Definition
Quantum mechanics is the theory of physics at the atomic and subatomic scale. It replaces the deterministic trajectories of classical mechanics with a wave function ψ that evolves smoothly under the Schrödinger equation and whose squared magnitude gives the probability of finding a particle in any given configuration when it is measured.
Its core ingredients are wave-particle duality, the uncertainty principle, and the probabilistic interpretation of measurement outcomes.
Why it matters
How it works
The state of a quantum system is described by a complex-valued wave function ψ — a vector in a Hilbert space. Observables are represented by Hermitian operators. The expected value of an observable in a state is ⟨ψ|Ô|ψ⟩, and the possible measurement outcomes are the operator's eigenvalues. Between measurements, ψ evolves unitarily according to the Schrödinger equation iℏ ∂ψ/∂t = Ĥψ, where Ĥ is the Hamiltonian operator.
Measurement is the conceptually unique step. The Born rule says the probability of finding outcome a is |⟨a|ψ⟩|². After the measurement, in the Copenhagen picture, the system is "collapsed" into the corresponding eigenstate. Other interpretations (many-worlds, consistent-histories) reframe what this collapse means without changing the predictions.
Two phenomena drive the strangeness. Superposition allows a system to be in a linear combination of states at once: an electron can be at two locations until measured. Entanglement correlates particles so that measuring one instantly determines properties of the other, even if they are spatially separated — but, as Bell's theorem clarified, no faster-than-light signaling is possible.
In practice quantum mechanics is computational. You write down the Hamiltonian for your system — atom, molecule, solid, quantum gate — solve or approximate the Schrödinger equation, and read off probabilities. The results match experiment with extraordinary precision while leaving deep philosophical questions about reality, measurement, and observers unresolved.