Wave function

In <a href="/facts/Quantum_physics/BRc3Mzgr">quantum physics</a>, a wave function (or wavefunction) is a mathematical description of the <a href="/facts/Quantum_state/7jbxd7Dx">quantum state</a> of an isolated <a href="/facts/Quantum_system/BRc3Mzgr">quantum system</a>. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital <a href="/facts/Psi_(letter)/AUdz70Lq">psi</a>, respectively). Wave functions are <a href="/facts/Complex_number/2w7mqI7e">complex-valued</a>. For example, a wave function might assign a complex number to each point in a region of space. The <a href="/facts/Born_rule/XAsF5Ndf">Born rule</a> provides the means to turn these complex <a href="/facts/Probability_amplitude/ryPgJvs8">probability amplitudes</a> into actual probabilities. In one common form, it says that the <a href="/facts/Squared_modulus/Jzpq0R0z">squared modulus</a> of a wave function that depends upon position is the <a href="/facts/Probability_density_function/zvfybna4">probability density</a> of <a href="/facts/Measurement_in_quantum_mechanics/VY7jPfP0">measuring</a> 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 <a href="/facts/Operator_(quantum_mechanics)/rCJw8UFz">quantum operators</a>, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.
Wave functions can be <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> of variables other than position, such as <a href="/facts/Momentum/TTkGv5eY">momentum</a>. 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 <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a>. Some particles, like <a href="/facts/Electron/L0KBo5cW">electrons</a> and <a href="/facts/Photon/mFNhAEzA">photons</a>, have nonzero <a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a>, 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 <a href="/facts/Isospin/mnksCIaP">isospin</a>. 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 <a href="/facts/Column_matrix/pPuRr9Xk">column matrix</a> (e.g., a 2 × 1 column vector for a non-relativistic electron with spin 1⁄2).
According to the <a href="/facts/Superposition_principle/GKGs9C92">superposition principle</a> of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>. 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 <a href="/facts/Born_rule/XAsF5Ndf">Born rule</a>, relating transition probabilities to inner products. The <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a> determines how wave functions evolve over time, and a wave function behaves qualitatively like other <a href="/facts/Wave/fU0HH6St">waves</a>, such as <a href="/facts/Water_wave/7IsEqwJp">water waves</a> or waves on a string, because the Schrödinger equation is mathematically a type of <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a>. This explains the name "wave function", and gives rise to <a href="/facts/Wave%25E2%2580%2593particle_duality/Au3UlcwQ">wave–particle duality</a>. However, the wave function in quantum mechanics describes a kind of physical phenomenon, as of 2023 still open to different <a href="/facts/Interpretations_of_quantum_mechanics/WEjlzVPl">interpretations</a>, which fundamentally differs from that of <a href="/facts/Classic_mechanical/gGnINVPE">classic mechanical</a> waves.

Wave function open-in-new

Wave function