  The physical origin is investigated of Robin boundary conditions for wave
functions at an infinite reflecting wall. We consider both Schr\"odinger and
phase-space quantum mechanics (a.k.a. deformation quantization), for this
simple example of a contact interaction. A non-relativistic particle moving
freely on the half-line is treated as moving on the full line in the presence
of an infinite potential wall, realized as a limit of a Morse potential. We
show that the wave functions for the Morse states can become those for a free
particle on the half-line with Robin boundary conditions. However, Dirichlet
boundary conditions (standard walls) are obtained unless a mass-dependent fine
tuning (to a reflection resonance) is imposed. This phenomenon was already
observed for piece-wise flat potentials, so it is not removed by smoothing. We
argue that it explains why standard quantum walls are standard. Next we
consider the Wigner functions (the symbols of both diagonal and off-diagonal
density operator elements) of phase-space quantum mechanics. Taking the
(fine-tuned) limit, we show that our Wigner functions do reduce to the expected
ones on the half-line. This confirms that the Wigner transform should indeed be
unmodified for this contact interaction.
