In general, apart from mathematical standpoint, physically what causes the discontinuity of the derivative of wavefunction at infinitely high discontuinity of potential, but not in the case of a finite one.
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Qmechanic
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user157588
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Not strictly correct : How about the particles momentum reverses on bombarding a hard wall? – Sunyam Oct 19 '17 at 15:51
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The mathematical standpoint is discussed in my Phys.SE answer here. – Qmechanic Oct 19 '17 at 18:32
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Physically, when the wave function has a boundary condition involving an infinite potential, there is a discontinuity in the derivative because the the wave function must immediately vanish at the boundary (because you will never find the particle in the potential), and it may not do so smoothly, usually creating a corner. For a finite potential, the particle may be found inside it, even if it's not very likely - so the wave function is not required to immediately drop to zero and instead smoothly transitions to an exponential.
Billy Kalfus
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