What is de Broglie's hypothesis of matter waves and why was it proposed?

In one line

De Broglie proposed that every moving particle has a wave associated with it, with wavelength \(\lambda = h/p\), just as light waves show particle behaviour.

Explained simply

Light was known to behave both as a wave (interference) and as a particle (photoelectric effect). In 1924 de Broglie asked: if nature is symmetric, why shouldn't particles like electrons also behave as waves? He suggested that a moving electron has a "matter wave" whose wavelength depends on its momentum. A heavy, fast object has an extremely tiny wavelength, so we never notice it; an electron's wavelength is about the size of an atom, so its wave nature becomes important.

How it works

1. For a photon, energy \(E = h\nu\) and also \(E = mc^2\) (equivalent mass), so momentum \(p = mc = h\nu/c = h/\lambda\).
2. So for a photon, \(\lambda = h/p\).
3. De Broglie assumed the same relation holds for any material particle of mass \(m\) moving with speed \(v\): \(\lambda = h/(mv)\).
4. Confirmation: Davisson and Germer (1927) observed diffraction of electrons from a nickel crystal, with the measured wavelength matching \(h/p\).
5. Consequence: electron microscopes use electron waves, which have much shorter wavelengths than visible light and so give higher resolution.

Key formula

\(\lambda = \dfrac{h}{p} = \dfrac{h}{mv}\). \(\lambda\): de Broglie wavelength (m), \(h = 6.63\times10^{-34}\) J s, \(p\): momentum (kg m/s), \(m\): mass (kg), \(v\): speed (m/s, non-relativistic).

Quick example

A 0.15 kg cricket ball at 30 m/s: \(\lambda = 6.63\times10^{-34}/(0.15\times30) = 1.47\times10^{-34}\) m, far too small to detect. An electron at \(10^6\) m/s: \(\lambda = 6.63\times10^{-34}/(9.11\times10^{-31}\times10^6) = 7.28\times10^{-10}\) m = 7.28 Å, comparable to atomic spacing, so electron diffraction is observable.

Common mistakes

• Thinking only charged particles have matter waves; all moving particles do.
• Using \(\lambda = h/(mc)\) for particles; use their actual speed \(v\).
• Calling matter waves electromagnetic waves.

How MU asks it

Asked as "State de Broglie hypothesis" or "Explain the concept of matter waves" (3 marks), often followed by a numerical. Write the reasoning from photon momentum, give \(\lambda = h/mv\), mention Davisson-Germer confirmation, and compare a macroscopic object with an electron.