Manifestation of the divergence between antisymmetrized-molecular-dynamics and container pictures of $^{9}$Be via ${}^{9}$Be($p,pn$)${}^{8}$Be knockout reaction.

The $^{9}$Be nucleus is famous for its nuclear-molecular-orbit structure as a consequence of the $\alpha$-cluster formation. The valence neutron in the ground state of $^{9}$Be is theoretically predicted to occupy the $\pi$-orbital but the properties of the molecular orbitals have not been established by the experimental observables. Furthermore, divergence exists between the theoretical descriptions of ${}^{9}$Be from different perspectives, $\textit{i.e.}$, the antisymmetrized molecular dynamics and the container picture of cluster dynamics. To settle this discrepancy, we propose to probe the spatial extension of the $\pi$-orbit neutron in the $^{9}$Be nucleus via the ${}^{9}$Be($p,pn$)${}^{8}$Be knockout reaction. The $^{9}$Be target is described by two different sophisticated microscopic models, the antisymmetrized molecular dynamics (AMD) and Tohsaki-Horiuchi-Schuck-Ropke (THSR) wave functions. The corresponding reduced width amplitudes (RWAs) in the $^{8}$Be$+n$ channel are extracted from both the AMD and THSR wave functions, and they are found to describe drastically different valence-nucleon motion, which shows the theoretical ambiguity in describing the $\pi$-orbitals in $^{9}$Be. Using the RWAs as input, the physical observables of the ${}^{9}$Be($p,pn$)${}^{8}$Be knockout reaction are predicted by the distorted-wave impulse approximation (DWIA) framework. The magnitudes of the triple-differential cross sections (TDX) are found to be highly sensitive to the RWA input. It is concluded that the ${}^{9}$Be($p,pn$)${}^{8}$Be knockout reaction could provide a feasible probing for the spatial extension of the $\pi$-orbital in the $^{9}$Be nucleus.