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\begin{document}

 In 1913, Bohr obtains a general scheme for the explanation of spectra by completing the quantum hypothesis of light by a new postulate incompatible with classical notions: the \htmladdnormallink{quantization}{http://planetphysics.org/encyclopedia/WignerWeylMoyalQuantizationProcedure.html} of the energy levels of atoms.

According to Bohr, the atom does not behave as a classical system capable of exchanging energy in a continuous manner. It can exist only in a certain number of \emph{stationary states} or quantum states each having a well-defined energy. One says that the energy of the atom is quantized. It can vary only by jumps, each jump corresponding to a transition from one state to another.

This postulate allows us to specify the mechanism of absorption or emission of light through quanta. In the presence of light an atom of energy $E_i$ may undergo a transition to a state of higher energy $E_j(E_i)$ by absorbing a photon $hv$ provided that the total energy is conserved, namely

$$ hv = E_j - E_i $$

Similarly, the atom can undergo a transition to a state of lower energy $E_k(<E_i)$ by emitting a photon $hv$ whose frequency satisfies the relation

$$ h v = E_i-E_k $$

If the atom finds itself in its lowest energy state (ground state) it cannot radiate and remains stable.

In this way an explanation is found for the existence of spectral lines characteristic of each atom and satisfying the Rydberg-Ritz combination principle: the spectral terms are equal, to within a factor of $h$, to the energies of the quantum states of the atom. In particular, for the case of the hydrogen atom, one rediscovers the Balmer \htmladdnormallink{formula}{http://planetphysics.org/encyclopedia/Formula.html} by assuming that the energy levels are given by the formula

\begin{equation}
E_n = -h \frac{R}{n^2}  \,\,\,\,\,\,\, (n=1,2,3,...,\infty)
\end{equation}

Another confirmation of the quantization of atomic energy levels is furnished by the experiment of Franck and Hertz on the inelastic \htmladdnormallink{collisions}{http://planetphysics.org/encyclopedia/Collision.html} between electrons and atoms (1914). The experiment consists in bombarding atoms by monoergic electrons and in measuring the \htmladdnormallink{kinetic energy}{http://planetphysics.org/encyclopedia/KineticEnergy.html} of the scattered electrons. From this one deduces by subtraction the quantity of energy absorbed in the collision by the atoms. Let $E_0, E_1,E_2, ...$ be the sequence of quantized energy levels of the atoms, $T$ the kinetic energy of the incident electrons. Under the conditions of the experiment, the atoms of the target are practically all in their ground state. As long as $T$ lies below the difference $E_1-E_0$ between the energy of the ground state and that of the first excited state, the atom cannot absorb energy and all collisions are elastic. As soon as $T > E_1 - E_0$, inelastic collisions can occur in which the electron loses a quantity of energy equal to $E_1-E_0$ and the atom goes into its first excited state. This is exactly what is found experimentally. One similarly observes collisions with excitation of the second excited state as soon as $T>E_2-E_0$, and so on.

Hence the quantization of atomic energy levels appears as experimental fact. This property is not peculiar to atoms. Progress of experimentation, especially in the field of spectroscopy, has shown that quantization is found in the case of molecules and of more complex systems of particles as well. We thus face a very general property of matter which classical corpuscular theory is unable to explain.

\subsection{References}

[1] Messiah, Albert. "Quantum Mechanics: \htmladdnormallink{volume}{http://planetphysics.org/encyclopedia/Volume.html} I." Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public domain work [1].

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