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A quantum system can exist in a superposition of multiple states simultaneously, written as |ψ⟩ = Σ cₙ|φₙ⟩ where |cₙ|² are probabilities. Unlike classical systems where the system definitely is in one state, a superposed quantum state genuinely exhibits properties of all constituent states until measured. Superposition is fundamental to quantum mechanics and has no classical analogue.
You already know that quantum states live in a vector space, and that a state |ψ⟩ can be expanded in any complete basis {|φₙ⟩} as |ψ⟩ = Σ cₙ|φₙ⟩. Superposition is what this expansion means physically: the system is not "secretly" in one of the basis states while we remain ignorant — it genuinely occupies all states simultaneously, weighted by the coefficients cₙ. This is the sharpest departure from classical physics, and it demands a careful re-examination of what "state" means.
A classical coin lying flat is definitely heads or definitely tails — we might not know which, but one of those descriptions is true. A quantum coin in superposition is different: before measurement it is neither heads nor tails, and the description |ψ⟩ = c₀|heads⟩ + c₁|tails⟩ is complete — there is nothing more to say. The evidence for this is interference. If the system were merely in a classical mixture (heads with probability |c₀|², tails with probability |c₁|²), probabilities for overlapping paths would simply add. But quantum amplitudes add before squaring, producing interference fringes that only appear when both terms are simultaneously "present." The double-slit experiment is the iconic demonstration: each particle goes through both slits simultaneously, and the wave-like interference pattern cannot be explained unless the particle was in a superposition of both paths.
The coefficients cₙ are probability amplitudes — complex numbers whose squared magnitudes |cₙ|² give the probability of finding the system in state |φₙ⟩ upon measurement. The normalization condition Σ|cₙ|² = 1 ensures probabilities sum to one. Before measurement, the superposition is the complete description. Upon measurement, the state collapses to a single eigenstate: the system is now definitely in some |φₖ⟩ with probability |cₖ|², and all other amplitudes vanish. This collapse is instantaneous and irreversible, and it is the source of the quantum measurement problem you will study next.
A crucial subtlety: superposition is basis-dependent. A state that is a superposition in the energy eigenbasis may be an eigenstate in the position basis, and vice versa. The Schrödinger equation governs how superpositions evolve in time — deterministically, linearly, and coherently, preserving the full superposition — until measurement disrupts it. This tension between deterministic evolution and probabilistic collapse is at the heart of quantum foundations. But the computational power of superposition is already clear: a quantum system of n two-level particles lives in a 2ⁿ-dimensional Hilbert space, and superposition allows it to occupy all 2ⁿ directions simultaneously — the basis of quantum computing.