A wave function or wavefunction is a probability amplitude in quantum mechanics describing how a particle or a system of particles behaves in a given quantum state. The probability amplitude is distributed as a waveform through space and time, the unifying concept of wave-particle duality, hence the name wavefunction. Generally it is a function of position space and time, or momentum space and time, or rotation, that returns the probability amplitude of a position or momentum for a subatomic particle. Sometimes the wavefunction may not depend on time, but will always depend on space. As a function of a space, it maps the possible states of the system into the field of complex numbers. The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time.
The most common symbols are ψ or Ψ (greek lower and capital psi, used interchangeably).
The square of the (complex valued) modulus |ψ|2 is equal to the probability density (not just the probability) of finding a particle in an infinitesimal space element surrounding a point in space and time; this is the Born interpretation of the wavefunction (see below).
The SI units for ψ are unusual, containing a square root of a metre: m-d/2, where d = dimension of space, d = 1 for one dimension etc. The reason for the square root is so that the units of probability density |ψ|2 are m-d, as should be since a probability is clearly dimensionless, and it is probability per unit space.
For example, in an atom with a single electron, such as hydrogen or ionized helium, the wave function of the electron provides a complete description of how the electron behaves, for a given quantum state. It can be decomposed into a series of atomic orbitals which form a basis for the possible wave functions. For multi-electron atoms (more than one electron) or any system with multiple particles, the underlying space is the possible configurations of all the electrons and the wave function describes the probability amplitude of those configurations.
Simple examples of wave functions are in common quantum mechanics problems; the particle in a box and the free particle (or a particle in an infinitely large box).