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To find the expectation value of momentum

  1. Apr 13, 2012 #1
    1. The problem statement, all variables and given/known data

    At time t=0 a particle is described by a one dimensional wavefunction
    (capital)ψ(x,0)= (2a/)^(1/4) e^(-ikx)e^(-ax^2)
    (three lines)=(2a/)^(1/4) e^(-ikx-ax^2)--------equation 1
    k and a are real positive constants

    2. Relevant equations

    I think this is the one
    <p subscript(x)> = ∫(from-inf to inf)[(capitalψ)*(x,t)(-i(hbar)(∂/∂x))capitalψ(x,t)]dx
    for normalization ∫from-inf to inf e^-x^2 dx =√π
    and ∫from-inf to inf e^-ibx dx=√π *e^(-b^2/4) where b is a real constant



    3. The attempt at a solution

    ----
    I verified that equation 1 is normalized
    by completing the square and letting t=u/a I then integrate upon which I arrive at √(π/a) *e^(k^2/4a)

    I know that *e(k^2/4a) is a phase factor which is irrelevant to the equation
    so
    integration of e^(-ikx-ax^2)=√(π/a)

    since ∫-inf to inf IΨI^2dx=(Ψ*)(Ψ)=Ae(-ax^2)*Ae(-ax^2)=(A^2)e(-2ax^2)
    and if e(-2ax^2)=√(π/2a)
    then A^2=√(π/2a)
    hence A=(2a/π)^(1/4)

    Was it necessary to normalize
    before I begin in attempting to answer the question

    attempting to find<p subscript(x)>
    <psubscriptx> =∫(Ψ*)(-ihbar(∂/∂x))(Ψ)=(e(-ax^2))(-ihbar((1/2)e^(-4ax^2))(e(-ax^2))
    and here is where I am really worried
    I dont know if for <psubscriptx> I am plugging in correctly
    any ideas or advice?

    ----
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Apr 13, 2012
  2. jcsd
  3. Apr 13, 2012 #2

    fzero

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    That is a real quantity, so it's not just a phase factor.
    It looks like it.

    You didn't compute the derivative in the last expression.
     
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