On the relationship between quantum and classical chemistry
The question of the relationship between quantum and classical theory is still debatable. N. Bohr believed that, no matter how far the phenomena went beyond the framework of the classical physical explanation, all experimental data should be described using classical concepts.
The rationale for this is simply to ascertain the exact meaning of the word "experiment". The word "experiment" we point to a situation where we can tell others what exactly we did and what exactly we learned. Therefore, the experimental setup and the results of observations should be described unambiguously in the language of classical physics. "
It's easy to see that the above rationale does not stand up to criticism. The results of the experiments express the predictions of the quantum theory, and therefore their description refers to it, and not to the classical theory. The objection of Bohr is that the description of the results of the experiments is unambiguous, and not probabilistic. But this argument is not entirely correct either. The results of a one-time experiment do describe a very definite event. But the fact is that the results of repeated experiments can not be described without involving probability ideas. There is another argument in favor of the classical theory. Macroscopic objects behave differently than microscopic objects, in particular, they do not participate in the processes of interference and diffraction. But this argument also misses the target. According to quantum mechanics, phenomena in a mixed state behave differently than coherent processes. But this does not mean that they fall out of their scope of quantum theory.
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In chemistry very often operate with so-called combined quantum-classical methods. The system is divided into two parts, smaller, as a rule, relatively small, described quantum mechanical. It can be, for example, a group of atoms. Its environment, for example a solvent, is considered an object described in a classical way. It remains unclear how right it is to combine two different ways of describing - quantum and classical. Let's consider this question in a systematic way.
There was a time when only classical theory ( T kl) was known to physicists and chemists. In the first quarter of the XX century. it turned out that there is a class of phenomena that can only be comprehended by means of the quantum theory (D). Immediately arose the question of the compatibility of T kl and Tkv. This kind of question was not something unexpected for researchers. They knew that the new theory surpasses the outdated concept. In the framework of the latter concept, some features of the phenomena studied were not taken into account, therefore, it is insufficient. But the situation with the ratio Tkl and Tkv was unique. Many physicists, among whom N. Bohr was most famous, decided that classical physics adequately describes macroscopic manifestations, and therefore it is impeccable with regard to them. But this opinion was not confirmed. The thing is that macrosystems are the result of interactions of microobjects, which, by definition, are described by quantum mechanics. Consequently, the status of macroobjects can be determined on the basis of quantum mechanics. But in this case the classical theory will be reduced to quantum theory.The appearance of quantum mechanics, argues Grigoriev, Myakishev and Shirokov, reasonably, can not be regarded as a refutation of the mechanics of classical mechanics. Newton's mechanics, which perfectly describes the motion of macroscopic bodies, appears as an approximation to quantum mechanics, the latter being able to determine when this approximation is good. The relationship between physical quantities in the classical theory can be derived from the quantum theory as approximate relations between the mean values of these quantities. Classical mechanics can be considered as the limiting case of quantum mechanics, as the first, most crude approximation to it, valid provided that the potential energy varies little at the de Broglie wavelength. "
As you can see, these three authors reason in a fundamentally different manner than N. Bohr. For them, in any form, classical physics is not the key to understanding quantum mechanics. They quite definitely proceed in their reasoning not from the principle of complementarity, but from the postulate of the wave function. We can say that their analysis is conceptual. In essence, it is correct, but some concepts used by them, in particular, such as "approximation", "limiting case", "derivation from quantum mechanics classical", it is advisable to clarify. It seems to the author that there is such an opportunity, especially in connection with the idea of a scientific-theoretical series and system.
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Consider three relationships:
( 12 . 2 )
The series (12.2) characterizes the appearance of the quantum theory in the process of solving some problems of the classical theory. It arose as an alternative to the classical theory. There was no other way, because the apparatus of the classical theory was recognized as insufficient for understanding certain phenomena, for example, the spectra of atoms. Quantum theory seems to have abolished the classical theory. This was the first impression. When they penetrated into the essence of the matter, the most perspicacious conceptually, the authors began to consider classical theory as an approximation to the quantum concept. They interpreted the content of the classical theory from the quantum viewpoint. It is this circumstance that is mapped in the series (12.3). Arrow = & gt; in contrast to the arrow expresses not overcoming the problems, but the process of interpretation. Now the rights of scientific citizenship of the classical theory are recognized, but it acts as an approximation to the quantum theory. To some extent, the opposition between the two theories persists. As it turned out, and largely due to research in the field of chemistry, the unity between quantum and classical theory can be more intimate than previously thought. In a schematic form this circumstance expresses the relation (12.4). Now the classical theory completely ceases to be opposed to the quantum theory.
At first glance, it seems that this is not possible at all. But with the closest analysis, it turns out, at least, that the opposition of classical and quantum theory, as a rule, assumes unnecessarily harsh forms. Usually the circumstance that is not taken into account is as follows.
Classical theory deals with mixed states. But quantum theories are drawn to similar conditions. Consequently, in one respect they are completely identical to each other. If we bear in mind only this relation, then the understanding of the classical theory as an approximation to the quantum conception becomes clear. Let us illustrate this with an example of the so-called combined quantum-classical approximation.
The active subsystem is described quantum-mechanically using a set of basis functions, and the environment by means of specially parametrized classical potentials. From here comes the name of all applied methods - combined quantum-classical ... & quot ;. But is it really a question of the quantum-classical method? Claiming his presence, are not they attempting to combine alternative, that is, conflicting theories? The answer to this question depends on the understanding of classical chemistry. If it appears in the image , then there is no contradiction. But in this case, strictly speaking, we are not talking about the quantum-classical method, but about the method of partial abstraction from coherent states. The researchers do not simply add to the classical description the classical one, which inevitably would lead to numerous contradictions, but interpret phenomena based on the concept of the scientific-theoretical system. The essence of the matter is expressed by him.
Thus, classical chemistry can be understood in different ways. First, attempts may be made to explain pure states through mixed ones. Such attempts are doomed to failure, because the potential of classical chemistry is clearly insufficient for conceptual comprehension of the specifics of pure states. Secondly, classical chemistry can be interpreted as a theory of mixed states. It is in this case that it is consistently consistent with quantum chemistry. Thirdly, classical chemistry can be understood as a description of some averaged states. This kind of interpretation is not entirely correct, for the true meaning of classical chemistry consists in its representation not of averaged, but of mixed states. Fourthly, classical chemistry can be interpreted as the limiting case of quantum chemistry. This case deserves special discussion.
According to the so-called "correspondence principle" of the theory, the validity of which is established for one or another subject area, with the emergence of new, more general theories, they are not eliminated as something false, but retain their significance for the previous domain as the limiting form and particular case of new theories " ;
The term compliance principle was introduced into science by the same N. Bohr in 1923}}, but for the first time its content became known in connection with the creation of a special theory of relativity. In some formulas of this theory, giving the speed of light (c) of an infinitely large value () leads to formulas of Newtonian mechanics. Usually this circumstance is expressed as follows: if the speed of light tends to infinity (), then the formulas of the special theory of relativity pass to the formulas of classical mechanics. In continuation of this methodology, it is asserted that when the Planck constant approaches zero (), the quantum theory formulas become classical formulas, for example, the Schrödinger equation becomes the Hamilton-Jacobi equation.
But in a conceptual sense, the situation is not as simple as it seems to supporters of the correspondence principle. The point is that the said transition does not have the conceptual content that is attributed to it. In fact, they argue that the mixed state is the limit of the pure state. But this contradicts quantum mechanics, according to which the transition from a pure state to a mixed one is the result of a certain type of interaction leading to decoherence, rather than the passage to the limit . This transition does not explain the difference between classical physics and quantum physics, but merely fixes it in a purely formal way.
The meaning of the correspondence principle was carefully analyzed by SV Illarionov. He convincingly showed that the rectilinear application of the limiting transition often leads to interpretational difficulties. In particular, Illarionov considers the so-called stationary Schrödinger equation (the potential V does not depend on time):
The transition turns the Schrodinger equation into an equation that is satisfied only for ψ = 0, which is equivalent to the absence of a physical system. As applied to the Schrodinger equation, the transition is not meaningless, but only if it uses a substitution:
and already in the equation for the function S the limit transition is made. Its meaning lies in the fact that the dimensionless quantity λ/L, tends to zero, where λ is the de Broglie wavelength, and L is the characteristic size of the system.
Disagreeing with the principle of correspondence in his usual sense, SV Illarionov, nevertheless, did not reject it. He was convinced that this principle is vital for the consistent formulation of the quantum theory. In accordance with this conviction, he formulates the principle of correspondence as the principle of limitations. The meaning of the principle of limitations is that the transition from the old theory to the new one is treated as the introduction of new limitations in the structure of the theory while preserving the limitations of the old theory. " The new theory is a generalization of the old. Thus, in the transition from classical mechanics to quantum mechanics, classical physical variables (for example, momentum and coordinate) are replaced by the momentum and coordinate operators, respectively. These physical variables are limited, because it turns out, with respect to what phenomena are appropriate to use them.
In Illarionov's interpretation, the dynamics of scientific knowledge appears as an ascent from old theories to new ones. One of the advantages of the principle of limitations he saw was that the principle is not directed backwards, that is, from the new theory to the old one, and "forward." The Restriction Principle directs research to a search for that fundamental in the old theory that must be preserved and used in the process of generalization when creating a new theory. "
Of course, Illarionov's analysis is of considerable interest. But he is not in all satisfactory. The drawback of the so-called principle of limitations is that the emphasis is on moving forward, on the old theory. Meanwhile, both aspects are important, and the movement from old theories to new ones, and in the opposite direction.
According to the author, the comparison of the principle of limitations with the principle of unity of the scientific and theoretical series and the system does not testify in his favor. The scientific-theoretical series acts as the overcoming of certain problems. If this is done, then the interpretation of the deed allows us to find out the weaknesses of the old theory and realize its limitations. The subject of restrictions appears, but only as a result of previous work. The notion of a special principle of limitations would be appropriate if it were conceptually preceding other principles. In other words, transduction would begin with him. But this is exactly what is not. K. Popper is right, scientists concentrate their efforts on solving problems. All the rest is clarified in the process of this activity. If the theory functions without significant disruptions, then the topic of its limitations does not arise at all. Thus, according to the author, Illarionov is persuasive in criticizing the correspondence principle in the form of limit transitions. Rehabilitation of the same correspondence principle in the form of representations about limitations, in fact, did not take place.
1. Classical chemistry is not a special case of quantum chemistry. A particular case of quantum chemistry can only be its variety, but not classical chemistry.
2. Classical chemistry receives an adequate interpretation, if it is interpreted by quantum chemistry. After that, it loses its classic features.
3. The researcher should be guided by the interpretational series of chemical theories, which includes the quantum mechanical interpretation of classical chemistry.
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