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
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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.
that would be |x(t)> = e^{-iHt} |x(0)> .. this is standard notation
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sometimes you will see |x ; t> with a semi-colon but the comma is reserved for tensor products
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Ok, so replace my ,, with ;; The argument stands. (and by “wave function” I meant “prob amplitude”, i.e. ψ(x,t), sorry) The wf that satisfies Schrödinger is the “vector” Ψ(t) = ψ(•,t), which is not the same as |ψ>, but related to it through the t-dep basis.
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