  Non-Archimedean mathematics is an approach based on fields which contain
infinitesimal and infinite elements. Within this approach, we construct a space
of a particular class of generalized functions, ultrafunctions. The space of
ultrafunctions can be used as a richer framework for a description of a
physical system in quantum mechanics. In this paper, we provide a discussion of
the space of ultrafunctions and its advantages in the applications of quantum
mechanics, particularly for the Schr\"{o}dinger equation for a Hamiltonian with
the delta function potential.
