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Energy State Probability Particle in a Box

  1. Sep 21, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that the probability of obtaining En for a particle in a box with wave function

    [itex]\Psi[/itex](x) = [itex]\sqrt{\frac{30}{L^{5}}}[/itex](x)(L-x) for 0 < x < L

    and [itex]\Psi[/itex](x) = 0 for everywhere else

    is given by

    |cn|2 = 240/(n6[itex]\pi[/itex]6)[1-(-1)n]2

    2. Relevant equations

    cn = [itex]\int[/itex][itex]\psi[/itex][itex]^{*}_{n}[/itex][itex]\Psi[/itex](x)dx

    The probability is cn squared.

    Shouldn't have to use eigenvalues and eigenfunctions.

    3. The attempt at a solution

    I used the integral from (2) and used the given uppercase Psi and used the sqrt(2/L)sin(n*pi*x/L) lowercase psi (conjugate), from 0 to L.

    The integral quickly turned messy with integration by parts and such.

    I would like to know if I am on the right track here... If I am, I will just work through the integral until I get the right answer.

    I'm hoping there is a much easier way to do this.


    P.S. Sorry this is a repost from another section. I wasn't getting any responses from the Intro Physics section so thought I would try here.
  2. jcsd
  3. Sep 21, 2011 #2


    User Avatar
    Homework Helper

    Yep, you're on the right track. Yes, the integral takes a little bit of work, but it is of the form
    [tex]\int(\text{polynomial in }u)\sin u\;\mathrm{d}u[/tex]
    and those integrals are doable.
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