de Broglie’s Matter Waves

In 1924 Louis de Broglie proposed a bold symmetry: just as Einstein and Compton had shown that light (classically a wave) carries momentum $p = h/\lambda$ , perhaps matter (classically particles) also exhibits wave-like behaviour characterised by a wavelength. His postulate reads

$\lambda = \frac{h}{p},$

where $h = 6.626 \times 10^{-34}\ \text{J·s}$ is Planck's constant and $p$ is the relativistic momentum of the particle. For non-relativistic speeds, $p = mv$ .

This single equation — later confirmed by electron diffraction experiments (Davisson and Germer, 1927; G. P. Thomson, 1927) — placed particles and waves on equal footing and planted the seed for Schrödinger's wave mechanics.

Where the formula comes from

De Broglie's reasoning was elegant. For a photon, Einstein's relation gives $E = pc$ , and Planck's relation gives $E = h\nu = hc/\lambda$ . Combining these:

$pc = \frac{hc}{\lambda} \implies p = \frac{h}{\lambda} \implies \lambda = \frac{h}{p}.$

De Broglie simply assumed the same relation holds for any particle, replacing the photon momentum with the classical momentum $p = mv$ for a massive body.

Electrons accelerated through a voltage

A common scenario: an electron (charge $e = 1.602 \times 10^{-19}\ \text{C}$ , mass $m_e = 9.109 \times 10^{-31}\ \text{kg}$ ) is accelerated from rest through a potential difference $V$ . All the electrical potential energy converts to kinetic energy:

$K = eV = \frac{p^2}{2m_e} \implies p = \sqrt{2m_e eV}.$

Substituting into de Broglie's relation:

$\lambda = \frac{h}{\sqrt{2m_e eV}}.$

For $V = 100\ \text{V}$ this works out to approximately $1.23\ \text{Å}$ ( $1.23 \times 10^{-10}\ \text{m}$ ), which is comparable to atomic spacings in a crystal — the reason electron beams can diffract from crystal planes and produce interference patterns just like X-rays.

The wave-particle picture

The de Broglie wavelength is not a classical wave; it encodes the quantum phase of the particle's wavefunction. In double-slit experiments with electrons, the interference fringes appear even when electrons are sent one at a time — each particle interferes with itself, guided by its wavefunction $\psi(x)$ whose spatial oscillation period is exactly $\lambda = h/p$ .

This duality is not a curiosity confined to electrons. The same relation has been verified for neutrons, atoms, and even large molecules such as buckminsterfullerene ( $\text{C}_{60}$ , $60$ carbon atoms), confirming that wave-particle duality is a universal feature of quantum mechanics.

Try it

This is a numerical exercise — your code should return a number. An electron is accelerated through $100\ \text{V}$ . Return its de Broglie wavelength in metres.

Run your code to see the quantum state.