Stationary States in a Potential Well

1 9. Conclusions Bibliography Glossary Bohr hypotheses: Set of hypotheses that Bohr introduced to explain the stability of the atom. Coulomb potential: Potential that holds the electrons attached to the nucleus. Creation and Annihilation operators: Mathematical objects that create or annihilate particles. Dynamical phase: It is the factor that contains the time dependence of the wave function. Hydrogen atom: The simplest atom in nature, consisting of one electron revolving around a proton. Laplace operator: A partial differential operator that contains the second partial derivatives with respect to the space coordinates. Magnetic quantum number: A quantum number that is associated to the direction of projection of the of the angular momentum Orbital quantum number: The number associated to the quantization of the orbital motion Principal quantum number: The number associated to the radial solution of the Schrödinger equation and defines the energy of the allowed orbits. Quantized electron orbits: The electrons can move in the atom only in very specific orbits. Quantum Harmonic oscillator: The quantum analogous model for a potential whose restoring force is proportional to the displacement. Quantum Mechanics: Theory of the laws that rule the microscopic world, i.e. at the atomic sizes and less. Rutherford planetary atomic model: model to describe the atom in which the nucleus is at the center and the electrons move around it. Schrödinger equation: The differential equation that describes the movement of atomic particles. Square well potential: A one-dimensional model of a potential with vertical walls Standing wave: The solutions to the Schrödinger equation that are stationary. Zeeman effect: The effect of splitting the electronic energy levels when an atom is immersed in a magnetic field. Summary In the early days of the 20th century a set of important observations in atomic and molecular physics could not be explained on the basis of the laws of classical physics. One of the main findings was the emission of light by excited atoms with very particular frequencies. To explain those findings a new development in physics was necessary, now known as quantum mechanics. In particular, the concept of stationary states was introduced by Niels Bohr, in 1913, in order to explain those observations and the stability of atoms. According to E.C. Kemble (1929), the existence of discrete atomic and molecular energy levels brought into mechanics a new kind of atomicity superposed on the atomicity of electrons and protons. We review here in a historical context …