Two Electrons in a Box: Energies

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.



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Energy level estimates for an infinitely deep -unit-square well perturbed by the potential are made using variational theory (red) and first- and second-order perturbation theory (green). Energy levels are shown for the parameter in the domain .
The variational basis functions are the normalized forms of the nontrivial members of the set with and limited to a maximum of . The and state symmetries contain the plus sign while the and state symmetries contain the minus sign. For the and symmetries the sum is even. For the and symmetries the sum is odd. The zero of the potential is defined as the bottom of the box.
Snapshot 1: if the full range of energies is displayed, there is some loss of detail
Snapshot 2: judicious choices for options "plot minimum energy" and "plot maximum energy" let you isolate any region of interest
Snapshot 3: second-order perturbation theory can be a good approximation
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