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3-D harmonic oscillator expectation value

  1. Aug 20, 2014 #1

    dyn

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    1. The problem statement, all variables and given/known data
    The Hamiltonian for the 3-D harmonic oscillator in spherical polar coordinates is given in the question.The question then asks : using the trial wavefunction ##ψ=e^(-αr) ## show that


    2. Relevant equations

    ##<ψ|H|ψ>/<ψ|ψ> = ([itex]\hbar[/itex]α)^2/2m + 3mω^2/2α^2##

    The following integral is also given## ∫ x^nexp(-ax) = n!/a^(n+1) ##for n≥0 with limits from 0 to ∞

    3. The attempt at a solution

    I applied the Hamiltonian to ψ. The [itex]\theta[/itex] and [itex]\phi[/itex] terms drop out. I then perform the integral of ∫ ψ^*Hψ but I have a term containing 1/r and the given integral doesn't apply for n<0
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 21, 2014 #2

    vela

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    Please show your work.
     
  4. Aug 24, 2014 #3

    nrqed

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    What did you use for the measure of integration dV? It contains ## r^2 dr##, did you forget that ##r^2##?
     
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