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How to determine a Limits of Integration of Wave Packet

  1. Nov 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider a force-free particle of mass m described, at an instant of time t = 0, by
    the following wave packet:
    [itex]
    \begin{array}{l}
    0 \ \mathrm{for} \ |x| > a + \epsilon \\
    A \ \mathrm{for} \ |x| ≤ a \\
    -\frac{A}{\epsilon} (x − a − \epsilon) \ \mathrm{for} \ a < x ≤ a + \epsilon \\
    \frac{A}{\epsilon}(x + a + \epsilon) \ \mathrm{for} \ − a − \epsilon ≤ x < a \\
    \end{array}
    [/itex]
    where a, ε, and a normalization constant A are all positive numbers. Calculate mean
    values and variances of the position and momentum operators [itex] x , x^{2} , \sigma_{x} \ and \ p_{x}[/itex] .



    2. Relevant equations
    [itex]
    1=\int_{-\infty}^{\infty} |\psi(x,t)|^{2}
    [/itex]

    3. The attempt at a solution
    I want to determine normalization constant A. I don't know what kind of integration limits i should use for the case:
    [itex] A \ \mathrm{for} \ |x| ≤ a [/itex].
    Do you have any ideas ? Thanks in advance !
     
  2. jcsd
  3. Nov 11, 2011 #2

    dextercioby

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    Science Advisor
    Homework Helper

    You have a lot of intervals and need to use the fact that the integral will be a sum of integrals for each interval. So write all the integrals, compute them all the find A. Then compute all other items.
     
  4. Nov 11, 2011 #3
    Ok , so far i have:
    [itex]\int_{??}^{??} A^{2} dx (?) -\frac{A^{2}}{\epsilon^{2}}\int_{a}^{a+\epsilon} (x-a-\epsilon)^{2} dx +\frac{A^{2}}{\epsilon^{2}}\int_{-a-\epsilon}^{a} (x+a+\epsilon)^{2} dx =1.[/itex] My question is what are the integration limits for the first integral ?
     
  5. Nov 11, 2011 #4

    dextercioby

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    Homework Helper

    Where did you pick the first one from ? Is there an interval where the wavefunction is 1 ?
     
  6. Nov 12, 2011 #5
    So, should i include first integral ? That was my problem/question ?
     
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