a) Assume two operators $A$ and $B$.
1) Assume $$[A,B]=0 $$ and $$ ψ= \sum c_n u_n ~~~~\text a~ wavefunction~ describing~ the~ state~ of~ the~ system $$ with $$Aψ=a_n u_n $$ $$Bψ=b_n u_n$$ If we take two measurements, one after the other we have $$ABψ=a_n b_n u_n $$
2)Now let's assume that $$[A,B] \neq 0 $$If first we take $$Αψ=a_n u_n$$ and then we measure with $B$, shall we have $$B(a_n u_n)=b_m v_m $$ a different eigenvector of the system , that is: $$ψ-->a_n u_n --> b_m v_m $$
Does this result comes from the fact that for $B$, the vector $$u_n$$ is a superposition with some probabilities $P_n$ to have a result, after the act of B on $$u_n,$$ the $$b_m v_m$$ eigenvector of B?
b) Also, assume that we make a measurement with $A$. After a time $t$ from the measurement the system under consideration will return to a superposition state.
Does that happen because of the uncertainty principle or because of the internal interactions of the system? If the first is there a mathematical formulation to explain the evolution from the eigenstate to the superposition state? If the second, can we in theory represent the interactions through operators and the evolution that concludes to the superposition state (and if so, do these operators produce eigenstates of the system or different states)?
Note:I find the two parts of the question relevant because the subject under consideration is about operators and how they act on a system, or if you like is about understanding if every interaction can be described by operators and what a sum of measurements in a row can has as a result on the system. But if you think that the two parts should be two questions, please comment and answer only to the first part. I shall post the second part independently.