Solving the collective Schrödinger equation to study fission rates

Spontaneous fission proceeds from the ground state by pure quantum tunneling through a fission barrier, and because the rate is exponential in the barrier action, half-lives span some thirty orders of magnitude across the chart. The tunneling probability is almost always evaluated with the Wentzel-Kramers-Brillouin (WKB) approximation, whose accuracy has not been assessed systematically across a wide range of nuclei. This thesis provides that assessment by solving the one-dimensional collective Schrödinger equation numerically along microscopic fission paths and quantifying the deviation of the WKB half-lives from the exact ones. A dedicated solver integrates the collective Schrödinger equation, with its coordinate-dependent inertia, using a Runge-Kutta RK5(4) scheme; the transmission coefficient is extracted from the transmitted amplitude after truncating the barrier to impose plane-wave asymptotics. The physical input is the BSkG3 energy-density-functional dataset (MOCCa code): potential energy surfaces and perturbative cranking inertias, from which least-action paths are extracted using the quadrupole moment as the collective coordinate. Applied to more than 3200 nuclei with available BSkG3 fission paths at a zero-point energy of 0.5 MeV above the ground state, WKB stays within about 0.2 orders of magnitude of the exact result for the large majority, consistently across half-lives spanning more than 200 orders of magnitude. The residual offset was traced to a boundary-reflection effect at the barrier feet, produced by the physically-motivated truncation of the barrier. About 1.3% of the nuclei depart by more than half an order of magnitude, and by up to 3-4, through resonances from quasi-bound states of the isomeric well, a genuinely quantum effect that WKB cannot capture by construction. No significant correlations are found between the deviation and different fission barrier features such as A, Z, height or length. The sensitivity of the deviation to the choice of the zero-point energy was studied, revealing a distribution peaking at 0.3-0.4 orders of magnitude below about 1.5 MeV, settling near zero above roughly 2 MeV. Apart from resonances, the deviation is thus bounded below one order of magnitude already at a few MeV, making it only mildly sensitive to this choice; however, the absolute half-lives are not, given their enormous span and exponential behavior, and the proper choice of this energy remains an open discussion. Overall, the semiclassical treatment is not the limiting approximation in current large-scale SF predictions: the entire WKB-Schrödinger spread sits comfortably within the deviation of state-of-the-art EDF half-lives from experiment. The exact solver nonetheless matters for individual nuclei near a resonance, and offers a natural route to multi-dimensional fission paths, where the notion of a unique path breaks down and tunneling amplitude can branch across competing valleys.