Correspondence principle
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the correspondence principle states that the behavior of systems described by the theory of
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 chemistry, ...
(or by the
old quantum theory The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory ...
) reproduces
classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
in the limit of large
quantum numbers In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be kno ...
. In other words, it says that for large
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
s and for large
energies In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...
, quantum calculations must agree with classical calculations. The principle was formulated 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. B ...
in 1920, though he had previously made use of it as early as 1913 in developing his model of the atom. The term codifies the idea that a new theory should reproduce under some conditions the results of older well-established theories in those domains where the old theories work. This concept is somewhat different from the requirement of a formal
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
under which the new theory reduces to the older, thanks to the existence of a deformation parameter. Classical quantities appear in quantum mechanics in the form of
expected values In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of observables, and as such 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 ...
(which predicts the time evolution of the expected values) lends support to the correspondence principle.


Quantum mechanics

The rules of quantum mechanics are highly successful in describing microscopic objects,
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, and ...
s and
elementary particles In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, anti ...
. But ''macroscopic systems,'' like springs and
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
s, are accurately described by classical theories like
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 classical ...
and
classical electrodynamics Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fie ...
. If quantum mechanics were to be applicable to macroscopic objects, there must be some limit in which quantum mechanics reduces to classical mechanics. ''Bohr's correspondence principle demands that classical physics and quantum physics give the same answer when the systems become large''.
Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
referred to the principle as "Bohrs Zauberstab" (Bohr's magic wand) in 1921. The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the
classical limit The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
. Bohr provided a rough prescription for the correspondence limit: it occurs ''when the quantum numbers describing the system are large''. A more elaborated analysis of quantum-classical correspondence (QCC) in wavepacket spreading leads to the distinction between robust "restricted QCC" and fragile "detailed QCC". "Restricted QCC" refers to the first two moments of the probability distribution and is true even when the wave packets diffract, while "detailed QCC" requires smooth potentials which vary over scales much larger than the wavelength, which is what Bohr considered. The post-1925 new quantum theory came in two different formulations. In
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
, the correspondence principle was built in and was used to construct the theory. In the Schrödinger approach classical behavior is not clear because the waves spread out as they move. Once the Schrödinger equation was given a probabilistic interpretation, Ehrenfest showed that Newton's laws hold on average: the quantum statistical expectation value of the position and momentum obey Newton's laws. The correspondence principle is one of the tools available to physicists for selecting quantum theories corresponding to
reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only imaginary. The term is also used to refer to the ontological status of things, indicating their existence. In physical terms, r ...
. The
principles of quantum mechanics Principles of Quantum Mechanics is a textbook by Ramamurti Shankar. The book has been through two editions. It is used in many college courses around the world. Contents # Mathematical Introduction # Review of Classical Mechanics # All Is Not ...
are broad: states of a physical system form a
complex vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
and
physical observables In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
are identified with
Hermitian operators In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
that act on this
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit.


Other scientific theories

The term "correspondence principle" is used in a more general sense to mean the reduction of a new
scientific theory A scientific theory is an explanation of an aspect of the natural world and universe that has been repeatedly tested and corroborated in accordance with the scientific method, using accepted protocols of observation, measurement, and evaluatio ...
to an earlier scientific theory in appropriate circumstances. This requires that the new theory explain all the phenomena under circumstances for which the preceding theory was known to be valid, the "correspondence limit". For example, *Einstein's
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 o ...
satisfies the correspondence principle, because it reduces to classical mechanics in the limit of velocities small compared to the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
(example below); *
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 ...
reduces to
Newtonian gravity Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
in the limit of weak gravitational fields; * Laplace's theory of
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
reduces to Kepler's when interplanetary interactions are ignored; * Statistical mechanics reproduces thermodynamics when the number of particles is large; * In biology, chromosome inheritance theory reproduces Mendel's laws of inheritance, in the domain that the inherited factors are protein coding
genes In biology, the word gene (from , ; "...Wilhelm Johannsen coined the word gene to describe the Mendelian units of heredity..." meaning ''generation'' or ''birth'' or ''gender'') can have several different meanings. The Mendelian gene is a ba ...
. * In mathematical economics, as formalized in ''
Foundations of Economic Analysis ''Foundations of Economic Analysis'' is a book by Paul A. Samuelson published in 1947 (Enlarged ed., 1983) by Harvard University Press. It is based on Samuelson's 1941 doctoral dissertation at Harvard University. The book sought to demonstrate a ...
'' (1947) by
Paul Samuelson Paul Anthony Samuelson (May 15, 1915 – December 13, 2009) was an American economist who was the first American to win the Nobel Memorial Prize in Economic Sciences. When awarding the prize in 1970, the Swedish Royal Academies stated that he "h ...
, the correspondence principle and other postulates imply testable predictions about how the equilibrium changes when parameters are changed in an economic system. In order for there to be a correspondence, the earlier theory has to have a domain of validity—it must work under ''some'' conditions. Not all theories have a domain of validity. For example, there is no limit where Newton's mechanics reduces to Aristotle's mechanics because Aristotle's mechanics, although academically dominant for 18 centuries, does not have any domain of validity (on the other hand, it can sensibly be said that the falling of objects through the air ("natural motion") constitutes a domain of validity for ''a part of'' Aristotle's mechanics).


Examples


Bohr model

If an electron in an atom is moving on an orbit with period , classically the electromagnetic radiation will repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit does not decay very much in one cycle, the radiation will be emitted in a pattern which repeats every period, so that the Fourier transform will have frequencies which are only multiples of . This is the classical radiation law: the frequencies emitted are integer multiples of . In quantum mechanics, this emission must be in quanta of light, of frequencies consisting of integer multiples of , so that classical mechanics is an approximate description at large quantum numbers. This means that the energy level corresponding to a classical orbit of period must have nearby energy levels which differ in energy by , and they should be equally spaced near that level, \Delta E_n = \frac \,. Bohr worried whether the energy spacing should be best calculated with the period of the energy state E_n, or E_, or some average—in hindsight, this model is only the leading semiclassical approximation. Bohr considered circular orbits. Classically, these orbits must decay to smaller circles when photons are emitted. The level spacing between circular orbits can be calculated with the correspondence formula. For a Hydrogen atom, the classical orbits have a period determined by
Kepler's third law In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...
to scale as . The energy scales as , so the level spacing formula amounts to \Delta E \propto \frac \propto E^. It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut. The angular momentum of the circular orbit scales as . The energy in terms of the angular momentum is then E \propto \propto . Assuming, with Bohr, that quantized values of are equally spaced, the spacing between neighboring energies is \Delta E \propto - \frac \approx - \frac \propto - E^. This is as desired for equally spaced angular momenta. If one kept track of the constants, the spacing would be , so the angular momentum should be an integer multiple of , L = \frac = n \hbar ~ . This is how Bohr arrived at his
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
. Since only the level ''spacing'' is determined heuristically by the correspondence principle, one could always add a small fixed offset to the quantum number— could just as well have been . Bohr used his physical intuition to decide which quantities were best to quantize. It is a testimony to his skill that he was able to get so much from what is only the leading order approximation. A less heuristic treatment accounts for needed offsets in the ground state L2, cf.
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 ...
.


One-dimensional potential

Bohr's correspondence condition can be solved for the level energies in a general one-dimensional potential. Define a quantity which is a function only of the energy, and has the property that : = T This is the analogue of the angular momentum in the case of the circular orbits. The orbits selected by the correspondence principle are the ones that obey for integer, since : \Delta E = E_ - E_n = (J_ - J_n) = \,\Delta J This quantity is canonically conjugate to a variable which, by the Hamilton equations of motion changes with time as the gradient of energy with . Since this is equal to the inverse period at all times, the variable increases steadily from 0 to 1 over one period. The angle variable comes back to itself after 1 unit of increase, so the geometry of phase space in coordinates is that of a half-cylinder, capped off at = 0, which is the motionless orbit at the lowest value of the energy. These coordinates are just as canonical as , but the orbits are now lines of constant instead of nested ovoids in space. The area enclosed by an orbit is
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
under canonical transformations, so it is the same in space as in . But in the coordinates, this area is the area of a cylinder of unit circumference between 0 and , or just . So is equal to the area enclosed by the orbit in coordinates too, : J = \int_0^T p \, dt ~. The quantization rule is that the action variable is an integer multiple of .


Multiperiodic motion: Bohr–Sommerfeld quantization

Bohr's correspondence principle provided a way to find the semiclassical quantization rule for a one degree of freedom system. It was an argument for the old quantum condition mostly independent from the one developed by
Wien en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST ...
and Einstein, which focused on adiabatic invariance. But both pointed to the same quantity, the action. Bohr was reluctant to generalize the rule to systems with many degrees of freedom. This step was taken by
Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
, who proposed the general quantization rule for an
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
system, : J_k = h n_k. \, Each action variable is a separate integer, a separate quantum number. This condition reproduces the circular orbit condition for two dimensional motion: let be polar coordinates for a central potential. Then is already an angle variable, and the canonical momentum conjugate is , the angular momentum. So the quantum condition for reproduces Bohr's rule: : \int_0^ L d\theta = 2\pi L = n h. This allowed Sommerfeld to generalize Bohr's theory of circular orbits to elliptical orbits, showing that the energy levels are the same. He also found some general properties of quantum angular momentum which seemed paradoxical at the time. One of these results was that the z-component of the angular momentum, the classical inclination of an orbit relative to the z-axis, could only take on discrete values, a result which seemed to contradict rotational invariance. This was called ''space quantization'' for a while, but this term fell out of favor with the new quantum mechanics since no quantization of space is involved. In modern quantum mechanics, the principle of superposition makes it clear that rotational invariance is not lost. It is possible to rotate objects with discrete orientations to produce superpositions of other discrete orientations, and this resolves the intuitive paradoxes of the Sommerfeld model.


The quantum harmonic oscillator

Here is a demonstration of how large quantum numbers can give rise to classical (continuous) behavior. Consider the one-dimensional
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最 ...
. Quantum mechanics tells us that the total (kinetic and potential)
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
of the oscillator, , has a set of discrete values, :E=(n+1/2)\hbar \omega, \ n=0, 1, 2, 3, \dots ~ , where is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the oscillator. However, in a classical harmonic oscillator such as a lead ball attached to the end of a spring, we do not perceive any discreteness. Instead, the energy of such a macroscopic system appears to vary over a continuum of values. We can verify that our idea of
macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenomena an ...
systems falls within the correspondence limit. The energy of the classical harmonic oscillator with
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 amplit ...
, is : E = \frac . Thus, the quantum number has the value : n = \frac - \frac = \frac -\frac If we apply typical "human-scale" values = 1 kg, = 1
rad RAD or Rad may refer to: People * Robert Anthony Rad Dougall (born 1951), South African former racing driver * Rad Hourani, Canadian fashion designer and artist * Nickname of Leonardus Rad Kortenhorst (1886–1963), Dutch politician * Radley R ...
/ s, and = 1 m, then ≈ 4.74×1033. This is a very large number, so the system is indeed in the correspondence limit. It is simple to see why we perceive a continuum of energy in this limit. With = 1 rad/s, the difference between each energy level is ≈ 1.05 × 10−34 J, well below what we normally resolve for macroscopic systems. One then describes this system through an emergent
classical limit The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
.


Relativistic kinetic energy

Here we show that the expression of
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 accele ...
from
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 o ...
becomes arbitrarily close to the classical expression, for speeds that are much slower than the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, .
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's mass-energy equation : E = \frac c^2 ~, where the velocity, is the velocity of the body relative to the observer, m_0 is the ''rest'' mass (the observed mass of the body at zero velocity relative to the observer), and is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. When the velocity vanishes, the energy expressed above is not zero, and represents the ''rest'' energy, : E_0 = m_0 c^2. \ When the body is in motion relative to the observer, the total energy exceeds the rest energy by an amount that is, by definition, the ''kinetic'' energy, : T = E - E_0 = \frac \ - \ m_0 c^2 ~. Using the approximation ( 1 + x )^n \approx 1 + nx for , x, \ll 1 , we get, when speeds are much slower than that of light, or , \begin T &=m_0 c^2 \left( \frac - 1 \right) \\ &= m_0 c^2 \left( \left( 1 - v^2/c^2 \right) ^ - 1 \right) \\ &\approx m_0 c^2 \left( (1 - (-\begin \frac \end )v^2/c^2) - 1 \right) \\ &= m_0 c^2 \left( \begin \frac \end v^2/c^2 \right) \\ &= \begin \frac \end m_0 v^2 ~, \end which is the Newtonian expression for
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 accele ...
.


See also

* Quantum decoherence *
Classical limit The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
*
Classical probability density The classical probability density is the probability density function that represents the likelihood of finding a particle in the vicinity of a certain location subject to a potential energy in a classical mechanical system. These probability den ...
* Leggett–Garg inequality


References

{{DEFAULTSORT:Correspondence Principle Quantum mechanics Theory of relativity Philosophy of physics Theoretical physics Principles Metatheory