Identical particles in quantum mechanics display indistinguishability, a principle stating that two or more identical particles cannot be distinguished from one another, even in principle. This leads to the concept of quantum superposition, where particles occupy multiple states simultaneously, a fundamental feature of quantum mechanics.
The phenomenon where identical particles appear to occupy opposite locations simultaneously can be interpreted through the lens of quantum entanglement and superposition. Quantum entanglement occurs when particles become correlated in such a way that the state of one particle is directly related to the state of another, no matter the distance between them. This results in a highly non-intuitive probability distribution that cannot be explained by classical statistics.
The position of particles in quantum mechanics does not have definite boundaries but rather is described by probabilities. Mathematically, particles are expressed as wave functions, solutions to the Schrödinger equation, which describe the probability amplitude of a particle's position, momentum, and other physical properties. Particles such as electrons obey Fermi-Dirac statistics, while bosons, such as photons, follow Bose-Einstein statistics, illustrating two different behaviors of indistinguishable particles due to their spin characteristics.
If identical particles are considered fermions, the Pauli exclusion principle dictates that no two fermions can occupy the same quantum state simultaneously, indirectly allowing them to appear in different locations. In the case of bosons, being able to occupy the same state, they can manifest coherent states across distinct locations, facilitated by their symmetric wave functions.
The notion of identical particles being in opposite places also involves a delay-choice or interpretation attribution due to the absence of classical locality and determinism. Quantum mechanics fundamentally alters the framework whereby particles possess specific trajectories or histories, further challenging classical intuitions about separability and individuality. The behavior and probabilistic nature of quantum systems starkly contrast with classical expectations found in macroscopic physical systems.