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WOLFRAM|DEMONSTRATIONS PROJECT

Two Electrons in a Box: Wavefunctions

​
interaction potential height, V
9
select a symmetry type
singlet
A
1
states
singlet
B
1
states
triplet
A
2
states
triplet
B
2
states
select state
{1,1}
display choice
ψ
ψ × ψ
The one-dimensional analog of the Coulomb potential,
1/r
, is the absolute value function,
|x|
. Two electrons (at locations
x
and
y
) restricted to a one-dimensional
π
-unit-width box repel each other with a potential proportional to
|y-x|
. The one-dimensional Schrödinger equation is mathematically identical to the corresponding two-dimensional Schrödinger equation for a single electron moving in a
π
-unit-square box experiencing the potential
V(1-|y-x|)
with
V
a strength parameter. The potential has
C
2v
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
sin(mπx)sin(nπy)
,
1≤m,n≤7
. The Demonstration shows the potential together with either the wavefunction
ψ(x,y)
or its square
ψ×ψ
for various values of the potential parameter. If the display choice is for
ψ×ψ
, the one-dimensional density function
1
∫
0
2
ψ
(x,y)dy
is shown in red on the
y=0
surface of the displayed cube.
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