Bohr-Sommerfeld Quantization Rule Research Articles

The Maslov index for composite power-law potentials: The case of the first Exton potential

Using the exact solution of the one-dimensional Schrödinger equation, we demonstrate that the Maslov index involved in the Bohr-Sommerfeld semiclassical quantization rule for the first Exton potential depends on the intensity parameters of the potential. As a function of the intensity of the potential term proportional to x−3/2, it varies within a rather wide range, from −1 to 0, adopting −1/6 when this term and its conjugate term ∼x−1, are absent, which is the case when the Exton potential reduces to the inverse-square-root potential.

Asymptotic theory of not completely integrable soliton equations.

We develop the theory of transformation of intensive initial nonlinear wave pulses to trains of solitons emerging at asymptotically large time of evolution. Our approach is based on the theory of dispersive shock waves in which the number of nonlinear oscillations in the shock becomes the number of solitons at the asymptotic state. We show that this number of oscillations, which is proportional to the classical action of particles associated with the small-amplitude edges of shocks, is preserved by the dispersionless flow. Then, the Poincaré-Cartan integral invariant is also constant, and therefore, it reduces to the quantization rule similar to the Bohr-Sommerfeld quantization rule for a linear spectral problem associated with completely integrable equations. This rule yields a set of "eigenvalues," which are related to the asymptotic solitons' velocities and their characteristics. It is implied that the soliton equations under consideration give modulationally stable solutions; therefore, these "eigenvalues" are real. Our analytical results agree very well with the results of numerical solutions of the generalized defocusing nonlinear Schrödinger equation.

Coulomb Problem for Classical Spinning Particles

We consider the motion of a weakly relativistic charged particle with an arbitrary spin in central potential e/r in terms of classical mechanics. We show that the spin–orbital interaction causes the precession of the plane of orbit around the vector of total angular momentum. The angular velocity of precession depends on the distance of the particle from the centre. The effective potential for in-plane motion is central, with the corrections to Coulomb terms coming from spin–orbital interaction. The possible orbits of a quantum particle are determined by the Bohr–Sommerfeld quantization rule. We give examples of orbits corresponding to small quantum numbers, which were obtained by numerical integration of equations of motion. The energies of stationary states are determined by spin–orbital interaction.

The Self-Deployment of Worlds on the Other Side of the Big Bang

String theory and quantum loop gravity have given us a lot of hope and have achieved some success in trying to build a quantum theory of gravity. However, we still don’t have definitive answers. In this paper, we substantiate the idea of the existence of a Primary Parent Particle, constituents of the medium of primary matter before the Big Bang and describe its structure, consisting of three Beginnings united in the form of Borromeo rings. We derived a generalization of the Bohr-Sommerfeld quantization rules on the multidimensional case and considered problems of the embedding spaces into the each other based on the Nash theorems. Assuming that at the distances occupied by the triad we can consider the space Euclidean flat, we introduce quantum mechanical equations describing them. The problems of constructing the theory of quantum gravity and Dark matter radiation are considered separately. As part of our approach, we describe the acceleration of the Universe created by Dark Energy during expansion. We criticize principle of dualism, which can be used only for linear theories, but is not applicable to nonlinear theories. We investigated various distributions of dark matter densities and the gravitational potentials induced by them in galaxies and velocities for rotation curves. The main goal in the article was to show self-deployment of the Universe and its origin before Big Bang.

On the Stable Eigenvalues of Perturbed Anharmonic Oscillators in Dimension Two

We study the asymptotic behavior of the spectrum of a quantum system which is a perturbation of a spherically symmetric anharmonic oscillator in dimension 2. We prove that a large part of its eigenvalues can be obtained by Bohr-Sommerfeld quantization rule applied to the normal form Hamiltonian and also admit an asymptotic expansion at infinity. The proof is based on the generalization to the present context of the normal form approach developed in [BLM20b] (see also [PS10]) for the particular case of $T^d$.

A Theoretical Study on Vibrational Energies of Molecular Hydrogen and Its Isotopes Using a Semi-classical Approximation

This study aims to apply a semi-classical approach using some analytically solvable potential functions to accurately compute the first ten pure vibrational energies of molecular hydrogen (H2) and its isotopes in their ground electronic states. This study also aims at comparing the accuracy of the potential functions within the framework of the semi-classical approximation. The performance of the approximation was investigated as a function of the molecular mass. In this approximation, the nuclei were assumed to move in a classical potential. The Bohr-Sommerfeld quantization rule was then applied to calculate the vibrational energies of the molecules numerically. The results indicated that the first vibrational transition frequencies (v1ß0) of all hydrogen isotopes were consistent with the experimental ones, with a minimum percentage error of 0.02% for ditritium (T2) molecule using the Modified-Rosen-Morse potential. It was also demonstrated that, in general, the Rosen-Morse and the Modified-Rosen-Morse potential functions were better in terms of calculating the vibrational energies of the molecules than Morse potential. Interestingly, the Morse potential was found to be better than the Manning-Rosen potential. Finally, the semi-classical approximation was found to perform better for heavier isotopes for all potentials applied in this study.

Accuracy of Morse and Morse-like oscillators for diatomic molecular interaction: A comparative study

This study aims at comprehensively evaluating and comparing the goodness-of-fit of several potential energy curves (PECs), which are Morse, Manning-Rosen, Rosen-Morse and Modified-Rosen-Morse oscillators, to the corresponding experimental Rydberg-Klein-Rees (RKR) or RKR-style data of diatomic molecules. In this work, we also determined vibrational energies of the diatomics based on Bohr-Sommerfeld quantization rule. The goodness-of-fit to RKR data of 34 electronic states of 17 diatomics and the accuracy of the calculated vibrational energies were compared for the four oscillators and reported simultaneously for the first time in this study. These results showed that some oscillators demonstrated better goodness-of-fit than the others. Overall, the (Modified)-Rosen-Morse oscillators generated more accurate vibrational levels for most molecules studied, while Manning-Rosen oscillator was generally better for molecules in their excited electronic states. Although all oscillators performed well for more classical molecules, the Morse oscillator was superior to the other three in describing the ground states of 12C16O and 28Si32S. Particularly, it reproduced all 38 observed experimental ground state vibrational levels of 12C16O with error below 0.79% and even accurately predicted four states unobserved experimentally.

Fractal geometry and the mapping of Efimov states to Bloch states.

Efimov states are known to have a discrete real-space scale invariance; working in momentum space we identify the relevant discrete scale invariance for the scattering amplitude defining its Weierstrass function as well. Through the use of the mathematical formalism for discrete scale invariance for the scattering amplitude we identify the scaling parameters from the pole structure of the corresponding zeta function; its zeroth-order pole is fixed by the Efimov physics. The corresponding geometrical fractal structure for Efimov physics in momentum space is identified as a ray across a logarithmic spiral. This geometrical structure also appears in the physics of atomic collapse in the relativistic regime connecting it to Efimov physics. Transforming to logarithmic variables in momentum space we map the three-body scattering amplitude into Bloch states and the ladder of energies of the Efimov states are simply obtained in terms of the Bohr-Sommerfeld quantization rule. Thus through the mapping the complex problem of three-body short-range interaction is transformed to that of a noninteracting single particle in a discrete lattice.

Bohr–Sommerfeld Quantization Rule in the Case of a Two-Dimensional Motion of a Quantum Particle in the Field of Decreasing Power-Like Potential

We study the finite two-dimensional motion of a quantum particle in the field of the potential $$V(r) = - {{V}_{0}}{{r}^{{ - \beta }}}$$ with parameters $${{V}_{0}} > 0$$ and $$\beta \in (0,2)$$ . An analysis of an asymptotic equation equivalent to the Bohr–Sommerfeld quantization rule is given. As a result, a simple and explicit approximation for the energies of weakly-bound states of this particle has been obtained.

Semiclassical approach in classical representation

Semiclassical approach is developed in the classical representation. It is proved that the semiclassical quantization condition in the classical representation coincides with the Bohr–Sommerfeld quantization rule in the initial coordinate representation.

The one dimensional semi-classical Bogoliubov-de Gennes Hamiltonian with PT symmetry: generalized Bohr-Sommerfeld quantization rules

We present a method for computing first order asymptotics of semiclassical spectra for 1-D Bogoliubov-de Gennes (BdG) Hamiltonian from Supraconductivity, which models the electron/hole scattering through two SNS junctions. This involves: 1) reducing the system to Weber equation near the branching point at the junctions; 2) constructing local sections of the fibre bundle of microlocal solutions; 3) normalizing these solutions for the “flux norm” associated to the microlocal Wronskians; 4) finding the relative monodromy matrices in the gauge group that leaves invariant the flux norm; 5) from this we deduce Bohr-Sommerfeld (BS) quantization rules that hold precisely when the fibre bundle of microlocal solutions (depending on the energy parameter E) has trivial holonomy. Such a semi-classical treatement reveals interesting continuous symetries related to monodromy.

Quantization of Horizon Area of Kerr–Newman–de Sitter Black Hole**Supported by the CSIR, Visiting Associate in Inter University Centre for Astronomy and Astrophysics, Pune, India

Using the adiabatic invariant action and applying the Bohr–Sommerfeld quantization rule and first law of black hole thermodynamics, a study of the quantization of the entropy and horizon area of a Kerr–Newman–de Sitter black hole is carried out. The same entropy spectrum is obtained in two different coordinate systems. It is also observed that the spacing of the entropy spectrum is independent of the black hole parameters. Also, the corresponding quantum of horizon area is in agreement with the results of Bekenstein.

An Exact Version of the Egorov Theorem for Schrödinger Operators in $$L^{2}({\mathbb {T}})$$ L 2 ( T )

We provide an exact version of the Egorov Theorem for a class of Schrodinger operators in $$L^2({\mathbb {T}})$$ , where $${\mathbb {T}}={\mathbb {R}}/2\pi {\mathbb {Z}}$$ is the one-dimensional torus. We show that the classical Hamiltonian, after the symplectic transformation to action coordinates, can be composed with a toroidal semiclassical $$\psi $$ do in order to recover the Schrodinger operator. This result turns out to be strictly related to the Bohr-Sommerfeld quantization rules as well as to the inverse spectral problem and the periodic homogenization of Hamilton–Jacobi equations.

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.

Semiclassical theory of Landau levels and magnetic breakdown in topological metals

The Bohr-Sommerfeld quantization rule lies at the heart of the modern semiclassical theory of a Bloch electron in a magnetic field. This rule is predictive of Landau levels and quantum oscillations for conventional metals, as well as for a host of topological metals which have emerged in the recent intercourse between band theory, crystalline symmetries and topology. The essential ingredients in any quantization rule are connection formulae that match the semiclassical (WKB) wavefunction across regions of strong quantum fluctuations. Here, we propose (a) a multi-component WKB wavefunction that describes transport within degenerate-band subspaces, and (b) the requisite connection formulae for saddlepoints and type-II Dirac points, where tunneling respectively occurs within the same band, and between distinct bands. (a-b) extend previous works by incorporating phase corrections that are subleading in powers of the field; these corrections include the geometric Berry phase, and account for the orbital magnetic moment and the Zeeman coupling. A comprehensive symmetry analysis is performed for such phase corrections occurring in closed orbits, which is applicable to solids in any (magnetic) space group. We have further formulated a graph-theoretic description of semiclassical orbits. This allows us to systematize the construction of quantization rules for a large class of closed orbits (with or without tunneling), as well as to formulate the notion of a topological invariant in semiclassical magnetotransport -- as a quantity that is invariant under continuous deformations of the graph. Landau levels in the presence of tunneling are generically quasirandom, i.e., disordered on the scale of nearest-neighbor level spacings but having longer-ranged correlations; we develop a perturbative theory to determine Landau levels in such quasirandom spectra.

Application of path-integral quantization to indistinguishable particle systems topologically confined by a magnetic field

We demonstrate an original development of path-integral quantization in the case of a multiply connected configuration space of indistinguishable charged particles on a 2D manifold and exposed to a strong perpendicular magnetic field. The system occurs to be exceptionally homotopy-rich and the structure of the homotopy essentially depends on the magnetic field strength resulting in multi loop trajectories at specific conditions. We have proved, by a generalization of the Bohr-Sommerfeld quantization rule, that the size of a magnetic field flux quantum grows for multi loop orbits like $(2k+1)\frac{h}{c}$ with the number of loops $k$. Utilizing this property for electrons on the 2D substrate jellium we have derived upon the path integration a complete FQHE hierarchy in excellent consistence with experiments. The path-integral has been next developed to a sum over configurations, displaying various patterns of trajectory homotopies (topological configurations), which in the nonstationary case of quantum kinetics reproduces some unclear formerly details in the longitudinal resistivity observed in experiments.

Geometric Phase and Orbital Moment in Quantization Rules for Magnetic Breakdown.

The modern semiclassical theory of a Bloch electron in a magnetic field encompasses the orbital magnetization and geometric phase. Beyond this semiclassical theory lies the quantum description of field-induced tunneling between semiclassical orbits, known as magnetic breakdown. Here, we synthesize the modern semiclassical notions with quantum tunneling-into a single Bohr-Sommerfeld quantization rule that is predictive of magnetic energy levels. This rule is applicable to a host of topological solids with unremovable geometric phase, that also unavoidably undergo breakdown. A notion of topological invariants is formulated that nonperturbatively encode tunneling, and is measurable in the de Haas-van Alphen effect. Case studies are discussed for topological metals near a metal-insulator transition and overtilted Weyl fermions.

Semiclassical Asymptotics of Solutions to Hartree Type Equations Concentrated on Segments

We study the nonlinear eigenvalue problem for two-dimensional Hartree type equations with selfaction potentials possessing logarithmic singularity and depending on the distance between points. To find a series of asymptotic eigenvalues, we derive a counterpart of the Bohr–Sommerfeld quantization rule. The corresponding asymptotic eigenfunctions are localized near a plane segment.

Spectroscopy of the Rotating Kaluza-Klein Spacetime via Revisited Adiabatic Invariant Quantity

In this paper, we have investigated the spectroscopy of the rotating Kaluza-Klein spacetime by applying Bohr–Sommerfeld quantization rule and the first law of thermodynamics. we derived the expression of the adiabatic invariant quantity in the dragged–Painleve coordinate system. Then, via revisited adiabatic invariant quantity, we derive the area and entropy spectra of the spacetime. We obtained the area spectrum of the Kaluza-Klein spacetime is ${\Delta } A=8\pi {l_{P}^{2}}$ , and the entropy spectrum is ΔS = 2π. This result is consistent with the Bekenstein’s original result, which imply the entropy and horizon area are discrete and equidistant for the spacetime.

On the atomic-number similarity of the binding energies of electrons in filled shells of elements of the periodic table

An expression for the binding energies of electrons in the ground state of an atom is derived on the basis of the Bohr–Sommerfeld quantization rule within the Thomas–Fermi model. The validity of this relation for all elements from neon to uranium is tested within a more perfect quantum-mechanical model with and without the inclusion of relativistic effects, as well as with experimental binding energies. As a result, the ordering of electronic levels in filled atomic shells is established, manifested in an approximate atomic-number similarity. It is proposed to use this scaling property to analytically estimate the binding energies of electrons in an arbitrary atom.