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Harmonic oscillator coherent state wavefunction

  1. Aug 5, 2015 #1
    Hi, I am trying to find the wavefunction of a coherent state of the harmonic oscillator ( potential mw2x2/2 ) with eigenvalue of the lowering operator: b.

    I know you can do this is many ways, but I cannot figure out why this particular method does not work.

    It can be shown (and you can find this easily on the internet) that the eigenvalue b evolves as:

    b(t)=b0e-iwt


    What I did was find the coherent state wavefunction u(x,t) by using the eigenfunction equation with the lowering operator a:

    a[u(x,t)]=b*u(x,t)

    with a=(mwx+(h-bar)*d/dx)/sqrt(2mw(h-bar))

    that gives

    u(x,t)=const*e-(1/l2)*(x-l*b)2

    where l=sqrt(2(h-bar)/mw)



    now, when I put in the time evolution of b I get:

    u(x,t)=const*e-(1/l2)*(x-lb0e-iwt)2

    (I plugged in b(t) from before in for b)



    This state does not satisfy the schodinger equation, for one thing it cannot be normalized because the integral over all x of the norm squared of the wavefunction varies in time.

    This confuses me because b(t) comes from treating the coherent state as a superposition of harmonic oscillator energy eigenstates, which come from the schrodinger equation. Since the schrodinger equation conserves the integral over all x of probability density, why do I get a state which does not do so from harmonic oscilator states (and thus, by extension, the schrodinger equation)?

    Thanks so much in advance, I have done this over several times over the last day and cannot find out anything I did wrong nor a solution online.
     
    Last edited: Aug 5, 2015
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  3. Aug 6, 2015 #2

    vanhees71

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  4. Aug 6, 2015 #3
    Thanks,

    But I was wondering if someone knows why the method I described does not work.
     
  5. Aug 8, 2015 #4

    stevendaryl

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    You said in your original post:

    Could you give a URL to where this is shown? It doesn't make sense to me.
     
  6. Aug 8, 2015 #5

    vanhees71

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    This relation is true in the Heisenberg picture of time evolution, and one must not mix these pictures as in the original posting. I have a treatment of the problem in my QM lecture notes, but only in German, but there are many formulae; so perhaps it's possible to understand the calculations:

    http://theory.gsi.de/~vanhees/faq/quant/node51.html
     
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