Third-order Integrator Research Articles

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.

Use of the WKB Method for Obtaining Energy Eigenvalues

A simple, general method is derived for evaluating the second- and third-order WKB energy integrals by rewriting the integrals having nonintegrable singularities in terms of derivatives, with respect to the energy, of integrals having integrable singularities. As an example, it is shown that the higher-order WKB integrals vanish for the one-dimensional linear harmonic oscillator. A calculation of some eigenvalues using this method is made for potentials of the form V(x)=λx2ν and the results are compared to the ``exact'' results obtained from a numerical integration of the Schrödinger equation. It is observed that inclusion of the third-order integral improves the accuracy of WKB eigenvalues.