WOLFRAM|DEMONSTRATIONS PROJECT

2D Analog for Coulomb Degeneracies

It is a remarkable fact that the nonrelativistic Coulomb (hydrogenlike atom) have sets of degenerate eigenfunctions (

(3s

), and so on, whose members can exhibit completely different geometrical shapes. It was first shown by Vladimir Fock in 1935 that the Coulomb problem has the same energy spectrum as a free particle moving on the surface of a four-dimensional hypersphere. Thus the bound states of the hydrogen atom exhibit the same

SO(4)

symmetry as rotations in four dimensions. For the physical three-dimensional problem, this is a "hidden symmetry", which can be attributed to the existence of an additional constant of the motion—the Runge–Lenz vector. In principle, the symmetry can be exhibited by stereographic projection of degenerate hyperspherical harmonics onto a three-dimensional space.

This Demonstration considers a more accessible "toy" model for a two-dimensional hydrogen atom, in which the degenerate orbitals can be obtained by stereographic projection of spherical harmonics on a three-dimensional sphere onto a plane. The sphere can be rotated about its vertical axis by an angle of inclination

. This generates different linear combinations of tesseral harmonics

ℓm

(θ,ϕ)

. For example,

(θ,ϕ)=

(cosθ)

can be transformed into

(θ,ϕ)=cosϕ

(cosθ)

when

is varied from 0 to

π/2

. Positive regions of the functions are shown in blue, and negative regions in green. The tesseral harmonic is projected onto the plane tangent to the south pole of the sphere. For clarity, the sphere and plane are separated in the graphic. The contours in the plane are also multiplied by an exponential factor

-r/2

to show a greater range of magnitudes. The key aspect of the Demonstration is how the nodes of the tesseral harmonics, separating positive and negative regions of the wavefunction, project onto the plane. For

Θ=0

, we obtain a two-dimensional

-orbital, with the great-circle node of the tesseral harmonic transformed into the radial node of the

-orbital. For

Θ=π/2

, we obtain the angular node characteristic of a

-orbital. Intermediate values of

give various

hybrid orbitals, which are also degenerate in the free atom.