# Quantum mechanics

# Planck-Einstein equations

### Complément :

A video about the double slit paradox

### Fondamental : De Broglie's hypothesis

Let us consider particles of mass , conversely to photons which have no mass.

A photon is both a wave and a corpuscle.

With this idea in mind, De Broglie proposed in 1923 the association of material particles and a wavelength given by :

Where :

is the quantity of movement (impulsion) of the particle and is Planck's constant.

### Attention : Planck-Einstein equations

Let us consider a material particle of mass , velocity , nonrelativistic quantity of movement and energy .

The relations between wave description and corpuscular description are called Planck-Einstein equations :

By using the pulsation and the wave vector :

With :

We can say that any atomic or subatomic particle is also a wave : it is the famous particle-wave duality of elementary particles in quantum mechanics.

**Some order of magnitude :** (see rpn.univ-lorraine.fr)

Electrons accelerated with a voltage of , .

### Complément : Wave associated to a free particle

The wave associated to the particle, if it moves on the axis can be described as a plane wave :

The phase velocity is :

The group velocity is :

Or :

and , so :

The energy of the free particle is equal to its kinetic energy :

Hence :

Thus, the group velocity (the velocity of a wave pack centered on a pulsation ) is naturally identified to the velocity of a quantum particle.