How Do You Solve the Harmonic Oscillator Commutator Problem?

[LagrangeEuler]

Homework Statement

$H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2x^2$
Show that
$[H,[H,x^2]]=(2\hbar\omega)^2x^2-\frac{4\hbar^2}{m}H$

Homework Equations

$[x,p]=i\hbar$

The Attempt at a Solution

I get
$[H,x^2]=-\frac{i\hbar}{m}(px+xp)$
what is easiest way to solve this problem?

 

[vela]

Show your work.

 

[matematikuvol]

$[H,x^2]=[\frac{p^2}{2m},x^2]=\frac{1}{2m}[p^2,x^2]=p[p,x^2]+[p,x^2]p$
$[p,x^2]=-2i\hbar x$
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.

 

[LagrangeEuler]

First part is same. But its to hard. I tried
$$[H,[H,x^2]]=[H,Hx^2-x^2H]=H[H,x^2]-[H,x^2]H=H(-2i\hbar)x+2i\hbar xH$$

 

[LagrangeEuler]

I solve it.