results in a wide gaussian (bell-curve-like) distribution for the photons. In the double-slit case, these two uncertainties interfere with one another to create the beloved interference pattern. But in both cases, the pattern results from the uncertainty in momentum resulting from the certainty (albiet, narrowed down to two possibilities in the double slit case) in position. If not for the uncertainty principle, the double slit experiment would just result in two, well defined bright spots behind the slits.

    Why do the two uncertainties in a double slit not simply add up to one, wide blob of uncertainty? That has to do with the Schrodinger Equation and the fact that the wave function represents probability amplutides, not probabilities themselves. The probabilities are derived by taking the complex conjugate of the wave function, thereby eliminating the imaginary component. But in the case of a double slit, the phase difference in the photons going through the two different slits is reflected in the wave function before the complex conjugate is taken (and thereby the actual probability is derived). So that when you do find the probability density function for the two-slit setup, the phase difference between the photons from each slit is reflected in the probability density function as an interference pattern and, sure enough, that's what we see when we do the experiment.

    If someone else wants to correct anything I said or explain it more clearly be my guest. :)
     
  5. Mar 20, 2008 #4

    Ken G

    User Avatar
    Gold Member

    That all sounds good to me, though it might not be necessary to think so mathematically if you are not used to complex numbers. I might add that in a sense the "deeper source" to both the uncertainty principle and the interference pattern is just wave mechanics. The only time you need quantum mechanics is if you want to ask what a single quantum will do, but if you just want to understand the overall pattern and think of any spreading as due to "uncertainty", then the whole thing also works with water waves going between two breaks in a jetty. In that case the uncertainty principle comes from the way every excitation of the water can serve as a source of new waves, and those new waves are seen to propagate out unless they destructively interfere (which they do quite a lot if the wavelength is short). That same destructive interference causes the pattern you see.

    This connects to quantum mechanics when we connect, as peter0302 said, the wave amplitudes you see on the water with probabilities of the quantum showing up in those places.
     
  6. Mar 21, 2008 #5
    Can someone confirm something that's been bothering me?

    For double-slit interference to be visible to the naked eye, the light source needs to be a laser, or something collimated, correct? If you try it with a flashlight, you'll just get a blob on the other end, right?

    Can we equate the collimated nature of laser light with a well-defined momentum which becomes ill-defined as soon as we simultaneously try to isolate it to a well-defined position (aka the slits)?
     
  7. Mar 21, 2008 #6

    DrChinese

    User Avatar
    Science Advisor
    Gold Member

    There is a different reason that non-coherent light does not generate a double slit pattern when it is something like a flashlight. The beam is not in phase and so there is a tremendous amount of destructive interference. The fringe pattern is cancelled out. If you could send those same photons through the double slit apparatus a photon at a time, the pattern would emerge.
     
  8. Mar 22, 2008 #7

    Ken G

    User Avatar
    Gold Member

    Not to be combative, but it's not a question of phase, I would say, it's bandwidth. Lasers have coherent phase, but that's not necessary for a two-slit pattern, because the location of the nulls doesn't depend on the initial phase anyway. In other words, you could superimpose a thousand different laser beams and you'd still get the same pattern-- when they all produce nulls individually, they will also do that in tandem. But if the bandwidth is not tight, you are superimposing different wavelengths that do have nulls at different places, so you won't get the pattern. In short, you need a specific color to the beam, even sharper than the eye can discern. That holds even if you send the photons through one at a time, so the flashlight wouldn't work even at such a low intensity.
     
    Last edited: Mar 22, 2008
  9. Mar 22, 2008 #8
    You will have an interference pattern (but much more complicated) even with many colours, provided there is a phase relation between all the various colours.
     
  10. Mar 22, 2008 #9

    Ken G

    User Avatar
    Gold Member

    Why would there be a phase relation between the different colors? Perhaps a multicolored hologram? The OPer might think you are saying a flashlight would have such a relation.
     
  11. Mar 22, 2008 #10
    Actually what I was getting at was simply whether the light needed to be parellel, and if that had anything to do with a "certain" momentum translating into an "uncertain" position when detected.
     
  12. Mar 22, 2008 #11

    Ken G

    User Avatar
    Gold Member

    Yes, I was referring only to what DrChinese said. You are correct that the light being parallel connects to the pattern we see (and here lightarrow's comments are again relevant-- nonparallel yet patterned sources could yield their own different and possibly more complicated patterns). Also, the overall spreading of the diffraction pattern, regardless of whether one gets fringes or not, is indeed an example of the uncertainty principle. But I would have said it the other way-- knowledge about the lateral positioning as the light exits the slits translates into uncertainty in the lateral momentum, which in turn leads to spreading of the final position.
     
    Last edited: Mar 22, 2008
  13. Mar 22, 2008 #12
    Actually I think you can generate interference patters with white light. From red to blue is a narrow enough bandwidth provided your source is small enough and far enough away to get out of the near field range and into a far field arrangement. One example; sunlight from the sun is far enough away but the sun is too wide as a source making it a far field source. But in an eclipse during the phase when there is just thin crescent of light, I’ve seen walking interference patterns vibrating within the sunlight on the ground during a eclipse at the ABC islands a several years ago. That’s why they recommend laying out a white sheet so you can see the effect if the sky is clear enough.

    I think there are other more down to earth examples of white light interference patterns as well.
     
  14. Mar 22, 2008 #13

    Ken G

    User Avatar
    Gold Member

    That's interesting, the bandwidth is about the same as the frequency, for sunlight, so the variance in the k vector is as large as the k vector itself. To get significant diffraction, one needs a large dot product between the k vector and the slit width, but to get fringes, you also need a small dot product between the variance in k and the slit width. So you can probably use sunlight if you are willing to compromise the spreading of the pattern, but the traditional "two-slit pattern" is probably not possible, I would expect.
     
  15. Mar 22, 2008 #14
    What you mean like Young's slit experiment, I'm pretty sure he wasn't using lasers. :smile:
     
  16. Mar 22, 2008 #15
    I was thinking the same thing. The electrons in Young's experiment were neither collimated nor in coherent phase. They were pretty random as I recall. So I think Dr.Chinese is right that if a flashlight emitted one photon at a time, (which, by definition, would be one precise color at a time) you'd get the pattern.

    Which is why I continue to be astounded by the fact that you don't get the pattern from entangled photons unless you take affirmative steps to destroy the position certainty of the twin.
     
  17. Mar 22, 2008 #16
    Just to be sure are you talking about the double slit quantum eraser experiment?

    You would it is called the Feynman double slit experiment which produces interference patterns when you fire one photon at a time.
     
    Last edited: Mar 22, 2008
  18. Mar 22, 2008 #17
    I am indeed. That and the Dopfer thesis as well.
     
  19. Mar 22, 2008 #18
    https://www.physicsforums.com/showthread.php?t=158413&highlight=double+slit+eraser

    Yeah its been done here and its explainable in quantum theory.

    http://arxiv.org/PS_cache/quant-ph/pdf/9903/9903047v1.pdf

    Still very interesting though.

    https://www.physicsforums.com/showthread.php?t=106070

    Mathematically, it turns out that the integral of the wave is well explained by theory. So the behaviour of the entangled photons is exactly what we would expect, given the conditions in the experiment.

    It's much more adequately explained on those links and in much better detail.

     
    Last edited: Mar 22, 2008
  20. Mar 22, 2008 #19
    Oh, I know what the math and quantum theory say. I just still find it remarkable. :)
     
  21. Mar 22, 2008 #20

    Ken G

    User Avatar
    Gold Member

    Young did use a coherent source of light, though not a laser:
    http://micro.magnet.fsu.edu/primer/java/interference/doubleslit/ says
    "In order to test his hypothesis, Young devised an ingenious experiment. Using sunlight diffracted through a small slit as a source of coherent illumination, he projected the light rays emanating from the slit onto another screen containing two slits placed side by side."
    As I said, the phase relation in a laser is irrelevant to the interference pattern, but the sharp color and the collimated direction are very relevant. As for entangled photons, the behavior you cite is exactly what quantum mechanics predicts. It would be more astounding if our macroscopic intuition trumped quantum mechanics in that single application but not all the other surprising things particles do.
     
    Last edited: Mar 22, 2008
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Uncertainty principle and double-slit interference?
Loading...