Bohr-Sommerfeld Quantization Condition Research Articles

Effect of nuclear deformation onα-decay half-lives

We systematically investigate the influence of nuclear deformations on the $\ensuremath{\alpha}$-decay half-lives of the deformed medium, heavy, and superheavy nuclei for $106\ensuremath{\leqslant}A\ensuremath{\leqslant}273$ and $52\ensuremath{\leqslant}Z\ensuremath{\leqslant}110$ from the ground state to ground state $\ensuremath{\alpha}$ transitions within the framework of the Wentzel--Kramers--Brillouin (WKB) method by considering the Bohr-Sommerfeld quantization condition. The deformed Wood-Saxon form in a phenomenological way and the deformed Coulomb potential in a special form have been used in order to take into account all deformation effects in the calculations and the preformation of the parent nuclei have also been regarded. Calculations have been conducted for the spherical nuclei in order to present clearly the effects of the deformations on the half-lives. We point out that all considerations have improved the results and very good agreement has been obtained with experimental data.

Traveling Wave Reactor and Condition of Existence of Nuclear Burning Soliton-Like Wave in Neutron-Multiplying Media

Physical fundamentals of traveling wave reactor are considered. We show that the condition of existence of nuclear burning soliton-like wave in a neutron-multiplying medium is determined in general by two conditions. The first condition (necessary) is determined by relationship between the equilibrium concentration and critical concentration of active (fissionable) isotope that is a consequence of the Bohr–Sommerfeld quantization condition. The second condition (sufficient) is set by the so-called Wigner quantum statistics, or more accurately, by a statistics of the Gaussian simplectic ensembles with respect to the parameter that describes the squared width of burning wave front of nuclear fuel active component.

The semiclassical limit of eigenfunctions of the Schrödinger equation and the Bohr–Sommerfeld quantization condition, revisited

The semiclassical limit of eigenfunctions of the Schrödinger equation and the Bohr–Sommerfeld quantization condition, revisited

Multi-instantons and exact results IV: Path integral formalism

This is the fourth paper in a series devoted to the large-order properties of anharmonic oscillators. We attempt to draw a connection of anharmonic oscillators to field theory, by investigating the partition function in the path integral representation around both the Gaussian saddle point, which determines the perturbative expansion of the eigenvalues, as well as the nontrivial instanton saddle point. The value of the classical action at the saddle point is the instanton action which determines the large-order properties of perturbation theory by a dispersion relation. In order to treat the perturbations about the instanton, one has to take into account the continuous symmetries broken by the instanton solution because they lead to zero-modes of the fluctuation operator of the instanton configuration. The problem is solved by changing variables in the path integral, taking the instanton parameters as integration variables (collective coordinates). The functional determinant (Faddeev–Popov determinant) of the change of variables implies nontrivial modifications of the one-loop and higher-loop corrections about the instanton configuration. These are evaluated and compared to exact WKB calculations. A specific cancellation mechanism for the first perturbation about the instanton, which has been conjectured for the sextic oscillator based on a nonperturbative generalized Bohr–Sommerfeld quantization condition, is verified by an analytic Feynman diagram calculation.

Poles of intégrale tritronquée and anharmonic oscillators. Asymptotic localization from WKB analysis

Poles of intégrale tritronquée are in bijection with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We show that the poles are well approximated by solutions of a pair of Bohr–Sommerfeld quantization conditions (the Bohr–Sommerfeld–Boutroux (B–S–B) system): the distance between a pole and the corresponding solution of the B–S–B system vanishes asymptotically.

Momentum-resolved Landau-level spectroscopy of Dirac surface state inBi2Se3

We investigate Dirac fermions on the surface of the topological insulator Bi2Se3 using scanning tunneling spectroscopy. Landau levels (LLs) are observed in the tunneling spectra in a magnetic field. In contrast to LLs of conventional electrons, a field independent LL appears at the Dirac point, which is a hallmark of Dirac fermions. A scaling analysis of LLs based on the Bohr-Sommerfeld quantization condition allowed us to determine the dispersion of the surface band. Near the Fermi energy, fine peaks mixed with LLs appear in the spectra, which may be responsible for the anomalous magneto-fingerprint effect [J. G. Checkelsky et al., Phys. Rev. Lett. 103, 246601 (2009)].

Wentzel-Kramers-Brillouin approach and quantum corrections to classical dynamics in the Josephson problem

We apply a many-body Wentzel-Kramers-Brillouin (WKB) approach to determine the leading quantum corrections to the semiclassical dynamics of the Josephson model, describing interacting bosons able to tunnel between two localized states. The semiclassical dynamics is known to divide between regular oscillations and self-trapped oscillations where the sign of the imbalance remains fixed. In both cases, the WKB wave functions are matched to Airy functions, yielding a modified Bohr-Sommerfeld quantization condition. At the critical energy dividing normal and self-trapped oscillations, the WKB wave functions should instead be matched to parabolic cylinder functions, leading to a quantization formula that is not just the Bohr-Sommerfeld formula, and recovering the known logarithmic quantum break times at this energy. This work thus provides another illustration of the usefulness of the WKB approach in certain many-body problems.

Poles of tritronquée solution to the Painlevé I equation and cubic anharmonic oscillator

The distribution of poles of zero-parameter solution to Painleve I, specified by P. Boutroux as integrale tritronquee, is studied in the complex plane. This solution has regular asymptotics −\( \sqrt {{z \mathord{\left/ {\vphantom {z 6}} \right. \kern-\nulldelimiterspace} 6}} \) + O(1) as z → ∞, | arg z| 4π/5 it is a meromorphic function with regular asymptotic distribution of poles at infinity. This fact together with numeric simulations for |z| < const allowed B. Dubrovin to make a conjecture that all poles of the integrale tritronquee belong to this sector. As a step to prove this conjecture, we study the Riemann-Hilbert (RH) problem related to the specified solution of the Painleve I equation. It is “undressed” to a similar RH problem for the Schrodinger equation with cubic potential. The latter determines all coordinates of poles for the integrale tritronquee via a Bohr-Sommerfeld quantization conditions.

Calculation of the characteristic functions of anharmonic oscillators

The energy levels of quantum systems are determined by quantization conditions. For one-dimensional anharmonic oscillators, one can transform the Schrödinger equation into a Riccati form, i.e., in terms of the logarithmic derivative of the wave function. A perturbative expansion of the logarithmic derivative of the wave function can easily be obtained. The Bohr–Sommerfeld quantization condition can be expressed in terms of a contour integral around the poles of the logarithmic derivative. Its functional form is B m ( E , g ) = n + 1 2 , where B is a characteristic function of the anharmonic oscillator of degree m, E is the resonance energy, and g is the coupling constant. A recursive scheme can be devised which facilitates the evaluation of higher-order Wentzel–Kramers–Brioullin (WKB) approximants. The WKB expansion of the logarithmic derivative of the wave function has a cut in the tunneling region. The contour integral about the tunneling region yields the instanton action plus corrections, summarized in a second characteristic function A m ( E , g ) . The evaluation of A m ( E , g ) by the method of asymptotic matching is discussed for the case of the cubic oscillator of degree m = 3 .

Systematics ofα-decay half-lives around shell closures

We present a systematic calculation of $\ensuremath{\alpha}$-decay half-lives of even-even heavy and superheavy nuclei in the framework of the preformed $\ensuremath{\alpha}$ model. The microscopic $\ensuremath{\alpha}$-daughter nuclear interaction potential is calculated by double-folding the density distributions of both $\ensuremath{\alpha}$ and daughter nuclei with a realistic effective Michigan three-Yukawa nucleon-nucleon interaction, and the microscopic Coulomb potential is calculated by folding the charge density distributions of the two interacting nuclei. The half-lives are found to be sensitive to the density dependence of the nucleon-nucleon interaction and the implementation of the Bohr-Sommerfeld quantization condition inherent in the Wentzel-Kramers-Brillouin approach. The $\ensuremath{\alpha}$-decay half-lives obtained agree reasonably well with the available experimental data. Moreover, the study has been extended to the newly observed superheavy nuclei. The interplay of closed-shell effects in $\ensuremath{\alpha}$-decay calculations is investigated. The $\ensuremath{\alpha}$-decay calculations give the closed-shell effects of known spherical magicities, $Z=82$ and $N=126$, and further predict enhanced stabilities at $N=152,162$, and $184$ for $Z=100,108$, and $114$, owing to the stability of parent nuclei against $\ensuremath{\alpha}$ decays. It is worth noting that the aim of this work is not only to reproduce the experimental data better, but also to extend our understanding of $\ensuremath{\alpha}$-decay half-lives around shell closures.

ON THE RIEMANN HYPOTHESIS, AREA QUANTIZATION, DIRAC OPERATORS, MODULARITY, AND RENORMALIZATION GROUP

Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert–Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line: sn = ½ + iρn. A detailed analysis of a one-dimensional Dirac-like operator with a potential V(x) is given that reproduces the spectrum of energy levels En = ρn, when the boundary conditions ΨE (x = -∞) = ± ΨE (x = +∞) are imposed. Such potential V(x) is derived implicitly from the relation [Formula: see text], where the functional form of [Formula: see text] is given by the full-fledged Riemann–von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the [Formula: see text] terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr–Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large [Formula: see text] has a one-to-one correspondence with the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the renormalization group program.

Exoticαdecays around theN=126magic shell

We investigate the $\ensuremath{\alpha}$-decay half-lives of the exotic $N=125,126,127$ isotones by the generalized density-dependent cluster model (GDDCM) in combination with the microscopic two-level model. The decay widths are calculated using the overlap integral of the quasibound state wave function, the scattering state wave function, and the difference of potentials, instead of using the simple semiclassical WKB method along with the Bohr-Sommerfeld quantization condition. The $\ensuremath{\alpha}$-preformation factors are evaluated by the $Z$-dependent formula based on the two-level model, where the closed-shell effect is included. The calculated half-lives of $\ensuremath{\alpha}$ transitions to both ground states and excited states are found to be in good agreement with the experimental data.

Quantization of black hole entropy from quasinormal modes

Using the new physical interpretation of quasinormal modes proposed by Maggiore, we calculate the quantum spectra of entropy for various types of non-rotating black holes with no charge. The spectrum is obtained by imposing Bohr-Sommerfeld quantization condition to the adiabatic invariant quantity. We conjecture that the spacing of entropy spectrum is equidistant and is independent of the dimension of spacetime. However, the spacing of area spectrum depends on gravity theory. In Einstein's gravity, it is equally spaced, otherwise it is non-equidistant.

Semiclassical resonances for a two-level Schrödinger operator with a conical intersection

We study the resonant set of a two-level Schrodinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr-Sommerfeld quantization condition and an asymptotic description of the set of resonances as a distorted lattice.

Semiclassics for superheavy elements

A semiclassical treatment for the fusion process near the Coulomb barrier is carried out. The interaction barrier is calculated by using the asymmetric two centre shell model and it fits nicely to a quartic double barrier. The generalized Bohr–Sommerfeld quantization condition for quartic oscillator gives an eigenvalue spectrum. The energy level in that spectrum, just below the continuum domain, refers to the maximum excitation energy E max ∗ carried by the resulting compound nucleus and fall fairly well within the observed limit. The shell structure of the colliding partners plays an important role in fixing the E max ∗ . Further, the phenomenon of mass transfer between the reaction partners is studied by solving the time dependent Schrodinger equation in η degree of freedom. These results confirm that a vigorous mass transfer occurs in case of doubly magic reaction partners and the resulting compound nucleus is formed with lesser energy. Calculations are made for superheavy elements 256 102No and 258 104Rf.

High-energy localized eigenstates of an electronic resonator in a magnetic field

We present a semiclassical analysis of the high-energy eigenstates of an electron inside a closed resonator. An asymptotic method of the construction of the energy spectrum and eigenfunctions, localized in the small neighborhood of a periodic orbit, is developed in the presence of a homogeneous magnetic field and arbitrary scalar potential. The isolated periodic orbit is confined between two interfaces which could be planar, concave or even convex. Such a system represents a quantum electronic resonator, an analog of the well-known high-frequency optical or acoustic resonator with eigenmodes called ‘bouncing ball vibrations’. The first step in the asymptotic analysis involves constructing a solitary localized asymptotic solution to the Schrödinger equation (electronic Gaussian beam—wavepackage). Then, the stability of a closed continuous family of periodic trajectories confined between two reflecting surfaces of the resonator boundary was studied. The asymptotics of the eigenfunctions were constructed as a superposition of two electronic Gaussian beams propagating in opposite directions between two reflecting points of the periodic orbits. The asymptotics of the energy spectrum are obtained by the generalized Bohr–Sommerfeld quantization condition derived as a requirement for the eigenfunction asymptotics to be periodic. For one class of periodic orbits, localized eigenstates were computed numerically by the finite element method using FEMLAB and proved to be in a very good agreement with those computed semiclassically.

Critical view of WKB decay widths

A detailed comparison of the expressions for the decay widths obtained within the semiclassical WKB approximation using different approaches to the tunneling problem is performed. The differences between the available improved formulas for tunneling near the top and the bottom of the barrier are investigated. Though the simple WKB method gives the right order of magnitude of the decay widths, a small number of parameters are often fitted. The need to perform the fitting procedure remaining consistently within the WKB framework is emphasized in the context of the fission model based calculations. Calculations for the decay widths of some recently found superheavy nuclei using microscopic alpha-nucleus potentials are presented to demonstrate the importance of a consistent WKB calculation. The half-lives are found to be sensitive to the density dependence of the nucleon-nucleon interaction and the implementation of the Bohr-Sommerfeld quantization condition inherent in the WKB approach.

Quantum dot potentials: Symanzik scaling, resurgent expansions, and quantum dynamics

This article is concerned with a special class of the ``double-well-like'' potentials that occur naturally in the analysis of finite quantum systems. Special attention is paid, in particular, to the so-called Fokker-Planck potential, which has a particular property: the perturbation series for the ground-state energy vanishes to all orders in the coupling parameter, but the actual ground-state energy is positive and dominated by instanton configurations of the form exp(-a/g), where a is the instanton action. The instanton effects are most naturally taken into account within the modified Bohr-Sommerfeld quantization conditions whose expansion leads to the generalized perturbative expansions (so-called resurgent expansions) for the energy values of the Fokker-Planck potential. Until now, these resurgent expansions have been mainly applied for small values of coupling parameter g, while much less attention has been paid to the strong-coupling regime. In this contribution, we compare the energy values, obtained by directly resumming generalized Bohr-Sommerfeld quantization conditions, to the strong-coupling expansion, for which we determine the first few expansion coefficients in powers of g^(-2/3). Detailed calculations are performed for a wide range of coupling parameters g and indicate a considerable overlap between the regions of validity of the weak-coupling resurgent series and of the strong-coupling expansion. Apart from the analysis of the energy spectrum of the Fokker-Planck Hamiltonian, we also briefly discuss the computation of its eigenfunctions. These eigenfunctions may be utilized for the numerical integration of the (single-particle) time-dependent Schroedinger equation and, hence, for studying the dynamical evolution of the wavepackets in the double-well-like potentials.

Semiclassical description of a sixth-order quadrupole boson Hamiltonian

A sixth-order quadrupole boson Hamiltonian is treated through a time dependent variational principle approach choosing as trial function a coherent state with respect to zeroth b 0 † and second b 2 † + b −2 † components of the quadrupole bosons. The coefficients involved in the model Hamiltonian are chosen so that the classical effective potential energy term has two distinct minima. The classical system is described by two degrees of freedom and has two constants of motion. The equation of motion for the radial coordinate is analytically solved and the resulting trajectories are extensively studied. One distinguishes three energy regions exhibiting different types of trajectories. When one passes from the region characterized by two wells to the region of energies higher than the maximum value of the effective potential the trajectories period exhibits a singularity which reflects in fact a phase transition. The classical trajectories are quantized by a constraint of the integral action similar to the well known Bohr–Sommerfeld quantization condition. The semiclassical spectra corresponding to the two potential wells have specific properties. The tunneling process through the potential barrier is also studied. It is a remarkable fact that the transmission coefficients exhibit jumps in magnitude when the intrinsic momentum acquires certain values. It is an open question whether such discontinuities in the transmission coefficients show up also when the final states are not bound. Semiclassical energy levels are compared also with the exact eigenvalues of the initial Hamiltonian, obtained through a diagonalization procedure. The present approach suggests that the energy levels in the laboratory frame can be written as a sum of energy associated to the intrinsic degrees of freedom and a rotational term obtainable through a projection method. This is numerically confirmed for the ground band.

One-dimensional Coulomb-like problem in deformed space with minimal length

Spectrum and eigenfunctions in the momentum representation for 1D Coulomb potential with deformed Heisenberg algebra leading to minimal length are found exactly. It is shown that correction due to the deformation is proportional to square root of the deformation parameter. We obtain the same spectrum using Bohr-Sommerfeld quantization condition.