# Quantum harmonic oscillator. Finding operators

Problem:

I'm trying to verify that $p_H(T)$ and $x_H(T)$ satisfy the following equations, (by solving the Heisenberg equation):

$x_H(t)=x_H(0)cos(\omega t)+(1/m\omega)p_H(0)sin(\omega t)$

$p_H(t)=-m\omega x_H(0)sin(\omega t)+ p_H(0)cos(\omega t)$.

Hamiltonian for the quantum Harmonic oscillator:

$H_H(t)=(1/2m)p_H^2(t)+(k/2)x^2_H(t)$.

Heisenberg equation:

$\partial/\partial t(O_H(t)=(i/\hbar)[H_H(t),O_H(t)]$.

Attempt:

To derive $x_H(t)$ and $p_H(t)$ I need to first compute $[H_H(t),x_H(t)]$ and $[H_H(t),p_H(t)]$.

The solution to this problem states that: $[H_H(t),x_H(t)]=(1/2m)[p^2_H(t),x_H(t)]$, which can be reduced to $(1/2m)(-i\hbar 2)p_H(t)$.

However, I cannot see how $[H_H(t),x_H(t)]=(1/2m)[p^2_H(t),x_H(t)]$ is obtained. As I would have thought that this expression would consider $[(1/2m)p_H^2(t)+(k/2)x^2_H(t),x_H(t)]$.

I presume this is because a step is missed and I am not following the maths.

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$x_H$ must always commute with $x_H$ (or any higher powers), $[x_H,x_H]=x_H^2−x_H^2=0$. –  Ondřej Černotík Apr 6 '13 at 12:40