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.
![[Snapshot]](HTMLImages/index.en/thumbnail_1.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_2.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_3.gif)
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