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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that would be |x(t)> = e^{-iHt} |x(0)> .. this is standard notation
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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“vector” in the sense of the “dot product” over positions: i hbar d/dt <x,t|ψ> = Σ_y <x;t| H |y;t> <y;t|ψ> i.e. i hbar d/dt Ψ(t) = h Ψ(t) (h = matrix elements of H in the position basis)
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