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Wave Particle Duality
Wave particle duality is a phenomina that is apparent in matter, (albeit in small scales) that particles can behave like waves.
This was first made prevelant in the infamous double slit experiment. The interest as to the behavour of matter on the quantum scale was first sparked by the debate of whether light was in fact particle of wave. A wave-particle dual nature was soon found with the electron. When physists first realised the implications of wave particle duality many were sceptical including Einstein.
The significance of this fact is that the behaviour of light and matter can be explained through the use of a differential equation which represents a wave function.
The Schrodinger equation.
This ability to describe reality in the form of waves is one of the true mysteries of quantum mechanics. This wave function represents the probability of finding a given particle at a given point. Particles behave according to the probability of these laws and exhibit the wave properties. In other words, the probability of a particle being in any location is a wave, but the actual physical appearance of that particle isn't.