Here are the isosurfaces (psi=0.0055 red, psi=-0.0055 blue) of the fifteen lowest energy electron eigenmode wave functions of a hydrogen atom. Probability density of the electron equals [abs(psi(x,y,x)]^2, and here the wave functions have been chosen to be real. The 15 lowest eigenvalues (energies E) and eigenmodes (wave functions) are displayed. Also shown are quantum numbers (n,l), which were determined a posteriori. The calculations were performed in a 30 a.u. cube with homogeneous Dirichlet boundary conditions. The finite element mesh consisted of 40^3 bilinear elements.

• 1st eigenmode, E=-0.4865: jpg (n=1,l=0)
• 2nd eigenmode, E=-0.1268: jpg (n=2,l=1)
• 3rd eigenmode, E=-0.1268: jpg (n=2,l=1)
• 4th eigenmode, E=-0.1268: jpg (n=2,l=1)
• 5th eigenmode, E=-0.1215: jpg (n=2,l=0)
• 6th eigenmode, E=-0.0527: jpg (n=3,l=2)
• 7th eigenmode, E=-0.0527: jpg (n=3,l=2)
• 8th eigenmode, E=-0.0527: jpg (n=3,l=2)
• 9th eigenmode, E=-0.0501: jpg (n=3,l=2)
• 10th eigenmode, E=-0.0501: jpg (n=3,l=2)
• 11th eigenmode, E=-0.0461: jpg (n=3,l=1)
• 12th eigenmode, E=-0.0461: jpg (n=3,l=1)
• 13th eigenmode, E=-0.0461: jpg (n=3,l=1)
• 14th eigenmode, E=-0.0395: jpg (n=3,l=0)
• 15th eigenmode, E=-0.0174: jpg (n=4,l=3)

The system is described by Scrödinger equation (i.e., non-relativistic approximation, no spin). The Coulomb potential V=-1/r was slightly modified near the origin in order to remove the divergence. The so-called Hartree atomic units are employed: Planck constant hbar = electron charge e = electron mass m_e = Bohr radius a0 = 1. The atomic unit a.u. or Bohr radius a0 is in SI units 0.0529 nm. In ideal hydrogen atom the energies are E(n) = -1/(2n^2), where the main quantum number n is a positive integer. In SI units this corresponds to E = - 13.605 eV / n^2 (infinite proton mass) or E = - 13.595 eV / n^2 (physical proton mass).

However, because here the potential is not exactly of Coulomb type, the accidental degeneracy of hydrogen atom is lifted: energy E depends also on the orbital quantum number l, E = E(n,l). Furthermore, the boundary conditions - atom in a finite cube - partially break the l-degeneracy, which below can be observed from the fact that energies of eigenmodes 6-8 and 9-10 are not identical. (Actually, this system, a finite cube with finite elements of similar cubic shape, is special because here the numerical method does not further break the energy degeneracy.)