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The Uncertainty Principle
 
Light has both wave and particle properties. Its wave properties manifest themselves during propagation of light (interference, diffraction). Its particle-like properties are significant during interactions of light with matter (the photoelectric effect, emission and absorption of light by atoms.)
The properties of a photon as an energy particle E and momentum p are interconnected with its wavelike properties (frequency ν and wavelength λ), as shown by the following equation:
E = hν ; p = hν/c = h/λ ,
where h = 6.63∙10-34 J s is Planck's constant.
According to de Broglie's hypothesis, every body with mass m and moving with velocity v should have a wavelength
λ = h/mv = h/p.
The wavelike aspects are most prominent in elementary particles. Due to their small mass, de Broglie wavelength appears to be comparable with distances between adjacent atoms in crystals. Diffraction occurs when a particle beam interacts with the crystal lattice. For example, electrons with energy 150 eV have a wavelength λ ≈ 10-10 m. Note that distances between atoms in crystals are of the same order. This explains why an electron beam will be scattered on a crystal like a wave, i.e. according to diffraction laws.
To illustrate the wave nature of particles, a model utilizing a beam of electrons (or other particles) passing through a slit of width Δx is frequently used. According to wave theory, after diffraction on the slit, the beam broadens at angular divergence θ ≈ λ/Δx. In accordance with the particle theory, the broadening of the beam after passing through the slit is explained by the appearance of a certain transverse momentum in a particle. The uncertainty of this transverse momentum is
Δpx ≈ θp ≈ (λ/Δx)p ≈ h/Δx.
The relation
ΔpxΔx ≈ h
is known as the Heisenberg uncertainty principle. This principle explains the properties of an elementary object as a wave in terms of its properties as a particle.
The experiment whereupon an electron beam passes through two slits may be an even better illustration of the wave properties of particles. Such an experiment is an analog of the two-slit interference experiment that was introduced by T. Young.

Change the width of the slits and alternate their number. Note that the diffraction image in the experiment with two slits is not a sum of diffraction images from separate slits. This quantum behavior of the experimental material cannot be explained by classical physics.

 
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