I want to show that K=K(x,x′,t−t′)=∑βexp[−iEβℏ(t−t′)] satisfies the Schrödinger equation H|ψ⟩=iℏ∂t|ψ⟩ with respect to x and t, where the β's are the Eigenstates of the Hamiltonian and satisfy ∑|β⟩⟨β|=1. So I started calculating and got iℏ∂tK=...=⟨x|exp[−iEβℏ(t−t′)]|x′⟩. But here I am stuck since I am not that familiar with QM and bra-ket notation/relations. What is the next step or which fact do I have to use to show the claim?
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