I Can we operate with several operators at once on a state?

[Tio Barnabe]

It's known that the time-translation operator is $\exp(-i Ht)$ and the space-translation operator is $\exp(i (p \cdot x))$. The former causes a time-translation for a state vector whereas the latter causes a space-translation.

Can we operate with the two operators on the state vector? Like $\exp(-i Ht) \exp(i p \cdot x)$.
What would be the interpretation? That the "particle" is moving in both space and time?

 

[PeterDonis]

Can we operate with the two operators on the state vector?

Sure. Operators on a vector space can be composed, which is what you are describing. The thing you have to be careful of is that many pairs of operators do not commute--i.e., the result you get when you compose them depends on the order in which they are applied. (I don't think that's the case for your particular example, though.)

What would be the interpretation? That the "particle" is moving in both space and time?

No, that you have obtained a state vector that represents a particle at a different point in space at a different instant of time.