What key mathematical representation is used to describe quantum superposition, and what do its components signify?

Quantum superposition is mathematically described using a wave function. The wave function, typically denoted by the Greek letter psi (Ψ), is a mathematical function that encapsulates all the information about the state of a quantum system. It describes the probability amplitude of finding the particle in a specific state. In Dirac notation, we can express the wave function as a linear combination of basis states: |Ψ⟩ = Σ cᵢ|φᵢ⟩. Here, |Ψ⟩ represents the overall quantum state, and |φᵢ⟩ are the basis states, which are the set of possible states the system can occupy (e.g., the spin-up and spin-down states of an electron, or different energy levels of a molecule). The coefficients cᵢ are complex numbers, and their squared magnitudes, |cᵢ|², represent the probability of measuring the system in the corresponding basis state |φᵢ⟩. The sum of all |cᵢ|² must equal 1, reflecting that the particle must be found in one of the possible states. Importantly, the complex phase of the coefficients cᵢ encodes interference effects between the different basis states. The superposition principle means that before a measurement is made, the quantum system exists in a combination of all the basis states simultaneously, with each state contributing to the overall wave function according to its coefficient. The wave function is not directly observable; it's a probability amplitude whose square gives the probability density. The act of measurement forces the system to collapse into one of the definite basis states, with the probability of collapsing into a particular state given by the square of the absolute value of its corresponding coefficient in the wave function.