The Classical Correspondence Principle

Consider the isothermal equations governing an anisotropic, non-aging medium, given by (1.8.9, 11) and the boundary conditions (1.8.15). It will be assumed in this section that the boundary regions / = 1,2, 3 are time-independent. 

Equations (1.8.19), (2.1.1) are formally identical to the equations governing the elastic problem, in FT form, where the regions B and the specified functions C/(r, t), rf/(r, t) are the same but where the elastic moduli are replaced by the complex moduli /i y /(cu). If solutions to the elastic problem are known, one can obtain a solution of the corresponding viscoelastic problem by replacing the moduli by the complex moduli in the expressions for the displacements and stresses and calculating the inverse transforms. This observation is the content of the Classical Correspondence Principle, which allows us to use the vast catalogue of known elastic solutions to generate viscoelastic solutions. 

We will now spell out this procedure in more detail. Let t), a[y (r, 01 

It follows immediately from Fourier s Integral Theorem (Sect. A3.1) that if CO, t ) is independent of co at any point r, then w,(r, t) and t) are equal at that point. In particular, this gives that the boundary values for the elastic solution are the same as those for the viscoelastic solution, as has been assumed. If any elastic solution is independent of the moduli at any point, then the elastic and viscoelastic solutions will be the same. 

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The Classical Correspondence Principle was enunciated in reasonably general form by Read (1950) and Lee (1955) among others and discussed rigorously by Sternberg (1964) for the more general non-isothermal case. Sternberg (1964) reviews the older literature in some detail. Tao (1966) discusses correspondences between elastic and viscoelastic inertial problems in terms of Laplace transforms, essentially generalizing the work of Lee (1955). 

The conclusion is also valid for viscoelastic bodies - if the non-inertial approximation applies. This follows immediately by invoking the Classical Correspondence Principle. Our object in this section is to generalize the result to the case of two viscoelastic bodies in contact. 

I. Viscoelastic Solutions in Terms of Elastic Solutions. The fundamental result is the Classical Correspondence Principle. It is based on the observation that the time Fourier transform (FT) of the governing equations of Linear Viscoelasticity may be obtained by replacing elastic constants by corresponding complex moduli in the FT of the elastic field equations. It follows that, whenever those regions over which different types of boundary conditions are specified do not vary with time, viscoelastic solutions may be generated in terms of elastic solutions that satisfy the same boundary conditions. In practical terms this method is largely restricted to the non-inertial case, since then a wide variety of elastic solutions are available and transform inversion is possible. 

The first boundary value problem could also have been solved by invoking the Classical Correspondence Principle,... 

The problem of a stationary crack which has always been open is relatively trivial. Since it is covered by the Classical Correspondence Principle, we expect... 

We will write down the displacement-traction relationship on the boundary that will form the basis of the considerations of this chapter. This is essentially the solution of the stress boundary value problem, discussed in Sect. 3.2 in the plane case. We shall neglect surface shear, however, so that the required relationship is a generalization to Viscoelasticity of the classical Boussinesq relationship. Its form follows directly from the elastic result by invoking the Classical Correspondence Principle. A more explicit derivation may be found in Hunter (1961) and also Golden (1978), who includes a shear traction term. Letting... 

These equations have the same form as the corresponding elastic relations with moduli replaced by complex moduli, as required by the Classical Correspondence Principle discussed in Sect. 2.1. Relation (7.1.15 b), which will be used in Sect. 7.2, was written down by Golden (1979b) from the elastic results of Eason (1965). More general results are derived by Golden (1982 b). 

During the seventies, the work on non-inertial problems was consolidated. The main purpose of the present volume is to present a coherent, unified development of this topic, in particular of those problem classes which are not covered by the Classical Correspondence Principle. There has also been some progress on inertial problems. Typically however, to make progress on such problems it is necessary either to confine one s attention to the most idealized configurations or to introduce some approximation. Also, the mathematical techniques used have been generally rather sophisticated. We briefly discuss this work in the last chapter, and derive certain results by comparatively elementary methods. 

In chapter 1, the properties of the viscoelastic functions are explored in some detail. Also the boundary value problems of interest are stated. In chapter 2, the Classical Correspondence Principle and its generalizations are discussed. Then, general techniques, based on these, are developed for solving non-inertial isothermal problems. A method for handling non-isothermal problems is also discussed and in chapter 6 an illustrative example of its application is given. Chapter 3 and 4 are devoted to plane isothermal contact and crack problems, respectively. They utilize the general techniques of chapter 2. The viscoelastic Hertz problem and its application to impact problems are discussed in chapter 5. Finally in chapter 7, inertial problems are considered. 

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,... 

We mention the obvious fact that for a given system and temperature the number of partial waves needed is a function of the separation R for which g(R) is to be evaluated. Large R require many partial waves at large R the pair distribution function g(R) is, therefore, largely classical (correspondence principle). The classical and quantal pair distribution functions g(R) differ most significantly at separation less than Rm, the position of the potential minimum. Quantum computations of the pair distribution function may, therefore, be restricted to relatively small separations. For larger separations classical or semi-classical expressions may be employed which will be sketched below. In this way, the number of partial waves required for pair distribution functions need not be much greater than for line shape computations. 

It is less obvious how to assign quantum variables. While, for instance, the total classical angular momentum of a collision fragment is clearly defined, in the case of small fragments product rotational quantum numbers are typically sought. The standard procedure for linear molecules is to assume rigid rotor conditions and to solve the usual correspondence principle relations to obtain the quasiclassical product quantum numbers (restricted to integer values). 

One conclusion that can be reached from the early work on effective potentials [1,21-23], the work of Cao and Voth [3-8], as well as the centroid density-based formulation of quantum transition-state theory [42-44,49] is that the path centroid is a particularly useful variable in statistical mechanics about which to develop approximate, but quite accurate, quantum mechanical expressions and to probe the quantum-classical correspondence principle. It is in this spirit that a general centroid density-based formulation of quantum Boltzmann statistical mechanics is presented in the present section. This topic is the subject of Paper I, and the emphasis in this section is on analytic theory as opposed to computational approaches (cf. Sections III and IV). 

The Table 1.1 further illustrates the Bohr quantum-classical correspondence principle. 

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 fifth postulate and its corollary are extremely important concepts. Unlike classical mechanics, where everything can in principle be known with precision, one can generally talk only about the probabilities associated with each member of a set of possible outcomes in quantum mechanics. By making a measurement of the quantity A, all that can be said with certainty is that one of the eigenvalues of /4 will be observed, and its probability can be calculated precisely. However, if it happens that the wavefiinction corresponds to one of the eigenfunctions of the operator A, then and only then is the outcome of the experiment certain the measured value of A will be the corresponding eigenvalue. 

The classical model predicts that the largest probability of finding a particle is when it is at the endpoints of the vibration. The quantum-mechanical picture is quite different. In the lowest vibrational state, the maximum probability is at the midpoint of the vibration. As the quantum number v increases, then the maximum probability approaches the classical picture. This is called the correspondence principle. Classical and quantum results have to agree with each other as the quantum numbers get large. 

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. 

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