In wave-particle duality, the term "wave" refers to a mathematical description of the behavior of particles at the quantum level. It does not imply the presence of a physical wave in the classical sense, such as a water wave or a sound wave.
In quantum mechanics, particles are described by wave functions, which are mathematical functions that assign a probability amplitude to each possible state of the particle. The square of the wave function gives the probability density of finding the particle in a particular state. The wave function is typically represented by a complex-valued function, and its behavior is governed by an equation known as the Schrödinger equation.
The wave function allows us to calculate probabilities for various outcomes of measurements or observations. However, when a measurement is made on a particle, its wave function "collapses" into a specific state, corresponding to the measured property of the particle, such as its position or momentum. This collapse is sometimes referred to as the "particle" aspect of wave-particle duality, as it manifests as a localized particle-like behavior during measurements.
The wave-like behavior of particles is observed in phenomena such as diffraction and interference. These effects arise from the interference of different possible paths or states of the particle, giving rise to characteristic wave patterns. These patterns can be observed, for example, in experiments involving electrons or photons passing through slits or interfering with each other.
So, in wave-particle duality, the "wave" refers to the mathematical description of the particle's behavior, represented by the wave function, which exhibits wave-like properties and describes the probabilistic nature of particles at the quantum level.