# Two Electrons in a Box: Wavefunctions

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 (x,y)dy is shown in red on the surface of the displayed cube.

1/r

|x|

x

y

π

|y-x|

π

V(1-|y-x|)

V

C

2v

sin(mπx)sin(nπy)

1≤m,n≤7

ψ(x,y)

ψ×ψ

ψ×ψ

1

∫

0

2

ψ

y=0