The correspondence principle between classical and quantum quantities

Questions of the reestablishment of a classical quantity from the known quantum operator are investigated in this article, in that case when the operators are constructed according to the rule proposed in article [15]. The existence of a twofold limiting transition is proved, in which the investigated rule satisfies the requirements of the correspondence principle.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 18–23, June, 1974.The author expresses his deep gratitude to V. V. Kuryshkin for suggesting the topic and for systematic aid in the work.     

Lyapunov generation of entanglement and the correspondence principle

We show how a classically vanishing interaction generates entanglement between two initially nonentangled particles, without affecting their classical dynamics. For chaotic dynamics, the rate of entanglement is shown to saturate at the Lyapunov exponent of the classical dynamics as the interaction strength increases. In the saturation regime, the one-particle Wigner function follows classical dynamics better and better as one goes deeper and deeper in the semiclassical limit. This demonstrates that quantum-classical correspondence at the microscopic level does not require coupling to a large number of external degrees of freedom.     

Energetic correspondence principle in gravitation theory

The correspondence between various quantities in the general theory of relativity in Newtonian theory is analyzed. The analysis is carried out in a proper reference frame of a single Fermi observer. It turns out that only the Papapetrou pseudotensor, of the quantities considered, satisfies the correspondence principle.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 92–95, September, 1976.     

Pictures of quantum nuclear rotation beyond the correspondence principle

We analyze the time evolution of simple nuclear rotational wave packets (WP) called circular, linear or elliptic, depending on squeezing parameter -, assuming that E = EP₀I(I + 1). The scenario of fractional revivals found by Averbukh and Perelman is adapted to symmetric WP and compared to that which holds for asymmetric WP. In both cases various shapes are identified under these lines in particular many cases of cloning. "Mutants" WP are found most often. Finally the time evolution of a WP formed by Coulomb excitation on ²³⁸U and calculated by semiclassical theory is also presented.     

On the potentialx 2N and the correspondence principle

Eigenenergies and frequencies are obtained for a particle oscillating in the potential (1/2)k ^N × ^2N , wherek is a constant,x is displacement, andN is a real number. These potentials include the harmonic oscillator (N = 1) and the square well (N = ). Then ^th eigenenergy has the formA _N n ^(N) , where(N) = 2N/(N + 1), andA _N is independent ofn. Application is made to the correspondence principle for the potentialsN > 1 and it is concluded the classical continuum is not obtained in Bohr's limitn . Complete correspondence is attained in Planck's limith 0, so that for these configurations the limitsh 0 andn are not equivalent. A classical analysis of these potentials is included which reveals the relation log _E (/ _N ) = (N – 1)/2N between frequencyv and energyE, where the constant _N is independent ofE.     

Bohr correspondence principle for large quantum numbers

Periodic systems are considered whose increments in quantum energy grow with quantum number. In the limit of large quantum number, systems are found to give correspondence in form between classical and quantum frequency-energy dependences. Solely passing to large quantum numbers, however, does not guarantee the classical spectrum. For the examples cited, successive quantum frequencies remain separated by the incrementhI ^?1, whereI is independent of quantum number. Frequency correspondence follows in Planck's limit,h → 0. The first example is that of a particle in a cubical box with impenetrable walls. The quantum emission spectrum is found to be uniformly discrete over the whole frequency range. This quality holds in the limitn → ∞. The discrete spectrum due to transitions in the high-quantum-number bound states of a particle in a box with penetrable walls is shown to grow uniformly discrete in the limit that the well becomes infinitely deep. For the infinitely deep spherical well, on the other hand, correspondence is found to be obeyed both in emission and configuration. In all cases studied the classical ensemble gives a continuum of frequencies.     

对应原理两种表述的讨论

对<大学物理>杂志以前刊发的一篇文章中关于对应原理的两种表述及综合运用的部分论点提出了不同的看法和新的见解,并阐述了对普朗克对应原理和玻尔对应原理的理解.     

Heisenberg对应原理下氢原子1/r矩阵元的量子-经典对应

通过详细计算表明,在准经典情况下,氢原子1r的矩阵元的量子力学结果与它的Heisenberg矩阵元近似相等,在经典极限下,它们相同. 关键词： 量子力学 对应原理     

Conditions for applicability of the Newtonian approximation of the theory of relativity and the correspondence principle

The assertion of correspondence between the theory of relativity and Newtonian mechanics, and the resultant condition of applicability of the latter, has only mathematical meaning. Analysis of the basic principles of both theories is used to show their physical incompatibility while the conditions for the applicability of the Newtonian approximation of the theory of relativity are found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 46–48, July, 1991.     

Rotordynamic analysis using the Complex Transfer Matrix: An application to elastomer supports using the viscoelastic correspondence principle

Numerous methods are available to calculate rotordynamic whirl frequencies, including analytic methods, finite element analysis, and the transfer matrix method. The typical real-valued transfer matrix (RTM) suffers from several deficiencies, including lengthy computation times and the inability to distinguish forward and backward whirl. Though application of complex coordinates in rotordynamic analysis is not novel per se, specific advantages gained from using such coordinates in a transfer matrix analysis have yet to be elucidated. The present work employs a complex coordinate redefinition of the transfer matrix to obtain reduced forms of the elemental transfer matrices in inertial and rotating reference frames, including external stiffness and damping. Application of the complex-valued state variable redefinition results in a reduction of the 8×8 RTM to the 4×4 Complex Transfer Matrix (CTM). The CTM is advantageous in that it intrinsically separates forward and backward whirl, eases symbolic manipulation by halving the transfer matrices’ dimension, and provides significant improvement in computation time. A symbolic analysis is performed on a simple overhung rotor to demonstrate the mathematical motivation for whirl frequency separation. The CTM?s utility is further shown by analyzing a rotordynamic system supported by viscoelastic elastomer rings. Viscoelastic elastomer ring supports can provide significant damping while reducing the cost and complexity associated with conventional components such as squeeze film dampers. The stiffness and damping of a viscoelastic damper ring are determined herein as a function of whirl frequency using the viscoelastic correspondence principle and a constitutive fractional calculus viscoelasticity model. The CTM is then employed to obtain the characteristic equation, where the whirl frequency dependent stiffness and damping of the elastomer supports are included. The Campbell diagram is shown, demonstrating the CTM?s ability to intrinsically separate synchronous whirl direction for a non-trivial rotordynamic system. Good agreement is found between the CTM results and previously obtained analytic and experimental results for the elastomer ring supported rotordynamic system.     

Remark on the variational principle in the AdS/CFT correspondence for the scalar field

We discuss how the variational principle can be used as a criterion for choosing, among scalar field actions implying the same equation of motion, the appropriate one for the AdS/CFT correspondence.     

Spectrum of a quantized black hole,correspondence principle,and holographic bound

An equidistant spectrum of the horizon area of a quantized black hole does not follow from the correspondence principle or from general statistical arguments. On the other hand, such a spectrum obtained in loop quantum gravity (LQG) either does not comply with the holographic bound or requires a special choice of the Barbero-Immirzi parameter for the horizon surface, distinct from its value for other quantized surfaces. The problem of distinguishability of the edges in LQG is discussed, with the following conclusion: Only under the assumption of partial distinguishability of the edges can the microcanonical entropy of a black hole be made both proportional to the horizon area and satisfying the holographic bound.     

The general caloron correspondence

We outline in detail the general caloron correspondence for the group of automorphisms of an arbitrary principal G-bundle Q over a manifold X, including the case of the gauge group of Q. These results are used to define characteristic classes of gauge group bundles. Explicit but complicated differential form representatives are computed in terms of a connection and Higgs field.     

The maximum entropy principle as a consequence of the principle of Laplace

The maximum entropy principle states that the probability distribution which best represents our information is the one which maximizes the entropy with the given evidence as constraints. We prove that this principle is implied from the Laplace principle of equiprobabilities applied to the setS of allN-term sequences of results which are compatible with the given evidence. We generalize to the information gain method of Kullback.