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Suppose that we have a reasonable potential distribution. Must a wave function which satisfies the Schroedinger equation, always be twice differentiable at all points? Or are there counterexamples?

Riemann
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    Delta potential: https://physics.stackexchange.com/q/720463/247642 – Roger V. Dec 07 '22 at 16:08
  • It says there that the derivative can experience a jump. So then the solution is not twice differentiable. How do you then interpret the SE at that point, which contains a second derivative? – Riemann Dec 08 '22 at 20:12
  • It is a boundary condition - like other boundary conditions. Note that physically jumps and discontinuities are always aporoximations. – Roger V. Dec 09 '22 at 05:22
  • Does that mean that in an actial physical system, functions are not only twice differentiable but infinitely differentiable at all points? – Riemann Dec 09 '22 at 11:26
  • Wave functions are mathematical objects that exist only in human mind - the real physical systems do not know anything about how we describe them. So the correct question is whether wave functions that are allowed in theory can have discontinuous derivative. These are allowed, but are usually understood as limiting cases - e.g., delta potential can be viewed as a limit of a rectangular potential of width $a$ and height $V_0$: $a\rightarrow 0, V_0\rightarrow\pm\infty, V_0a=const$. It is a tedious but useful exercise to rederive the boundary condition this way. – Roger V. Dec 09 '22 at 11:32

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