Bohr-Sommerfeld Quantization Condition Research Articles

Adiabatic Bohr-Sommerfeld calculations for the hydrogenic Stark effect

A general prescription is presented for the semiclassical calculation of quantized energy levels of a class of parametric quantum systems. Starting with the Bohr-Sommerfeld quantization conditions of an exactly solvable system in parabolic coordinates, the calculation evolves the classical equations of motion while adiabatically introducing a nonzero parameter. We present accurate results for the quadratic Stark effect in addition to the linear Stark effect, an improvement over previous semiclassical studies.

Instantons in quantum mechanics and resurgent expansions

Certain quantum mechanical potentials give rise to a vanishing perturbation series for at least one energy level (which as we here assume is the ground state), but the true ground-state energy is positive. We show here that in a typical case, the eigenvalue may be expressed in terms of a generalized perturbative expansion (resurgent expansion). Modified Bohr–Sommerfeld quantization conditions lead to generalized perturbative expansions which may be expressed in terms of nonanalytic factors of the form exp(−a/g), where a>0 is the instanton action, and power series in the coupling g, as well as logarithmic factors. The ground-state energy, for the specific Hamiltonians, is shown to be dominated by instanton effects, and we provide numerical evidence for the validity of the related conjectures.

Semiclassical approach to Regge poles trajectories calculations for nonsingular potentials: Thomas–Fermi type

A simple semiclassical approach, based on the investigation of anti-Stokes line topology, is presented for calculating Regge poles for nonsingular (Thomas–Fermi type) potentials, namely potentials with singularities at the origin weaker than order −2. The anti-Stokes lines for Thomas–Fermi potentials have a more complicated structure than those of singular potentials and require careful application of complex analysis. The explicit solution of the Bohr–Sommerfeld quantization condition is used to obtain approximate Regge poles. We introduce and employ three hypotheses to obtain several terms of the Regge pole approximation.Please note that the pdf of this article was replaced with a corrected version on 29 June 2004. Minor changes have been made to pages 6950–6953.

Bohr-Sommerfeld quantization condition for the Gross-Pitaevskii equation.

The discrete spectrum of the nonlinear eigenvalue problem associated to the one-dimensional Gross-Pitaevskii equation with a smooth potential is studied in the quasiclassical limit. We particularly focus on the corrections to the Bohr-Sommerfeld quantization rule for the excited energy levels due to the nonlinearity. Explicit predictions are obtained analytically for these corrections and are supported by numerical computations.

Noncompact Heisenberg spin magnets from high-energy QCD III. Quasiclassical approach

The exact solution of the noncompact SL(2, C) Heisenberg spin magnet reveals a hidden symmetry of the energy spectrum. To understand its origin, we solve the spectral problem for the model within quasiclassical approach. In this approach, the integrals of motion satisfy the Bohr–Sommerfeld quantization conditions imposed on the orbits of classical motion. In the representation of the separated coordinates, the latter wrap around a Riemann surface defined by the spectral curve of the model. A novel feature of the obtained quantization conditions is that they involve both the α- and β-periods of the action differential on the Riemann surface, thus allowing us to find their solutions by exploring the full modular group of the spectral curve. We demonstrate that the quasiclassical energy spectrum is in a good agreement with the exact results.

Semiclassical approach for calculating Regge-pole trajectories for singular potentials

A simple semiclassical approach, based on the investigation of the anti-Stokes line topology is presented for calculating Regge-poles trajectories for singular potentials, viz. potentials more singular than ${r}^{\ensuremath{-}2}$ at the origin. It uses the explicit solution of the Bohr-Sommerfeld quantization condition with the proviso that the positions of two turning points of the effective potential responsible for the Regge poles be relatively close together. We also demonstrate that due to this closeness the Regge trajectories asymptotically approach parallel equidistant straight lines with a slope of $\mathrm{cot}(\ensuremath{\varphi}/m),$ m being the power and $\ensuremath{\varphi}$ the argument of the coefficient of the potential. Illustrative results are presented for the polarization and Lennard-Jones potentials.

The relativistic hydrogen-like atom in 2+1 dimensions

In (2+1)-dimensional space the exact solutions of the Dirac equation with cutoff Coulomb potential that takes into account the nucleus size is obtained in the under critical region of the discrete spectrum. Solutions of the Dirac equation in a strong external field arc obtained in the WKB approximation. A field is considered strong if the electron binding energy exceeds 2mc 2 and the descrete spectrum levels may be lowered into the lower continuum. The wave functions in the classically allowed and forbidden regions are found and the conditions for matching them on transition through the turning point are obtained. The WKB method is applied to the following problems: 1) generalization of the Bohr-Sommerfeld quantization conditions with allowance for relativistic effects and the spin in 2+1 dimensions: 2) finding energy and width of the quasistationary level in the lower continuum.

Exact resolution method for general 1D polynomial Schrödinger equation

The stationary 1D Schr\"odinger equation with a polynomial potential $V(q)$ of degree N is reduced to a system of exact quantization conditions of Bohr-Sommerfeld form. They arise from bilinear (Wronskian) functional relations pairing spectral determinants of (N+2) generically distinct operators, all the transforms of one quantum Hamiltonian under a cyclic group of complex scalings. The determinants' zeros define (N+2) semi-infinite chains of points in the complex spectral plane, and they encode the original quantum problem. Each chain can now be described by an exact quantization condition which constrains it in terms of its neighbors, resulting in closed equilibrium conditions for the global chain system; these are supplemented by the standard (Bohr-Sommerfeld) quantization conditions, which bind the infinite tail of each chain asymptotically. This reduced problem is then probed numerically for effective solvability upon test cases (mostly, symmetric quartic oscillators): we find that the iterative enforcement of all the quantization conditions generates discrete chain dynamics which appear to converge geometrically towards the correct eigenvalues/eigenfunctions. We conjecture that the exact quantization then acts by specifying reduced chain dynamics which can be stable (contractive) and thus determine the exact quantum data as their fixed point. (To date, this statement is verified only empirically and in a vicinity of purely quartic or sextic potentials $V(q)$.)

Quasiclassical approximation with the centrifugal potential excluded

We develop a modification of the WKB method (the modified quantization method, or MQM) for finding the radial wave functions. The method is based on excluding the centrifugal potential from the quasiclassical momentum and changing correspondingly the phase in the Bohr-Sommerfeld quantization condition. MQM is used to calculate the asymptotic coefficients at zero and at infinity. We use the examples of power-law and funnel potentials to show that MQM not only dramatically broadens the possibilities of studying the energy spectrum and the wave functions analytically but also ensures accuracy to within a few percent even when one calculates states with a radial quantum number n r ∼1, provided that the angular momentum l is not too large. We also briefly discuss the possibility of generalizing MQM to the relativistic case (the spinless Salpeter equation).

Ballistic transport and spin-orbit interaction of two-dimensional electrons on a cylindrical surface

The components of the ballistic magnetoconductance tensor of a two-dimensional electron gas placed on a cylindrical sector are calculated for various geometries. For a quasiclassical system a method is proposed for finding the conductance based only on the Bohr-Sommerfeld quantization condition and not requiring a knowledge of the matrix elements of the velocity. The effect of curvature of the surface on the spin-orbit interaction in a two-dimensional electron gas is investigated. As examples, the microwave absorption and longitudinal conductance of a hollow cylindrical wire are calculated, and also the conductance of a cylindrical sector. There are qualitative differences from planar systems, in particular the relative sign of the curvature and the spin-orbit coupling constant becomes important.

Semiclassical limits of the spectrum of Harper's equation.

Harper's equation, a model for magnetic field effects in lattices, can be analyzed using semiclassical methods when the commensurability parameter $\ensuremath{\beta}$ is small. We discuss an effective Hamiltonian ${\stackrel{^}{H}}_{\mathrm{eff}}$ describing a subset of the spectrum, for which the rational limit $\ensuremath{\beta}\ensuremath{\rightarrow}p/q$ is also a semiclassical limit; we give the first two terms of the expansion of ${\stackrel{^}{H}}_{\mathrm{eff}}$ in powers of $\ensuremath{\beta}\ensuremath{-}p/q$. We derive a Bohr-Sommerfeld quantization condition, involving a Berry phase correction, and an equation for the bandwidth when $\ensuremath{\beta}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}p}_{1}/{q}_{1}$ with ${q}_{1}$ large. We also discuss the dynamics of ${\stackrel{^}{H}}_{\mathrm{eff}}$ under infinitesimal gauge transformations of the rational Bloch states.

Semiclassical study of quantum scattering on the line

We study the well-known problem of 1-d quantum scattering by a potential barrier in the semiclassical limit. Using the so-called exact WKB method and semiclassical microlocal analysis techniques, we get a very precise and complete description of the scattering matrix, in particular when the energy is very close to a unique, quadratic maximum of the potential. In our one-dimensional setting, we also recover the Bohr-Sommerfeld quantization condition for the resonances generated by such a maximum.

Phase-integral approach to quantal two- and three-dimensional isotropic anharmonic oscillators.

A two- or three-dimensional quantal isotropic anharmonic oscillator is treated by means of the phase-integral method of Fr\"oman and Fr\"oman. The generalized Bohr-Sommerfeld quantization condition for the radial wave function is expressed in terms of complete elliptic integrals up to the fifth order of the phase-integral approximation. The quantization condition is solved numerically, and energy levels are obtained for various quantum numbers. Comprison with numerically exact results is also made.

Geometric phases in the asymptotic theory of coupled wave equations.

Traditional approaches to the asymptotic behavior of coupled wave equations have difficulties in the formulation of a consistent version of the Bohr-Sommerfeld quantization conditions. These difficulties can be circumvented by using the Weyl calculus to diagonalize the matrix of wave operators. In analyzing the diagonalized wave equations, geometric phases enter in an important way, especially in the development of Bohr-Sommerfeld quantization rules. It turns out that a version of Berry's phase is incorporated into the symplectic structure in the ray phase space, influencing the classical Hamiltonian orbits, the construction of solutions to the Hamiltonian-Jacobi equation, and the computation of action integrals. Noncanonical coordinates in the ray phase space are useful in carrying out these calculations and in making the construction of eigenvalues and wave functions manifestly gauge invariant.

An alternative formulation of the WKB approximation in quantum mechanics

An expansion of Schrodinger's theory in powers of h is presented. It has the property that the standard WKB results, together with the Bohr-Sommerfeld quantization condition are obtainable by a zeroth-order perturbation in h, rather than by a first-order one (in contrast with the usual expansion methods). It also makes the connection between the (classical) Hamilton-Jacobi equation and the WKB approximation more transparent.

State densities and ionization equilibrium of atoms in dense plasmas.

The semiclassical Bohr-Sommerfeld quantization condition is used to derive an approximate analytical expression for the state density of the hydrogen atom in a dense plasma. An ion-sphere model with an infinitely high potential wall is assumed. The expression leads to a universal curve that spans all values of the electron density. The curve is continuous and smooth over the entire energy range, starting from the hydrogenic state density for low-lying bound states and approaching the plane-wave state density in the high-energy limit of the continuum. The number of bound states is approximately proportional to the inverse of the square root of the electron density. Integration of the state density over the Boltzmann distribution of the electronic energy results in an ionization equilibrium relation, leading to modified Saha's equation. The correction factor for this modified equation is a function of both the electron temperature and the electron density, and is expressed as a universal function of the ion coupling parameter.

Wave packets, Maslov indices, and semiclassical quantization

The Bohr-Sommerfeld quantization condition, as refined by Keller and Maslov, reads I=(n + m/4)h, where I is the classical action, n is the quantum number, and where m is the Maslov index, an even integer. The occurrence of the integers n and m in this formula is a reflection of underlying topological features of semiclassical quantization. In particular, the work of Arnold and others has shown that m/2 is a winding number of closed curves on the classical symplectic group manifold, Sp(2N). Wave packets provide a simple and elegant means of establishing the connection between semiclassical quantization and the homotopy classes of Sp(2N), as well as a practical way of calculating Maslov indices in complex problems. Topological methods can also be used to derive general formulas for the Maslov indices of invariant tori in the classical phase space corresponding to resonant motion.

Improvement of the generalized quantal Bohr–Sommerfeld quantization condition

AbstractThe quantal quartic oscillator, characterized by two real and two complex conjugate transition points (simple zeros), is studied by means of the phase-integral method developed by Fröman and Fröman, and various quantization conditions are obtained. The main results, obtained in §3·3 and §4, are summarized below.A correction to the generalized Bohr–Sommerfeld quantization condition, due to the complex conjugate transition points, is obtained from estimates of the. F-matrix for a path passing above the complex conjugate transition points. This quantization condition is closely related to the quantization condition for a double-oscillator. The correction is shown to be rigorously valid when the distance between the real transition points is large compared to the distance between the complex conjugate ones, which may lie at an unspecified distance from each other.At sufficiently large distances from the cluster of four transition points the solutions of the Schrödinger equation can be characterized by the Stokes constants between anti-Stokes lines emerging from the cluster in six directions towards infinity. There are initially six non-trivial such Stokes constants in the problem under consideration, but, from the results in previous papers by N. Fröman and the present authors, these Stokes constants are linked by a number of algebraic relations, so that one or two Stokes constants suffice to describe the solutions far away from the cluster of transition points. The quantization condition is expressed in terms of these Stokes constants. Approximate values of the relevant Stokes constants are given in the three limiting cases when all four transition points coalesce (and the first-order phase-integral approximation is used), when the distance between the real transition points is sufficiently large (extreme double-oscillator), and in the Bohr-Sommerfeld limit.

The canonical description and Bohr-Sommerfeld quantization condition for the fractional quantum Hall effect system

In this note, by introducing the canonical description which is one dimensional in nature, we establish the Bohr-Sommerfeld quantization condition for the fractional quantum Hall effect system. The generic and model independent physical implications of this condition are explored. The guiding center representations of the N-body Schrödinger equation for the system is also derived.

The energy eigenvalues of hyperbolical potential functions

The energy eigenvalues E(υ) of the hyperbolical potential functions V(r)± = hcA{δ - σ(coth α(r - r 0))±1}2 were determined with a semiclassical procedure (the Bohr-Sommerfeld quantization condition) and a quantum-mechanical method (the Schrodinger equation). The resulting term values G(υ) = E(υ)/hc are different: G(υ) = D e - A{δσ(s + υ + 1/2)-1 - (s + υ + 1/2)}2 (for υ = 0, 1, 2, 3, …). D e is the spectroscopic dissociation energy in cm-1. s = σ for the semiclassical and s = (σ2 + ¼)1/2 for the quantum-mechanical solutions. The energy eigenvalues of V(r)± include all known energy functions which one obtains from exact quantum-mechanical or semi-classical solutions. Hence V(r)± is not only a better description for the potential energy of a molecular ‘vibration’ than the Morse function, it represents also perfectly intermolecular interactions and includes with the Kratzer energy eigenvalues the Rydberg terms of an electron in an atom. The ‘vibration’ of a two atomic molecule is discussed in space and in a ...