Bohr-Sommerfeld Quantization Research Articles

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.

Quasi-Classical Calculation of Eigenvalues by Maslov Quantization Condition

The Maslov quantization condition is a condition for Lagrangian submanifolds which is regarded as a mathematical extension of the Bohr-Sommerfeld quantization condition. In this survey note, we apply the Maslov quantization condition to several concrete Schr odinger operators and quantize invariant Lagrangian submanifolds of their classical systems. We see the quasi-classical energy levels are equal to the quantum ones for these operators and also the number of Lagrangian submanifolds is equal to the multiplicities of eigenvalues for these operators.

Alpha and cluster decay of some deformed heavy and superheavy nuclei

The alpha decay half-lives of some deformed superheavy nuclei and cluster decay half-lives of some deformed heavy nuclei were calculated in framework of WKB semiclassical method and by including Bohr–Sommerfeld quantization condition. Deformed double-folding nuclear and Coulomb potentials, and centrifugal potential have been considered for effective potential. Two forms of deformed surface diffuseness were used for calculations of alpha decay half-lives. In most of cases the calculated alpha decay half-lives with proposed deformed surface diffuseness were in better agreement with experiment in comparison with constant one. By using available experimental data, the cluster preformation probabilities were determined. Predicted cluster decay half-lives with inclusion of two different cluster preformation probability formulas were compared with those values of two empirical formulas. It was revealed the significant role of preformation probability in the cluster decay half-life calculations.

Spectral Portraits in the Semi-Classical Approximation of the Sturm-Liouville Problem with a Complex Potential

We study the problem on the limit behavior of the discrete spectrum of the Sturm-Liouville problem − ε y ″ ( x ) + P ( x λ ) y ( x ) = 0 y ( a ) = y ( b ) = 0 , provided that the physical parameter ϵ tends to zero. It is assumed that λ is the spectral parameter, the function P is polynomial on x with analytic coefficients on λ varying on a domain G in the complex λ-plane ℂ. The case P(x, λ) = p(x) - λ with complex valued function p corresponds to the usual linear spectral problem. Boundary conditions are formed by arbitrary complex numbers a, b or ±∞. We shall show that in this case the eigenvalues are concentrated along the so-called limit spectral graph as ϵ → 0. We define three type of curves, forming this graph and find the asymptotic formulae for the eigenvalue distribution along the curves of various types. For the case P(x, λ) = p(x) - λ with real polynomial p(x) these formulae coincide with well known Bohr-Sommerfeld quantization formulae. Non-self-adjoint Sturm-Liouville problems are often found in mathematical physics. We demonstrate this considering the well-known in hydromechanics Orr-Sommerfeld spectral problem.

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.

Exactness of Bohr–Sommerfeld quantisation for two non-central potentials

In this paper we demonstrate the integrability of the Hamilton–Jacobi equation for two non-central potentials in spherical polar coordinates, and present complete solutions for the classically bound orbits. We then show that the semiclassical method of Bohr–Sommerfeld quantisation exactly reproduces the bound state spectra of the corresponding quantum mechanical Schrödinger equations. One of these potentials has previously been analysed in parabolic coordinates; the results for the other are, to the authors’ best knowledge, original.

Improved nucleus–nucleus folding potential with a repulsive core due to the change of intrinsic kinetic energy

The change in the internal kinetic energy (KE) of two interacting nuclei grows by increasing their density overlap. We investigated to what extent the double-folding potential based on the density dependent M3Y nucleon–nucleon interaction can be improved by considering this repulsive KE. We adopted the α-decay of , the 24Ne decay of 232U, and the fusion cross section as examples of reactions involving density redistribution. We performed the α and cluster decay calculations based on solving the Schrödinger equation to find the wavefunction and the penetration probability. Upon normalizing the calculated KE by applying the Bohr–Sommerfeld quantization condition, the necessary missed pocket is physically developed in the internal region of the total folding potential. The folding potential improved by considering the pivotal repulsive KE reduced the uncertainty inherent in the decay calculations at small internuclear distances. The calculated fusion cross section is substantially improved and its steep falloff at sub-barrier energies is reasonably reproduced.

Regge trajectories for the mesons consisting of different quarks

By applying the Bohr–Sommerfeld quantization approach to the quadratic form of the spinless Salpeter-type equation (QSSE), we show that the obtained Regge trajectories for the mesons consisting of unequally massive quarks take the form M^2=beta left( {c_l}l+{pi }n_r+c_0right) ^{2/3}+c_1, which have the same form as the Regge trajectories for charmonia and bottomonia. Then we apply the obtained Regge trajectories to fit the spectra of the strange mesons, the heavy-light mesons (the D, D_s, B and B_s mesons) and the bottom-charmed mesons. The fitted Regge trajectories are in agreement with the experimental data and the theoretical predictions, which demonstrates that the newly proposed Regge trajectories can be applied universally to the light mesons, the heavy-light mesons and the heavy mesons. By fitting the spectra of the mesons composed of different quarks, the concavity of these Regge trajectories are illustrated, which is of cardinal significance for the potential models.

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.

Calculation of proton radioactivity half-lives

We have investigated the proton radioactivity half-lives of 45 proton emitters within the framework of the Wentzel–Kramers–Brillouin (WKB) method by considering the Bohr–Sommerfeld quantization condition. The orientation angle dependent Woods–Saxon, Coulomb, and spin–orbit potentials and centrifugal potential have been used in calculations. We observed that using the orientation angle dependent formalism decreases the values of calculated half-lives.

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.

Some applications of the most general form of the higher-order GUP with minimal length uncertainty and maximal momentum

In this paper, using the new type of D-dimensional nonperturbative Generalized Uncertainty Principle (GUP) which has predicted both a minimal length uncertainty and a maximal observable momentum,1 first, we obtain the maximally localized states and express their connections to [P. Pedram, Phys. Lett. B 714, 317 (2012)]. Then, in the context of our proposed GUP and using the generalized Schrödinger equation, we solve some important problems including particle in a box and one-dimensional hydrogen atom. Next, implying modified Bohr–Sommerfeld quantization, we obtain energy spectra of quantum harmonic oscillator and quantum bouncer. Finally, as an example, we investigate some statistical properties of a free particle, including partition function and internal energy, in the presence of the mentioned GUP.

Systematics of α-decay fine structure in odd-mass nuclei based on a finite-range nucleon–nucleon interaction

A systematic study on α-decay fine structure is presented for odd-mass nuclei in the range 83≤Z≤92. The α-decay partial half-lives and branching ratios to the ground and excited states of daughter nuclei are calculated in the framework of the Wentzel–Kramers–Brillouin (WKB) approximation with the implementation of the Bohr–Sommerfeld quantization condition. The microscopic α-daughter potential is obtained using the double-folding model with a realistic M3Y-Paris nucleon–nucleon (NN) interaction. The exchange potential, which accounts for the knock-on exchange of nucleons between the interacting nuclei, is calculated using the finite-range exchange NN interaction which is essentially a much better approximation as compared to the zero-range pseudo-potential adopted in the usual double-folding calculations. Our calculations of α-decay fine structure have been improved by considering the preformation factor extracted from the recently proposed cluster formation model on basis of the binding energy difference. The computed partial half-lives and branching ratios are compared with the recent experimental data and they are in good agreement.

Theoretical predictions for α -decay chains of Og118290−298 isotopes using a finite-range nucleon-nucleon interaction

The $\ensuremath{\alpha}$-decay half-lives of the recently synthesized superheavy nuclei (SHN) are investigated by employing the density dependent cluster model. A realistic nucleon-nucleon ($\mathit{NN}$) interaction with a finite-range exchange part is used to calculate the microscopic $\ensuremath{\alpha}$-nucleus potential in the well-established double-folding model. The calculated potential is then implemented to find both the assault frequency and the penetration probability of the $\ensuremath{\alpha}$ particle by means of the Wentzel-Kramers-Brillouin (WKB) approximation in combination with the Bohr-Sommerfeld quantization condition. The calculated values of $\ensuremath{\alpha}$-decay half-lives of the recently synthesized Og isotopes and its decay products are in good agreement with the experimental data. Moreover, the calculated values of $\ensuremath{\alpha}$-decay half-lives have been compared with those values evaluated using other theoretical models, and it was found that our theoretical values match well with their counterparts. The competition between $\ensuremath{\alpha}$ decay and spontaneous fission is investigated and predictions for possible decay modes for the unknown nuclei $_{\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{1em}{0ex}}118}^{290\ensuremath{-}298}\mathrm{Og}$ are presented. We studied the behavior of the $\ensuremath{\alpha}$-decay half-lives of Og isotopes and their decay products as a function of the mass number of the parent nuclei. We found that the behavior of the curves is governed by proton and neutron magic numbers found from previous studies. The proton numbers $Z=114$, 116, 108, 106 and the neutron numbers $N=172$, 164, 162, 158 show some magic character. We hope that the theoretical prediction of $\ensuremath{\alpha}$-decay chains provides a new perspective to experimentalists.

Quasinormal modes and quantization of area/entropy for noncommutative BTZ black hole

We investigate the quasinormal modes and area/entropy spectrum for the noncommutative BTZ black hole. The exact expressions for QNM frequencies are presented by expanding the noncommutative parameter in horizon radius. We find that the noncommutativity does not affect conformal weights (h_{L}, h_{R}), but it influences the thermal equilibrium. The intuitive expressions of the area/entropy spectrum are calculated in terms of Bohr–Sommerfeld quantization, and our results show that the noncommutativity leads to a nonuniform area/entropy spectrum. We also find that the coupling constant xi , which is the coupling between the scalar and the gravitational fields, shifts the QNM frequencies but not influences the structure of area/entorpy spectrum.

Passage through a potential barrier and multiple wells

Consider the semiclassical limit, as the Planck constant $\hbar\ri 0$, of bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. We show that, for each eigenvalue of the Schr\"odinger operator, the Bohr-Sommerfeld quantization condition is satisfied at least for one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of a potential barrier. We show that, at least from one side, the barrier fixes the phase of wave functions in the same way as a potential barrier of infinite width. On the other hand, it turns out that for each well there exists an eigenvalue in a small neighborhood of every point satisfying the Bohr-Sommerfeld condition.

Regge trajectories for heavy quarkonia from the quadratic form of the spinless Salpeter-type equation

In this paper, we present one new form of the Regge trajectories for heavy quarkonia which is obtained from the quadratic form of the spinless Salpeter-type equation (QSSE) by employing the Bohr-Sommerfeld quantization approach. The obtained Regge trajectories take the parameterized form M^2={beta }({c_l}l+{pi }n_r+c_0)^{2/3}+c_1, which are different from the present Regge trajectories. Then we apply the obtained Regge trajectories to fit the spectra of charmonia and bottomonia. The fitted Regge trajectories are in good agreement with the experimental data and the theoretical predictions.

Revealing the Topology of Fermi-Surface Wave Functions from Magnetic Quantum Oscillations

The modern semiclassical theory of a Bloch electron in a magnetic field now encompasses the orbital magnetic moment and the geometric phase. These two notions are encoded in the Bohr-Sommerfeld quantization condition as a phase ($\lambda$) that is subleading in powers of the field; $\lambda$ is measurable in the phase offset of the de Haas-van Alphen oscillation, as well as of fixed-bias oscillations of the differential conductance in tunneling spectroscopy. In some solids and for certain field orientations, $\lambda/\pi$ are robustly integer-valued owing to the symmetry of the extremal orbit, i.e., they are the topological invariants of magnetotransport. Our comprehensive symmetry analysis identifies solids in any (magnetic) space group for which $\lambda$ is a topological invariant, as well as identifies the symmetry-enforced degeneracy of Landau levels. The analysis is simplified by our formulation of ten (and only ten) symmetry classes for closed, Fermi-surface orbits. Case studies are discussed for graphene, transition metal dichalchogenides, 3D Weyl and Dirac metals, and crystalline and $\mathbb{Z}_2$ topological insulators. In particular, we point out that a $\pi$ phase offset in the fundamental oscillation should \emph{not} be viewed as a smoking gun for a 3D Dirac metal.

Defect in the Joint Spectrum of Hydrogen due to Monodromy.

In addition to the well-known case of spherical coordinates, the Schrödinger equation of the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, the z component of the angular momentum, and an operator involving the z component of the quantum Laplace-Runge-Lenz vector obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. The precise value of the energy above which monodromy is observed depends on the distance of the focus points of the spheroidal coordinates. The presence of monodromy means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number n and the magnetic quantum number m correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multivalued.