Chapter 5

Quantum mechanics

Heisenberg's indeterminacy equations

FondamentalHeisenberg's space uncertainty principle

At a given time, the measure of the position and the quantity of movement on the axis both show fundamental indeterminacies.

These are written and .

They verify Heisenberg's indeterminacy equation :

This inequality shows that a quantum state does not give a perfect knowledge of this state in a classical point of view.

The notion of trajectory disappears in quantum mechanics.

Heisenberg's space uncertainty principle

A link to the website "Culture sciences physiques" ("Fenêtre ouverte sur la physique quantique") : click HERE

ComplémentHeisenberg's indeterminacy principle explains the diffraction of light

Let us consider the diffraction of a light beam by a thin slit of width .

The photon has travelled through a slit.

We can write :


Heisenberg's indeterminacy principle gives :

The indeterminacy on the quantity of movement is given by the vertical coordinate :

Hence the diffraction angle :

The order of magnitude of the angle is the same as the one given by the diffraction theory.

Diffraction can be interpreted thanks to Heisenberg's uncertainty principle.

ExempleMinimum energy of a particle in an infinite potential well

Let us consider a particle of mass in a "quantum box" : the potential energy of the particle is nil in the well (­ ) and infinite elsewhere.

The particle cannot exit the potential well.

The maximum indeterminacy on the position of the particle is approximately :

The indeterminacy principle gives the minimum uncertainty for  :

The minimum energy of the particle is then :

So :

Heisenberg's inequalities can determine the minimum energy of containment.

FondamentalHeisenberg's time indeterminacy principle

Let  be the indeterminacy on the measure of an energy and the duration of the measure of this energy. Then :

  • If the duration of the interaction was nil, then would be infinite and the energy of the particle would not be determined.

  • Conversely, a fundamental state with an infinite lifetime has a perfectly determined energy.

  • This inequality asserts that the energy conservation principle cannot be verified. The uncertainty is  during a time  such as .

  • With this principle can be explained the concept of virtual particles, messenger of fundamental interactions.

    Such a particle of mass (and of energy linked to its mass, ) can exist during a period of time given by the indeterminacy principle :

Feynman diagram

The previous picture shows a Feynman diagram of the interaction between two electrons : the oblique lines represent the moving electrons and the wavy lines represent the virtual exchanged photon.


A video about the indeterminacy principle

Heisenberg's Uncertainty Principle Explained
Bohr's model
Minimum energy of an harmonic oscillator