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As you said, you want that your wave function is normalized. So, it is not correct that the solution outside the wall is $$\psi=Cexp(\mu x)+Dexp(-\mu x).$$ In fact this function diverges for $x\rightarrow\pm\infty.$ In order to keep your wave function normalized, you must use one solution for $x<-a$ and another one for $x>a.$ For $x<-a$, you ...

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From a physics point of view, you have some kind of mass-diffusion problem. It is true that a mathematician will name it "heat equation" because the classical problem with the operator $\partial_t - \Delta$ is the heat equation. Homogeneous Neumann BC is indeed appropriate to model a "no flux" condition at the boundaries, be it a flux of heat or of mass. A ...

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Introduction $\def\ph{\varphi}\def\vr{\vec{r}}\def\eps{\varepsilon}\def\rmr{{\rm r}}\def\pd{\partial}\def\l{\left}\def\r{\right}\def\ltag#1{\tag{#1}\label{#1}}\def\div{\operatorname{div}}\def\grad{\operatorname{grad}}\def\nR{{\mathbb{R}}}$ You do not need a combination of a surface charge and a volume charge to re-produce a point-charge field outside a ...

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Here is our interpretation of OP's question: We are essentially talking about the asymptotic form of positive energy scattering states to the time-independent Schrödinger equation (TISE) for the two free regions $x \to \pm \infty$. (As OP notes, scattering states are not normalizable, and therefore do not belong to the Hilbert space. Nevertheless they can be ...

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