The Bohr Atom
In the early 20th century it became apparent that classical electrodynamics and mechanics could not account for many phenomena, particularly the absorption and emission of radiation. For example the hydrogen atom emits an infinite series of sharp spectral lines. The hydrogen atom consists of a single electron revolving around the nucleus. According to classical electrodynamics an electrons accelerated in this manner should radiate continuously, gradually lose energy and finally plunge into the nucleus. Yet the orbit seems to be very stable. The electron maintains the same energy in spite of frequent collisions between atoms. By analogy, suppose a star passed near the solar system. The planetary orbits around the sun could not remain stable under these conditions. They would be permanently altered. There was no way to account for the consistency in the energy of electrons.
Another dilemma was the existence of maxima in the intensity as a function of wavelength of black-body radiation. The intensity at any temperature begins to decrease in the ultra-violet region. Classical theory predicted a continuous increase in intensity into the high-energy ultra-violet region. Max Planck was able to solve this problem by postulating discrete, finite quanta of energy. From this came the relationship between energy and frequency, the proportionality constant h being called Planck’s constant.
Einstein took this a step further and postulated that light consists of quanta (corpuscles) of energy hn. In 1905 he successfully explained the photoelectric effect with the light quanta concept. If ultra-violet light falls on a alkali metal surface in high vacuum, the metal becomes positively charged, giving off electrons. The energy of the electrons can be measured. The velocity of the electrons depends on the frequency of the light, not the intensity. The energy was found to be proportional to the frequency minus a constant characteristic of the metal. This constant was the amount of energy required to remove it from the metal. This is expressed as
where A is the energy with which the electron was held in the atom.
The structure of the hydrogen atom can be approached by assuming that the electron is held in its orbit by equating the force required to accelerate it into a circular orbit with the coulombic attraction between nucleus and electron.
The total energy of the electron is given by the sum of the potential and kinetic energies.
As mev2=e2/r, then, by substitution,
This will give the energy as a function of r. The problem here is that the radius r and thus the energy E could have any value. This is the planetary model in which the energy would continuously decrease until r goes to zero. To accommodate the known behavior of the atom, Bohr postulated that the angular momentum of the electron could have only certain values. He proposed that the angular momentum was an integral multiple of h/2p.
The resultant formula for the Bohr radius is
where n is the quantum number. This model explained the spectral lines of the hydrogen atom as transitions of the electron from one energy level to another.
DEH
is the difference in energy between the initial state I and the final state II for a
transition of the electron. The wave number is the inverse of the wave length. 
. The wave number is given by
Below is a plot of transitions from n=1 to n=2,3,4,.... with the wave numbers and energy increasing to the left. This corresponds to the absorption spectrum of hydrogen when the gas is exposed to light containing all the wave-lengths. The right-most line represents the absorption of the energy required to excite an electron from the ground state E0 to the next highest level E1at 82,259cm-1. The next line to the left corresponds to the transition from the ground state to E2 at 97,492cm-1. The lines get closer and closer to the maximum wave number of 109,678cm-1,at which point the electron has left the atom.
When heated, hydrogen gas will emit radiation. The highest energy series of the emission spectrum matches the absorption spectrum and is called the Lyman series. The emission spectrum also includes other series that don't occur in the absorption spectrum. The Lyman series is in the ultraviolet. The next series (with a similar pattern) is in the visible and ultraviolet region and is called the Balmer series. The Paschen series is in the infrared region.
The proposal that the electrons could only exist in discrete stationary states with energies E0, E1, E2, ... allowed for an interpretation of the observed emission and absorption lines in the hydrogen spectrum. The frequency of a spectral line corresponds to certain energy. That energy corresponds to a transition from one discrete energy level to another. An atom can only absorb a photon with an energy that matches the difference between two energy levels in the atom. Thus the transition E1 - E0 = hn1, E2 - E0 = hn2, ....where E0 represents the lowest energy state in the absence of any excitement. It would follow from the model that once an electron absorbed a photon with, for example, the energy to excite it from the ground state E0 to the third energy level E3, there would be several possible ways in which it gives up its excess energy. It can fall all the way back to E0 and emit hn30, or it can go to E2 and then to E0, emitting hn32 and then hn20, and so forth. The diagram below illustrates the different paths.
Then hn30 should equal hn31 + hn10, and hn32+ hn20 and hn32, + hn21 + hn10. This is the pattern that is observed. The theory was very successful but it was not the whole story. For example it could not predict the behavior of atoms in an electric field. Nor could it explain the spectral details of atoms larger than hydrogen. Such things required the methods of wave mechanics.