The wave-particle duality of particles in quantum mechanics is a fundamental concept that states that particles can exhibit both wave-like and particle-like characteristics. This duality is captured by the wave-like behavior of particles described by their wave-functions and the particle-like behavior observed in experiments such as particle detections.
The Heisenberg uncertainty principle is a key consequence of this wave-particle duality. It states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle can be known simultaneously. The most well-known form of the uncertainty principle is the position-momentum uncertainty principle, which states that the more precisely the position of a particle is known, the less precisely its momentum can be known, and vice versa.
The uncertainty principle arises from the wave-like nature of particles. When a particle is described by a wave-function, it is spread out over a range of possible positions or momenta. The more localized the wave-function is in position space, the broader it becomes in momentum space, and vice versa. This relationship is mathematically captured by the Fourier transform, which relates the position and momentum representations of a wave-function.
When we make a measurement to determine the position of a particle, for example, we are effectively collapsing the wave-function to a specific position. However, since the wave-function is spread out, there is inherent uncertainty in the momentum associated with that particle. Similarly, if we try to measure the momentum of a particle precisely, the wave-function must be spread out in position space, leading to uncertainty in the position.
In essence, the Heisenberg uncertainty principle arises due to the wave-like behavior of particles and the fact that their wave-functions are spread out over a range of possible states. It sets a fundamental limit to the precision with which certain pairs of complementary properties, such as position and momentum, can be simultaneously known.