Bohr's Correspondence Principle Research Articles

Is Bohr's correspondence principle just Hankel's principle of permanence?

No, but the paper argues that Bohr understood his correspondence principle, or at least an aspect of that principle expressed by the notion of rational generalization, as grounded in Hankel's principle of permanence, adapted to new historical and theoretical contexts. This is shown to illuminate some otherwise obscure aspects of Bohr's approach to quantum theory, as well as a seemingly strange criticism against this approach, due to Feyerabend and Bohm.

Manifestation of classical instability in the quantum density of states of a double well potential

Within Bohr's correspondence principle, quantum theory should reproduce the classical world when ħ→0. In practice, obtaining the complete spectrum is challenging since the discrete energies come close to each other before passing to continuum. However it is possible to identify signatures that the spectrum is approaching the continuum limit. In this work, for ħ→0, we focus our attention on the spectrum of the 1D quartic double well and find signatures of the classical instability in the quantum spectrum near the critical energy, and explore how the tunneling phenomenon remains only at energies close to this critical energy.

Classical Love number for quantum black holes

We present a method for comparing the classical and quantum calculations of the electric quadrupolar Love number $k_2$ and show that our previous derivation of the quantum Love number of a quantum blackhole matches exactly the classical calculation of $k_2$ when quantum expectation values are replaced by the corresponding classical quantities, as dictated by the Bohr correspondence principle. The standard derivation of $k_2$ for classical relativistic stars relies on fixing boundary conditions on the surface of the star for the Einstein equations in the presence of an external perturbing field. An alternative method for calculating $k_2$ uses properties of the spectrum of the non-relativistic fluid modes of the star. We adopt this alternative method and use it to derive an effective description of the interior modes in terms of a collection of driven harmonic oscillators characterized by different frequencies and amplitudes. We compare these two classical methods and find that most of the interior information can be integrated out, reducing the problem of calculating $k_2$ to fixing a single boundary condition for the perturbed Einstein equations on the surface of the deformed star. We then determine this single boundary condition in terms of the spectrum of the object and proceed to identify the relationship between classical quantities and quantum expectation values and to verify the agreement between the results of the effective classical calculation and the quantum calculation.

A novel kind of equations linking the quantum dynamics and the classical wave motions based on the catastrophe theory

Considering that the catastrophe theory could describe quantitatively any phase transition process, we adopt the folding and cusp catastrophe types as the potential functions in the Schrödinger equation to attempt to link the quantum dynamics and the classical wave motions. Thus, through the dimensionless analysis a novel kind of partial differential equations is derived out. When the scaling parameter of the novel equation is equal to the Planck's constant, this equation becomes a detailed time-independent Schrödinger equation, from which Bohr correspondence principle can be found. On the other hand, when the scaling parameter tends to zero, this equation could degenerate to the classical Helmholtz equation. Therefore, this novel kind of equations could describe quantitatively the variation process of the wave functions from the macroscopic level to the quantum size.

FUNCTORS AND SPACES IN IDEMPOTENT MATHEMATICS

Idempotent mathematics is a branch of mathematics in which idempotent operations (for example, max) on the set of reals play a central role. In recent decades, we have seen intensive research in this direction. The principle of correspondence (this is an informal principle analogous to the Bohr correspondence principle in the quantum mechanics) asserts that each meaningful concept or result of traditional mathematics corresponds to a meaningful concept or result of idempotent mathematics. In particular, to the notion of probability measure there corresponds that if Maslov measure (also called idempotent measure) as well as more recent notion of max-min measure. Also, there are idempotent counterparts of the convex sets; these include the so-called max-plus and max min convex sets. Methods of idempotent mathematics are used in optimization problems, dynamic programming, mathematical economics, game theory, mathematical biology and other disciplines. In this paper we provide a survey of results that concern algebraic and geometric properties of the functors of idempotent and max-min measures.

THE DARK MATTER AND ENERGY IN THE DE SITTER WORLD

It is shown that dark matter and energy are cosmological quantum effects. De Sitter world is considered as a cosmological model. It is shown that in the de Sitter world, gravity and anti-gravity are different states of the Wigner elementary quantum system. It has been proven that in the Minkowski world, which is the limiting case of the de Sitter world, antigravity can be excluded. Moreover, it is shown that the Wigner - Inönü limit of the de Sitter model to the Minkowski world plays the role of Bohr's correspondence principle in quantum mechanics. A plan for further research is proposed.

Quantization method and Schrödinger equation of fractional time and their weak effects on Hamiltonian: Phase transitions of energy and wave functions

Fractional time quantum mechanics (FTQM) is a method of describing the time evolution of quantum dynamics based on fractional derivatives. For any potential, we obtain the modeling method to describe quantum systems with fractional time consistent with fundamental quantum physics laws, which makes fractional time effects naturally enter quantum mechanics. The method is only using the start states, not based on usually directly replacing the integer derivatives by fractional ones or parallel introductions of standard models. In the process, we solve three open problems perplexing the studies on FTQM: What is the essential quantization method? How does one represent the fractional time Hamiltonian while retaining physical significance? How does one avoid the violations of current models of FTQM for many fundamental quantum physics laws? Then, a FTQM framework is built by amalgamating two quantization methods under a unified foundation of fractional time. The framework contains the quantization method, the Hamiltonian, the Hamilton operator, the Schrödinger equation, the energy correspondence relation, the Bohr correspondence principle and the time-energy uncertainty relation. And the effects of fractional time are revealed: containing historical information of particle’s motions; representing weak actions of Hamilton operator. An example is provided, and analytic expressions of the energy and wave functions are obtained. These account for the distinct nonlinear phenomena: the phase transition of energy and wave functions, which can not be revealed in the previous methods. The phase transitions cause some classical physical effects and phenomena: (1) energy gaps filled by energy levels; (2) increase in particle orbits; (3) a famous bound states in continuum (BICs) firstly found in FTQM; (4) a new explanation of the discrete energy levels from the perspective of energy level filling.

Nongeometric states in a holographic conformal field theory

In the AdS$_3$/CFT$_2$ correspondence, we find some conformal field theory (CFT) states that have no bulk description by the Ba\~nados geometry. We elaborate the constraints for a CFT state to be geometric, i.e., having a dual Ba\~nados metric, by comparing the order of central charge of the entanglement/R\'enyi entropy obtained respectively from the holographic method and the replica trick in CFT. We find that the geometric CFT states fulfill Bohr's correspondence principle by reducing the quantum KdV hierarchy to its classical counterpart. We call the CFT states that satisfy the geometric constraints geometric states, and otherwise non-geometric states. We give examples of both the geometric and non-geometric states, with the latter case including the superposition states and descendant states.

WKB approximation for a deformed Schrodinger-like equation and its applications to quasinormal modes of black holes and quantum cosmology

In this paper, we use the WKB approximation method to approximately solve a deformed Schrodinger-like differential equation: [−ħ2∂ξ2g2(−iħα∂ξ)−p2(ξ)]ψ(ξ)=0, which are frequently dealt with in various effective models of quantum gravity, where the parameter α characterizes effects of quantum gravity. For an arbitrary function g(x) satisfying several properties proposed in the paper, we find the WKB solutions, the WKB connection formulas through a turning point, the deformed Bohr–Sommerfeld quantization rule, and the deformed tunneling rate formula through a potential barrier. Several examples of applying the WKB approximation to the deformed quantum mechanics are investigated. In particular, we calculate the bound states of the Pöschl–Teller potential and estimate the effects of quantum gravity on the quasinormal modes of a Schwarzschild black hole. Moreover, the area quantum of the black hole is considered via Bohr's correspondence principle. Finally, the WKB solutions of the deformed Wheeler–DeWitt equation for a closed Friedmann universe with a scalar field are obtained, and the effects of quantum gravity on the probability of sufficient inflation is discussed in the context of the tunneling proposal.

Quantum-classical correspondence in the vicinity of periodic orbits

Quantum-classical correspondence in chaotic systems is a long-standing problem. We describe a method to quantify Bohr's correspondence principle and calculate the size of quantum numbers for which we can expect to observe quantum-classical correspondence near periodic orbits of Floquet systems. Our method shows how the stability of classical periodic orbits affects quantum dynamics. We demonstrate our method by analyzing quantum-classical correspondence in the quantum kicked top (QKT), which exhibits both regular and chaotic behavior. We use our correspondence conditions to identify signatures of classical bifurcations even in a deep quantum regime. Our method can be used to explain the breakdown of quantum-classical correspondence in chaotic systems.

Tales from the prehistory of Quantum Gravity

The main purpose of this paper is to analyse the earliest work of L\'eon Rosenfeld, one of the pioneers in the search of Quantum Gravity, the supposed theory unifying quantum theory and general relativity. We describe how and why Rosenfeld tried to face this problem in 1927, analysing the role of his mentors: Oskar Klein, Louis de Broglie and Th\'eophile De Donder. Rosenfeld asked himself how quantum mechanics should \textit{concretely} modify general relativity. In the context of a five-dimensional theory, Rosenfeld tried to construct a unifying framework for the gravitational and electromagnetic interaction and wave mechanics. Using a sort of "general relativistic quantum mechanics" Rosenfeld introduced a wave equation on a curved background. He investigated the metric created by what he called `quantum phenomena', represented by wave functions. Rosenfeld integrated Einstein equations in the weak field limit, with wave functions as source of the gravitational field. The author performed a sort of semi-classical approximation obtaining at the first order the Reissner-Nordstr\"om metric. We analyse how Rosenfeld's work is part of the history of Quantum Mechanics, because in his investigation Rosenfeld was guided by Bohr's correspondence principle. Finally we briefly discuss how his contribution is connected with the task of finding out which metric can be generated by a quantum field, a problem that quantum field theory on curved backgrounds will start to address 35 years later.

Quantization of plasmon in two-dimensional square quantum dot system

Plasmon in quantum dot system is one of the most notable research topics in the field of optoelectronics. With the development of nanotechnology, plasmon in nano-structure has received considerable attention due to its potential applications in future natural science areas. To better understand the quantum effect and the properties of plasmon, in this paper we use the linear response theory and the tight-binding approximation to investigate the collective response of charge in a twodimensional square quantum dot system. The results show that when the frequency of the external field equals the frequency of the plasmon, there are strong charge collective oscillations in the quantum dot system, accompanied by great energy absorption and near-field enhancement. Owing to the quantization of plasmon, the collective charge oscillations in a two-dimensional square quantum dot system are found at different frequencies. The number of quantum modes of plasmon increases with the size and electron number of square quantum dots increasing, this behaviour of quantum mode of plasmon is similar to the one of phonon. The reasons for this behaviour are as follows. First, with the increase of quantum dot size, there are more energy levels around the fermi energy, and the electrons can jump from more energy levels to the outside of fermi circle, so there are more collective excitation frequencies (i.e., more quantum modes of plasmon) in a larger size system. Second, with the increase of electron number in quantum dots, there are more energy levels occupied by electrons, so there are more quantum modes of plasmon too. Furthermore, the size dependence of plasmon shows that with the increase of quantum dot size, the frequency interval between two neighbouring modes of plasmon is smaller, and the discrete modes of plasmon will gradually display quasi-continuous characteristic and transform gradually into the classical continuous modes of plasmon, and the frequency spectrum of plasmon turns into the classical dispersion relation. Such a characteristic is in accord with Bohr's correspondence principle, implying that the quantum plasmon and classical plasmon are gradually unified in a macroscopic size. The dependence of plasmon on the size and electron number of quantum dots also show that with the increase of the quantum dot size, the frequencies of the plasmon is red-shifted and the excitation intensity of the plasmon increases; with the increase of the electron number in quantum dot, the frequency of the plasmon is blue-shifted and the excitation intensity of the plasmon increases.

Bohr's Correspondence Principle for the Renormalized Nelson Model

In the mid Sixties Edward Nelson proved the existence of a consistent quantum field theory that describes the Yukawa-like interaction of a non-relativistic nucleon field with a relativistic meson field. Since then it is thought, despite the renormalization procedure involved in the construction, that the quantum dynamics should be governed in the classical limit by a Schr\"odinger-Klein-Gordon system with Yukawa coupling. In the present paper we prove this fact in the form of a Bohr correspondence principle. Besides, our result enlighten the nature of the renormalization method employed in this model which we interpret as a strategy that allows to put the related classical Hamiltonian PDE in a normal form suitable for a canonical quantization.

ON THE EXISTENCE OF A NON-RELATIVISTIC HERMITIAN TIME OPERATOR

The existence of a hermitian time operator is proposed in the framework of non-relativistic quantum mechanics.The Heisenberg equation of motion is shown to yield constant rate of flow of time.It is shown to yield results consistent with classical expectations,when Bohr's correspondence principle is invoked.Further,it is shown that a massless free particle with non-zero momentum experiences timelessness in its own frame of reference whereas a negatively massive particle perceives forwardness of time.Moreover,application to quantum harmonic oscillators shows instantaneous transition of electrons from one eigenstate to another.

Diffusion behavior of Cr diluted in bcc and fcc Fe: Classical and quantum simulation methods

We characterize the atomic mobility behavior driven by vacancies, in bcc and fcc Fe−Cr diluted alloys, using a multi-frequency model. We calculate the full set of the Onsager coefficients and the tracer self and solute diffusion coefficients in terms of the mean jump frequencies. The involved jump frequencies are calculated using a classical molecular static (CMS) technique. For the bcc case, we also perform quantum calculations based on the density functional theory (DFT). There, we show that, in accordance with Bohr's correspondence principle, as the size of the atomic cell (total number of atoms) is increased, quantum results with DFT recover the classical ones obtained with CMS calculations. This last ones, are in perfect agreement with available experimental data for both, solute and solvent diffusion coefficients. For high temperatures, in the fcc phase where no experimental data are yet available, our CMS calculations predict the expected solute and solvent diffusion coefficients.

On Quasi-Normal Modes, Area Quantization and Bohr Correspondence Principle

In (Int. Journ. Mod. Phys. D 14, 181 2005), the author Khriplovich verbatim claims that “the correspondence principle does not dictate any relation between the asymptotics of quasinormal modes and the spectrum of quantized black holes” and that “this belief is in conflict with simple physical arguments”. In this paper we analyze Khriplovich’s criticisms and realize that they work only for the original proposal by Hod, while they do not work for the improvements suggested by Maggiore and recently finalized by the author and collaborators through a connection between Hawking radiation and black hole (BH) quasi-normal modes (QNMs). This is a model of quantum BH somewhat similar to the historical semi-classical model of the structure of a hydrogen atom introduced by Bohr in 1913. Thus, QNMs can be really interpreted as BH quantum levels (the “electrons” of the “Bohr-like BH model”).Our results have also important implications on the BH information puzzle.

Quantum monodromy and molecular spectroscopy

Monodromy (or once round) is a classical property of integrable dynamical systems in two or more degrees of freedom, which imposes a characteristic pattern on the quantum mechanical eigenvalue distribution. This article explains the connection by showing how the presence of an isolated critical point of the Hamiltonian leads to a classical action function that is multi-valued with respect to energy and angular momentum. Consequently, by the Bohr correspondence principle between actions and quantum numbers, there can be no uniquely defined global system of quantum numbers. Implications for the interpretation of highly excited molecular spectra are brought out by reference to quasi-linear molecules, which transfer one degree of freedom from rotational to vibrational motion during the excitation process. Emphasis is placed on the simplest examples, while a brief resumé of the wide scope of the quantum monodromy phenomenon is given in the final section.

Classical and semiclassical theories of atom scattering from corrugated surfaces

A theory based in the semiclassical eikonal approximation is developed to describe energy transfer in the collision of an atomic projectile with a surface which is either ordered or disordered. This theory is extended from the quantum mechanical regime to the classical regime of complete quantum decoherence via the Bohr correspondence principle of large numbers of excited quanta. In the quantum mechanical regime, the theory reproduces the well-known eikonal approximation for elastic collisions, provides a simple and useful expression for single-phonon inelastic scattering, and leads to further expressions for multiple-phonon transfers. In the classical limit, the theory produces an expression that includes the effects of surface corrugation in addition to the excitations of large numbers of phonons. This theory shows that a simple measurement of the most probable intensity of energy-resolved scattering spectra taken as a function of surface temperature, with all other experimentally controllable parameters held fixed, can be used to extract the surface corrugation amplitude. Comparisons with data for Ar scattering from the molten metals Ga and In show good agreement with the measured energy-resolved spectra, the in-plane angular distributions, the out-of-plane angular distributions, and produces values for the corrugation amplitudes that range from 10% to 30% of the average interparticle spacing.

QUANTIZATION OF ENTROPY SPECTRA OF BLACK HOLES

From the quasinormal modes (QNM) of black holes, we obtain the quantizations of the entropy and horizon area of black holes via Bohr–Sommerfeld quantization, based on Bohr's correspondence principle. For this, we identify the appropriate action variable of the classical system corresponding to a black hole. By considering the BTZ black holes in topologically massive gravity as well as Einstein gravity, it is found that the spectra of not the horizon areas but the entropies of black holes are equally spaced. We also propose that other characteristic modes of black holes, which are non-QNM or holographic QNM, can be used in quantization of entropy spectra just like QNM. From these modes, it is found that only the entropy spectrum of the warped AdS3 black hole is equally spaced as well. Furthermore, by considering a scattering problem in a black hole, we propose that the total transmission modes and total reflection modes of black holes can be regarded as characteristic modes of black holes and result in the equally spaced entropy of the Kerr and Reissner–Nordström black holes. Finally, we conclude that there is a universal behavior that the entropy spectra of various black holes are equally spaced.

Quantum tunneling as a classical anomaly

Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.