Chapter 5

# Heisenberg's indeterminacy equations

## Fondamental : Heisenberg'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

## Complément : Heisenberg'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 :  Diffraction

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.

## Exemple : Minimum 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.

## Fondamental : Heisenberg'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.

## Complément :

A video about the indeterminacy principle

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