Bohr’s correspondence principle

On the basis of Bohr s correspondence principle, Heisenberg postulated that a comparable set of quantum-mechanical canonical relations must exist, and... 

For the experienced practitioner of atomic physics there appears to be an enigma right at this point. What does nonlinear chaos theory have to do with linear quantum mechanics, so successful in the classification of atomic states and the description of atomic dynamics The answer, interestingly, is the enormous advances in atomic physics itself. Modern day experiments are able to control essentially isolated atoms and molecules to unprecedented precision at very high quantum numbers. Key elements here are the development of atomic beam techniques and the revolutionary effect of lasers. Given the high quantum numbers, Bohr s correspondence principle tells us that atoms are best understood on the basis of classical mechanics. The classical counterpart of most atoms and molecules, however, is chaotic. Hence the importance of understanding chaos in atomic physics. 

Durran, R., Neate, A., Truman, A. The divine clockwork Bohr s correspondence principle and Nelson s stochastic mechanics for the atomic elliptic state. J. Math. Phys. 49,1 2 (2008). [electronic] URL http //arxiv.org/abs/0711.0157 http //arxiv.org/abs/0711.0157... 

At these high energies there are a sufficient number of energy and angular momentum states involved that one is approaching the classical limit" of Bohr s Correspondence Principle and this is why one can anticipate the success of such a semi-classical picture. 

In the various forms of the Maxwell equation it will be noticed that Planck s constant h has cancelled between numerator and denominator. It appears therefore as if the equations might be independent of the quantum theory cmd they were, of course, obtained by Maxwell before this theory was develop. However, it is to be remembered that the results of the present chapter depend on the particles being very q arsely distributed over the quantum states.t Under the same conditions the separation of the translational states is very small compared to kT, and therefore the translational energy is virtually continuous, as was supposed by Maxwell. This is an aspect of the fact that quantum behaviour converges towards classical behaviour at high values of the quantum numbers, in accordance with Bohr s correspondence principle. 

Bohr s building principle now comes into play. The very nearly self-evident hypothesis is this the normal state of the third electron (in lithium) will correspond in the value of the principal quantum number to what would have been the first excited level in hydrogen. In other words, even in the unexcited state, the third electron of lithium, its optical or valency electron, has n = 2. 

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

Equation 4.7 is the Bohr postulate, that electrons can defy Maxwell s laws provided they occupy an orbit whose angular momentum (corresponding to an orbit of appropriate radius) satisfies Eq. 4.7. The Bohr postulate is not based on a whim, as most textbooks imply, but rather follows from (1) the Plank equation Eq. 4.3, AE = hv and (2) starting with an orbit of large radius such that the motion is essentially linear and classical physics applies, as no acceleration is involved, then extrapolating to small-radius orbits. The fading of quantum-mechanical equations into their classical analogues as macroscopic conditions are approached is called the correspondence principle [11]. 

Bonino brought forward a further contribution to the theory of infrared spectra of organic liquids by incorporating the Bohr-Sommerfeld quantum conditions, including the correspondence principle of Bohr as well. This paved the way toward establishing a correlation between the physical and chemical image of molecules in the study of infrared spectra. From this series of papers on infrared spectroscopy, one can already observe the interdisciplinary character of Bonino s thought. In a lecture delivered some years later, Bonino offered these reflections on his chosen field of research ... 

This conjecture did not feature explicitly in Bohr s original argument, which he based on a correspondence principle, and only emerged in later work. 

When first confronted with the oddities of quantum effects Bohr formulated a correspondence principle to elucidate the status of quantum mechanics relative to the conventional mechanics of macroscopic systems. To many minds this idea suggested the existence of some classical/quantum limit. Such a limit between classical and relativistic mechanics is generally defined as the point where the velocity of an object v —> c, approaches the velocity of light. By analogy, a popular definition of the quantum limit is formulated as h —> 0. However, this is nonsense. Planck s constant is not variable. 

In what precedes we have contrasted the statements of classical atomic mechanics with the results of experimental research. The latter, on Bohr s interpretation, give discrete energy levels with the values En = —hRIn, where R stands for a constant determined experimentally. It is of course the business of the new atomic mechanics to explain the value found for the Balmer tium. The liii( . of approach to the solution of the problem is indicated by the principle which we stated at the outset (p. 91)—the correspondence principle. 

The intensity of the spectral line is the product of two factors, the irumber of excited atoms and the radiating strength J of an individual atom, which we have just calculated. Thus, with regard to the conditions of excitation of lines, those ideas in Bohr s theory which are brilliantly verified by experiment are just the ideas which are retained in their entirety in the wave mechanics. The latter theory adds a more exact calculation of the intensity J of the individual elementary act, depending on evaluation of the integrals occurring in the matrix elements, while on this question Bohr s theory could only with difficulty make a few statements, with the help of very considerable use of the correspondence principle. 

Besides the diseontinuous states there are also wstates forming a continuous range (with positive energy) they correspond to the hyperbolic orbits of Bohr s theory. The jumps from one hyperbola to another or to a stationary state give rise to the emission of the continuous X-ray spectrum emitted when electrons are scattered or caught by nuclei. The intensity of this spectrum has been calculated by Kramers (1923) from the standpoint of Bohr s theory by a very ingenious application of the correspondence principle. His... 

TABLE 1.1 Check of the Correspondence Principle for Asymptotic Bohr s Hydrogen Atom Levels (White, 1934 Putz et al., 2010)... 

As mentioned in chapter 7, when Bohr presented his method for ascribing electron shells, most physicists were puzzled by the manner in which he obtained his results. Letters to Bohr following the pubhcation of his theory of the periodic system in Nature magazine contain passages such Ernest Rutherford s Everybody is eager to know whether you can fix the rings of electrons by the correspondence principle or whether you have recourse to the chemical facts to do so. And fixsm... 

W Pauli, letter to N. Bohr, February 21, 1924, quoted in Bohr-Pauli Correspondence, Collected Papers of Niels Bohr, edited by J. Rud Nielsen, vol. 5, North-HoUand Publishing, Amsterdam, 1981.Translation on p. 412—414.This quotation is all the more remarkable because, as argued below, it was Pauh s own exclusion principle, formulated a few months later, that seemed to reinstate the notion of individual electrons in stationary states. The notion of individual electrons in individual stationary states was finally refuted wdth the advent of quantum mechanics. Only the atom as a whole possesses stationary states.The distinction is rather important for the physics of many-electron systems. 

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