Any one who has seen a Neon sign has observed the phenomena of spectral emission by a gas. Different types of gases glow with different colours. The visual coloration which we observe is generated from emissions at very specific radation wave lengths, characteristic of the type of gas being stimulated to emit.
This phenomena was observed in the 19th century but with no physical explanation was put forward as to why this phenomena occured. In 1885 Johann Balmer a Swiss mathematician published a paper in which he unveiled a mathematical pattern in the observed spectral line app_id=266412s for hydrogen.
Lambda = hm^{2}/(m^{2}  n^{2}).
Lambda was the observed wave length for a spectral line app_id=266412 of hydrogen, while h=365.46 nm, n=2 and and m could be set to 3, 4, 5 or 6 in order to calculate a known spectral line app_id=266412 of hydrogen. By setting m equal to 7 he successfully predicted the location of a then, as yet unobserved spectral line app_id=266412 for hydrogen. The prediction was later confirmed to be accurate.
In 1888 a Swedish physicist Johannes Rydberg presented a more general formulation of the Balmer relation that could be used to predict the location of spectral line app_id=266412s for alkali elements as well as for hydrogen. This is referred to as the Rydberg formula.
1/Lambda = R Z^{2} (1/n^{2}  1/m^{2}) Where  R is the Rydberg constant for the element;
 Z is the atomic number for the element;
 Lambda is the wavelength of light for the spectral line app_id=266412;
 n and m are integers where n is less than m.
It was in 1913 that Niels Bohr published a model that provided a physical explanation for Rydberg relation. In his model for the hydrogen atom, electrons could only take on fixed orbits around the hydrogen nucleus, where the radius of the n^{th} orbit needed to be:
r_{n}=r_{0}n^{2} where r_{0} is a combined function of the electronic charge, the mass of the electron, the permitivity of free space and Planck's constant.
Using classicial physics assumptions he calculated the energy of an electron in the n^{th} orbit to be equal to:
E_{n}=E_{0} / n^{2}
where E_{0} is another function of the electronic charge, the mass of the electron, the permitivity of free space and Planck's constant. E_{0} was calculated to be 13.6055 eV.
Now to see the relation between this model and the Rydberg formula consider the amount of energy released as an electron moves from the m^{th} down to the n^{th} level and consider as well the photovoltaic effect where Einstein demonstrated that light was emitted in energy packets equal to Planck's Constant (h) times the frequency (f) of the emitted radiation.
Delta_E = E_{0}(1/m^{2}1/n^{2}) = h.f = h.c / Lambda
OR 1/Lambda = (E_{0} / (h.c)).(1/n^{2}1/m^{2})
Once the constants are evaluated this becomes the equivalent of the Ryberg Formula.
We now understand that when:  n is the quantum number for the final energy state of the electron; and
 m is the quantum number for its initial energy state.
The energy released in the transition (Delta_E) is converted into a photon with frequency 'Delta_E / h' corresponding to the Rydberg Line which results when we substitute the quantum numbers 'n' and 'm' for the values of the integer parameters in the Rydberg formula.
