Question

Show for stationary eigenstates of the Schrodinger equation in one dimension.

Qualitatively explain why the eigenfunction ψ(x) is smooth, i.e., its first derivative dψ/dx is continuous, even when the potential V (x) has a discontinuity (a finite jump). Hints: use the delta function

Answer 1

We know derivative of a discontinous function is derac delta. Now if first derivative will become discontinous at a point a then its derivative i.e.    will become derac delta function about that point . Now we need in Scrondiger eq to solve wavefunction. Here    is which is creating problem in caculation. Hence must be continous for accepted wavefunction in quantum mechanics.

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