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Free particle in one dimension

  1. May 23, 2014 #1
    Hi,
    I´m not sure if my way of tackling a question, probably it's a trivial problem, but it's important for me to get it right so any help will be greeted.
    The question is as follows:

    Problem: consider a particle in a one-dimensional system. The wave function ψ(x) is as follows:
    ψ(x)= 0 for (-∞<x<0),
    ψ(x)= 1/√a for (0<x<a)
    ψ(x)= 0 for (a<x<∞)

    i) if the kinetic energy is measured what is the most probable value?
    ii) which values of the momentum can never be found when measuring it?

    My reasoning:
    i) I use the operator for kinetic energy: K= -[itex]\frac{h^2}{2m}[/itex][itex]\frac{∂^2}{∂x^2}[/itex]
    which when applied: ∫ψ(x)*Kψ(x)dx gives me zero.
    If this was right I assume the most probable value of the kinetic energy is zero.
    ii) I have no clue whatsoever what the question means.

    Thanks and forgive my bad english, regards from Spain!
     
  2. jcsd
  3. May 23, 2014 #2

    DrClaude

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    Staff: Mentor

    Hi angass, welcome to PF!

    Is that correct? Because this is not a continuous function, and therefore not a valid wave function.
     
  4. May 27, 2014 #3
    This looks suspiciously like a griffiths or leibowitz homework problem....

    You want to take express psi(x) in a fourier basis (as a sum of e^i(kx-wt)). Doing the fourier transform will be a simple integral from 0 to a with 1/a^1/2 as a constant.

    i) You're forgetting the edges at 0 and a. The derivative is not zero there. Once you express this function as fourier expansion, you'll see you don't get zero anymore.

    ii) apply the definition of p = i hbar d/dx to the fourier transform. The answer should drop out. I expect some of the coefficients for specific k terms in the fourier transform will be zero. (the fourier basis is an eigen value of the momentum and kinetic energy operator)
     
  5. May 28, 2014 #4

    bhobba

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    Science Advisor
    Gold Member

    Same here mate.

    To the OP if it is we have a homework section.

    But if it isn't - Google is your friend eg:
    http://www.colorado.edu/physics/TZD/PageProofs1/TAYL07-203-247.I.pdf [Broken]

    Thanks
    Bill
     
    Last edited by a moderator: May 6, 2017
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