Metodos Matematicos da Fisica

From a classical point of view, an electron and a proton are bodies of unknown shape and size, but with well-known inertial (or gravitational) and electric charges. While their electric charges differ only in sign, their gravitational charges are quite different (the proton is 1836 more intense).

Coulomb established the electric force inspired by Newton's gravitational force: both have a strength varying with the inverse square of the distance between source and test body and are directly proportional to the product of their charges. For atomic distances (of the order of an Angstrom), the electric force between a proton (source) and an electron (test) is about 80 billionth of a Newton, \(10^{39}\) stronger than the gravitational force. The hydrogen atom is made up of a proton and an electron. The mutual (electrical) attraction between a proton and an electron at a distance of $0.53\,\mathring{A}$ ($0.53\times 10^{-10}\,$m; Bohr radius) has an intensity of only $8.24\times 10^{-8}\,$N. How can such a tiny force be responsible for the integrity of one of the most important atoms in this universe?

The Newtonian programme establishes the existence of a force resulting from the spatial variation of potential energy in both the gravitational and electrical cases. This potential energy can be interpreted as the energy expended to bring a test body from very far away to a certain distance from the source. For the hydrogen atom, this energy, $4.36\times 10^{-18}\,$J (or $27.2\,$eV; electron volt), is essentially electrical.

How is a hydrogen atom formed? If possible, is it enough to bring together a proton and an electron, previously separate bodies, to form a new object, the hydrogen atom? Once they are placed together, what prevents the electron from moving indefinitely closer to the proton? Were all the hydrogen atoms in the universe formed at the same time? What information do we have about the hydrogen atom?

We can safely say that a hydrogen atom is a system without any precedent in the classical (Newtonian) description. The most striking new feature of this system is the impossibility of absorbing any amount of energy. An atom (in general) can absorb energy delivered by photons and then emit it. Classically, there is nothing to stop a system from absorbing/emitting any amount of energy. Surprisingly, an atom (in general) only absorbs certain amounts of energy and only emits these same amounts of absorbed energy. Absorption/emission processes are said to be discretised or quantised. The energies available for absorption/emission in an atom make up its energy spectrum. Each atom has a unique absorption/emission spectrum, like a kind of fingerprint. It is assumed that all atoms of the same type have the same spectrum. These spectra can be modelled mathematically, i.e. there is order. If there is order, there are laws.

To begin to better understand this new system, called the hydrogen atom, let's investigate what Newtonian mechanics has to say about it. The proton, because it has a greater inertial charge, can be seen as the source and the electron as the test body (with an electric charge of opposite sign). The proton, the source, creates an electrical potential that varies with the inverse of the distance. The electron, the test body, when placed in the presence of this electric potential, feels the Coulomb electric force (attractive) which varies with the inverse of the square of the distance in the direction of the line passing through these two bodies.his is the same scenario as the solar system formed by the Sun (source) and the Earth (test body) interacting via the Newtonian gravitational force. So all we have to do is swap the inertial charges for electric charges, and the gravitational constant for the electric constant, to fully utilise the Newtonian description of a solar system made up of two bodies. It's worth bearing in mind that the inertial charge must be kept in the inertial part of Newton's second law, which contains the variation in linear momentum.

If it has a force, it has trajectories, according to Newton's second law: \begin{equation} \vec{F}_{e}=\dot{\vec{p}},\quad \vec{p}=m\vec{v},\; \dot{\vec{p}}=\frac{d\vec{p}}{dt},\end{equation} where $m$ is the electron's mass and \begin{equation} \vec{F}_{e}=-K\frac{\hat{r}}{r^2},\quad K=C_{e}e^2,\end{equation} is the electric force ($e$ is the unit of electric charge). A force like this, radial, with spherical symmetry, is called a central force. A central force that varies with the inverse square of the radial distance implies three conserved quantities: the mechanical energy $E$, the angular momentum vector $\vec{L}$ and the eccentricity vector $\vec{M}$, \begin{equation} E=\frac{1}{2}mv^2-\frac{K}{r},\quad \vec{L}=\vec{r}\times\vec{p},\quad K(\vec{M}+\hat{r})=\vec{v}\times\vec{L},\end{equation} respectively. These three conserved quantities lead, without the need to solve any differential equations, to the trajectory \begin{equation} r(\theta)=\frac{L^2}{mK}\frac{1}{1+M\cos\theta},\quad M^2=1+\frac{E}{\varepsilon},\; \varepsilon=\frac{1}{2}m\frac{K^2}{L^2},\end{equation} which represents a conic section (in polar coordinates $r$ and $\theta$), with an eccentricity $M$ and a major semi-axis \begin{equation} a=-\frac{K}{2E}.\end{equation}

As a student, I was told that this planetary model doesn't serve to describe a hydrogen atom due to the emission of radiation by an accelerated electric charge. I didn't yet know the principle of equivalence established by Einstein. An electric charge with a constant acceleration does not emit radiation. Among the conic sections is the circle, with zero eccentricity, $M=0$ which implies $E=-\varepsilon$. This circular orbit has a constant acceleration modulus, which guarantees the stability of this planetary hydrogen atom. Thus, the energy $E$ of the electron in this circular orbit of radius $a$ is \begin{equation} E=-\frac{1}{2}m\frac{K^2}{L^2},\quad a=\frac{L^2}{mK}.\end{equation}