Chapter 5

Quantum mechanics

Bohr's model

FondamentalBohr's assumptions

The hydrogen atom is composed of a proton O (of mass and electric charge ) and of an electron P (of mass and charge ).

The electron travels around the proton in a circular orbit of radius .

The proton is supposed stationary and at the origin of a Galilean referential (that of the laboratory).

The only force considered here is the Coulomb interactive force between the two charges.

Bohr's assumptions are :

  • Steady orbits exist (also called stationary orbits) for which the electron does not radiate any energy.

  • The electron absorbs or radiates energy only when travelling from one stationary orbit to another.

How can these stationary orbits be characterized ?

Bohr proved that, through conditions imposed by his assumptions, stationary orbits were those for which the kinetic moment of the electron was an integer multiple of a constant , equal to , where is planck's constant.

Hence, if is the kinetic moment of the electron (or angular momentum) on a stationary orbit, in relation to the proton then :

Where is a positive integer.

FondamentalMechanical study

Let's determine a relation between the radius of the trajectory and the speed of the electron.

Newton's second law applied to the electron in the Galilean referential of the nucleus of the atom gives, when projected on radius  :

Hence :

The kinetic energy , the potential energy and the mechanical energy of the electron can be expressed as functions of and constants and  :

FondamentalQuantification of angular momentum, radii and atom energies

To interpret the spectral rays of the hydrogen atom, Bohr's model constrains the electron to have a quantified kinetic moment (angular momentum) :

Where is a natural integer different from zero, and is planck's constant.

Let us determine the radii, written , of the orbits allowed.

The kinetic moment of the electron in relation to the nucleus placed in O is :

The condition of quantification of the kinetic moment constrained by N. Bohr leads to a quantification of radii (now written ) of the allowed orbits.

Indeed, the expression :

gives :

represents the radius of the orbit of the electron for .

The state of the electron associated to is called fundamental state, and is called Bohr's radius.

It can be proven that the energy of the electron is quantified and can be written as :

Actually :

With :

represents the mechanical energy of the electron in its fundamental state.

Numerically :

ExempleExample of Balmer's sequence

By creating an electric discharge in a light bulb full of hydrogen gas (or dihydrogen), these molecules dissociate themselves and the excited atoms emit electromagnetic radiation of different wavelengths.

Physicist J. Balmer determined experimentally that the wavelengths of these radiations which are located in the domain of visible light, verified this phenomenological relation : ( is a positive integer)

Where is Rydberg's constant.

It is equal to .

Bohr's model can help determine the theoretical value of this constant  :

When the hydrogen atom goes from a level defined by the quantum number to the state that corresponds to the quantum number , it emits a photon whose energy is given by :

Consequently, the wavenumber (defined as the inverse of the wavelength of the emited radiation) is :

Rydberg's constant can be interpreted as the quantity :

Its theoretical value corresponds very well to the experimental value determined by J. Balmer.

In conclusion, Bohr's model gives the right levels of energy for the hydrogen atom.

However, the classical representation of an electron moving on a circular orbit revolving around the proton has been proved wrong by quantum mechanics.

The latter gives the image of an electron that is not located and of which can only be known the probability of presence, “materialized” by a cloud of probability (atomic orbitals) surrounding the nucleus.

Structure of the emission spectrum of the hydrogen atom

ComplémentWave interpretation of Bohr's atom model

The wave of the electron can be represented as a wave running around the orbit as it would run on a rope.

After a large number of periods, the wave will, in most cases self­destruct by interference with itself.

Wave interpretation of Bohr's atom model

In order for the interferences not to be destructive the length of the orbit must be a multiple integer of the wavelength of the electron.

is the radius of the orbit :

Considering that :

We find that the angular momentum is quantified :

The electronic wave can help interpret the quantification of the angular momentum.

Black-body Radiation
Additional exercises