Matter and wave-particle duality

I always liked Quantum Mechanics :) & Louis de Broglie was indeed one brilliant chap!

• De Broglie had intended a career in humanities, and received his first degree in history. Afterwards, though, he turned his attention toward mathematics and physics and received a degree in physics. With the outbreak of the First World War in 1914, he offered his services to the army in the development of radio communications.

• Louis de Broglie was born to a noble family in Dieppe, Seine-Maritime, younger son of Victor, 5th duc de Broglie. He became the 7th duc de Broglie upon the death without heir in 1960 of his older brother, Maurice, 6th duc de Broglie, also a physicist. He did not marry. When he died in Louveciennes, he was succeeded as duke by a distant cousin, Victor-François, 8th duc de Broglie.

Matter and wave-particle duality

The fundamental idea of Louis de Broglie's 1924 thesis was the following: 

"The fact that, following Einstein's introduction of photons in light waves, one knew that light contains particles which are concentrations of energy incorporated into the wave, suggests that all particles, like the electron, must be transported by a wave into which it is incorporated... My essential idea was to extend to all particles the coexistence of waves and particles discovered by Einstein in 1905 in the case of light and photons." "With every particle of matter with mass m and velocity v a real wave must be 'associated'", related to the momentum by the equation:

where λ is the wavelength, h is the Planck constant, p is the momentum, m is the rest mass, v is the velocity and c is the speed of light in a vacuum.

The de Broglie relations

The de Broglie equations relate the wavelength λ to the momentum p, and frequency f to the total energy E (including its rest energy) of a particle:

where h is Planck's constant. The two equations can be equivalently written as

using the definitions is the reduced Planck's constant (also known as Dirac's constant, pronounced "h-bar"), k = 2π / λ is the angular wavenumber, and ω = 2πf is the angular frequency. In each pair, the second is also referred to as the Planck-Einstien relation, since it was also proposed by Planck and Einstein.

Using the relativistic mass formula from special relativity

m = γm₀

allows the equations to be written as^[4]

where m₀ is the particle's rest mass, v is the particle's velocity, γ is the Lorentz factor, and c is the speed of light in a vacuum. See group velocity for details of the derivation of the de Broglie relations. Group velocity (equal to the particle's speed) should not be confused with phase velocity (equal to the product of the particle's frequency and its wavelength). In the case of a non-dispersive medium, they happen to be equal, but otherwise they are not.

Experimental confirmation

Elementary particles

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target. The angular dependence of the reflected electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for x-rays. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be only exhibited by waves. Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter. When the de Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.

This was a pivotal result in the development of quantum mechanics. Just as the photoelectric effect demonstrated the particle nature of light, the Davisson-Germer experiment showed the wave-nature of matter, and completed the theory of wave-particle duality. For physicists this idea was important because it means that not only can any particle exhibit wave characteristics, but that one can use wave equations to describe phenomena in matter if one uses the de Broglie wavelength.

Louis de Broglie
Born	15 August 1892 Dieppe, France
Died	19 March 1987 (aged 94) Louveciennes, France
Nationality	French
Fields	Physics
Institutions	Sorbonne University of Paris
Alma mater	Sorbonne
Doctoral advisor	Paul Langevin
Doctoral students	Jean-Pierre Vigier Alexandru Proca
Known for	Wave nature of electrons de Broglie wavelength
Notable awards	Nobel Prize in Physics (1929)