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Verify Ψ is solution of quantum oscillator using H operator

  1. Feb 20, 2016 #1
    1. The problem statement, all variables and given/known data
    verify that Ψ(x) = ( 1/a√π)½ exp(-(x2/2a2))
    is a solution to the TISE for linear harmonic oscillator. Where a = √(hbar/mw). and V(x) = ½ mw2x2.

    2. Relevant equations
    HΨ=EΨ
    E_n = (n+½)hbar*w


    3. The attempt at a solution

    I've started by differentiating the wave function twice to find http://www5a.wolframalpha.com/Calculate/MSP/MSP2111e793aff6db9408500002d959h09e9180hfg?MSPStoreType=image/gif&s=28 [Broken].
    Then when I apply the Hamiltonian operator to the wave function I cant see a way this would cancel down to (n+½)hbar*w. Have I gone about this the right way at all?

    Thanks
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Feb 20, 2016 #2

    Dr Transport

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    Think about what [itex] n [/itex] is for that wave function
     
  4. Feb 20, 2016 #3
    pretty sure n=0, ive seen the full version of the wave function i have before and n=0 would make it cancel to what I have now but im still a little lost
     
  5. Feb 20, 2016 #4

    Dr Transport

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    if that is the [itex] n [/itex] for that wave function, then that is the equation you must solve....
     
  6. Feb 20, 2016 #5
    so E should equal 1/2 hbar*w but I cant find a way to make the Schrodinger equation I've written to cancel to that. Maybe I'm not understanding
     
  7. Feb 20, 2016 #6

    Ray Vickson

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    Show your complete, final, expression for ##H \psi##, so we can tell if you have made an algebraic error or not. Right now, we cannot say where your problem lies.
     
    Last edited by a moderator: May 7, 2017
  8. Feb 20, 2016 #7
    hey. since I last posted I think i got a solution.
    my expression was
    H = -hbar^2/2m d^2/dx^2 + 1/2 mw^2x^2

    I took the second derivative of psi and wrote it as (x^2/a^4 - 1/a^2)*psi and then moved everything onto the left hand side with

    hbar^2/2m (x^2/a^4 - 1/a^2)psi + (E-1/2mw^2x^2)psi=0

    then collected x^2 terms and constant terms and found

    E - hbar^2/2ma^2 = 0
    so substituting a back into the equation it reduced down to (hbar*w)/2
     
  9. Feb 21, 2016 #8
    So I had to do the same problem but this time with

    2xe*e^((-x^2)/(2a^2))

    This time when I collected terms I ended with only one x^3,x^2,x,constant term each. Am I right in thinking this wave function is not a solution then? To the schroedinger equation and linear harmonic oscillator
     
  10. Feb 21, 2016 #9

    Ray Vickson

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    What, exactly, is your final expression for ##H \psi##? Write out all the details!
     
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