classical limit of quantum mechanics
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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 experime ...
to approximate or "recover"
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
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, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
postulate called the correspondence principle was introduced to
quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ...
by
Niels Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922 ...
: 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, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
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 method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...
). 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.) Thus typically, quantum commutators (equivalently, Moyal brackets) reduce to Poisson brackets, in a group contraction. In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, due to Heisenberg's
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
, an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
can never be at rest; it must always have a non-zero
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
, 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 number In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an impo ...
s 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 which 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 chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mech ...
. Quantum mechanics and classical mechanics are usually treated with entirely different formalisms: quantum theory using Hilbert space, and classical mechanics using a representation in phase space. One can bring the two into a common mathematical framework in various ways. In the
phase space formulation The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...
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 and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajectorie ...
upon quantization. In a crucial paper (1933),
Dirac Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety o ...
explained how classical mechanics is an
emergent phenomenon In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
of quantum mechanics:
destructive interference In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
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.)


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 Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
. 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 regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
), 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 Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, 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 is much smaller than its size and the sizes of the problem addressed. See Newtonian limit. *Wave optics might also be regarded as a deformation of
ray optics Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of '' rays''. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstan ...
for deformation parameter . *Likewise,
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...
deforms to statistical mechanics with deformation parameter .


See also

* Classical probability density *
Ehrenfest theorem The Ehrenfest theorem, named after Paul Ehrenfest, an Austrian theoretical physicist at Leiden University, relates the time derivative of the expectation values of the position and momentum operators ''x'' and ''p'' to the expectation value of th ...
*
Madelung equations The Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation, written in terms of hydrodynamical variables, similar to the Navier–Stokes equations of flu ...
*
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, which ...
*
Quantum chaos Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mech ...
* Quantum decoherence * Quantum limit *
Quantum realm Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
*
Semiclassical physics Semiclassical physics, or simply 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 ...
*
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 method for finding approximate solutions to linear differential equations with spatially varying coefficients. It is typically used for a semiclassical calculation in quantum mecha ...


References

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