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I Expectation value of momentum for free particle

  1. Jun 14, 2016 #1
    Hello!
    Could somebody please tell me how i can compute the expectation value of the momentum in the case of a free particle(monochromatic wave)? When i take the integral, i get infinity, but i have seen somewhere that we know how much the particle's velocity is, so i thought that we can get it from the expectation value of its momentum.

    Also, Wikipedia states(here: https://en.wikipedia.org/wiki/Free_particle#Measurement_and_calculations -- just scroll down a bit) that one CAN do the calculation via the integral and get hbar*k.

    Thanks!
     
  2. jcsd
  3. Jun 14, 2016 #2

    A. Neumaier

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    The result depends on the state in which you compute the expectation. In a Gaussian state the result is finite.
     
  4. Jun 14, 2016 #3
    Hello! No, no, i am talking about a monochromatic wave(single k).
     
  5. Jun 14, 2016 #4

    A. Neumaier

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    For an unnormalizable wave function such as ##|k\rangle## you cannot talk about expectations, let alone compute them!
     
  6. Jun 14, 2016 #5
    So, why does Wikipedia(the link i have provided) state that it is hbar*k?
     
  7. Jun 14, 2016 #6

    A. Neumaier

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    You must calculate the expectation for wave packets (such as Gaussians), and then take the limit of infinite spread in position, and zero spread in momentum.
     
  8. Jun 14, 2016 #7
    Gaussian with infinity variance gives a plane wave or a constant(in space) function?
     
  9. Jun 14, 2016 #8

    A. Neumaier

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    ##\psi(x)=e^{\pm ik\cdot x-x^2/2\sigma^2}## gives (after normalization, and with the correct sign) a moving Gaussian wave packet with momentum ##\langle p\rangle=\hbar k## and approaches a plane wave in the limit ##\sigma\to\infty##
     
  10. Jun 14, 2016 #9
    Thanks!
     
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