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Harmonic oscilator Tamvakis

  1. Jun 22, 2013 #1
    1. The problem statement, all variables and given/known data
    ## H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2##
    Show that
    ##[H,[H,x^2]]=(2\hbar\omega)^2x^2-\frac{4\hbar^2}{m}H##


    2. Relevant equations
    ##[x,p]=i\hbar##


    3. The attempt at a solution
    I get
    ##[H,x^2]=-\frac{i\hbar}{m}(px+xp)##
    what is easiest way to solve this problem?
     
  2. jcsd
  3. Jun 22, 2013 #2

    vela

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    Show your work.
     
  4. Jun 24, 2013 #3
    ##[H,x^2]=[\frac{p^2}{2m},x^2]=\frac{1}{2m}[p^2,x^2]=p[p,x^2]+[p,x^2]p##
    ##[p,x^2]=-2i\hbar x##
    from that
    ##[H,x^2]=-2i\hbar px-2i\hbar xp##
    from that
    ##[H,[H,x^2]]=p[\frac{p^2}{2m},-2i\hbar x]+[\frac{p^2}{2m},-2i\hbar x]p+p[\frac{1}{2}m\omega^2x^2,-2i\hbar x]+[\frac{1}{2}m\omega^2x^2,-2i\hbar p]x##
    from that
    ##[H,[H,x^2]]=-4p^2\frac{\hbar^2}{m}-4m\omega^2i\hbar x##
    you don't get this result.
     
    Last edited: Jun 24, 2013
  5. Jun 24, 2013 #4
    First part is same. But its to hard. I tried
    [tex][H,[H,x^2]]=[H,Hx^2-x^2H]=H[H,x^2]-[H,x^2]H=H(-2i\hbar)x+2i\hbar xH[/tex]
     
  6. Jun 25, 2013 #5
    I solve it.
     
    Last edited: Jun 25, 2013
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