This means that the action of the nth orbit, where n is an integer, must be nh, and one can then show that the radius of the nth orbit must be
From classical dynamics, Bohr knew that the total energy, kinetic plus potential, of a particle moving in a circular orbit is negative because the negative potential energy of the orbit exceeds the positive kinetic energy. The total energy is also inversely proportional to the radius of this orbit. He therefore placed the energy of the electron in the nth orbit equal to
multiplying the negative of the reciprocal of the radius by e2/2 for dimensional reasons. If the electron jumps from orbit number n to orbit number k, its energy changes by the amount or this change appears in the form of a single quantum of energy, a photon, which is either emitted or absorbed. If k is larger than n, the photon is absorbed, but if k is less than n, the photon is emitted.
Thus, one arrives at Bohr's formula for the reciprocal of the wavelength l of the photon that is emitted when the electron jumps down from orbit n to orbit k by equating the negative of the above formula to the energy of the photon h. This gives the equation
The quantity is called the Rydberg constant R, after the Swedish physicist Robert Johannes Rydberg. If k is placed equal to 2, this formula is completely equivalent to Balmer's formula, and one obtains all the Balmer lines by then placing n equal to 3, 4, 5, and so on successively (transitions of electrons from higher orbits to the second orbit).
If k is placed equal to 1 and n is given the values 2, 3, 4, and so on (transitions of the electrons to the lowest orbit), one obtains the so-called Lyman series of lines that lie in the ultraviolet. Placing k equal to 3, 4, and 5, and then letting n take on all larger integer values obtain other series of lines, such as the Paschen, the Brackett, and the Pfund series, lying in the infrared.
These series of lines constitute the entire spectrum of the hydrogen atom, but the Bohr formula gives only the major, gross features of the spectrum. Careful spectroscopic analysis shows that the actual lines have a fine structure resulting from three elements; the ellipticity of the electronic orbits, the spin of the electron, and the spin of the proton. In practice, one must also take into account any stray magnetic and electric fields that may be present, and the fact that hydrogen is generally a mixture of ordinary atomic hydrogen, heavy atomic hydrogen, and molecular hydrogen. Furthermore, all these particles are moving about randomly so that random Doppler effects, named after the Austrian physicist and mathematician Christian Johann Doppler, are also present.