# Conceptual questions on unitarity and time evolution

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1. Dec 18, 2015

### "Don't panic!"

From a physical perspective, is the reason why one requires that the norm of a state vector (of an isolated quantum system) is conserved under time evolution to do with the fact that the state vector contains all information about the state of the system at each given time (i.e. the probabilities of it having a particular energy, momentum, etc..) and so when it is evolved in time, although the individual probabilities of each observable will change, the total probability will always be conserved, since there is no external influence on the system and so the set of allowed values for each observable will not increase/decrease. That is, the observables of the evolved quantum system must assume values (with a certain probability) from the original set of values (that they could "choose from" at the initial time) ?! (sorry, I feel I haven't worded this part in the most articulate way).

Additionally, does the composition of two evolution operators, i.e. $$U(t_{2},t_{0})=U(t_{2},t_{1})U(t_{1},t_{0})$$ follow from the requirement that quantum evolution is Markovian, that is, that one can obtain the same results by knowing the state of a system at a given instant in time as one would obtain from knowing complete evolution of a system? For example, Say one observer knew the state of a quantum system at an initial time $t_{0}$ to be $\lvert\psi (t_{0})\rangle$. The system is then allow to evolve to its state at some later time $t_{2}$, $\lvert\psi (t_{2})\rangle=U(t_{2},t_{0})\lvert\psi (t_{0})\rangle$. Another observer doesn't know what state the system was in at time $t_{0}$, but does know the state of the system at some time, $t_{1}$, $\lvert\psi (t_{1})\rangle$ (with $t_{0}<t_{1}<t_{2}$). Again, the system evolves from its state at time $t_{1}$, $\lvert\psi (t_{1})\rangle$, to its evolved state at time $t_{2}$, described by this observer by $\lvert\psi (t_{2})\rangle=U(t_{2},t_{1})\lvert\psi (t_{1})\rangle$. Since the final state is the same for both observers, and the first observer will also be able to determine the evolved state at $t_{1}$, $\lvert\psi (t_{1})\rangle=U(t_{1},t_{0})\lvert\psi (t_{0})\rangle$ (since they know the state of the system at the earlier time $t_{0}$), it follows that $$\lvert\psi (t_{2})\rangle=U(t_{2},t_{1})\lvert\psi (t_{1})\rangle=U(t_{2},t_{1})U(t_{1},t_{0})\lvert\psi (t_{0})\rangle=U(t_{2},t_{0})\lvert\psi (t_{0})\rangle\\ \Rightarrow\quad U(t_{2},t_{1})U(t_{1},t_{0})=U(t_{2},t_{0})$$

Would this be a correct description at all?

2. Dec 18, 2015

### Staff: Mentor

Wigners Theorem

Thanks
Bill

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3. Dec 18, 2015

### "Don't panic!"

So is the idea that the outcome of an experiment should be independent of the time that it was carried out? In my example, is the point that if one experimenter starts the experiment with a system (call it A) in some particular state at some earlier time and then another experimenter starts an experiment at a later time in which the system they are considering (call it B) is in an identical state to the state that the system A has evolved to at this later time, then from this point both should evolve (assuming the experimental conditions are identical) such that their final states, at some much later time, are identical?

4. Dec 18, 2015

### Staff: Mentor

What you wrote is rather convoluted.

Its much easier to view as simply a change in coordinates which fairly obviously shouldn't change the physics, in particular it shouldn't change the probabilities from the Born rule. It would be very very weird if simply changing the where the origin of your coordinate system is or the velocity its moving changed that..

While its rather obvious in actuality you are invoking the POR - Principle Of Relativity:
https://en.wikipedia.org/wiki/Principle_of_relativity

Thanks
Bill

5. Dec 18, 2015

### "Don't panic!"

Sorry I realise it is quite convoluted, but I was really trying to put in words what is conceptually going on, and why $U(t_{2},t_{0})=U(t_{2},t_{1})U(t_{1},t_{0})$, in terms of relating a quantum state at some initial time to it's evolved states at later times.