At the beginning of the 20th century, physicists could not correctly describe the behavior of very small particles such as electrons, atomic nuclei, and molecules. The behavior of these particles is correctly described by a set of physical laws that we call Quantum Mechanics.

At the beginning of the century, a small number of physicists, among whom we can mention Bohr, Einstein, Born, Dirac, Schrödinger, Heisember, De Broglie, Jordan, Pauli, contributed to mathematically formalize the Theory that was practically complete at the end of the decade. 1920's

The study of Quantum Mechanics can be carried out following two different paths. The first way consists of analyzing those physical problems that Classical Mechanics is incapable of solving and that, however, were correctly interpreted by Quantum Mechanics. We can say:

- The Black Body Spectral Radiation Law
- The photoelectric effect.
- The heat capacities of solids.
- The atomic spectrum of the hydrogen atom.
- The Compton Effect

The second way that we can follow is the axiomatic one. We start from some fundamental postulates from which results on the behavior of microscopic physical systems are deduced. These results are contrasted with the experiment, being able to observe the greater or lesser agreement between the theory and the experimental data, which provides a direct measure of the goodness of the theory.

In this section we will address the study of Quantum Mechanics from the axiomatic point of view. The best known formulations are the Schrödinger formalism which is based on the wave description of matter. The Heisenberg and Dirac formalism employs algebra of vectors, operators, and matrices. Schrödinger showed that both formalisms are equivalent and can be used interchangeably.

The study of Quantum Mechanics can be complex and not very motivating at first, since it starts from some postulates that may seem strange, capricious and difficult to understand. This initial sensation should not discourage us since the application of the theory to practical problems (particle in a box, harmonic oscillator, rigid rotor) will allow us to see the simplicity with which this theory works.