Wave function

A wave function in quantum mechanics is a mathematical object that represents a particular quantum state of a specific isolated system of one or more particles. A single wave function describes the entire system, covering at once all the particles in it. For the one state, however, there are many different wave functions, each giving its respective representative description. Each of the different representatives contains all the information that can be known about the state, the different versions being mutually interconvertible by one-to-one mathematical transformations. A wave function can be interpreted as a probability amplitude. All quantities associated with measurements, such as the average momentum of a particle, can be derived from the wave function. It is a central entity in quantum mechanics, including for example quantum field theory. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi).

For a specific system, there are many possible representations, that is to say, complete sets of commuting observables, and many suitable coordinate systems, continuous as well as discrete. These may be freely chosen. For the particular state, there is one representative wave function, a complex-valued function of the system's degrees of freedom, that belongs to the chosen representation and coordinate system. By a postulate of quantum mechanics, such observables are Hermitian linear operators on the space of states, representing physical observables, for example position, momentum and spin, that can, in principle, be simultaneously or jointly measured with arbitrary precision. There is at least one such complete set of observables for which the state is a simultaneous eigenstate.