Quantum Chemistry for Atoms, Molecules and Solids: Relativistic Energy Levels for Hydrogen Atom

schematic energy-level diagram

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In nonrelativistic quantum mechanics, the energy levels of the hydrogen atom are given by the formula of Bohr and Schrödinger, , expressed in hartrees (assuming the appropriate correction for the reduced mass of the electron). The energy depends only on the principal quantum number and is -fold degenerate (including electron spin). In Dirac's relativistic theory, this degeneracy is partially resolved and the energy is found to depend as well on the angular-momentum quantum number . To second order in the fine-structure constant , the hydrogen energy levels are given by . In Dirac's theory, levels such as and remain degenerate. The discovery of the Lamb shift showed that these two levels were actually split by 1057.8 MHz. This was a major stimulus for the development of quantum electrodynamics in the 1950s. The Lamb shift, significant only for -states), raises the energy by approximately . The relativistic and radiative correction to hydrogen energy levels can therefore be written , to third order in . In this Demonstration, you can conceptually vary the fine-structure constant from 0 to its actual value, or equivalently the speed of light from to 1 (meaning m/s), to show the transition from nonrelativistic to relativistic energies for quantum numbers , and . The energies are expressed in MHz (1 hartree = MHz).