Bohr’s Atom 179
the periodic motion and the frequency of the subsequent electromagnetic
radiation are identical. Finally, in classical mechanics, an electron must
orbit the nucleus in an infinite number of paths, differing by only an
infinitesimal amount of energy. In Bohr’s scheme, however, the number
of orbits is severely limited by restricting the allowed orbits to those for
which the angular momentum is equal to an integer times Planck’s
constant, h, divided by 2π. If we represent the angular momentum by L,
then L = l h/2π where l is an integer. The angular momentum of the
electron is equal approximately to the product of its momentum times the
radius of its orbit. This definition is exact if the orbit is a perfect circle.
By placing this restriction on the angular momentum, Bohr was able
to obtain Balmer’s formulae for the radiated frequencies of the hydrogen
atom. Bohr was also able to calculate Rydberg’s constant, Ry and showed
that it is simply related to the mass of the electron, me, the charge of the
electron, e and Planck’s constant, h, by the formula
Ry = 2 π^2 me e^4 /h^3. This result, in which one of the fundamental constants
of nature was related to the others, was a great success and insured the
acceptance of Bohr’s model.
This model not only explained Balmer’s formula for the hydrogen
atom but it also explained Ritz’s combination principle. Let us label the
quantum states or energy levels of the atom by E 1 , E 2 , ... , En where E 1
is the energy of the ground state, E 2 is the energy of the first excited
state, ... , and En is the energy of the (n-1)th excited state. (See Fig. 19.1).
Here, we refer to the higher energy orbits of the electrons of the atom as
excited states. These electrons have absorbed energy, but do not retain
the additional energy very long. They shortly lose the excess energy by
radiating one or more photons as they drop back to the ground state.
Bohr showed that the energy of the nth level, En, is equal to –hRy/n^2.
Bohr’s scheme correctly explained the spectroscopic rules of Balmer
and Ritz. Ritz’s combination principle follows from the existence of
atomic energy levels, the conservation of energy and Bohr’s frequency
condition. The model was severely limited, however, in the number of
predictions it could make. For instance, there was no means of
calculating the relative intensity of various spectral lines, which
experimentally differed from each other. Bohr’s theory was also unable
to predict the polarization of the light radiated by the atom. (The
polarization indicates in which direction the oscillating electric field of
the photon is aligned). Finally, not all the spectral lines indicated by the
model actually occur experimentally.