Two Electrons in a Box: Wavefunctions

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.

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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.
The set of basis functions can be divided into four sets with symmetry appropriate to the potential:
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