In quantum mechanics, what mathematical object describes the state of a system?

In quantum mechanics, a wave function describes the state of a system. Usually written with the Greek letter psi, it is a mathematical object that contains the information needed to calculate the probabilities of possible measurement results.

For a particle described in position space, the wave function is generally a complex-valued function of position and time. The square of its absolute value, written |ψ|², gives the probability density for finding the particle at a particular location. A physically meaningful wave function must be normalized so that the probabilities of all possible outcomes add up to one.

The Schrödinger equation describes how a wave function changes over time. Wave functions can also be combined, producing superpositions in which a system has several possible measurement outcomes before observation. This mathematical feature helps explain interference and many quantum effects.

The word wave can cause confusion. A wave function is not necessarily a physical ripple like a water wave, and its interpretation depends on the formulation of quantum theory. In more abstract language, a quantum state is a vector in Hilbert space; a wave function is that state expressed in a chosen representation.