Question: Which mathematical function provides information about the probability of amplitude of position, momentum, and other physical properties?
About the article "Quantum mechanics", section "Abstract".

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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.
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. 
In theoretical physics, the Veneziano amplitude refers to the discovery made in 1968 by Italian theoretical physicist Gabriele Veneziano that the Euler gamma function, when interpreted as a scattering amplitude, has many of the features needed to explain the physical properties of strongly interacting mesons, such as symmetry and duality.
Conformal symmetry was soon discovered.
The formula is the following:  formula_1 .
"k"n is a vector (such as a four-vector) referring to the momentum of the nth particle.
Γ is the gamma function.
This discovery can be considered the birth of string theory, as the discovery and invention of string theory came about as a search for a physical model which would give rise to such a scattering amplitude.  
In chemistry, a molecular orbital (or MO) is a mathematical function describing the wave-like behavior of an electron in a molecule.
This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region.
The term "orbital" was introduced by Robert S. Mulliken in 1932 as an abbreviation for "one-electron orbital wave function".
At an elementary level it is used to describe the "region" of space in which the function has a significant amplitude.
Molecular orbitals are usually constructed by combining atomic orbitals or hybrid orbitals from each atom of the molecule, or other molecular orbitals from groups of atoms.
They can be quantitatively calculated using the Hartree–Fock or self-consistent field (SCF) methods. 
In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors").
Formally, these reside in a complex separable Hilbert space—variously called the "state space" or the "associated Hilbert space" of the system—that is well defined up to a complex number of norm 1 (the phase factor).
In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space.
The exact nature of this Hilbert space is dependent on the system—for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes.
Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint) linear operator acting on the state space.
Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.
If the operator's spectrum is discrete, the observable can attain only those discrete eigenvalues.
In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function, also referred to as state vector in a complex vector space.
This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments.
For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time.
Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, with accuracy.
For instance, electrons may be considered (to a certain probability) to be located somewhere within a given region of space, but with their exact positions unknown.
Contours of constant probability, often referred to as "clouds", may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability.
Heisenberg's uncertainty principle quantifies the inability to precisely locate the particle given its conjugate momentum.
According to one interpretation, as the result of a measurement the wave function containing the probability information for a system collapses from a given initial state to a particular eigenstate.
The possible results of a measurement are the eigenvalues of the operator representing the observable—which explains the choice of "Hermitian" operators, for which all the eigenvalues are real.
The probability distribution of an observable in a given state can be found by computing the spectral decomposition of the corresponding operator.
Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute.
The probabilistic nature of quantum mechanics thus stems from the act of measurement.
This is one of the most difficult aspects of quantum systems to understand.
It was the central topic in the famous Bohr-Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments.
In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied.
Newer interpretations of quantum mechanics have been formulated that do away with the concept of "wavefunction collapse" (see, for example, the relative state interpretation).
The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity.
For details, see the article on measurement in quantum mechanics.
Generally, quantum mechanics does not assign definite values.
Instead, it makes a prediction using a probability distribution; that is, it describes the probability of obtaining the possible outcomes from measuring an observable.
Often these results are skewed by many causes, such as dense probability clouds.
Probability clouds are approximate, but better than the Bohr model, whereby electron location is given by a probability function, the wave function eigenvalue, such that the probability is the squared modulus of the complex amplitude, or quantum state nuclear attraction.
Naturally, these probabilities will depend on the quantum state at the "instant" of the measurement.
Hence, uncertainty is involved in the value.
There are, however, certain states that are associated with a definite value of a particular observable.
These are known as eigenstates of the observable ("eigen" can be translated from German as meaning "inherent" or "characteristic").
De.pons.eu</ref>In the everyday world, it is natural and intuitive to think of everything (every observable) as being in an eigenstate.
Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence.
However, quantum mechanics does not pinpoint the exact values of a particle's position and momentum (since they are conjugate pairs) or its energy and time (since they too are conjugate pairs); rather, it provides only a range of probabilities in which that particle might be given its momentum and momentum probability.
Therefore, it is helpful to use different words to describe states having "uncertain" values and states having "definite" values (eigenstates).
Usually, a system will not be in an eigenstate of the observable (particle) we are interested in.
However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or "generalized" eigenstate) of that observable.
This process is known as wavefunction collapse, a controversial and much-debated process that involves expanding the system under study to include the measurement device.
If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of the wavefunction collapsing into each of the possible eigenstates.
For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position "x"0 (neither an eigenstate of position nor of momentum).
When one measures the position of the particle, it is impossible to predict with certainty the result.
It is probable, but not certain, that it will be near "x"0, where the amplitude of the wave function is large.
After the measurement is performed, having obtained some result "x", the wave function collapses into a position eigenstate centered at "x".
The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian (the operator corresponding to the total energy of the system) generates the time evolution.
The time evolution of wave functions is deterministic in the sense that - given a wavefunction at an "initial" time - it makes a definite prediction of what the wavefunction will be at any "later" time.
During a measurement, on the other hand, the change of the initial wavefunction into another, later wavefunction is not deterministic, it is unpredictable (i.e., random).
A time-evolution simulation can be seen here.
Wave functions change as time progresses.
The Schrödinger equation describes how wavefunctions change in time, playing a role similar to Newton's second law in classical mechanics.
The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity (like a classical particle with no forces acting on it).
However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain with time.
This also has the effect of turning a position eigenstate (which can be thought of as an infinitely sharp wave packet) into a broadened wave packet that no longer represents a (definite, certain) position eigenstate.
Some wave functions produce probability distributions that are constant, or independent of time—such as when in a stationary state of constant energy, time vanishes in the absolute square of the wave function.
Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions.
For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus (Fig.
1) (note, however, that only the lowest angular momentum states, labeled "s", are spherically symmetric).
The Schrödinger equation acts on the "entire" probability amplitude, not merely its absolute value.
Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states.
This gives rise to the "wave-like" behavior of quantum states.
As it turns out, analytic solutions of the Schrödinger equation are available for only a very small number of relatively simple model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion, and the hydrogen atom are the most important representatives.
Even the helium atom—which contains just one more electron than does the hydrogen atom—has defied all attempts at a fully analytic treatment.
There exist several techniques for generating approximate solutions, however.
In the important method known as perturbation theory, one uses the analytic result for a simple quantum mechanical model to generate a result for a more complicated model that is related to the simpler model by (for one example) the addition of a weak potential energy.
Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces only weak (small) deviations from classical behavior.
These deviations can then be computed based on the classical motion.
This approach is particularly important in the field of quantum chaos.