Repeated measurements on a quantum system interacting with other quantum systems

[accdd]

In quantum mechanics if I repeat a measurement of the same observable in succession I get the same quantum state if it is not a degenerate state.
If I make the system under consideration interact with another quantum system and meanwhile keep measuring it what happens?
Does the system not interact because it is being measured? Does the system behave as if it were classical? All answers are welcome

 

[Morbert]

Assuming the measurement is a projective measurement:

In quantum mechanics if I repeat a measurement of the same observable in succession I get the same quantum state if it is not a degenerate state.

Why the caveat about degeneracy? The projector $P_i$ is idempotent ( $P_iP_i\dots P_i|\psi\rangle = P_i|\psi\rangle$ ) even if eigenvalue $i$ has degenerate eigenstates.

If I make the system under consideration interact with another quantum system and meanwhile keep measuring it what happens?
Does the system not interact because it is being measured? Does the system behave as if it were classical? All answers are welcome

This corresponds to the scenario $P_i P_i \dots P_i U |\psi\rangle|\phi\rangle$? We know that $P_iP_i\dots P_i = P_i$ like above.

 

[Peter Morgan]

if I repeat a measurement of the same observable in succession I get the same quantum state

You only get the same result and the same state if you repeat precisely the same measurement. Put another way, how close the new measurement is to the old measurement determines how likely it is that you will get the same result. If there's an evolution between the two measurements that modifies the state so that the result of the same measurement would be different, that will be equivalent to there being no such evolution but with the subsequent measurement being different: this is directly comparable to the difference between the Schrödinger picture and the Heisenberg picture of states and measurements.
Sequential measurement is a thing in the literature. My recent article in JPhysA 2022, "The collapse of a quantum state as a joint probability construction", https://doi.org/10.1088/1751-8121/ac6f2f, on arXiv as https://arxiv.org/abs/2101.10931, discusses sequential measurements as part of a wider discussion.
Your one observable case can be thought of as a special case of a commutative algebra of observables. There is a point of view in which a commutative algebra of observables can be thought of as in some sense "classical", however if it were that simple we would not still be talking about interpretations of QM. You could try a paper by Tsang&Caves in Phys. Rev. X 2012, Ref [17] in the article I just mentioned, "Evading quantum mechanics: engineering a classical subsystem within a quantum environment", which develops the idea of Quantum Non-Demolition meeasurement from inside QM, however there are many approaches that come under the general heading "Modal Interpretations" that develop similar ideas.

 

[Lord Jestocost]

In quantum mechanics if I repeat a measurement of the same observable in succession I get the same quantum state if it is not a degenerate state.
If I make the system under consideration interact with another quantum system and meanwhile keep measuring it what happens?
Does the system not interact because it is being measured? Does the system behave as if it were classical? All answers are welcome

Maybe, the quantum zeno effect might be of interest:

“The quantum Zeno effect has an interesting history. It was first understood by von Neumann [3], who proved that any given quantum state $\phi$ can be “steered” into any other state $\psi$, by applying a suitable series of measurements. If $\phi$ and $\psi$ coincide (modulo a phase factor), the evolution yields, in modern language, a quantum Zeno effect.”

From “Quantum Zeno dynamics and quantum Zeno subspaces” by Paolo Facchi, Giuseppe Marmo and Saverio Pascazio (Journal of Physics: Conference Series, Volume 196, SUDARSHAN: SEVEN SCIENCE QUESTS 6–7 November 2006)