= Potential-step threshold wavefunction
{title2=$\psi(x)=2A\cos(kx)\ (x\leq0),\quad\psi(x)=2A\ (x>0)$}
For a positive <step potential> and left incidence, take $E\to V_0^+$ at fixed incident <probability amplitude> $A$. The transmitted <wavenumber> tends to zero while the incident wavenumber tends to $k=\sqrt{2mV_0}/\hbar$. <Wavefunction matching at a finite potential step> gives reflected amplitude $A$ and transmitted amplitude $2A$, producing the spatial <wavefunction> in the title. The left <probability density> is $4|A|^2\cos^2kx$ and the right density $4|A|^2$. The transmitted <probability current> still tends to zero because its velocity tends to zero. Over adjacent intervals of length $2\pi/k$, the right integrated density is twice the left. These ratios use finite local integrals of a non-normalizable stationary scattering state.
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