Correspondence principles

Heller E J 1983 The correspondence principle and intramolecular dynamics Faraday Discuss. Cham. Soc. 75 141-53... 

The procedure described above is an application of the time-temperature correspondence principle. By shifting a set of plots of modulus (or compliance) versus time (or frequency) at any temperature (subscript 1) along the log t axis, we obtain the value of that mechanical property at another time and temperature (subscript 2). Using the shear modulus as an example, the time-temperature correspondence principle states... 

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 next question which arises concerns the explicit forms of f(x) and Our previous work in Section 11.1 (correspondence principle... 

Consider next the current operator ju(x). The correspondence principle suggests that its form is j ( ) = — (e/2)( ( )yB, (a )]. Such a form foijn(x) does not satisfy Eq. (11-477). In fact, due to covariance, the spectral representation of the vacuum expectation value of 8u(x)Av(x), where ( ) is an arbitrary four-vector, is given by... 

The original semiclassical version of the centrifugal sudden approximation (SCS) developed by Strekalov [198, 199] consistently takes into account adiabatic corrections to IOS. Since the orbital angular momentum transfer is supposed to be small, scattering occurs in the collision plane. The body-fixed correspondence principle method (BFCP) [200] was used to write the S-matrix for f — jf Massey parameter a>xc. At low quantum numbers, when 0)zc —> 0, it reduces to the usual non-adiabatic expression, which is valid for any Though more complicated, this method is the necessary extension of the previous one adapted to account for adiabatic corrections at higher excitation... 

The third problem is like the confusion caused in MT by maintaining the concept of the Ether. Most practitioners of QM think about microscopic systems in terms of the principles of QM probability distributions, superposition principle, uncertainty relations, complementarity principle, correspondence principle, wave function collapse. These principles are an approximate summary of what QM really is, and following them without checking whether the Schrddinger equation actually confirms them does lead to error. 

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,... 

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

Using correspondence principles between photochemistry, electrochemistry and ordinary chemistry tells us that the redox properties of sulfones and sulfoxides could photochemically manifest themselves in two main directions. 

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 (2.23) is the quantum-mechanical analog of the classical definition of momentum, p = mv = m(Ax/At). This derivation also shows that the association in quantum mechanics of the operator (h/i)(d/dx) with the momentum is consistent with the correspondence principle. 

These interference patterns are wonderful manifestations of wave function behavior, and are not found in classical electronics or electrodynamics. Since the correspondence principle tells us that quantum and classical systems should behave similarly in the limit of Planck s constant vanishing, we suspect that adequate decoherence effects will change the quantum equation into classical kinetics equations, and so issues of crosstalk and interference would vanish. This has been... 

Criss, C.M. Cobble, J.W., "Thermodynamic Properties of High Temperature Aqueous Systems. IV Entropies of the Ions up to 200°C and the Correspondence Principles", JACS, 1964, 86,... 

Ac° estimated from "correspondence principle". This approach, proposed By Criss and Cobble (12), appears to be the most widely used one. It rests on what has been called the entropy correspondence principle which states that the partial molal entropy of an ion at temperature T is related to its partial molal entropy at 298°K by a relation... 

While the above comparisons are hardly conclusive, they do suggest caution in applying the correspondence principle as it presently exists to temperatures above 200°C. 

The (admittedly limited) comparisons shown in Figures 1 and 2 suggest that either the assumption of a constant AC or the correspondence principle leads to reasonable estimates of tire free energy of formation at temperatures up to 150°C to 200°C. Beyond this range, the extrapolations are in doubt. It appears that an overall assessment of the correspondence principle with emphasis on temperatures of 200°C and above, and with refinement of the Criss and Cobble parameters would be very desirable. 

Figure 2. Temperature dependence of molal ion product of water (a) Fisher and Barnes (36) (b) Sweeton et at. (35) (c) Sirota and Shviraev (31) (d) Correspondence Principle estimate of Lewis (91)...

Estimation Procedures. There are basically two ways which have been developed to deal with the fact that heat capacity terms are large in reactions involving ions. One is based on empirical relationships (the entropy correspondence principle) between ionic entropies at different temperatures which Criss and Cobble (62) developed and checked to 200 C. Lewis (63) has checked a number of its predictions against available experimental evidence and has found the method reasonably satisfactory for several... 

It is not a problem either for the protonation constant of S (i.e. the reciprocal of the second dissociation constant of H2S) some estimates of which are shown in Fig. 4. Neither Cobble s estimate (68), using the correspondence principle (curve a) nor Pohl s (69) extrapolation (curve b) using an empirical equation due to Harned and Embree (70) is showing any indication of the expected minimum in K. The extrapolation used by Khodakovskii et al (71) (curve c) is based on the more frequently used expression of Harned and Robinson (72) and a different selection of low temperature data. While their result looks more reasonable it is difficult to have much confidence in any of the results even up to 200 C. The apparent failure of the correspondence principle may arise as much from the choice of low temperature data as a failure of the relationship itself. 

Here, qj is the charge of electron j, r is the distance between electrons i and j. From the classical interaction energy we can extract the operator accounting for the electron-electron pair interaction g i,j) via the corresponding principle (c.p.) (21),... 

According to the correspondence principle the classical expression for the electron density p(r,t) can be converted to the quantum mechanical description by taking into account that the particle density is calculated by integration of the product of the iV-electron wave function 4 1 and its complex conjugate 4. We introduce the charge-weighted density by multiplication of the electron density with the electron charge,... 

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