semiclassical model of the hydrogen atom in which quantized values of angular momentum lead to the observed discrete energy levels. A full quantitative derivation of the observed spectrum was derived by Wolfgang Pauli in 1926 following development of quantum mechanics by Werner Heisenberg and others. Methods of production The only truly stable state of a hydrogen-like atom is the ground state with n = 1. The study of Rydberg states requires a reliable technique for exciting ground state atoms to states with a large value of n. Electron impact excitation Much early experimental work on Rydberg atoms relied on the use of collimated beams of fast electrons incident on ground-state atoms. Inelastic scattering processes can use the electron kinetic energy to increase the atoms' internal energy exciting to a broad range of different states including many high-lying Rydberg states, . Because the electron can retain any arbitrary amount of its initial kinetic energy, this process results in a population with a broad spread of different energies. Charge exchange excitation Another mainstay of early Rydberg atom experiments relied on charge exchange between a beam of ions and a population of neutral atoms of another species, resulting in the formation of a beam of highly excited atoms, . Again, because the kinetic energy of the interaction can contribute to the final internal energies of the constituents, this technique populates a broad range of energy levels. Optical excitation The arrival of tunable dye lasers in the 1970s allowed a much greater level of control over populations of excited atoms. In optical excitation, the incident photon is absorbed by the target atom, resulting in a precise final state energy. The problem of producing single state, mono-energetic populations of Rydberg atoms thus becomes the somewhat simpler problem of precisely controlling the frequency of the laser output, . This form of direct optical excitation is generally limited to experiments with the alkali metals, because the ground state binding energy in other species is generally too high to be accessible with most laser systems. For atoms with a large valence electron binding energy (equivalent to a large first ionization energy), the excited states of the Rydberg series are inaccessible with conventional laser systems. Initial collisional excitation can make up the energy shortfall allowing optical excitation to be used to select the final state. Although the initial step excites to a broad range of intermediate states, the precision