Wave Function

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.

Read more about Wave FunctionWavefunction Symmetry and Antisymmetry, Normalization Invariance, Wavefunctions As Vector Spaces, Ontology, Examples

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Semicircular Potential Well - Wave Function
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Multislice - Theory
... multislice algorithm is an approach to solving the Schrödinger wave equation The following section will include a mathematical formulation of the Multislice algorithm ... equation can also be represented in the form of incident and scattered wave as where is the Green’s function that represents the amplitude of the electron wave ... Hence for an incident plane wave of the form the Schrödinger equation can be written as We then choose the coordinate axis in such a way that the incident beam ...
Spin Contamination - Open-shell Wave Functions
... Within Hartree–Fock theory, the wave function is approximated as a Slater determinant of spin-orbitals ... In general, an N-electron Hartree–Fock wave function composed of Nα α-spin orbitals and Nβ β-spin orbitals can be written as where is the antisymmetrization operator ... This wave function is an eigenfunction of the total spin projection operator, Ŝz, with eigenvalue (Nα − Nβ)/2 (assuming Nα ≥ Nβ) ...
Bohmian - The Theory - The Ontology
... ontology of de Broglie-Bohm theory consists of a configuration of the universe and a pilot wave ... Thus, the ontology of pilot wave theory contains as the trajectory we know from classical mechanics, as the wave function of quantum theory ... So, at every moment of time there exists not only a wave function, but also a well-defined configuration of the whole universe ...
Spin Contamination - Elimination
... Given this, several approaches to remove or minimize spin contamination from UHF wave functions have been proposed ... The resulting wave function, while not completely free of contamination, dramatically improves upon the UHF approach especially in the absence of high order contamination ... Projected UHF (PUHF) annihilates all spin contaminants from the self-consistent UHF wave function ...

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