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Quantum Mechanics and the Hydrogen Atom

  1. Nov 3, 2011 #1
    Calculate the expectation value of the potential energy for an electron in a 1s orbital for a hydrogen atom





    Ive determined the potential energy operator to be V=-e2/4∏ε0r
    and a wave function of

    ψ= (1/4∏)1/2

    therefore i get
    <V> = ∫∫∫ψ*Vψr2sin∅drd∅dphi
    integrals from 0 to r, 0 to pi, 0 to 2pi


    not sure where to go from here.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 3, 2011 #2
    Nevermind, i got it.

    but if anyone is interested ill explain.

    For a 1s orbital of a hydrogen the wavefunction is ψ=root(1/∏ao3 e-r/ao)

    this gives
    ∫∫∫ψVψr2sinθdrdθd∅
    integrals are zero to 2pi, zero to pi, and zero to infinity.

    then factor out any terms that are not a function of r,θ, or∅.
    This gives several terms outside of the integral: ∫∫∫e-2r/aorsinθdrdθd∅

    then you can separate the integrals and evaluate. they were pretty easy to do.

    the final answer was <v> = -e2/4∏εoao
     
  4. Nov 4, 2011 #3

    dextercioby

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    Science Advisor
    Homework Helper

    You can use the virial theorem. The expectation value you're looking for is then half the ground state energy.
     
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