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Uncertainty Product for a hydrogen-like atom, need help

  1. May 12, 2013 #1
    1. The problem statement, all variables and given/known data

    Calculate the uncertainty product ΔrΔp for the 1s electron of a hydrogen-like atom with atomic number Z. (Hint: Use <p> = 0 by symmetry and deduce <p^2> from the average kinetic energy)

    2. Relevant equations

    All I have is the wavefunction. For a 1s, it takes the form:

    wavefunction = (Z/a0)^(3/2)*2exp(-Zr/a0)

    Where Z is the atomic number, a0 is the Bohr radius

    Other equations that I think I need include:

    The energy for a hydrogen-like atom: E = (-13.6 eV)(Z^2/n^2)

    Δr = √(<r^2> - <r>^2)
    Δp = √(<p^2> - <p>^2)

    <f> = ∫f(r)*P(r)dr

    That integral is from 0 to ∞, and the <> are supposed to denote averages.

    3. The attempt at a solution

    I can find Δr no problem. Just use the above formula, where f(r) is r^2 for <r^2> and f(r) is r for <r>^2. The problem is finding the uncertainty in momentum. The above formula for momentum uncertainty should reduce down to

    Δp = √(<p^2>)

    And that's where I'm stuck; the hint isn't helping me much.

    For another problem, I found the uncertainty in KE by finding the uncertainty in <U> for a hydrogen atom, and then used <KE> + <U> = <E>, where <E> = -13.6/n^2.

    I'm not sure what the potential for a hydrogen-like atom is. Also, how can I get KE into momentum? I'm just confused on this part as a whole.

    Any help is appreciated! Thank you!
     
  2. jcsd
  3. May 13, 2013 #2
    you may probably calculate <p2> by using
    <p2>=∫ψ*p2ψd3r,then apply first quantized form as p=-ih-∂/∂r and then you have to do some by part or a simple straightforward way without by part and done.On the other hand you might want to use
    k.E.=p2/2m
     
  4. May 18, 2013 #3
    Am I too late?

    It's no problem to calculate the expectation values of the position operators with a position-space wavefunction right? So it should be just as easy to calculate the expecation values of momemtum operators with a momentum-space wavefunction!

    If you find the momentum-space wavefunction, you are in the clear. My method completely ignores the hint though.
     
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