# Time Evolution of Heisenberg Operators $x$ and $x^2$

In the Heisenberg picture, operators evolve according to

$$\partial_t A = \frac{1}{i\hbar} [A,H].$$

My question is, does the following relation hold?

$$(X_H)^2 = (X^2)_H$$

The system (Hamiltonian) in mind is the 1D harmonic oscillator,

$$H = a^{\dagger}a + \frac 12$$

I would like to say that it does, but the Heisenberg equations of motion for $$x^2$$ and $$p^2$$ are very messy (and I believe nonlinear).

• So to be very clear, is your question if $X_H^2=(X^2)_H$? The subscript $H$ denotes here operator in Heisenberg picture. If so, you should write down the definition of e.g. $X_H=\ldots$ in terms of $X$ (the position operator in the Schrödinger picture). Oct 9, 2023 at 22:01
• Yes, that is my question. Are you saying something like the relation $X_H = U^{\dagger} X_S U$. In which case it is clear that $(X_H)^2 =(X^2)_H$ ? Oct 9, 2023 at 22:09
• Indeed. Consider to write an answer yourself. Oct 9, 2023 at 22:14

$$X_H = U^{\dagger}X_S U$$
$$(X_H)^2 = U^{\dagger}X_S U U^{\dagger}X_S U = U^{\dagger}(X_S)^2 U = (X^2)_H$$