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In general, for given choice of boundary conditions, it is important to adjust the action with compatible boundary terms/total divergence terms, in order to ensure the existence of the variational/functional derivative. As OP observes, the problem is (when deriving the Euler-Lagrange expression) that the usual integration by part argument fails if the ...

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To explicitly verify this, one solves the problem for a box of finite depth $V_0$. If you additionally assume the wavefunction and its first derivative to be continuous across the potential step, the solution becomes a matter of Solve the Schrödinger equation in the distinct regions in- and outside of the box. Match $\phi$ and $\phi'$ at the potential ...

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This is a typical case of a problem which is clear enough physically speaking, but mathematically messy. Where rigorous results are folkloristically employed to achieve some result which, actually, would need much more care in deriving it... But presumably, mathematical details would not change the physical picture. Here the difference between theoretical ...

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The uniqueness theorem actually stems from differential equation mathematics. If you have a complete set of 1) Diff eq. 2) Boundary conditions Then you have a single solution. This means also that if you found a solution that fulfils these conditions, it is the only solution you have. In the so called mirror problem what you ...

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I) The substitution $f=r\psi$ is the standard substitution to get a radial 3D problem to resemble a 1D problem, see e.g. Ref. 1. II) From the perspective of the normalization of the wavefunction $\psi(r)$, a $1/r$ singularity of $\psi(r)$ at $r=0$ is fine because $|\psi(r)|^2$ is suppressed by a Jacobian factor $r^2$ coming from the measure in 3D spherical ...

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