Bohr-Sommerfeld Quantization Condition Research Articles

Bohr–Sommerfeld quantization condition for self-adjoint Dirac operators

We study the eigenvalue problem for a self-adjoint 1D Dirac operator. It is known that, near an energy level where the square of the potential makes a simple well, the eigenvalues are approximated by a Bohr–Sommerfeld type quantization rule. A remarkable difference from the Schrödinger case appears in the Maslov correction term. This fact was recently found by Hirota [Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov–Shabat operator, J. Math. Phys. 58 (2017) 102–108] under the analyticity condition of the potential using a complex WKB method. In this paper, we approach this problem with a microlocal technique focusing on the asymptotic behavior of the eigenfunction along the characteristic set to generalize the result to [Formula: see text] potentials.

Exact WKB Analysis and TBA Equations for the Stark Effect

Abstract We apply the exact Wentzel–Kramers–Brillouin (WKB) analysis to a couple of 1D Schrödinger-type equations reduced from the Stark effect of hydrogen in a uniform electric field. By introducing Langer’s modification and incorporating the Stokes graphs, we prove the exactness of the Bohr–Sommerfeld quantization conditions for the Borel-resummed quantum WKB periods in the specific parameter regions of the electric field intensity and magnetic quantum number. It is also found these quantization conditions get modified with an additional suppressed contribution when the parameters vary beyond the specific regions. We also present thermodynamic Bethe ansatz (TBA) equations governing the quantum periods in the absence of Langer’s modification and discuss its wall-crossing and analytic continuation. Numerical calculations are conducted to compare the complex resonant frequencies from our quantization conditions against ones from the Riccati–Padé method; the TBA equations are also confirmed by comparing its expansions with all-order quantum periods.

Simple model for cluster radioactivity half-lives in trans-lead nuclei* *Supported in part by the National Natural Science Foundation of China (12175100, 11975132), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of the Education Bureau of Hunan Province, China (18A237), the Natural Science Foundation of Shandong Province, China (ZR2022JQ04), the Opening Project of Cooperative Innovation Center

In this study, considering the modified preformation probability to be , where and are the α-particle preformation probability and an adjustable parameter proposed by Wang et al. [Chin. Phys. C 45, 044111 (2021)], respectively, we extend a new simple model put forward by Bayrak [J. Phys. G 47, 025102 (2020)] to systematically study the cluster radioactivity half-lives of 28 trans-lead nuclei ranging from to , which is based on the Wentzel-Kramers-Brillouin approximation and Bohr–Sommerfeld quantization condition. For comparison, a universal decay law proposed by Qi et al. [Phys. Rev. C 80, 044326 (2009)], a three-parameter model-independent formula put forward by Balasubramaniam et al. [Phys. Rev. C 70, 017301 (2004)], and the semi-empirical model proposed by Tavares et al. [Eur. Phys. J. A 49, 1 (2013)] are used. Our calculated results reproduce the experimental data well, with a standard deviation of 0.818. Furthermore, we use this model to predict the cluster radioactivity half-lives of 51 possible cluster radioactive candidates whose cluster radioactivities are energetically allowed or observed but not yet quantified in NUBASE2020.

Exchange-correlation bound states of the triplet soft-sphere fermions by path-integral Monte Carlo simulations.

Path-integral Monte Carlo simulations in the Wigner approach to quantum mechanics has been applied to calculate momentum and spin-resolved radial distribution functions of the strongly correlated soft-sphere quantum fermions. The obtained spin-resolved radial distribution functions demonstrate arising triplet clusters of fermions, that is the consequence of the interference of exchange and interparticle interactions. The semiclassical analysis in the framework of the Bohr-Sommerfeld quantization condition, applied to the potential of the mean force corresponding to the same-spin radial distribution functions, allows to detect exchange-correlation bound states in triplet clusters and to estimate corresponding averaged energy levels. The obtained momentum distribution functions demonstrate the narrow sharp separated peaks corresponding to bound states and disturbing the Maxwellian distribution.

Systematic investigation of the alpha and cluster decay half-lives for heavy and superheavy nuclei within the shifted Deng-Fan potential

The half-lives of alpha decays for even-even nuclei with Z=60−118 and the cluster radioactivities for even-even emitters for which the experimental data are available, also the various cluster decays of even-even radium isotopes are calculated. Our calculations are within the framework of Wentzel-Kramers-Brillouin (WKB) method by considering the Bohr-Sommerfeld quantization condition based on the shifted Deng-Fan potential. The obtained results are compared with the squared Woods-Saxon and Cosh potentials and several theoretical models, including the universal model (UNIV), the unified fission model (UFM), the universal decay law (UDL), and Ni, Royer, Tavares and Balasubramaniam semiempirical formulas, as well as the experimental data. Good agreement between the experimental and the proposed potential model results, indicates that the shifted Deng-Fan potential is a suitable phenomenological potential to describe both alpha and cluster decays of radioactive nuclei. Also we observe the significant role of neutron magicity in the half-lives of cluster decays of radium isotopes at N=126 of daughter nuclei and the parent nucleus 214Ra.

Intrinsic quantum dynamics of particles in brane gravity

The Newtonian dynamics of particles in brane gravity is investigated. Due to the coupling of the particles’ energy–momentum tensor to the tension of the brane, the particle is semi-confined and oscillates along the extra dimension. We demonstrate that the frequency of these oscillations is proportional to the kinetic energy of the particle in the brane. We show that the classical stability of particle trajectories on the brane gives us the Bohr–Sommerfeld quantization condition. The particle’s motion along the extra dimension allows us to formulate a geometrical version of the uncertainty principle. Furthermore, we exhibited that the particle’s motion along the extra dimension is identical to the time-independent Schrödinger equation. The dynamics of a free particle, particles in a box, a harmonic oscillator, a bouncing particle, and tunneling are re-examined. We show that the particle’s motion along the extra dimension yields a quantized energy spectrum for bound states.

Structure and α decay for the neutron-deficient nuclei with 89≤Z≤94 in the density-dependent cluster model combined with a relativistic mean-field approach

The density-dependent cluster model combined with relativistic mean-field theory is used to explore the structure and $\ensuremath{\alpha}$ decay for the neutron-deficient nuclei with $89\ensuremath{\le}Z\ensuremath{\le}94$, including two newly discovered nuclei $^{207}\mathrm{Th}$ [Phys. Rev. C 105, L051302 (2022)] and $^{214}\mathrm{U}$ [Phys. Rev. Lett. 126, 152502 (2021)]. The effective nucleon-nucleon interactions and matter density distributions from the relativistic mean field are employed to construct the $\ensuremath{\alpha}$-daughter potential with a double-folding model. The Pauli blocking effect is considered by normalizing the strength of the $\ensuremath{\alpha}$-daughter potential with the Bohr-Sommerfeld quantization condition. The $\ensuremath{\alpha}$-preformation factor is calculated with the cluster formation model. The calculated $\ensuremath{\alpha}$-decay half-lives for the 106 observed nuclei with $89\ensuremath{\le}Z\ensuremath{\le}94$ are in excellent agreement with experimental data. Extending this model to the unknown nuclei $^{201--204}\mathrm{Ac}$, $^{205--206}\mathrm{Th}$, $^{209--210}\mathrm{Pa}$, $^{212--213,220}\mathrm{U}$, $^{215--218,221}\mathrm{Np}$, and $^{220--227}\mathrm{Pu}$, the evolution of the $N=126$ shell closure with neutron number is explored for the high-$Z$ isotopes. The available $\ensuremath{\alpha}$-decay energies, preformation factors, and $\ensuremath{\alpha}$-decay half-lives show a regular change with increasing neutron number. Especially, the robustness of the $N=126$ shell closure is shown up to the Pu isotopes.

Quantum field theoretic representation of Wilson surfaces. Part I. Higher coadjoint orbit theory

This is the first of a series of two papers devoted to the partition function realization of Wilson surfaces in strict higher gauge theory. A higher version of the Kirillov-Kostant-Souriau theory of coadjoint orbits is presented based on the derived geometric framework, which has shown its usefulness in 4-dimensional higher Chern-Simons theory. An original notion of derived coadjoint orbit is put forward. A theory of derived unitary line bundles and Poisson structures on regular derived orbits is constructed. The proper derived counterpart of the Bohr-Sommerfeld quantization condition is then identified. A version of derived prequantization is proposed. The difficulties hindering a full quantization, shared with other approaches to higher quantization, are pinpointed and a possible way-out is suggested. The theory we elaborate provide the geometric underpinning for the field theoretic constructions of the companion paper.

Classical and Quantum Spherical Pendulum

The seminal paper by Niels Bohr followed by a paper by Arnold Sommerfeld led to a revolutionary Bohr–Sommerfeld theory of atomic spectra. We are interested in the information about the structure of quantum mechanics encoded in this theory. In particular, we want to extend Bohr–Sommerfeld theory to a full quantum theory of completely integrable Hamiltonian systems, which is compatible with geometric quantization. In the general case, we use geometric quantization to prove analogues of the Bohr–Sommerfeld quantization conditions for the prequantum operators Pf. If a prequantum operator Pf satisfies the Bohr–Sommerfeld conditions and if it restricts to a directly quantized operator Qf in the representation corresponding to the polarization F, then Qf also satisfies the Bohr–Sommerfeld conditions. The proof that the quantum spherical pendulum is a quantum system of the type we are looking for requires a new treatment of the classical action functions and their properties. For the sake of completeness we have provided an extensive presentation of the classical spherical pendulum. In our approach to Bohr–Sommerfeld theory, which we call Bohr–Sommerfeld–Heisenberg quantization, we define shifting operators that provide transitions between different quantum states. Moreover, we relate these shifting operators to quantization of functions on the phase space of the theory. We use Bohr–Sommerfeld–Heisenberg theory to study the properties of the quantum spherical pendulum, in particular, the boundary conditions for the shifting operators and quantum monodromy.

Bound States of the Exchange—Correlation Excitons in the Uniform Electron Gas by the Monte Carlo Simulations

The modified path integral representation of Wigner functions and the new Monte Carlo approach has been suggested to account for the impact of the interparticle interaction on the Pauli exclusion principle of fermions. This approach also allows to calculate the momentum distribution functions and to reduce the “sign problem” that is inaccessible to the standard path integral Monte Carlo methods. The obtained pair electron–electron distribution functions for the “uniform electron gas” demonstrate the short-range quantum ordering of electrons associated with exchange-correlation excitons. The exchange-correlation exciton is caused by the interaction of electrons with positively charged exchange holes and the excluded volume effect. The developed approach allows one to study the density–temperature range of the exciton arising, existence, and decay. Using the potential of the mean force and semiclassical Bohr–Sommerfeld quantization condition, we have demonstrated the existence of bound states disturbing the Maxwellian distribution and estimated their average energy levels. The exchange-correlation excitons have not been observed earlier in the standard path integral Monte Carlo simulations.

Black Hole Quasinormal Modes and Seiberg–Witten Theory

We present new analytic results on black hole perturbation theory. Our results are based on a novel relation to four-dimensional {mathcal {N}}=2 supersymmetric gauge theories. We propose an exact version of Bohr-Sommerfeld quantization conditions on quasinormal mode frequencies in terms of the Nekrasov partition function in a particular phase of the Omega -background. Our quantization conditions also enable us to find exact expressions of eigenvalues of spin-weighted spheroidal harmonics. We test the validity of our conjecture by comparing against known numerical results for Kerr black holes as well as for Schwarzschild black holes. Some extensions are also discussed.

On the Bohr-Sommerfeld quantization condition and assault frequency in a semiclassical model for α decay

We study the impacts of the Bohr-Sommerfeld quantization condition and the assault frequency on the α decay half-life within the semiclassical model. The potential between the α particle and daughter nucleus is calculated by the double-folding model using the CDM3Y3 density-dependent nucleon-nucleon interaction with a finite-range exchange term. We show that the proper implementation of the Bohr-Sommerfeld condition leads to a considerable change of the calculated α decay half-life with certain forms of potential. We also propose an alternative treatment for the assault frequency based on the generalized oscillator potential. This description of assault frequency considerably improves the agreement between the calculated α decay half-lives and the experimental data.

Power-like potentials: From the Bohr—Sommerfeld energies to exact ones

For one-dimensional power-like potentials [Formula: see text], [Formula: see text], the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones as opposed to the negative parity states where the BSE remain above the exact ones for [Formula: see text] but below them for [Formula: see text]. The ground state BSE as function of [Formula: see text] are of the same order of magnitude as the exact energies for linear [Formula: see text], quartic [Formula: see text] and sextic [Formula: see text] oscillators but their relative deviation grows with [Formula: see text], reaching the value 4 at [Formula: see text]. For physically important cases [Formula: see text], for the 100th excited state BSE coincide with exact ones in 5–6 figures. It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correction [Formula: see text] (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function [Formula: see text] for all [Formula: see text]. It grows slowly with increasing [Formula: see text] for fixed quantum number [Formula: see text], while it decays with quantum number growth at fixed [Formula: see text]. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of [Formula: see text] (and [Formula: see text]).

Alpha decay study of Thorium isotopes using double folding model with NN interactions derived from relativistic mean field theory

The alpha decay half-lives of some Thorium isotopes have been calculated within the framework of the WKB semi-classical approximation, with the inclusion of the Bohr-Sommerfeld quantization condition. The effective potentials used are the double folding nuclear potential, Coulomb potential, and centrifugal potential. Apart from using the famous M3Y-Reid and M3Y-Paris nucleon-nucleon (NN) interactions, the microscopic NN interaction derived from relativistic mean field theory Lagrangian (R3Y), with different parametrizations, has also been used. The results obtained using the R3Y potentials are compared to the phenomenological M3Y-Reid and M3Y-Paris NN interactions. The R3Y based interactions are found to perform slightly better than the M3Y based interactions for density-independent double folding model. When density-dependence of the double folding model is used, the M3Y-Paris and R3Y-W give the best descriptions of the α−decay half-lives of the Thorium isotopes, while the M3Y-Reid gives the highest deviation from experimental data.

Meissner Effect: History of Development and Novel Aspects

The discovery of the Meissner (Meissner–Ochsenfeld) effect in 1933 was an incontestable turning point in the history of superconductivity. First, it demonstrated that superconductivity is an unknown before equilibrium state of matter, thus allowing to use the power of thermodynamics for its study. This provided a justification for the two-fluid model of Gorter and Casimir, a seminal thermodynamic theory founded on a postulate of zero entropy of the superconducting (S) component of conduction electrons. Second, the Meissner effect demonstrated that, apart from zero electric resistivity, the S phase is also characterized by zero magnetic induction. The latter property is used as a basic postulate in the theory of F. and H. London, which underlies the understanding of electromagnetic properties of superconductors. Here the experimental and theoretical aspects of the Meissner effect are reviewed. The reader will see that, in spite of almost nine decades age, the London theory still contains questions, the answers to which can lead to a revision of the standard picture of the Meissner state (MS) and, if so, of other equilibrium superconducting states. An attempt is made to take a fresh look at electrodynamics of the MS and try to work out with the issues associated with the description of this most important state of all superconductors. It is shown that the concept of Cooper’s pairing along with the Bohr–Sommerfeld quantization condition allows one to construct a semi-classical theoretical model consistently addressing properties of the MS and beyond, including non-equilibrium properties of superconductors caused by the total current. As follows from the model, the three “big zeros” of superconductivity (zero resistance, zero induction and zero entropy) have equal weight and grow from a single root: quantization of the angular momentum of paired electrons. The model predicts some yet unknown effects. If confirmed, they can help in studies of microscopic properties of all superconductors. Preliminary experimental results suggesting the need to revise the standard picture of the MS are presented.

Search for α-decay chains for superheavy nuclei with Z=125-127

We have studied the alpha-decay (AD) and spontaneous fission (SF) properties of the isotopes of Z=125, 126 and 127 superheavy nuclei. AD calculations have been performed over all the possible isotopes of Z=125-127 nuclei within the framework of Wentzel-Kramers-Brillouin (WKB) and Bohr-Sommerfeld quantization condition. To broaden the perspective of the calculations, we have also utilized from the analytic formula of Royer in the calculations of AD. In this way we believe that by using various approaches might be useful for the clarity of the predictions. Also, the SF half-lives have been computed by using formulae of Soylu and KPS. Decay modes for the isotopes are determined by using the results of different models. The decay modes obtained for Z=125 and Z=126 have been compared with the results of other studies, and α-decay chains have been presented for Z=127. Findings would provide opportunities for the future experimental researches.

A new potential model for alpha decay calculations

We systematically investigate the experimental α-decay half-lives of the 263 parent nuclei in the range of 52≤Z≤107 for the ground state to ground state α-transitions in terms of a molecular potential called the Morse potential by utilizing the Wentzel-Kramers-Brillouin (WKB) approximation and Bohr-Sommerfeld quantization condition. We find out the optimum parameters of the Morse potential to explain the α-decay half-lives of the parent nuclei and compare our results with the unified model for α-decay and α capture (UMADAC) and the Universal Decay Law (UDL). In calculations of the α-decay half-lives of the parent nuclei, we achieve better results than the UMADAC and UDL models in terms of few degrees of freedom.

Complex Eigenvalue Splitting for the Dirac Operator

We analyze the eigenvalue problem for the semiclassical Dirac (or Zakharov–Shabat) operator on the real line with general analytic potential. We provide Bohr–Sommerfeld quantization conditions near energy levels where the potential exhibits the characteristics of a single or double bump function. From these conditions we infer that near energy levels where the potential (or rather its square) looks like a single bump function, all eigenvalues are purely imaginary. For even or odd potentials we infer that near energy levels where the square of the potential looks like a double bump function, eigenvalues split in pairs exponentially close to reference points on the imaginary axis. For even potentials this splitting is vertical and for odd potentials it is horizontal, meaning that all such eigenvalues are purely imaginary when the potential is even, and no such eigenvalue is purely imaginary when the potential is odd.

Geometric phase and in-phase superpositions: A fresh perspective on interference in phase space

We apply geometric phase ideas to coherent states to shed light on interference phenomenon in the phase space description of continuous variable Cartesian quantum systems. The motivating idea is Pancharatnam's concept of two Hilbert space vectors being ’’in phase”. This leads to special sums and integrals of vectors which are locally in-phase superpositions. Geometric phase considerations naturally lead to the emergence of phase space areas. Applied to the overcomplete family of coherent states, we are led to preferred in-phase integral representations for various states of physical significance, such as the position, momentum and Fock states as well as the squeezed vacuum state. Interestingly, the Husimi-Kano Q function is maximized along the line of such superpositions. We also get a fresh perspective on the Bohr-Sommerfeld quantization condition. Finally, we use our exact integral representations to obtain asymptotic expansions for state overlaps and matrix elements, leading to phase space area considerations similar to the ones noted earlier in the seminal works of Schleich, Wheeler and collaborators, but now from a geometric phase perspective.

Alpha decay half lives of spherical even-even nuclei: Role of repulsive core

By adding repulsive nuclear potential, in Gaussian shape and double-folding form with different values of repulsive surface diffuseness parameter, to the attractive double-folding nuclear potential (M3Y) the effect of repulsive core on alpha decay half-lives and decay parameters of some even-even alpha emitters were analyzed. Calculations were accomplished using WKB approximation by considering Bohr-Sommerfeld quantization condition. The results showed calculated half-lives with M3Y lay between obtained lower and upper limits by M3Y plus repulsive potential with short-range repulsive diffuseness arep=0.01fm and long-range one arep=0.30fm. As result of crucial role of quantization factor an increasing trend of half-lives were observed by increase of arep. Also good coincidence between theory and experiment was observed for a distinct arep which highlights the formation of alpha particle near surface of parent nucleus.