It is the most widely accepted and used model in the study of the structure of atoms, molecules and the chemical reactivity of elements, due to the precision of its predictions and its relative simplicity.
This model is the evolution of several previous atomic models, such as the Rutherford model and the Bohr – Sommerfeld model, considered classical or semi-classical models.
Currently, there are theoretically more complete models than Schrödinger’s quantum mechanical model, such as the Dirac-Jordan model, which incorporates special relativity and is based on the Dirac wave equation. In this equation, spin, property of the electrons mentioned at the beginning, appears naturally.
There are also models based on quantum field theory, applied in high energy physics. These models are excellent for predicting the creation and annihilation of fundamental particles, the goal of this field of physics.
It is noteworthy that the most sophisticated theories converge to the same results as those of the Schrödinger equation, especially for light atoms.
Characteristics of the current atomic model
In the current view of the atom, based on non-relativistic quantum mechanics, there is no room for the concept of electronic orbits in the style of planetary systems.
However, the most widespread image of the atom is still that of a central positive nucleus and some points of negative electrical charge (the electrons), rotating in perfectly defined orbits around the central nucleus. But despite its roots, it no longer matches the current atomic model.
Figure 2 shows the old image and the current image of a helium atom in its lowest energy state (n = 1 and l = 0 level).
The classic image is useful to see that the nucleus contains two protons and two neutrons. Guaranteeing the neutrality of the atom, there are two electrons occupying the same energy level.
The rest is a far cry from reality, as the scale of the nucleus doesn’t even match that of the atom: the nucleus is 1/100000 times the size of the atom, but that’s where the atomic mass is concentrated.
Wave – particle duality
Classical mechanics states that every material particle has an associated wave, called the wave function . This is Louis De Broglie’s famous wave-particle duality .
In the current atomic model, the behavior of the electron at the scale of atoms is predominantly undulatory, while at the macroscopic level, like the electrons moving in the cathode ray tubes of old televisions, the corpuscular behavior predominates.
On the other hand, with photons, it is the opposite, in most optical phenomena (at the macroscopic level) they have a fundamentally undulatory behavior. And when they interact with atoms of matter, they behave like particles.
Because of this, the electrons around the nucleus are scattered in zones called atomic orbitals , whose shape and extent will depend on the energy level of the electrons and the angular momentum.
Both the energy and angular momentum of the electron around the nucleus have certain allowable values; therefore, they are quantized .
Schrödinger’s wave equation predicts which values of energy and angular momentum are allowed, as well as the wave function associated with each energy level and momentum.
The mathematical square of the wavefunction determines the orbitals, that is, the areas around the nucleus where electrons are most likely to be found.
To get a scaled image of the current atomic model, imagine that an atom has a diameter like a football field. The nucleus would be like an ant in the center of the field, but surprisingly 99.9% of the atomic mass.
On the other hand, electrons would be like phantom players diffused across the field, more likely to be found in midfield.
There are some alignments or allowable ways to occupy the field, which depend on the energy of the players (the electrons) and the amount of “rotation” or rotation around the center.
Postulates of the current atomic model
1.- The electron is characterized by its mass m, its spins s and for being the particle carrying an elementary negative charge (-e).
2.- Electrons have a dual behavior, simultaneous with wave particles, but depending on their energy and the scale of the phenomenon, one can be more predominant than the other.
3.- The electrons surround the positive atomic nucleus, in order to guarantee the electrical neutrality of the atom. Therefore, the number of electrons is equal to the number of protons; This is the atomic number , which gives the chemical and physical characteristics of each element.
4.- The interaction between electrons and nucleus is modeled by means of the Coulomb electrostatic potential V ( r ), to which the potential energy term is incorporated in the Hamiltonian operator.
5.- The term kinetic energy in the Hamiltonian operator is an operator constructed from the linear momentum operator, being the same:
p = – i ∂ / ∂ r
Where ħ is Planck’s constant divided by 2π.
6.- The Hamiltonian operator H = ( p⋅p) / 2m – and V ( r ) acts on the electron wave function Ψ ( r ).
7.- As the stationary solutions of the electronic wave function are searched, the time-independent Schrödinger equation is used:
H Ψ ( r ) = E Ψ ( r )
Where E represents the total energy of the electron.
8.- In atoms with several electrons, the interaction between them is not taken into account.
9.- When it comes to atoms with many electrons, the orbitals of the outermost electrons are modeled by the potential of the nucleus tracked by the innermost electrons, known as the Debye potential.
10.- Equation (7) has a solution for some discrete energy values, so that the famous Planck quanta arise naturally from the solutions of the Schrödinger equation.
11.- For each discrete value of E there is a wave function. But some solutions are degenerate depending on the value of the angular momentum L.
12.- The wave function is the product of a radial function, the azimuth function and the polar function.
13.- This wave function determines the allowed regions for the electron. The square of the wavefunction is the probability density of finding the electron at a given position, seen from the center of the atomic nucleus.
14.- The spin does not appear in the Schrödinger equation, but is incorporated into the atomic model using the Pauli principle:
The electron is a fermion with two possible states of rotation + ½ and -½.
Therefore, the same state characterized by the quantum numbers n, l, m of the Schrödinger equation can be occupied at most by 2 electrons with opposite rotations. In this way, the rotation becomes the fourth quantum number.
Influential Scientists in the Current Atomic Model
It sounds incredible, but most of the physicists who contributed to the current atomic model appear in the same photo. They met at the famous conferences sponsored by Ernest Solvay, a chemist and industrialist of Belgian origin, who became famous in the world of science.
They began to be carried out in 1911 and brought together the greatest scientists of the time, among them practically all those who contributed to the current atomic model.
The most famous of these conferences was held in Brussels in 1927 and this historic photograph was taken there:
- Peter Debye
- Irving Langmuir
- Martin Knudsen
- Auguste Piccard
- Max Planck
- William Lawrence Bragg
- Émile Henriot
- Paul Ehrenfest
- Marie Curie
- Hendrik Anthony Kramers
- Édouard Herzen
- Hendrik Antoon Lorentz
- Theophile of Donder
- Paul Adrien Maurice Dirac
- Albert Einstein
- Erwin Schrodinger
- Arthur Holly Compton
- Jules-Emile Verschaffelt
- Paul Langevin
- Louis-Victor de Broglie
- Charles-Eugene Guye
- Wolfgang Pauli
- Werner Heisenberg
- Max born
- Charles Thomson Rees Wilson
- Ralph Howard Fowler
- Léon Brillouin
- Niels Bohr
- Owen Williams Richardson