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Encyclopedia > Correspondence principle

In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers. Ehrenfest's theorem later justified and quantified this principle by showing how Newton's laws emerged from post-1925 or the "new" quantum mechanics. This is a discussion of a present category of science. ... Niels (Henrik David) Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in 1922. ... Year 1923 (MCMXXIII) was a common year starting on Monday (link will display the full calendar) of the Gregorian calendar. ... Fig. ... Classical mechanics (also called Newtonian mechanics) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. ...


The rules of quantum mechanics are highly successful in describing microscopic objects, such as atoms and elementary particles. On the other hand, experiments reveal that a variety of macroscopic systems (springs, capacitors, and so forth) can be accurately described by classical theories such as classical mechanics and classical electrodynamics. However, it is reasonable to believe that the ultimate laws of physics must be independent of the size of the physical objects being described. This is the motivation for Bohr's correspondence principle, which states that classical physics must emerge as an approximation to quantum physics as systems become "larger". For other uses, see Atom (disambiguation). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In the scientific method, an experiment (Latin: ex- periri, of (or from) trying) is a set of observations performed in the context of solving a particular problem or question, to support or falsify a hypothesis or research concerning phenomena. ... Helical or coil springs designed for tension A spring is a flexible elastic object used to store mechanical energy. ... See Capacitor (component) for a discussion of specific types. ... Classical mechanics (also called Newtonian mechanics) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ...


The conditions under which quantum and classical physics agree are referred to as the correspondence limit, or the classical limit. Bohr provided a rough prescription for the correspondence limit: it occurs when the quantum numbers describing the system are large, meaning either some quantum numbers of the system are excited to a very large value, or the system is described by a large set of quantum numbers, or both. A more elaborated analysis of quantum-classical correspondence (QCC) in time dependent wavepacket dynamics spreading process, leads to the distinction between robust "restricted QCC" and fragile "detailed QCC". See Ref.[2] and references therein. "Restricted QCC" refers to the first two moments of the probability distribution and survives even in the presence of diffraction, while "detailed QCC" requires smooth potentials as in the traditional formulation. The classical limit is the ability of a physical theory to approximate or recover classical mechanics when considered over special values of its parameters. ...


The correspondence principle is one of the tools available to physicists for selecting quantum theories corresponding to reality. The principles of quantum mechanics are fairly broad - for example, they state that the states of a physical system occupy a Hilbert space, but do not state what type of Hilbert space. The correspondence principle limits the choices to those that reproduce classical mechanics in the correspondence limit. For this reason, Bohr has argued that classical physics does not emerge from quantum physics in the same way that classical mechanics emerges as an approximation of special relativity at small velocities; rather, classical physics exists independently of quantum theory and cannot be derived from it. For other uses, see Reality (disambiguation). ... The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... Niels (Henrik David) Bohr (October 7, 1885 – November 18, 1962) was a Danish physicist who made fundamental contributions to understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in 1922. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... In physics, velocity is defined as the rate of change of displacement or the rate of displacement. ...

Contents

Other uses of the term

The term "correspondence principle" is also used in a more general philosophical sense to mean the reduction of a new hypothesized scientific theory to another scientific theory (usually a precursor to the former) which requires that the new theory explain all the phenomena under circumstances for which the preceding theory was known to be valid (the "correspondence limit"). In mathematics, theory is used informally to refer to a body of knowledge about mathematics. ...


For example, Einstein's theory of special relativity satisfies the correspondence principle, as it reduces to classical mechanics in the limit of small velocities in comparison to the speed of light (example below). Also, general relativity reduces to Newtonian gravitation in the limit of weak gravitational fields. For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... A line showing the speed of light on a scale model of Earth and the Moon, about 1. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Gravity is a force of attraction that acts between bodies that have mass. ...


In the field of Public Finance, the correspondence principle is used to define the provision of public goods and services by the geographic range of the positive and negative externalities caused by those goods and services. It is used in conjunction with the principle of subsidiarity, which devolves responsibility for providing public goods and services to the lowest level of government that can efficiently and equitably provide them. Subsidiarity is the idea that matters should be handled by the smallest (or, the lowest) competent authority. ...


In Paul Samuelson's classic economics text, Foundations of Economic Analysis, the term refers to the duality between determining stability of an economic equilibrium and deriving results in comparative statics. Paul Anthony Samuelson (born May 15, 1915, in Gary, Indiana) is an American neoclassical economist known for his contributions to many fields of economics, beginning with his general statement of the comparative statics method in his 1947 book Foundations of Economic Analysis. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Foundations of Economic Analysis is a book by Paul A. Samuelson published in 1947 (Enlarged ed. ... Price of market balance In economics, economic equilibrium is simply a state of the world where economic forces are balanced and in the abscence of external shocks the (equilibrium) values of economic variables will not change. ... Comparative statics is the comparison of two different equilibrium states, before and after a change in one of the variables. ...


Examples

Bohr Model

If an electron in an atom is moving on an orbit with period T, the electromagnetic radiation will classically repeat itself every orbital period. If the coupling to the electromagnetic field is weak, so that the orbit doesn't 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 1/T. This is the classical radiation law: the frequencies emitted are integer multiples of 1/T.


In quantum mechanics, this emission must be of quanta of light. The frequency of the quanta emitted should be integer multiples of 1/T 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 1/T must have nearby energy levels which differ in energy by h/T, and they should be equally spaced near that level. So the level spacing in a quantum system at large quantum numbers is mostly uniform, with spacing:

 Delta E = { hover T(E) }

This rule is the original form of the Bohr correspondence principle. Bohr worried whether the energy spacing 1/T should be best taken as the period of the energy state En or En + 1 or En − 1 or some average. Since this theory is only the leading semiclassical approximation, in hindsight there is no need to quibble.


Bohr considered circular orbits. These orbits will classically decay into smaller circular orbits, so they must also decay to smaller circles when they emit photons. The level spacing between circular orbits can be calculated with the correspondence formula. For a hydrogen atom, the classical orbits have a period T which is determined by Kepler's third law to scale as r3 / 2. The energy scales as 1/r, so the level spacing formula says that: Johannes Keplers primary contributions to astronomy/astrophysics were the three laws of planetary motion. ...

 Delta E = { 1 over r^{3over 2} } = E^{2 over 3}

It is possible to determine the energy levels by recursively stepping down orbit by orbit, but there is a shortcut. The angular momentum L of the circular orbit scales as  sqrt{r} . The energy in terms of the angular momentum is then

E propto {1over r} propto {1 over L^2}

Assuming that L is equally spaced, the spacing between neighboring energies would be

 Delta E = {1 over (L+1)^2 } - {1 over L^2} approx {d over dL} {1 over L^2} = - {2 over L^3} = -2 E^{2 over 3}

Which is what we want. So that the solution is to have equally spaced angular momentum. If you keep track of all the constants, the spacing in angular momentum is equal to hbar. So the angular momentum should be an integer multiple of hbar

 L = {nh over 2pi} = n hbar

This is how Bohr arrived at his model. Since only the level spacing is determined by the correspondence principle, you could always add a small fixed offset to the quantum number, so L could just as well have been (n + .338) / hbar. Bohr was guided by an intuition about which physical quantities were best to quantize semiclassically. The Bohr model of the hydrogen atom, where negatively charged electrons confined to atomic shells encircle a small positively charged atomic nucleus, and that an electron jump between orbits must be accompanied by an emitted or absorbed amount of electromagnetic energy hν. The orbits that the electrons travel in are...


Bohr's condition can be solved for the level energies in a general one dimensional system. Define a quantity J(E) which is a function only of the energy, and has the property that:

 {dJ over dE} = T

This is the analog of the angular momentum in the case of the circular orbits. The orbits selected by the correspondence principle are the ones that obey J=nh for n integer, since

 Delta E = E_{n+1} - E_n = {dE over dJ}(J_{n+1} - J_n) = {1 over T} Delta J

This quantity J is canonically conjugate to a variable θ which, by the Hamilton equations of motion changes with time as the gradient of energy with J. Since this is equal to the inverse period at all times, the variable θ increases steadily from 0 to 1 over one period. Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...


The angle variable comes back to itself after 1 unit of increase, so the geometry of phase space in J coordinates is that of a half-cylinder, capped off at J=0, which is the trivial orbit at E=0. These coordinates are just as canonical as x,p, but the orbits are now lines of constant J instead of nested ovoids in x-p space. The area enclosed by an orbit is invariant under canonical transformations, so it is the same in x-p space as in J-θ. But in the J,<theta> coordinates this area is easy to find, it is the area of a cylinder of unit circumference between 0 and J, or just J. In x,p coordinates, this is the area in phase space enclosed by the orbit. So J is equal to the area enclosed by the orbit: Liouvilles theorem has various meanings: In complex analysis, see Liouvilles theorem (complex analysis). ...

 J = int_0^T p {d x over dt} dt

The quantization rule is that the action variable J is an integer.


So Bohr's correspondence principle provided a way to find the semiclassical quantization rule for an arbitrary system. It was an argument for the quantum conditions mostly independent from the one developed by Wien, Planck and Einstein, which focused on adiabatic invariants. But both pointed to the same quantity. An adiabatic invariant is a property of a physical system which stays constant when changes are made slowly. ...


For multidimensional motions, the correspondence principle requires each of the idependent action variables to be integers. This extension allowed Arnold Sommerfeld to formulate a more accurate model of the hydrogen atom. He could solve this model even with a relativistic electron motion, and found the correct relativistic corrections to the spectral lines of hydrogen. This is very mysterious because the full quantum mechanical treatment of the hydrogen fine structure requires spin, which is not included in any semiclassical description. Still the semiclassical method gives the same formula as the Dirac equation, which is nowadays understood as a consequence of a hidden supersymmetry of the Dirac equation in a Coulomb field. Arnold Johannes Wilhelm Sommerfeld (December 5, 1868 in Königsberg, East Prussia – April 26, 1951 in Munich, Germany) was a German physicist who introduced the fine-structure constant in 1919. ... In atomic physics, the fine structure describes the splitting of the spectral lines of atoms. ... This article or section is in need of attention from an expert on the subject. ...


For chaotic systems, which do not have action angle variables, the description could not work, a point that troubled Einstein. This led him to introduce the quantum phase, which associated a phase of a quantum wave which was the solution of the classical Hamilton-Jacobi equation. Einstein's work was extended by Erwin Schrodinger, who modified the Hamilton Jacobi equation to get a reasonable equation that determined both the amplitude and phase. The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ... Erwin Schrödinger, as depicted on the former Austrian 1000 Schilling bank note. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...


The Bohr correspondence principle was extended by Hendrik Kramers and Werner Heisenberg to attempt to account for emission intensities of different states. This requires understanding the details of the orbit, which means including all the fourier coefficients of the motion. The Fourier coefficients of the dipole moment determines the classical emmission intensities, and these must have a quantum analog which have the right correspondence principle limit for large quantum numbers. They introduced matrices to describe these fourier coefficients in a joint publication, and this line of reasoning led Heisenberg to matrix mechanics. Hans Kramers (center) with George Uhlenbeck and Samuel Goudsmit, circa 1928. ... Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ...


Both approaches led to the same theory, which is modern quantum mechanics.


The quantum harmonic oscillator

We provide a demonstration of how large quantum numbers can give rise to classical behavior. Consider the one-dimensional quantum harmonic oscillator. Quantum mechanics tells us that the total (kinetic and potential) energy of the oscillator, E, has a set of discrete values: The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...

E=(n+1/2)hbar omega,  n=0, 1, 2, 3, dots

where omega, is the angular frequency 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. It has been suggested that this article or section be merged into Angular velocity. ... In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ...


We can verify that our idea of "macroscopic" systems fall within the correspondence limit. The energy of the classical harmonic oscillator with amplitude A, is It has been suggested that pulse amplitude be merged into this article or section. ...

 E = frac{m omega ^2 A^2}{2}

Thus, the quantum number has the value

 n = frac{E}{hbar cdot omega} - frac{1}{2} = frac{m omega A^2}{2hbar} -frac{1}{2}

If we apply typical "human-scale" values m = 1kg, omega, = 1 rad/s, and A = 1m, then n ≈ 4.74×1033. This is a very large number, so the system is indeed in the correspondence limit. The kilogram or kilogramme (symbol: kg) is the SI base unit of mass. ... Some common angles, measured in radians. ... Look up second in Wiktionary, the free dictionary. ... ‹ The template below (Unit of length) is being considered for deletion. ...


It is simple to see why we perceive a continuum of energy in said limit. With omega, = 1 rad/s, the difference between each energy level is hbar omegaapprox 1.05times 10^{-34}J, well below what we can detect. The joule (IPA: or ) (symbol: J) is the SI unit of energy. ...


Relativistic kinetic energy

Here we show that the expression of kinetic energy from special relativity becomes arbitrarily close to the classical expression for speeds that are much slower than the speed of light. The kinetic energy of an object is the extra energy which it possesses due to its motion. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... A line showing the speed of light on a scale model of Earth and the Moon, about 1. ...


Einstein's famous mass-energy equation Einstein redirects here. ...

 E = m c^2

represents the total energy of a body with relativistic mass

 m = frac{m_0} {sqrt{1 - v^2/c^2}}
where the velocity,  v  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  c  is the speed of light.

When the velocity  v  is zero, the energy expressed above is not zero and represents the rest energy: The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ... A line showing the speed of light on a scale model of Earth and the Moon, about 1. ... The rest energy of a particle is its energy when it is not moving relative to a given inertial reference frame. ...

 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 = m c^2 - m_0 c^2 = frac{m_0 c^2} {sqrt{1 - v^2/c^2}}  -  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  v ll c

 T  =m_0 c^2 left( frac{1} {sqrt{1 - v^2/c^2}} - 1 right)
=m_0 c^2 left( left( 1 - v^2/c^2 right) ^{-frac{1}{2}} - 1 right)
approx m_0 c^2 left( (1 - (-begin{matrix} frac{1}{2} end{matrix} )v^2/c^2) - 1 right)
=m_0 c^2 left( begin{matrix} frac{1}{2} end{matrix} v^2/c^2 right)
= begin{matrix} frac{1}{2} end{matrix} m_0 v^2

which is the Newtonian expression for kinetic energy. Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ...


References

[1] Weidner, Richard T., and Sells, Robert L. (1980) Elementary Modern Physics. ISBN 0-205-06559-7


[2] A. Stotland and D. Cohen, J. Phys. A 39, 10703 (2006).


  Results from FactBites:
 
Correspondence principle - Wikipedia, the free encyclopedia (711 words)
In physics, the correspondence principle is a principle, first invoked by Niels Bohr in 1923, which states that the behavior of quantum mechanical systems reduce to classical physics in the limit of large quantum numbers.
This is the motivation for Bohr's correspondence principle, which states that classical physics must emerge as an approximation to quantum physics as systems become "larger".
The principles of quantum mechanics are fairly broad - for example, they state that the states of a physical system occupy a Hilbert space, but do not state what type of Hilbert space.
correspondence principle - HighBeam Encyclopedia (375 words)
CORRESPONDENCE PRINCIPLE [correspondence principle] physical principle, enunciated by Niels Bohr in 1923, according to which the predictions of the quantum theory must correspond to the predictions of the classical theories of physics when the quantum theory is used to describe the behavior of systems that can be successfully described by classical theories.
Technically this principle means that the results of a quantum theory analysis of a problem that involves the use of very large quantum numbers must agree with the results of a classical physics analysis.
The correspondence principle provided an important theoretical basis for the development of a detailed correlation between the newer quantum theory and the classical physics that preceded it.
  More results at FactBites »

 
 

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