A wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of space and time. The laws of quantum mechanics (the Schrödinger equation) describe how the wave function evolves over time. The wave function behaves qualitatively like other waves, like water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality.
The most common symbols for a wave function are ψ or Ψ (lower-case and capital psi).
Although ψ is a complex number, |ψ|2 is real, and corresponds to the probability density of finding a particle in a given place at a given time, if the particle's position is measured.
The SI units for ψ depend on the system. For one particle in three dimensions, its units are m–3/2. These unusual units are required so that an integral of |ψ|2 over a region of three-dimensional space is a unitless probability (i.e., the probability that the particle is in that region). For different numbers of particles and/or dimensions, the units may be different (but they can be determined by dimensional analysis).
The wave function is central to quantum mechanics, because it is a fundamental postulate of quantum mechanics. It is the source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics—topics that continue to be debated even today.
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