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Is the reason why the time evolution operator is unitary based on purely physical arguments, i.e. that the physical processes that an isolated system undergoes shouldn't depend on any particular instant in time (homogeneity of time); thus two experimenters who conduct the same experiment starting from the same initial state, but at different times, should have the same probability amplitude for that state?! Or is there some mathematical argument as well?

Also, is the reason why the time evolution operator is linear implied by the superposition principle (as an arbitrary state can be expressed as a linear combination of basis states the operator should act linearly as otherwise the state as a whole would evolve differently to the superposition of states that it was initially represented by)?!

Will
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2 Answers2

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Time evolution is the exponential of the Hamiltonian, since the Hamiltonian is the generator of time-translation (equivalently: Energy is the charge of time translation).

As a physical observable corresponding to energy, the Hamiltonian has to be self-adjoint.

The exponential of a self-adjoint operator is unitary by Stone's theorem.

A "physical" argument is that time evolution should preserve whatever normalization we have chosen for our states (because the probability to find the state $\psi$ in $\phi$ at $t_0$ should be the same as finding the evolved state $\psi$ in the evolved state $\phi$ at $t_1$), i.e. it should preserve the inner product, i.e. it should be unitary.

ACuriousMind
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It is a consequence of the conservation of the total probability, that is, that $1=\langle a | a \rangle,$ being $|a\rangle$ the state in which your system is. As time makes the state evolve, the final state must also be normalized that way, so that the probability of finding it in the state it will be is one. An easy mathematical calculation leads to the fact that the adjoint of $U$ times $U$ ( $U$ is the time evolution operator ) must conserve distances. From that, that it must conserve any scalar product, and from that the unitarity. You can see this detailed in Leonard Susskind's freely available video of his lecture 9 on quantum entanglements.

César Tejeda Hernández
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