Bohrs Correspondence Principle

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,... 

At the limit of large quantum numbers, the results of quantum mechanics must agree with classical mechanics (Bohr correspondence principle). 

Therefore, the rule is that as much the quantum levels are higher as the quantum and classical frequencies approaches each other, establishing the so called Bohr correspondence principle between the quantum and classical worlds . 

An even more striking and practical form of Bohr correspondence principle may be unfold since we introduce the counter of the quantum transition states as ... 

The analysis of the Figure 3.8 clearly illustrates that since in classical interpretation of the motion in the harmonic potential the system has its maximum probability to be found at the position x when its velocity is minimum, i.e., at the maximum distance (at the amplitude) allowed by the oscillation, while in quantum motion this is certainly not the case of the system in its vibrational ground state (n = l) but only in the higher excited states (see the probability behavior for n = 10 and far above that) when the quantum probability of vibration becomes multiphed enough (by the quantum vibrational number) so that it shapes asymptotically to the classical potential of vibrational motion. Such behavior is nothing but the vibrational manifestation of the earlier discussed (Bohr) correspondence principle affirming that the quantum motion approaches the classical one in the very high levels of quantification. 

This example is suitable to illustrate the Bohr correspondence principle. It states that all regularities of quantum mechanics turn into the regularities of classical mechanics under the increasing quantum numbers. It is well known that the different levels of physical approximations are characteristic to certain areas of this science. Transformation from one area to another occurs not abruptly, but gradually. So, Newtonian mechanics becomes less and less exact when the velocity of particle motion increases, transforming into the relativistic one. We are interested here in the transition from quantum mechanics (in which quantization plays a fundamental role) to classical (in which the energy levels discontinuity is not observed). 

The absolute distance between levels increases at increasing j however, the relative values, AE/E vice versa, decrease. This corresponds to the Bohr correspondence principle. 

The validity of this classical interpretation is supported by an accurate estimate of the error arising from the Bohr-Sommerfeld correspondence principle. 

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. 

The next paper was by Dirac on the Theory of the Positron. In the following discussion Niels Bohr made a long intervention on the correspondence principle in connection with the relation between the classical theory of the electron and the new theory of Dirac. 

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]. 

It is known that the correspondence principle suggested by N. Bohr (1923) postulates that any theory pretending for better description and broader application range than the older one must include the latter in the form of marginal case . 

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. 

A simple demonstration of the correspondence principle is provided by the Bohr atomic model that allows for the transfer of an electron between neigh-... 

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. 

The Cognitive Correspondence Principle Named after the inspiration of Neils Bohr in tying quantum physics to Newtonian physics, this principle recognizes the mathematical hierarchy required in modeling complex systems and need to make simplifying assumptions. It states that each assumption must, in the limit, be found to be a valid truncation of a more precise mathematical description of the overall system. 

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. 

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