Question: Which mathematical function provides information about the physical properties of a particle?
About the article "Quantum mechanics", section "Abstract".

Candidate passages:
An Anamorphic Stretch Transform (AST) is a mathematical transformation in which analog or digital data is stretched and warped in a specific fashion such that after downsampling, the volume of data is reduced without loss of pertinent information.
The recipe for reshaping is prescribed by a mathematical function called "stretched modulation distribution" or "modulation intensity distribution" (not to be confused with a different function of the same name used in mechanical diagnostics).
A stretched modulation distribution is a 3D representation of a type of bilinear time–frequency distribution that describes the dependence of intensity or power on the frequency and time duration of the modulation.
It provides insight on how information bandwidth and data volume are modified upon dispersion in the time domain, or upon diffraction in the spatial domain.
It also provides a blueprint for compressing the data.
The mathematical transformation emulates propagation of the signal through a physical medium with specific dispersion and diffraction properties, but with an engineered nonlinear kernel. 
Quantum mechanics (QM – also known as quantum physics, or quantum theory) is a branch of physics which deals with physical phenomena at nanoscopic scales where the action is on the order of the Planck constant.
It departs from classical mechanics primarily at the "quantum realm" of atomic and subatomic length scales.
Quantum mechanics provides a mathematical description of much of the dual "particle-like" and "wave-like" behavior and interactions of energy and matter.
Quantum mechanics provides a substantially useful framework for many features of the modern periodic table of elements including the behavior of atoms during chemical bonding and has played a significant role in the development of many modern technologies.
In advanced topics of quantum mechanics, some of these behaviors are macroscopic (see macroscopic quantum phenomena) and emerge at only extreme (i.e., very low or very high) energies or temperatures (such as in the use of superconducting magnets).
For example, the angular momentum of an electron bound to an atom or molecule is quantized.
In contrast, the angular momentum of an unbound electron is not quantized.
In the context of quantum mechanics, the wave–particle duality of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons, and other atomic-scale objects.
The mathematical formulations of quantum mechanics are abstract.
A mathematical function, the wavefunction, provides information about the probability amplitude of position, momentum, and other physical properties of a particle.
Mathematical manipulations of the wavefunction usually involve bra–ket notation which requires an understanding of complex numbers and linear functionals.
The wavefunction formulation treats the particle as a quantum harmonic oscillator, and the mathematics is akin to that describing acoustic resonance.
Many of the results of quantum mechanics are not easily visualized in terms of classical mechanics.
For instance, in a quantum mechanical model the lowest energy state of a system, the ground state, is non-zero as opposed to a more "traditional" ground state with zero kinetic energy (all particles at rest).
Instead of a traditional static, unchanging zero energy state, quantum mechanics allows for far more dynamic, chaotic possibilities, according to John Wheeler.
The earliest versions of quantum mechanics were formulated in the first decade of the 20th century.
About this time, the atomic theory and the corpuscular theory of light (as updated by Einstein) first came to be widely accepted as scientific fact; these latter theories can be viewed as quantum theories of matter and electromagnetic radiation, respectively.
Early quantum theory was significantly reformulated in the mid-1920s by Werner Heisenberg, Max Born and Pascual Jordan, (matrix mechanics); Louis de Broglie and Erwin Schrödinger (wave mechanics); and Wolfgang Pauli and Satyendra Nath Bose (statistics of subatomic particles).
Moreover, the Copenhagen interpretation of Niels Bohr became widely accepted.
By 1930, quantum mechanics had been further unified and formalized by the work of David Hilbert, Paul Dirac and John von Neumann with a greater emphasis placed on measurement in quantum mechanics, the statistical nature of our knowledge of reality, and philosophical speculation about the role of the observer.
Quantum mechanics has since permeated throughout many aspects of 20th-century physics and other disciplines including quantum chemistry, quantum electronics, quantum optics, and quantum information science.
Much 19th-century physics has been re-evaluated as the "classical limit" of quantum mechanics and its more advanced developments in terms of quantum field theory, string theory, and speculative quantum gravity theories.
The name "quantum mechanics" derives from the observation that some physical quantities can change only in "discrete" amounts (Latin "quanta"), and not in a continuous ("cf."
analog) way.
In mathematical physics, in particular in effective field theory, a form factor is a function that gives the properties of a certain particle interaction without including all of the underlying physics.
It is measured experimentally when a theoretical calculation is unavailable or too difficult, see experimental particle physics. 
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable.
Many important properties of physical systems can be represented mathematically as matrix problems.
For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. 
Bohr, Heisenberg and others tried to explain what these experimental results and mathematical models really mean.
Their description, known as the Copenhagen interpretation of quantum mechanics, aimed to describe the nature of reality that was being probed by the measurements and described by the mathematical formulations of quantum mechanics.
The main principles of the Copenhagen interpretation are:  A system is completely described by a wave function, formula_15.
(Heisenberg)  How formula_15 changes over time is given by the Schrödinger equation.
The description of nature is essentially probabilistic.
The probability of an event – for example, where on the screen a particle will show up in the two slit experiment – is related to the square of the absolute value of the amplitude of its wave function.
(Born rule, due to Max Born, which gives a physical meaning to the wave function in the Copenhagen interpretation: the probability amplitude)  It is not possible to know the values of all of the properties of the system at the same time; those properties that are not known with precision must be described by probabilities.
(Heisenberg's uncertainty principle)  Matter, like energy, exhibits a wave–particle duality.
An experiment can demonstrate the particle-like properties of matter, or its wave-like properties; but not both at the same time.
(Complementarity principle due to Bohr)  Measuring devices are essentially classical devices, and measure classical properties such as position and momentum.
The quantum mechanical description of large systems should closely approximate the classical description.
(Correspondence principle of Bohr and Heisenberg) Various consequences of these principles are discussed in more detail in the following subsections.