In general for some smooth and bounded $\hat{V}$

$$ \left[\hat{V}(t_1), \hat{V}(t_2) \right] \neq 0 \text{ if } t_1 \neq t_2 $$

But what more can be said about commutators of this type? I am particularly interested in the case where the two times are "not far apart", say $$ |t_2 - t_1| \leq \epsilon $$

for some sufficiently small but not infinitesimal $\epsilon$. I know from other answers that if we define $$ A_t(\epsilon) = \left[\hat{V}(t+\epsilon), \hat{V}(t) \right] $$

then as $\epsilon \rightarrow 0, A_t(\epsilon)\rightarrow 0$. But I am interested in what bounds we can place on $\left[\hat{V}(t_1), \hat{V}(t_2) \right]$ provided $\hat{V}$ is smooth. My (very naive) approach was to note (letting $V_1=\hat{V}(t_1)$): $$ \begin{align} \left[V_1, V_2 \right]\cdot|t_2-t_1|^{-1} = (V_1-V_2)(V_1+V_2)|t_2-t_1|^{-1}-(V_1^2 -V_2^2)|t_2-t_1|^{-1} \end{align} $$

and if we say $||V||<C$ for $t\in[t_1, t_2]$, the above is roughly equal to (for small $|t_2-t_1|$

$$ \begin{align} \left[V_1, V_2 \right]\cdot|t_2-t_1|^{-1} = \frac{dV}{dt}(2C)-(V_1^2 -V_2^2)|t_2-t_1|^{-1} \end{align} $$

and I could probably go on simplifying. But clearly this is not a good way to place bounds, and I am interested in what clean results there might be.


In order to limit the possibilities, I am interested in 'common' Hamiltonians. Let use assume that:

  1. $H(t) = K + V(t)$, i.e. the only time dependence is in the potential
  2. $V(t) = g(t)\cdot V(x)$, the time dependence can be written as a time-dependent coupling strength for the smooth and bounded potential $V$. We can assume $g(t)$ is a polynomial of time.
  3. If it helps to have concrete examples, I am interested in the Newton potential where $\nabla^2V(x) = c_1 \cdot \psi^* \psi$, and also (but less so) the harmonic oscillator with a time varying frequency.

I am interested in results that assume more stringent conditions as long as it's clear what those are.

  • $\begingroup$ Sorry for the naive question, is then $V$ just an operator that depends on a parameter? Is there some characteristic of this dependence due to the interpretation of $t$ as time? Or are you just considering a smooth operator valued function $t\mapsto V(t)$? $\endgroup$ May 15, 2020 at 18:22
  • $\begingroup$ Do you have any conditions in mind to place on the Hamiltonian? If it can be arbitrary it seems difficult to say anything. $\endgroup$
    – knzhou
    May 15, 2020 at 18:27
  • $\begingroup$ @knzhou I will edit the question with some conditions. In general I am interested in common Hamiltonians $\endgroup$
    – gabe
    May 15, 2020 at 18:58
  • $\begingroup$ @user2723984 I don't see any characteristic that would depend on the interpretation as time, so you can consider it as a smooth parameter. Maybe someone else will chime in to say why that shouldn't be the case. $\endgroup$
    – gabe
    May 15, 2020 at 18:59
  • $\begingroup$ @gabe then I think this is essentially a math question, maybe the people at math SE can answer it better $\endgroup$ May 15, 2020 at 19:22


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.