Is there an accurate proof why wave function is continuous? I mean wave function as coefficient of eigen states when representing a state with eigen states, so, I am asking continuity of $\psi(x)=<\psi|x>$.
I understand that it is a function of $\mathcal{L}^2$, but how does it lead to $\psi(x)$ being of $C^2$?
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Zjjorsia
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It's rather math question than physics. Schrödinger equation calculates derivatives of $\psi()$ in time and coordinates domain. For doing that, not only wave function itself must be continuous, but also continuously differentiable, i.e. it's derivatives must be continuous as well. It's just how function application and calculus works. But sure you can check some proofs. This one was about continuous derivatives. – Agnius Vasiliauskas Oct 05 '23 at 13:44
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Possible duplicates: https://physics.stackexchange.com/q/262671/2451 and links therein. – Qmechanic Oct 05 '23 at 15:58
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Not a proof, no. But a reason.
The momentum and energy operators are derivatives of $\psi$. As $\psi$ approaches discontinuous, the derivatives get large, and expectation values of $p$ and $E$ approach infinity.
The form of the momentum operator follows from momentum being the generator of translation. See https://en.wikipedia.org/wiki/Momentum_operator
The form of the Energy operator can be derived from the fact that $\psi$ is a wave function. See https://en.wikipedia.org/wiki/Energy_operator.
mmesser314
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