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I have a problem in understanding why we consider the continuity of the Logarithmic derivative of the wave function at the boundary of the Scattering Potential? I understand that physical arguments require the wavefunction to be continuous, but why the logarithmic derivative?

$$\lim_{r \to a}\frac{r}{\phi} \frac{d}{dr} \phi,$$ where $a$ is the length scale of the scattering potential.

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    $\begingroup$ Continuity of the logarithmic derivative guarantees continuity of both the wave function and the derivative of the wave function, which is required in all cases of non-singular scattering potentials (delta-function potentials introduce discontinuities into the derivative of the wave function, but they're pretty unphysical). $\endgroup$
    – march
    Mar 4 '16 at 0:25
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For a set of wave functions $f_1$ and $f_2$, boundary conditions are $f_1=f_2\,$ and $f_1' = f_2'$. Combining these (division makes both equations equal to one) we get $f_1/f_1' = f_2/f_2$', which is the logarithmic derivative.

Function needs to be continuous and differentiable if it is to be integrable, and therefore normalizable (yes, I know my grammar is off). The logarithmic derivative is a convenient way of combining these things. It makes many calculations much easier as you automatically reduce four equations to two (in the common case of two boundaries, e.g. a symmetric well, you get four boundary equations)

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