Two Electrons in a Box: Wavefunctions

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The one-dimensional analog of the Coulomb potential, , is the absolute value function,
. Two electrons (at locations
and
) restricted to a one-dimensional
-unit-width box repel each other with a potential proportional to
. The one-dimensional Schrödinger equation is mathematically identical to the corresponding two-dimensional Schrödinger equation for a single electron moving in a
-unit-square box experiencing the potential
with
a strength parameter. The potential has
symmetry and the wavefunctions must share this symmetry. Approximate wavefunctions are found using the variational method with appropriate linear combinations of the 49 basis functions
,
. The Demonstration shows the potential together with either the wavefunction
or its square
for various values of the potential parameter. If the display choice is for
, the one-dimensional density function
is shown in red on the
surface of the displayed cube.
Contributed by: M. Hanson (July 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The potential function has a twofold rotation axis located at
. Functions that are symmetric with respect to this operation are designated as having
symmetry; if antisymmetric,
symmetry. The potential has a mirror plane,
, containing the
axis and the line
. The paired electron or singlet states must be symmetric with respect to
. The unpaired electron or triplet states are antisymmetric with respect to
. The
and
states are each subdivided into
,
,
, and
states. The states with subscript 1 are symmetric with respect to
. The states with subscript 2 are antisymmetric with respect to
. There is another mirror plane,
, containing the line
and the
axis.
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