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.