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Momentum eigenvalues and eigenfunctions

  1. Jan 25, 2014 #1
    1. The problem statement, all variables and given/known data

    For the following wave functions:
    ψ[itex]_{x}[/itex]=xf(r)
    ψ[itex]_{y}[/itex]=yf(f)
    ψ[itex]_{z}[/itex]=zf(f)

    show, by explicit calculation, that they are eigenfunctions of Lx,Ly,Lz respectively, as well as of L^2, and find their corresponding eigenvalues.

    2. Relevant equations

    I used:
    L[itex]_{x}[/itex]=-ih(y[itex]\partial/\partial z[/itex]-z[itex]\partial/\partial y[/itex])
    L[itex]_{y}[/itex]=-ih(z[itex]\partial/\partial x[/itex]-x[itex]\partial/\partial z[/itex])
    L[itex]_{z}[/itex]=-ih(x[itex]\partial/\partial y[/itex]-y[itex]\partial/\partial x[/itex])

    for solving:

    L[itex]_{z}[/itex]|ψ[itex]_{z}[/itex]>=lz|ψ[itex]_{z}[/itex]>
    L[itex]_{x}[/itex]|ψ[itex]_{x}[/itex]>=lx|ψ[itex]_{x}[/itex]>
    L[itex]_{y}[/itex]|ψ[itex]_{y}[/itex]>=ly|ψ[itex]_{y}[/itex]>


    and
    L^2|ψ>=l^2|ψ>

    where:
    L^2=L[itex]_{x}[/itex]^2+L[itex]_{y}[/itex]^2+L[itex]_{z}[/itex]^2

    3. The attempt at a solution

    For instance for Lz:
    ψ[itex]_{z}[/itex](r)=<r|z>=zf(r)

    L[itex]_{z}[/itex]|ψ[itex]_{z}[/itex]>=-ih(x[itex]\partial/\partial y[/itex]-y[itex]\partial/\partial x[/itex]) zf(r)=lz|ψ[itex]_{z}[/itex]>

    Is that correct?
     
  2. jcsd
  3. Jan 25, 2014 #2

    lightgrav

    User Avatar
    Homework Helper

    I think they want you to "explicitly" take partial derivatives of the function z f(r) .
     
  4. Jan 25, 2014 #3
    -ih (x([itex]\partial[/itex]z\[itex]\partial[/itex]y f(r)+z [itex]\partial[/itex]f(r)/[itex]\partial[/itex]y)-y([itex]\partial[/itex]z\[itex]\partial[/itex]x+z[itex]\partial[/itex]f(r)\[itex]\partial[/itex]x))=-ih (xz[itex]\partial[/itex]f(r)\[itex]\partial[/itex]y-yz [itex]\partial[/itex]f(r)\[itex]\partial[/itex]x)=-ih (x[itex]\partial[/itex]\[itex]\partial[/itex]y-y[itex]\partial[/itex]\[itex]\partial[/itex]x)zf(r)

    Which is the same as the initial expression.

    What am I missing?

    Thanks
     
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