  In scattering theory, the Wigner-Smith time delay, calculated through a
phaseshift derivative or its multichannel generalization, has been demonstrated
to measure the amount of delay or advance experienced by colliding particles
during their interaction with the scattering potential. Fetic, Becker, and
Milosevic argue that this concept cannot be extended to include
photoionization, viewed as a half-scattering experiment. Their argument is
based on the lack of information about scattering phaseshifts in the part of
the wavefunction (satisfying the ingoing-wave boundary condition) going to the
detector. This article aims to test this claim by examining a photodetachment
process in a simple 1D model with a short-range symmetrical potential. Using
time-dependent perturbation theory with a dipole interaction, the relevant
wavepacket of the outgoing particle is analyzed and compared to the free
wavepacket as a reference. Our findings confirm that, indeed, a time delay
arises in the liberated fragmentation wavepacket, which is expressed as an
energy derivative of the scattering phaseshift. Our study highlights that the
source of the phaseshift content in the wavepacket arriving at the detector is
the dipole matrix element, which is a direct consequence of imposing the
ingoing-wave boundary condition. We illustrate our results through numerical
simulations of both the non-free and free wavepackets. The amount of the
observed time delay is found to be half of that appearing in a typical
scattering experiment.
