The Schrödinger equation is E = p²/2m turned into an operator equation; the Klein–Gordon and Dirac equations do the same for the relativistic relation E² = p²c² + m²c⁴. All three give eigenvalues and eigenfunctions, but the dispersion relation changes, negative-energy solutions (antiparticles) and spin (four-component spinors) appear, and even the way boundary conditions are imposed changes. In ① you compare the dispersion relations of the three equations and the Dirac spinor, in ② you compare the levels of an infinite well (box), and in ③ you see how the density of states, Fermi energy, pressure, and exchange energy of the free-electron gas become “relativistic” with increasing density (negligible in metals, but in white dwarfs it becomes the Chandrasekhar limit that decides the fate of the star); in ④ you make the Thomas–Fermi model, the origin of density functional theory, relativistic (Vallarta–Rosen). For the equations and derivations, see the explanation page (in Japanese).
Companion page: Explanation (in Japanese) — Relativistic Quantum Mechanics for Readers Who Know Only Nonrelativistic Quantum Mechanics (Relativistic-QM-derivation.html) | Related: Elementary Particle Simulator, Lie Group Simulator | Mochizuki Research Group, Tokyo University of Science