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Ladder operators and a 1-D QHO

  1. Aug 22, 2012 #1
    1. The problem statement, all variables and given/known data
    ZPryy.jpg


    2. Relevant equations
    sSbrM.png


    3. The attempt at a solution
    I solved part a) correctly, I believe, giving me

    ψ = e[itex]^{-(√(km)/\hbar)x^{2}}[/itex]

    and a normalization constant A = ((π[itex]\hbar[/itex])/(km))[itex]^{-1/4}[/itex]

    I'm having difficulty with part b. I'm not exactly sure how I create a linear combination. I think I'm supposed to write the linear combination like so:

    a + a[itex]^{+}[/itex] = 2(k/2)[itex]^{1/2}[/itex]x

    (a + a[itex]^{+}[/itex])(2k[itex]^{-1/2}[/itex]) = x

    However, I am not sure how I use the operator methods to show that xψ0 is the first excited state.

    If anyone could nudge me in the right direction, I'd be eternally grateful! Thank you.
     
  2. jcsd
  3. Aug 22, 2012 #2

    vela

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    What do you get when you calculate ##x\psi_0## with the expression for x you have in terms of the raising and lowering operators?
     
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