Two Electrons in a Box: Energies
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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 
.
Contributed by: M. Hanson (August 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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
Permanent Citation
Time-Dependent Superposition of Particle-in-a-Box Eigenstates
Porscha McRobbie and Eitan Geva
Variational Method Applied to Stepped-Infinite-Square-Well Energies
M. Hanson
Time-Dependent Superposition of 2D Particle-in-a-Box Eigenstates
Porscha McRobbie and Eitan Geva
Degenerate Energy States
Douglas Kellis (Boise State University)
Energy Spectrum for a Finite Potential Well
Adam Strzebonski
Energy Levels of a Morse Oscillator
S. M. Blinder
Entanglement between a Two-Level System and a Quantum Harmonic Oscillator
Kwan-yuet Ho
Exact Numerical Solutions for the Stepped-Infinite-Square Well
M. Hanson
Quantum Scattering by a Rigid Sphere
S. M. Blinder
Scattering by a Symmetrical Eckart Potential
S. M. Blinder
