# Determining the state of a system

My textbook says:

"To determine the state of a system at a given instant, it suffices to perform on the system a set of measurements corresponding to a complete set of commuting observables (CSCO)"

This is my understanding. Please correct me if I'm wrong:

Let's say $$H$$ is an observable and CSCO on its own. Also its eigenstates ( $$|\psi_i\rangle$$ ) are discrete and non-degenerate and its eigenvalues are $$E_i$$.
I take $$N$$ systems that are in the same state $$|\psi\rangle$$ and make measurements for $$H$$. From the $$N$$ numbers I get, I can assign a probability $$p_i$$ to the result of each measurement, $$E_i$$. Then the state of the system will be: $$|\psi\rangle=\sum_i \sqrt{p_i} |\psi_i\rangle$$

Is this right or something else is intended?

Ref: Cohen-Tannoudji - vol. 1 - page 295

1. A short comment: If $$H$$ is an observable and a CSCO, then its eigenstates are necessarily non-degenerate. You wrote it as if those were separate requirements.
2. This is not correct. If you repeat your procedure with a very large number $$N$$ of measurements, then you can indeed figure out that the system state before the measurement was $$|\psi\rangle = \sum_i \mathrm e^{\mathrm i \varphi_i} \sqrt{p_i}\, |\psi_i\rangle ,$$ but you do not get any information about the phases $$\varphi_i$$.
3. What your textbook is trying to say is much simpler, I think: you know the state of the system after the measurement for certain. (If you measure the energy $$E_i$$, then the state afterwards is $$|\psi_i\rangle$$.)