wavefunction smoothness

Wavefunction smoothness is the requirement that a physical wavefunction join up neatly: wherever the potential is finite, both the wavefunction and its slope must be continuous, with no sudden jumps or sharp kinks. This is not an arbitrary aesthetic preference but a direct consequence of the Schrödinger equation itself, which contains a second derivative of the wavefunction.

The reasoning is clean. A jump in the wavefunction would make its first derivative infinite, and a kink in the slope would make the second derivative infinite. But the equation says that second derivative is proportional to the finite quantity (V − E) times the wavefunction. A finite right-hand side cannot equal an infinite left-hand side, so where V is finite the wavefunction has to be smooth.

These smoothness conditions are the practical glue used to solve real problems. In any situation built from regions with different potentials — a step, a well, a barrier — you solve the equation separately in each region and then demand that the pieces match in value and slope at every join. Those matching requirements are what determine the allowed energies and the relative sizes of the wave on each side.