The concept of a particle being a wave and having position and momentum simultaneously is a fundamental aspect of quantum mechanics. It is described by a concept known as wave-particle duality. According to quantum mechanics, particles such as electrons or photons can exhibit both particle-like and wave-like properties under different circumstances.
Wave-particle duality arises from the mathematical formalism of quantum mechanics, which describes the behavior of particles and waves at the microscopic scale. In this framework, particles are represented by wavefunctions, which are mathematical functions that describe the probability distribution of finding a particle in different states.
When a particle is in a state of superposition, it can exist in multiple states simultaneously. This superposition is represented by a wavefunction that exhibits wave-like properties, such as interference and diffraction. At this point, the particle behaves as a wave.
However, when a measurement is made to determine the position or momentum of the particle, the wavefunction collapses to a specific value corresponding to the observed measurement. This collapse of the wavefunction results in the particle behaving like a localized particle with a definite position and momentum.
It's important to note that the act of measurement disturbs the system and changes its state. This is known as the measurement problem in quantum mechanics. Before measurement, the particle can exist in a superposition of states, but after measurement, it is found in a specific state.
In summary, the wave-particle duality of quantum mechanics allows particles to exhibit both wave-like and particle-like properties. The wave-like behavior is associated with the particle's wavefunction, which describes the probability distribution of finding the particle in different states. The particle-like behavior emerges when a measurement is made, causing the wavefunction to collapse and revealing a specific position or momentum for the particle.