The explanation for exact spectral lines for each substance was provided by the quantum theory. In his 1913 model of the hydrogen atom Niels Bohr showed that the observed series of lines could be explained by assuming that electrons are restricted to atomic orbits in which their orbital angular momentum is an integral multiple of the quantity h /2π, where h is Planck's constant. The integer multiple (e.g., 1, 2, 3 …) of h /2π is usually called the quantum number and represented by the symbol n.
When an electron changes from an orbit of higher energy (higher angular momentum) to one of lower energy, a photon of light energy is emitted whose frequency ν is related to the energy difference Δ E by the equation ν = Δ E / h. For hydrogen, the frequencies of the spectral lines are given by ν = cR (1/ n f 2 - 1/ n i 2) where c is the speed of light, R is the Rydberg constant, and n f and n i are the final and initial quantum numbers of the electron orbits ( n i is always greater than n f ). The series of spectral lines for which n f = 1 is known as the Lyman series; that for n f = 2 is the Balmer series; that for n f = 3 is the Paschen series; that for n f = 4 is the Brackett series; and that for n f = 5 is the Pfund series. The Bohr theory was not as successful in explaining the spectra of other substances, but later developments of the quantum theory showed that all aspects of atomic and molecular spectra can be explained quantitatively in terms of energy transitions between different allowed quantum states.