WKB Quantization Condition Research Articles

Classical orbits in power-law potentials

The motion of bodies in power-law potentials of the form V(r)=λrα has been of interest ever since the time of Newton and Hooke. Aspects of the relation between powers α and ᾱ, where (α+2)(ᾱ+2)=4, are derived for classical motion and the relation to the quantum-mechanical problem is given. An improvement on a previous expression for the WKB quantization condition for nonzero orbital angular momenta is obtained. Relations with previous treatments, such as those of Newton, Bertrand, Bohlin, Fauré, and Arnold, are noted, and a brief survey of the literature on the problem over more than three centuries is given.

SPECTRAL INVERSE PROBLEM IN SUPERSYMMETRIC QUANTUM MECHANICS

The supersymmetric WKB quantization condition is used to study the so-called spectral inverse problem. Wavefunctions for the harmonic oscillator and hydrogen atom are obtained from the knowledge of their bound-state energy spectra. The analysis presented is based essentially on a repackaging of the conventional theory of integral equations.

Semiclassical quantization of integrable systems of few interacting anyons in a strong magnetic field.

We present a semiclassical theory of charged interacting anyons in a strong magnetic field. We derive the appropriate generalization of the WKB quantization conditions and determine the corresponding wave functions for nonseparable integrable anyonic systems. This theory is applied to a system of two interacting anyons, two interacting anyons in the presence of an impurity, and three interacting anyons. We calculate the dependence of the semiclassical energy levels on the statistical parameter and find regions in which this dependence follows very different patterns. The semiclassical treatment allows us to find the correlation between these patterns and the changes in the character of the classical motion of the system. We also test the accuracy of the mean-field approximation for low- and high-energy states of the three-anyon system.

An approximate analytic solution of the spectral inverse problem

An important problem of nonrelativistic quantum theory is: Given a bound-state energy spectrum, determine the space dependence of a potential function that will produce that spectrum. This is sometimes called the ‘‘spectral inverse’’ problem. Here, by use of the first-order WKB energy quantization condition, an approximate analytic solution of this problem is given for the case of one-dimensional symmetric oscillators and for a class of radial oscillators. This is done by reformulating the WKB quantization condition as an Abel integral equation that relates the quantum number to the potential’s space dependence. The resultant integral equation can be solved exactly. The harmonic oscillator and the hydrogen atom problems are used to demonstrate the method. The solution for the case of one-dimensional asymmetric oscillators is also briefly discussed and the Morse oscillator is used as an example. Finally, the history of the spectral inverse problem is reviewed and the conditions necessary for its exact solution are discussed. In particular, the conditions required for the inversion process to be unique are considered.

Exactness of supersymmetric WKB spectra for shape-invariant potentials

Potentials which have the property of “shape-invariance” are known to be exactly solvable. For all these potentials, it is proved that the supersymmetric WKB (SWKB) quantization condition (to leading order in ħ) also has the nice property of reproducing the exact bound-state spectra. These results are not modified even if the next higher-order correction in ħ is taken into account.

How good is the supersymmetry-inspired WKB quantization condition?

I show that unlike the standard lowest order WKB, the supersymmetry-inspired WKB quantization condition gives correct n-dependence for all the energy eigenvalues (except ground state) of the potential V( x) = A 2 x 6 + 3 Ax 2. I also display a number of other potentials for which SWKB is then shown to give exact eigenvalues. Finally, I conjecture that for the class of models given by V( x) = A 2 x 4 d+2 + (2 d + 1) Ax 2 d ( d = 0,1,2, …), the exact energy eigenvalues obey the SWKB-predicted energy dependence E n ∞ ( n + 1) (2 d+1)/( d+1) .

Semiclassical statistical mechanics of hard wall potentials via the transformed potential approach

A nonperturbative technique for treating the semiclassical statistical mechanics of hard wall systems based on the WKB quantization condition modified for a noninfinite interval is proposed. Application to some simple cases shows that the method is an improvement over a previous nonperturbative approach.

Semiclassical quantization of the relativistic Kepler problem

Semiclassical quantization conditions are applied to a variety of relativistic formulations of the problem of two spinless particles bound by a vector or scalar force. The WKB quantization conditions are found to yield the correct spectra whenever the equations of motion are separable. Miller's new quantum condition, making use of periodic orbits, is applied to almost circular orbits. It provides an approximate method of quantization for almost circular orbits in the action-at-a-distance formulation of relativistic charged particles interacting electromagnetically.

Second-order semiclassical calculations for diatomic molecules

The second-order WKB quantization condition is used to derive expressions for the second-order contributions to the vibrational energy levels and the rotational and centrifugal constants of a diatomic molecule. The second-order corrections to the RKR (Rydberg-Klein-Rees) turning points are also calculated. These second-order quantities are related to the quantities calculated from Kaiser's modified first-order quantization condition. Numerical calculations for the ground electronic state of carbon monoxide are compared with quantum-mechanical values obtained by numerical solution of the Schrödinger equation. In general, the second-order semiclassical values are in good agreement with the quantum-mechanical values, and can be computed by the present methods in a small fraction of the time.

RKR potentials and semiclassical centrifugal constants of diatomic molecules

Semiclassical formulas are obtained for the centrifugal constants D v and H v of a diatomic molecule by differentiation of the WKB quantization condition, and a numerical procedure is presented for the computation of these constants from the RKR turning points. A similar procedure also provides a simple and rapid method for the determination of the RKR turning points from the vibrational term values and rotational constants. Applications to carbon monoxide and iodine are discussed.

Use of the WKB Quantization Condition for Calculating Expectation Values

A systematic technique is developed for the calculation of expectation values for 1-dimensional problems by a direct use of the WKB quantization rule including the higher-order integrals without the introduction of wavefunctions. As an example of the technique we calculate the moments 〈rp〉 for the Coulomb potential for 2 ≥ p ≥ −3, for all bound states of the system. Our results are exact except for p = −3, in which case the inclusion of the higher-order integrals substantially improves the accuracy of the calculation.

Nonuniqueness of the Energy Correction in Application of the WKB Approximation to Radial Problems

An infinite set of transformations is given that convert a radial energy-eigenvalue problem into a 1-dimensional problem. Langer's transformations are included in this set. The transformations introduce no spurious, real-axis singularities into the effective potential, and yield wavefunctions that meet 1-dimensional boundary conditions and are asymptotically satisfactory for virtually all power law potentials. The first-order radial WKB quantization condition is transformation dependent through two parameters. It may be distinct from the Langer-Kemble condition obtained by substituting (l + ½)2 for l(l + 1) in the effective potential. Here, l is the rotational quantum number. For the vibrating rotator, a rough test of the usefulness of the (l + ½)2 substitution is described; flexibility advantages provided by the new transformations are pointed out.

Exact Quantization Conditions. II

By employing coordinate transformations, a generalized WKB quantization condition is derived which includes a modified WKB quantization rule and the related higher-order integrals. It is observed that a necessary condition for the first-order integral to give an exact quantization rule is the vanishing of the higher-order integrals. When this condition is satisfied, the resulting quantization condition is of the form of all previously known exact-quantization rules. The higher-order integrals are shown to vanish for certain cases of interest. An error in Paper I [C. Rosenzweig and J. B. Krieger, J. Math. Phys. 9, 849 (1968)] is noted and an examination of the higher-order integrals shows that a proposed quantization rule given there is not exact.

Application of a Higher-Order WKB Approximation to Radial Problems

The radial generalization of Dunham's one-dimensional WKB quantization condition, including second- and third-order corrections is derived using the Langer transformation. It is found that, although the first-order integral can be obtained from Dunham's results by substituting ${(l=\frac{1}{2})}^{2}$ for $l(l+1)$ in the effective potential, there is no choice of effective potential that leads to the correct second- and third-order integrals. It is suggested that all previous eigenvalue calculations using higher-order WKB approximations for the radial case should be reinvestigated. It is shown that the second- and third-order integrals identically vanish for the hydrogen atom and the three-dimensional harmonic oscillator, as expected.