The emission of scission neutrons from fissioning nuclei is of high practical interest. To study this process we have used the sudden approximation and also a more realistic approach that takes into account the scission dynamics. Numerically, this implies the solution of the bi-dimensional Schrödinger equation, both stationary and time-dependent. To describe axially symmetric extremely deformed nuclear shapes, we have used the Cassini parametrization. The Hamiltonian is discretized by using finite difference approximations of the derivatives. The main computational challenges are the solution of algebraic eigenvalue problems and of linear systems with large sparse matrices. We have employed appropriate procedures (Arnoldi and bi-conjugate gradients). The numerical solutions have been used to evaluate physical quantities, like the number of emitted neutrons per scission event, the primary fragments' excitation energy and the distribution of the emission points.