The absolute value function,

, can be used as a one-dimensional analog of the

potential in three dimensions. Two electrons constrained to one dimension (positions

and

) experience a repulsive potential proportional to

. The solutions to the Schrödinger equation for two electrons confined to a box are identical to those solutions for a single electron in a two-dimensional square well with the same but two-dimensional potential. The two-dimensional potential has

symmetry and is both symmetric with respect to the interchange of

and

* *(i.e. one-dimensional interchange of the two electrons) and symmetric with respect to the interchange

. The paired electron solutions (singlet states or states with symmetries

and

) must be symmetric with respect to the interchange of

and

. The unpaired solutions (triplet states or states with symmetries

and

) must be antisymmetric with respect to the interchange of

and

.

Variational theory (red) together with first- and second-order perturbation theory (green) approximations to the energy levels for a range of values of the potential strength parameter

in

are given in this Demonstration. The tick marks on the energy axis are labeled with the quantum numbers

for the

eigenvalues.