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De Broglie wavelength for (near) stationary objects.

  1. Dec 3, 2006 #1
    1. The problem statement, all variables and given/known data
    I must be missing out on some fundamental part of quantum mechanics, since i'm unable to quite grasp the following thought experiment. A 1kg ball is confined in a onedimensional box with a length of 1m. What is it's maximal De Broglie wavelength?

    2. Relevant equations
    [tex]\lambda=h/p[/tex] (De Broglie Hypothesis)
    [tex]\Delta x \Delta p \ge \frac{h}{4\pi}[/tex] (Heisenberg uncertainty principle)

    3. The attempt at a solution

    So the way i see it, it's maximal de broglie wavelength will occur when the momentum of the ball is at it's minimum. The momentum cannot be precisely determined to be zero since the ball is confined in finite space. Since it's [tex]\Delta x = 1\textrm{m}[/tex], the minimum in momentum will occur at [tex]\Delta p = \frac{h}{4\pi}[/tex].

    The wavelength at this momentum is :
    [tex]\lambda=\frac{h}{h/(4\pi)}} = 12.57m[/tex]

    Now this wavelength is larger than the dimensions of the confining space :confused: . Could someone shed some light on this? I'm getting a macroscopically tangible wavelength for an object at a very slow speed. Overall, how should i interpret de broglie wavelengths for objects with an extremely small momentum?
    Last edited: Dec 3, 2006
  2. jcsd
  3. Dec 3, 2006 #2
    I think you're going about the question the wrong way, because the UP gives the uncertainty in p, not the value of p and hence lambda. You want to use the fact that the probability drops to zero outside of this region (assuming the potential goes to infinity there). Also, the UP is [itex]\Delta x \Delta p \geq \hbar/2[/itex] not less than or equal to.
  4. Dec 3, 2006 #3
    I thought that using the smallest [tex]\Delta p[/tex] as a value for p would be correct since i want the ball to be as stationary as possible. I can't assume that [tex]p = 0[/tex] since i've got a finite [tex]\Delta x[/tex].

    I suppose my problem is that I may not decrease p so that the De Broglie wavelength exceeds the spatial dimensions of the container? Correct?

    Anyways, even if the momentum of the particle is large enough to fit the wavelength into the container, the near stationary ball still has a macroscopically large wavelength, which i'm not sure how to interpret. Does it mean that we could observe wave-effects if we viewed the ball long enough?

    (fixed the typo with the '<')
    Last edited: Dec 3, 2006
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