- #1
"Don't panic!"
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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?
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?