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# The Bohr atom

The Bohr atomic model (1913) is a landmark in the creation of quantum mechanics. Rutherford's atomic model was made up of electrons rotating around a charged nucleus of protons. The atom was described by the equations of Newtonian mechanics and those of the Maxwell-Lorentz theory of electrons.

But it contained a fundamental contradiction. According to these equations, an electron orbiting around the nucleus should radiate and lose energy. The electron should therefore spiral towards the nucleus while emitting light over a continuous spectrum of wavelengths.

Furthermore, since there was nothing to stabilise the electron orbits, the various impacts between atoms in a gas should rapidly destroy the orbits.

The theory therefore failed to reproduce two experimentally observed facts:

• the very existence of atoms;
• the existence established by spectroscopy of a discrete spectrum of wavelengths in the radiation that atoms are able to emit or absorb.

In the latter case, a very strange formula was established. Between two frequencies $\nu_{n}$ and $\nu_{m}$, characterising the lines in the spectrum of the hydrogen atom, there existed on subtracting them the following formula of Balmer:

$\nu_{nm}=cR_{H}(\frac{1}{m^2} - \frac{1}{n^2})$

where n and m are non-zero integers,$R_H$ the Rydberg constant, and c is the speed of light.

Everything happened as if there were energy states within the atom that could only vary in steps during the emission or absorption of light at a frequency $\nu_{nm}$. The lines were therefore associated with energies $E_m$ such that:

$E_m=-R(\frac{1}{m^2})$

Where $R$ is a constant that will be defined below.

Struck by this, Niels Bohr remembered Max Planck's result for the black body radiation formula. In 1900, Planck had shown that the radiation spectrum of a body in thermal equilibrium could be deduced. This could be done by adopting the assumption that matter can only absorb or emit light by packets of energy at a given $\nu_$ frequency. He had stated one of the most famous formulas in physics, with $E=mc^2$ , and this formula was written $E=h\nu$ where $h$is the famous Planck constant.

Bohr therefore adopted Planck's assumption and used it to postulate that the electrons in Rutherford's model of the atom could only circulate in quantified energy orbits $E_n$ . On jumping from one orbit to another, the electron could emit a discrete packet of light energy, a photon, with a frequency given by:

$E_m-E_n=h\nu_{nm}$

According to classical radiation theory, the emission frequency of radiation for a charged particle in a circular orbit should be the reciprocal of its period of rotation. It was therefore possible to determine the size of a hydrogen atom of hydrogen and calculate the constant $R=hcR_H$ from the mass of an electron, its charge and the Planck constant $h$. An astounding result.

The state of a hydrogen atom could thus be described by introducing integers n and m that were to be called "principal quantum numbers ". Similar formulas would be obtained for diatomic molecules modelled as rotating dumbbells, and above all for so called hydrogen-like atoms.

The energy levels and series of lines of the Bohr atom (credit: Guy Collin, professor emeritus, UQAC).

What would in future be called the Bohr model of the atom had some important strong features that were to play a determinant role in the discovery of the principles of quantum mechanics.

A - Firstly, it explained why the Ritz combination principle, so named after the person who discovered it, was obeyed. If we consider the frequencies of the spectral lines for the hydrogen atom, adding or subtracting these frequencies gives another possible spectral line for the atom. This is obvious if we consider Balmer's formula.

Let us take a transition $\nu_{14}$ with the emission of a photon and $\nu_{24}$, $\nu_{12}$, $\nu_{34}$ , $\nu_{13}$ , $\nu_{23}$ then $\nu_{14}$= $\nu_{24}$+ $\nu_{12}$= $\nu_{34}$ + $\nu_{13}$ = $\nu_{12}$ + $\nu_{34}$ + $\nu_{23}$ with emission of two and three photons.

B - Next, the space-time picture of the movement of an electron when it goes from one energy level to another is highly problematic. If it follows an intermediate trajectory between two orbits, why does it not radiate at that moment, violating the laws of the electrodynamics of charged particles? We are led to the picture of an instant jump from one orbit to another without any localisation in the intermediate space. The whole framework of physical principles and the equations of classical physics were upset.

C - Finally, when we consider the limit of transitions between two energy levels with large quantum numbers, say n and n+1, Balmer's formula gives something approaching a continuous spectrum. We obtain:

$\nu_{nn+1}=cR_{H}(\frac{(n+1)^ 2-n^2}{(n^2+n)^2})$

Which, when n becomes very large, tends to:

$\nu_{nn+1}=cR_{H}(\frac{2}{n^3 })$

Which does indeed correspond to an increasingly smaller and smaller difference between the emission and absorption frequencies of light for an electron in these high orbitals.

This is at the origin of Bohr's correspondence principle which states that at the limit of large quantum numbers for atomic systems we should find the formulas of classical physics. This principle makes it possible for the formulas for the intensity of radiation of quantum atoms to be connected with classical formulas and so guide theorists to an interpretation of the quantum equations by connecting them around their classical limits.

These three features were to be highly important in the discovery by Heisenberg of the matrix mechanics of quantum phenomena.

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