wavefunction

Wavefunction is a mathematical description of a quantum system's state. For a single nonrelativistic particle in three dimensions it is a complex-valued function ψ(r, t), where r denotes position and t time. The probability density of finding the particle near r at time t is |ψ(r, t)|^2, and the wavefunction is normalized so that ∫|ψ(r, t)|^2 d^3r = 1. The time evolution is governed by the Schrödinger equation iħ ∂ψ/∂t = Hψ, where H is the Hamiltonian operator. For N particles, ψ depends on all particle coordinates and evolves under the total Hamiltonian.

The wavefunction is not directly observable; instead, the Born rule provides probabilities for measurement outcomes derived from ψ. Observables correspond to operators, and the expectation value of an operator  is ⟨Â⟩ = ∫ ψ*(r, t)  ψ(r, t) d^3r. Global phase factors e^{iφ} do not affect physical predictions, so states related by a phase are equivalent. In systems with internal degrees of freedom, such as spin, the wavefunction can be a spinor or a set of coupled components. In momentum space, the wavefunction φ(p, t) is the Fourier transform of ψ(r, t).

For mixed states, the description uses a density operator rather than a single wavefunction. In relativistic contexts and quantum field theory, the notion of a wavefunction is supplanted by field operators and state vectors in a larger Hilbert space, with particle concepts becoming emergent in appropriate limits.