Planck-Einstein equations

Fondamental : De Broglie's hypothesis

Let us consider particles of mass \(m\), 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 \(\lambda\) given by :

\(\lambda = \frac {h}{p}\)

Where :

\(p=mv\)

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

Attention : Planck-Einstein equations

Let us consider a material particle of mass \(m\), velocity \(\vec v\), nonrelativistic quantity of movement \(\vec p = m \vec v\) and energy \(E\).

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

\(E = h\nu \;\;\;\;\;\;\;;\;\;\;\;\;\;\;p = \frac{h}{\lambda }\)

By using the pulsation and the wave vector :

\(E = \hbar \omega \;\;\;\;\;\;\;;\;\;\;\;\;\;\;p = \hbar k\)

With :

\(\hbar = \frac {h}{2\pi}=1,05.10^{-34}J.s\)

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 \(100\;V\), \(\lambda =0,12\;nm\).

Complément : Wave associated to a free particle

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

\(\underline {\psi} (x,t)=Ae^{i(kx -\omega t)}\)

The phase velocity is :

\(v_{\varphi}=\frac {\omega}{k}\)

The group velocity is :

\(v_{g}=\frac {d\omega}{dk}\)

Or :

\(\omega = E/\hbar\) and \(k=p/\hbar\), so :

\(v_{g}=\frac {dE}{dp}\)

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

\(E=\frac {1}{2}mv^2=\frac{p^2}{2m}\)

Hence :

\(\frac {dE}{dp}=\frac {p}{m}=v\)

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