= Wavefunction matching at a finite potential step
{title2=$\psi(a^-)=\psi(a^+),\quad\psi'(a^-)=\psi'(a^+)$}
For a finite jump in a one-dimensional <potential energy>, with constant <mass> and no <delta potential>, both the <wavefunction> and its first spatial <derivative> are continuous. Integrating the <Time-independent Schrödinger equation> across a vanishing interval makes the jump in $\psi'$ vanish, since the integral of $(V-E)\psi$ vanishes. A jump in $\psi$ itself would produce a derivative of a <Dirac delta function> in $\psi''$, with no matching term in the equation. The second <derivative> can jump. This matching allows transmission and <quantum tunnelling>; a finite step is not a hard wall.
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