Planck's concepts of quantization can be seen in Bohr's postulates 1 and 3. When the electron is in a stationary state, Bohr assumed that Newton's laws, Coulomb's law, and conservation of energy were valid. Bohr showed that the angular momentum of an electron with mass m traveling with speed v about a circular orbit of radius r is quantized as
where n is an integer and h is Planck's constant.
Also, he derived an expression for the radius of hydrogen from the electrostatic force (Coulomb's law) set equal to the centripetal force:
When n = 1, the radius is called the Bohr radius, which is the smallest orbit of hydrogen.
To find an expression for the total energy of the electron orbiting the atom, use the classical formula for total energy, then substitute r _{n }from above and v from the angular momentum to get the following:
When n = 1, the lowest energy state of the atom is called the ground state. The value of the ground state of hydrogen is −13.6 electron volts, which is in excellent agreement with the experimentally observed hydrogen ionization energy—the energy necessary to remove an electron in the ground state from an atom.
Combining this result with the equation in Bohr's postulate 2 yields
Because c = f λ, the equation becomes
From the preceding equation, the Rydberg constant may be calculated.
All of these constants are known, and the theoretical value for the Rydberg constant is the same as this derived R. This demonstrated agreement is remarkable, and it validated Bohr's postulates.
The Balmer series, found experimentally, can be explained by the Bohr model of the atom in the following way. Figure 1 is a diagram of the energy transitions possible for hydrogen.
Figure 1 
Energy transitions of a hydrogen atom, with the spectral series.


For the Balmer series, the hydrogen electron jumps from an initial excited state ( n = 3,4,5, …) to a final state at the n = 2 level. In so doing, it emits a photon with energy equal to the energy difference of the initial and final states. Other series indicated on Figure illustrate the other series of lines found by Theodore Lyman and Louis Paschen. This type of diagram is called an energy level diagram because it illustrates the discrete, allowed energy levels and the permissible transitions for the orbiting electron.
The next task was to suggest why only certain discrete energy levels are possible. De Broglie assumed that an orbit would be stable only if it contained a whole number or multiples of a whole number of electron de Broglie waves. Figure shows a representation of a standing circular wave of three wave lengths.
Figure 2 
Energy transitions of a hydrogen atom, with the spectral series.


The depicted orbit would be the permissible orbit with the quantum number of 3, that is, n = 3. This visual way of understanding quantization shows that the wave nature of matter is basic to a model of the atom. More complicated formulations of quantum physics that were developed later have built on these concepts.