Editing Wave function (section)

{{Short description|Mathematical description of quantum state}}
{{distinguish|Wave equation}}
[[File:Quantum harmonic oscillators animation.gif|thumb|upright=1.25|[[Quantum harmonic oscillator|Quantum harmonic oscillators]] for a single spinless particle. The oscillations have no trajectory, but are instead represented each as waves; the vertical axis shows the real part (blue) and imaginary part (red) of the wave function. Panels (A-D) show four different standing-wave solutions of the [[Schrödinger equation]]. Panels (E–F) show two different wave functions that are solutions of the Schrödinger equation but not standing waves.]]
[[File:Indeterminacy principle.gif|thumb|The wave function of an initially very localized free particle.]]

In [[quantum physics]], a '''wave function''' (or '''wavefunction''') is a mathematical description of the [[quantum state]] of an isolated [[quantum system]]. The most common symbols for a wave function are the Greek letters {{math|''ψ''}} and {{math|Ψ}} (lower-case and capital [[psi (letter)|psi]], respectively). Wave functions are [[complex number|complex-valued]]. For example, a wave function might assign a complex number to each point in a region of space. The [[Born rule]]<ref name=Born_1926_A /><ref name="Born_1926_B" /><ref>[[Max Born|Born, M.]] (1954).</ref> provides the means to turn these complex [[probability amplitude]]s into actual probabilities. In one common form, it says that the [[squared modulus]] of a wave function that depends upon position is the [[probability density function|probability density]] of [[measurement in quantum mechanics|measuring]] a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called ''normalization''. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply [[Operator (quantum mechanics)|quantum operators]], whose eigenvalues correspond to sets of possible results of measurements, to the wave function {{math|''ψ''}} and calculate the statistical distributions for measurable quantities.

Wave functions can be [[function (mathematics)|functions]] of variables other than position, such as [[momentum]]. The information represented by a wave function that is dependent upon position can be converted into a wave function dependent upon momentum and vice versa, by means of a [[Fourier transform]]. Some particles, like [[electron]]s and [[photon]]s, have nonzero [[Spin (physics)|spin]], and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as [[isospin]]. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for ''each'' possible value of the discrete degrees of freedom (e.g., z-component of spin). These values are often displayed in a [[column matrix]] (e.g., a {{math|2 × 1}} column vector for a non-relativistic electron with spin {{math|{{frac|1|2}}}}).

According to the [[superposition principle]] of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a [[Hilbert space]].  The inner product of two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the [[Born rule]], relating transition probabilities to inner products.  The [[Schrödinger equation]] determines how wave functions evolve over time, and a wave function behaves qualitatively like other [[wave]]s, such as [[water wave]]s 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]].  However, the wave function in quantum mechanics describes a kind of physical phenomenon, as of 2023 still open to different [[Interpretations of quantum mechanics|interpretations]], which fundamentally differs from that of [[classic mechanical]] waves.{{sfn|Born|1927|pp=354–357}}{{sfn|Heisenberg|1958|p=143}}<ref>[[Werner Heisenberg|Heisenberg, W.]] (1927/1985/2009). Heisenberg is translated by {{harvnb|Camilleri|2009|p=71}}, (from {{harvnb|Bohr|1985|p=142}}).</ref>{{sfn|Murdoch|1987|p=43}}{{sfn|de Broglie|1960|p=48}}{{sfn|Landau|Lifshitz|1977|p=6}}{{sfn|Newton|2002|pp=19–21}}