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I was wondering if anyone could clarify my understanding of unitary time evolution of quantum states, in particular for products of time evolution's:

Suppose we know state of a quantum system at [itex] t=t_{0}[/itex], given by [itex] \vert\psi\left(t_{0}\right)\rangle[/itex], then to determine its state at a later time [itex]t=t_{2}[/itex] we can first evolve the state from its configuration at [itex] t=t_{0}[/itex] to its configuration at [itex] t=t_{1}[/itex], i.e [tex]\vert\psi\left(t_{1}\right)\rangle= U\left(t_{1},t_{0}\right)\vert\psi\left(t_{0}\right)\rangle[/tex]

and then subsequently evolve this to its final state at [itex] t=t_{2}[/itex] via a further time evolution, [tex]\vert\psi\left(t_{2}\right)\rangle= U\left(t_{2},t_{1}\right)\vert\psi \left(t_{1}\right)\rangle= U\left(t_{2},t_{1}\right)U \left(t_{1},t_{0}\right)\vert\psi\left(t_{0}\right)\rangle[/tex]

Alternatively, we could immediately evolve the state from its initial configuration at [itex]t=t_{0}[/itex] to its final state at [itex]t=t_{2}[/itex] in the following manner, [tex]\vert\psi \left(t_{2}\right)\rangle= U\left(t_{2},t_{0}\right)\vert\psi \left(t_{0}\right)\rangle[/tex]

Given that time is homogeneous (as far as we know), the results of these two evolution's of the system should be equivalent (as the path we take through time in evolving the state should not affect its initial and final configurations), this implies that [tex]U\left(t_{2},t_{0}\right)= U\left(t_{2},t_{1}\right)U\left(t_{1},t_{0}\right)[/tex]

Is this a correct description?