The phenomenon you are referring to is known as "quantum superposition." According to quantum mechanics, particles at the microscopic level, such as electrons or photons, can exist in a superposition of multiple states simultaneously. This means that they can be in different positions, or other properties like momentum or spin, at the same time.
Mathematically, superposition is described using wave functions, which are mathematical representations of a particle's quantum state. A wave function assigns probabilities to different possible outcomes or states of the particle. In the case of position, the wave function describes the probability distribution of finding the particle at various locations.
The mathematical formalism used to describe superposition is called linear algebra. A superposition of states is represented by a linear combination of the individual states. For example, a particle can be in a superposition of being in two different positions, denoted as |position A⟩ and |position B⟩, which can be written as:
|superposition⟩ = α|position A⟩ + β|position B⟩
Here, α and β are complex numbers known as probability amplitudes, and the absolute squares of these amplitudes determine the probabilities of finding the particle in either position upon measurement.
Physically, superposition implies that until a measurement or observation is made, the particle does not have a definite position. It exists in a state that encompasses multiple possibilities. When a measurement is performed, the wave function collapses to a specific state, and the particle is found in a definite position with a corresponding probability.
Superposition is a fundamental aspect of quantum mechanics and has been experimentally observed in various systems, such as the famous double-slit experiment, where particles exhibit wave-like behavior and interfere with themselves when in superposition.