Final answer:
An antisymmetrical electron pair wave function forms due to the Pauli exclusion principle when two electrons in a box are allowed to interact. The combination of their individual wave functions must be antisymmetric on exchange to ensure compliance with the principle and the indistinguishability of electrons.
Step-by-step explanation:
When two electrons in a box originally separated by an infinite potential barrier are allowed to interact, the formation of an antisymmetrical pair wave function due to their interaction must be considered in accordance with Pauli's exclusion principle. The principle states that no two electrons can have the same set of quantum numbers. Initially, the electrons may be described by ground state wave functions, and upon removal of the barrier, to ensure the electrons are indistinguishable and abide by fermion exchange symmetry, an antisymmetric combination of the individual wave functions must be constructed.
Exchange symmetry inherent to electrons requires the total wave function of two electrons to be antisymmetric on exchange. The spatial and spin parts of the wave function must be combined in such a way that their overall wave function remains antisymmetric, ensuring compliance with the Pauli exclusion principle. When the electrons are confined to a potential box, the spatial part of their wave functions is described using sinusoidal functions (or exponential decay functions, depending on the context and the proximity of the walls).
The antisymmetric wave function implies that when one electron is found in a particular quantum state, the other must occupy a different state. This antisymmetry leads to fundamental behaviors in quantum systems such as the electronic structure of atoms and the formation of chemical bonds. Interacting electrons will self-organize into an antisymmetric state due to the fundamental requirements imposed by quantum mechanics and the intrinsic properties of electrons as fermions.