The state vector |ψ> does not evolve in time. What evolves in time is the wave function, ψ(x,t) := <x,t|ψ> |ψ> := Σ ψ(x,t) |x,t> 1 := Σ |x,t><x,t|, for all t
No, the state vector is time dependent |a,b> is short hand for |a>|b>.. there is no “time” operator
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The wave function is the state projected onto the position basis ψ(x,t) := <x|ψ(t)>
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The Hamiltonian H(X,t) is a function of dynamic vars X(t), whose eigvecs form a t-dependent eigbasis: X(t) |x,t> = x |x,t>, for each t. t is a param, not an eigval. The eigbasis & wavefunc evolve according to Schrödinger eqn. Their combination into |ψ> doesn’t.
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that would be |x(t)> = e^{-iHt} |x(0)> .. this is standard notation
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