At most explicit time-dependent operator

[carllacan]

Hi.

I have a little language problem. I'm studying in Germany, and my German is... nicht sehr gut, so I sometimes have problems understanding the exercises. The one I'm having issues right now has a part which says einen höchstens explizit zeitabhängigen Ope-rator I am Schrödingerbild.

My translation for this is "An at most explicitly time-dependent operator in Schrödinger picture." This doesn't make any sense: how can a Schrödinger-picture operator be time dependent? And what does it mean by "at most explicitly"

Thank you for your time.

PD: if you happen to speak German I'd would appreciate if you checked my translation.

PPD: I know I could simply ask the teacher, but... I should have made this exercise way earlier, and I'm kind of ashamed to openly tell him that I am doing it now, so I'd like to avoid it if possible.

 

[Bill_K]

A bit more context would help.

 

[carllacan]

There is not much. I am given the Hamiltonian for an Harmonic oscillator and I am told to find the time-dependence of $O_H= U^{\dagger}(t, t_0)O_SU(t, t_0)$, where $O_S$ is the mentioned "at most explicitly time-dependent operator in Schrödinger picture.", and U is the time-development operator.

Maybe I am to just state the Heisenberg equation of motion for O?

 

[Bill_K]

The Heisenberg equation of motion is $\frac{dO_H}{dt} = \frac{i}{\hbar}[H_H, O_H] + \frac{\partial O_S}{\partial t}$. I think he's saying, keep the $\frac{\partial O_S}{\partial t}$ term.

 

[carllacan]

The Heisenberg equation of motion is $\frac{dO_H}{dt} = \frac{i}{\hbar}[H_H, O_H] + \frac{\partial O_S}{\partial t}$. I think he's saying, keep the $\frac{\partial O_S}{\partial t}$ term.

I've never seen it in that form. I don't understand it, isn't the point of the Schrödinger picture that operators are time-independent?

 

[Bill_K]

I've never seen it in that form. I don't understand it, isn't the point of the Schrödinger picture that operators are time-independent?

Usually, but not always.

https://en.wikipedia.org/wiki/Heisenberg_picture#Mathematical_details

"The operator could have intrinsic time dependence if it represented the potential of a variable external field, or if it were the component of a tensor defined with respect to a rotating coordinate system."

 

[carllacan]

Usually, but not always.

https://en.wikipedia.org/wiki/Heisenberg_picture#Mathematical_details

Oh, thanks for the detail. That intrinsic time-dependence could be what is meant by "explicit zeit-abhängig". As I'm not told much more about O I can't get a closed form, but this is enough to start.

Thank you!