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The classical limit or correspondence limit is the ability of a
physical theory Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict natural phenomena. This is in contrast to experimental physics, which uses experi ...
to approximate or "recover"
classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior.


Quantum theory

A
heuristic A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...
postulate called the
correspondence principle In physics, a correspondence principle is any one of several premises or assertions about the relationship between classical and quantum mechanics. The physicist Niels Bohr coined the term in 1920 during the early development of quantum theory; ...
was introduced to quantum theory by
Niels Bohr Niels Henrik David Bohr (, ; ; 7 October 1885 – 18 November 1962) was a Danish theoretical physicist who made foundational contributions to understanding atomic structure and old quantum theory, quantum theory, for which he received the No ...
: in effect it states that some kind of continuity argument should apply to the classical limit of quantum systems as the value of the
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
normalized by the action of these systems becomes very small. Often, this is approached through "quasi-classical" techniques (cf.
WKB approximation In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to Linear differential equation, linear differential equations with spatially varying coefficients. It is typically used for a Semiclass ...
). More rigorously, the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than the reduced Planck constant , so the "deformation parameter" / can be effectively taken to be zero (cf.
Weyl quantization Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
.) Thus typically, quantum commutators (equivalently,
Moyal bracket In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a l ...
s) reduce to
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
s, in a group contraction. In
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, due to Heisenberg's
uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
, an
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
can never be at rest; it must always have a non-zero
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
, a result not found in classical mechanics. For example, if we consider something very large relative to an electron, like a baseball, the uncertainty principle predicts that it cannot really have zero kinetic energy, but the uncertainty in kinetic energy is so small that the baseball can effectively appear to be at rest, and hence it appears to obey classical mechanics. In general, if large energies and large objects (relative to the size and energy levels of an electron) are considered in quantum mechanics, the result will appear to obey classical mechanics. The typical occupation numbers involved are huge: a macroscopic harmonic oscillator with  = 2 Hz,  = 10 g, and maximum
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
0 = 10 cm, has  = , so that  ≃ 1030. Further see
coherent states In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmo ...
. It is less clear, however, how the classical limit applies to chaotic systems, a field known as quantum chaos. Quantum mechanics and classical mechanics are usually treated with entirely different formalisms: quantum theory using
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
, and classical mechanics using a representation in
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
. One can bring the two into a common mathematical framework in various ways. In the phase space formulation of quantum mechanics, which is statistical in nature, logical connections between quantum mechanics and classical statistical mechanics are made, enabling natural comparisons between them, including the violations of
Liouville's theorem (Hamiltonian) In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical mechanics, statistical and Hamiltonian mechanics. It asserts that ''the phase space, phase-space distribution functi ...
upon quantization. In a crucial paper (1933), Dirac explained how classical mechanics is an
emergent phenomenon In philosophy, systems theory, science, and art, emergence occurs when a complex entity has properties or behaviors that its parts do not have on their own, and emerge only when they interact in a wider whole. Emergence plays a central role ...
of quantum mechanics:
destructive interference In physics, interference is a phenomenon in which two coherent waves are combined by adding their intensities or displacements with due consideration for their phase difference. The resultant wave may have greater amplitude (constructive in ...
among paths with non- extremal macroscopic actions  »  obliterate amplitude contributions in the path integral he introduced, leaving the extremal action class, thus the classical action path as the dominant contribution, an observation further elaborated by Feynman in his 1942 PhD dissertation. (Further see
quantum decoherence Quantum decoherence is the loss of quantum coherence. It involves generally a loss of information of a system to its environment. Quantum decoherence has been studied to understand how quantum systems convert to systems that can be expla ...
.)


Time-evolution of expectation values

One simple way to compare classical to quantum mechanics is to consider the time-evolution of the ''expected'' position and ''expected'' momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the
Ehrenfest theorem The Ehrenfest theorem, named after Austrian theoretical physicist Paul Ehrenfest, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of the force F=-V'(x) on a m ...
. For a one-dimensional quantum particle moving in a potential V, the Ehrenfest theorem says :m\frac\langle x\rangle = \langle p\rangle;\quad \frac\langle p\rangle = -\left\langle V'(X)\right\rangle . Although the first of these equations is consistent with the classical mechanics, the second is not: If the pair (\langle X\rangle,\langle P\rangle) were to satisfy Newton's second law, the right-hand side of the second equation would have read :\frac\langle p\rangle =-V'\left(\left\langle X\right\rangle\right). But in most cases, :\left\langle V'(X)\right\rangle\neq V'(\left\langle X\right\rangle). If for example, the potential V is cubic, then V' is quadratic, in which case, we are talking about the distinction between \langle X^2\rangle and \langle X\rangle^2, which differ by (\Delta X)^2. An exception occurs in case when the classical equations of motion are linear, that is, when V is quadratic and V' is linear. In that special case, V'\left(\left\langle X\right\rangle\right) and \left\langle V'(X)\right\rangle do agree. In particular, for a free particle or a quantum harmonic oscillator, the expected position and expected momentum exactly follows solutions of Newton's equations. For general systems, the best we can hope for is that the expected position and momentum will ''approximately'' follow the classical trajectories. If the wave function is highly concentrated around a point x_0, then V'\left(\left\langle X\right\rangle\right) and \left\langle V'(X)\right\rangle will be ''almost'' the same, since both will be approximately equal to V'(x_0). In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least ''for as long as'' the wave function remains highly localized in position. p. 78 Now, if the initial state is very localized in position, it will be very spread out in momentum, and thus we expect that the wave function will rapidly spread out, and the connection with the classical trajectories will be lost. When the Planck constant is small, however, it is possible to have a state that is well localized in ''both'' position and momentum. The small uncertainty in momentum ensures that the particle ''remains'' well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories for a long time.


Relativity and other deformations

Other familiar deformations in physics involve: *The deformation of classical Newtonian into relativistic mechanics (
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
), with deformation parameter ; the classical limit involves small speeds, so , and the systems appear to obey Newtonian mechanics. *Similarly for the deformation of Newtonian gravity into
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, with deformation parameter Schwarzschild-radius/characteristic-dimension, we find that objects once again appear to obey classical mechanics (flat space), when the mass of an object times the square of the
Planck length In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: '' c'', '' G'', '' ħ'', and ''k''B (described further below). Expressing one of ...
is much smaller than its size and the sizes of the problem addressed. See
Newtonian limit In physics, the Newtonian limit is a mathematical approximation applicable to physical systems exhibiting (1) weak gravitation, (2) objects moving slowly compared to the speed of light, and (3) slowly changing (or completely static) gravitational f ...
. *Wave optics might also be regarded as a deformation of ray optics for deformation parameter . *Likewise,
thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
deforms to
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
with deformation parameter .


See also

* Classical probability density *
Ehrenfest theorem The Ehrenfest theorem, named after Austrian theoretical physicist Paul Ehrenfest, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of the force F=-V'(x) on a m ...
* Madelung equations *
Fresnel integral 250px, Plots of and . The maximum of is about . If the integrands of and were defined using instead of , then the image would be scaled vertically and horizontally (see below). The Fresnel integrals and are two transcendental functions n ...
*
Mathematical formulation of quantum mechanics The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, whic ...
* Quantum chaos *
Quantum decoherence Quantum decoherence is the loss of quantum coherence. It involves generally a loss of information of a system to its environment. Quantum decoherence has been studied to understand how quantum systems convert to systems that can be expla ...
*
Quantum limit A quantum limit in physics is a limit on measurement accuracy at quantum scales. Depending on the context, the limit may be absolute (such as the Heisenberg limit), or it may only apply when the experiment is conducted with naturally occurring qu ...
*
Semiclassical physics In physics, semiclassical refers to a theory in which one part of a system is described quantum mechanically, whereas the other is treated classically. For example, external fields will be constant, or when changing will be classically describ ...
*
Wigner–Weyl transform In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform (after Hermann Weyl and Eugene Wigner) is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödin ...
*
WKB approximation In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to Linear differential equation, linear differential equations with spatially varying coefficients. It is typically used for a Semiclass ...


References

* {{DEFAULTSORT:Classical Limit Concepts in physics Quantum mechanics Theory of relativity Philosophy of science Emergence