Correspondence principle

The quantum mechanical probability density of a particle is proportional to the square of the wave function of the matter wave at that point. For large quantum numbers the quantum mechanical probability density changes asymptotically into the classical one.

The correspondence principle was originally used to describe a relationship between terms used in classical physics and quantum mechanics . The term was coined in 1920 by Niels Bohr in the context of the older quantum theory . In this context it is also referred to as Bohr's principle of correspondence .

In his 1913 atomic model, Bohr was still based on a classical model, but was forced to formulate very restrictive conditions for the electron paths occurring in order to explain the discrete optical spectra observed . Nevertheless, the "older quantum theory" formulated in this way could not provide a complete theory of the spectra. For large quantum numbers , however, there were asymptotic formulas that corresponded to those of classical physics and could partially fill these explanatory gaps. In this sense, the correspondence principle served as a heuristic principle in the older quantum theory to describe the transition to classical physics (in this case electrodynamics ) for large quantum numbers.

In quantum mechanics, which emerged from 1925, the correspondence principle was also used to describe a heuristic method of relating quantum mechanical operators and their commutation relationships with those of classical mechanics .

In the philosophy of science (inspired by the example of quantum theory) the correspondence principle is understood to mean the relationship between different theories , usually an older and a newer one, on the same field of phenomena . It is about the basic concept of a theory hierarchy and development in the natural sciences . Correspondence principles are also used in this sense in other sciences such as crystallography .

The correspondence principle as a concept for the hierarchy of theories

The correspondence principle describes a certain relationship between an older scientific theory and a newer one with a larger range of validity. It is present when the newer theory comes to the same results as this on the area of validity of the older one. This type of theory development is typical and desirable in the natural sciences. In the social sciences , this was the subject of extensive epistemological struggles for positions that were justified by Imre Lakatos and Thomas S. Kuhn in terms of epistemology .

The newer theory in this case contains the older one as a borderline case and thus explains its earlier success. Furthermore, the new theory does not conflict with the older experimental findings. The newer theory can be structurally and conceptually completely different from the older one. This refutes the older theory in principle, but remains useful in its limited scope.

In the following, some significant examples from the history of science for the fulfillment of this correspondence principle are explained.

Newtonian physics and the theory of relativity

Although the theory of relativity introduces completely new ideas of space and time , its predictions pass into those of Newtonian physics when applied to our everyday sphere.

In the special theory of relativity , spatial and temporal distances depend on the state of motion of the observer. If the corresponding speeds are sufficiently small compared to the speed of light , the differences between these distances get below the experimental detection limit , so that the concepts of space and time of Newtonian physics, which are obsolete , can be applied. Likewise, the curvature of space due to the presence of masses and the dependence of the rate of clocks on their position in the gravitational field , as predicted in the general theory of relativity , can hardly be determined experimentally for sufficiently small areas of space, such as within our everyday radius of action. The relationship between special and general relativity also corresponds to the principle of correspondence.

Newtonian physics and quantum physics

The laws of Newtonian physics can be derived as a borderline case from those of quantum physics, although the latter are based on completely different and no longer clearly accessible concepts of matter and motion and although there are quantities in quantum mechanics (e.g. spin ) that are in do not occur in classical mechanics.

Quantum physics generally only allows probability forecasts for the value of a measured variable such as the location where an object will be located. It is therefore no longer any question regarding deterministic . If one calculates the so-called expected value , i.e. the mean value of this measured variable in the limit of infinitely frequent repetitions of the experiment , then it turns out, if the variable exists in classical physics, that it obeys the known equations of Newtonian physics ( Ehrenfest theorem ). If the rules of quantum physics are applied to macroscopic mechanical systems, the statistical spread of the measurement results becomes almost immeasurably small. Such systems correspond to i. a. a statistical ensemble of a large number of so-called pure quantum states with large quantum numbers . The deterministic character of classical physics for the macroscopic borderline case follows from quantum physics, although the latter is not itself deterministic.

Theory of Relativity, Quantum Physics and Quantum Gravity

One of the great problems of the theoretical structure of physics is currently that its two pillars, general relativity and quantum physics, do not fulfill the principle of correspondence in their relationship to one another. Both theories therefore have only a limited range of validity, so that today's physics cannot provide a complete description of nature. We are therefore looking for a theory of so-called quantum gravity that unites the theory of relativity and quantum physics by including both as borderline cases in the sense of the correspondence principle.

The correspondence principle in the older quantum theory

The older quantum theory combines the classical mechanics of quasi-periodic systems with additional assumptions , the most important of which is the restriction of the permissible orbits in phase space to those for which the quantization of the orbital angular momentum applies:

{\ displaystyle \ oint p \, \ mathrm {d} q = nh; \ quad n = 1,2, \ ldots}

With

the generalized impulse ${\ displaystyle p}$
the generalized coordinate ${\ displaystyle q}$
the principal quantum number ${\ displaystyle n}$
the Planck constant ${\ displaystyle h}$

or.

{\ displaystyle L = n \ hbar}

With

the orbital angular momentum ${\ displaystyle L}$
the reduced Planck quantum of action ${\ displaystyle \ textstyle \ hbar = {\ frac {h} {2 \ pi}}.}$

The correspondence principle now requires a connection between the coefficients of the Fourier expansion of the spatial coordinates with respect to time ${\ displaystyle D_ {k}}$

{\ displaystyle q (t) = \ sum _ {k} D_ {k} e ^ {\ mathrm {i} s_ {k} t}}

and the quantum-theoretically possible radiation transitions, as well as the intensity and polarization of the light emitted. So u. a. derive the spectroscopic selection rules by inferring from the disappearance of the nth Fourier component that the corresponding quantum jump by n units is impossible .

A condition that Bohr places on this correspondence is that of the approximate agreement with classical electrodynamics for large quantum numbers. This thus represents one of the epistemological correspondence principles described above.

The principle of correspondence in modern quantum mechanics

Following Heisenberg , the assignment of classical observables to their counterparts in the mathematical formulation of quantum mechanics, the operators on Hilbert spaces , is called correspondence. The classical theory in the application of the correspondence principle in this sense serves to find the physically meaningful equations of quantum mechanics - by adopting the algebraic form of the equations, whereby certain classical observables are replaced by the corresponding quantum mechanical operators. For example, by replacing the momentum variable with the corresponding momentum operators (and correspondingly for the position variable) from the classical energy equation, the Schrödinger equation arises . In the past, this assignment was sometimes referred to as Jordan's rule .

Correspondence in crystallography

Paul Niggli formulated the correspondence between crystal structure and morphology. On the one hand, the symmetry of the outer crystal faces (see point group ) is higher or equal to the symmetry of the crystal structure (see space group ). On the other hand, each outer crystal surface runs parallel to a family of lattice planes. Likewise, a crystal edge runs parallel to a family of lattice lines.

This morphological-structural correspondence also applies to all other properties of the crystal and was formulated as Neumann's principle by Woldemar Voigt in 1910. The symmetry of a property is higher than or equal to the symmetry of the crystal structure.

Web links

Alisa Bokulich: Bohr's Correspondence Principle. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
mikomma.de: Correspondence principle - animations of electrons in Rydberg states .

Individual evidence

↑ Niels Bohr : About the series spectrum of the elements. In: Journal of Physics . Vol. 2, No. 5, 1920, pp. 423-469, doi : 10.1007 / BF01329978 .
^ Karl Popper : The objective of empirical science. In: Hans Albert (Ed.): Theory and Reality. Selected essays on the science of the social sciences (= The Unit of Social Sciences. 2, ISSN 0424-6985 ). J. C. B. Mohr (Paul Siebeck), Tübingen 1964, pp. 75-86, here p. 84.
^ Max Jammer : The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York NY et al. 1966, p. 109.

[Bohr_1920-1] Niels Bohr : About the series spectrum of the elements. In: Journal of Physics . Vol. 2, No. 5, 1920, pp. 423-469, doi : 10.1007 / BF01329978 .

[2] Karl Popper : The objective of empirical science. In: Hans Albert (Ed.): Theory and Reality. Selected essays on the science of the social sciences (= The Unit of Social Sciences. 2, ISSN 0424-6985 ). J. C. B. Mohr (Paul Siebeck), Tübingen 1964, pp. 75-86, here p. 84.

[Jammer_1966-3] Max Jammer : The Conceptual Development of Quantum Mechanics. McGraw-Hill, New York NY et al. 1966, p. 109.