The Schrodinger equation in Hilbert space is expressed as : ∂∂tψ(t)=−iℏHψ(t).
Here ∂∂tψ(t)≡ψ′(t)≡limh→0[ψ(t+h)−ψ(t)h], and because ψ(t) is a Hilbert space vector, the limit is defined using convergence via the norm. (In other words, for any ϵ>0 there exists an hϵ>0, such that ||ψ′(t)−(ψ(t+h)−ψ(t)h)||<ϵ for all $|h|
But in the wavefunction realization (for a single particle), the Schrodinger equation is expressed as
∂∂tψ(x;t)=−iℏ[−ℏ22m∂2∂x2+V]ψ(x;t).
Here, however, ∂∂tψ(x;t) is a pointwise partial derivative with respect to t, and so the convergence depends only on each individual point x of ψ(x) separately.
If we take the abstract Hilbert space expression as definitive (axiomatically), then how can it be shown that the wavefunction realization actually expresses the same thing, given the different meanings of ∂∂t?
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