A long-standing problem and major shortcoming of semiclassical (WKB) bound-state wave functions is their divergence at the classical turning points. Presently available regularization schemes are accurate but rather complicated. It is shown how finite wave functions can be simply obtained by reorganizing the WKB perturbation expansion in powers of ℏ and retaining appropriate higher-order terms. The wave functions obtained by this method are compared with exactly known ones for the harmonic-oscillator and Morse potentials.A long-standing problem and major shortcoming of semiclassical (WKB) bound-state wave functions is their divergence at the classical turning points. Presently available regularization schemes are accurate but rather complicated. It is shown how finite wave functions can be simply obtained by reorganizing the WKB perturbation expansion in powers of ℏ and retaining appropriate higher-order terms. The wave functions obtained by this method are compared with exactly known ones for the harmonic-oscillator and Morse potentials.

American Journal of PhysicsOct 1, 19919 citations

Potential Scattering in the Generalized WKB Approximation.

C.K. Chan

Jan 1, 19800 citations

Semiclassical analysis of the Efimov energy spectrum in the unitary limit

Rajat K. Bhaduri, M. Brack, M. V. N. Murthy

We demonstrate that the ( $s$ -wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cutoff radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wave functions, regularized near the classical turning point by Langer's generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest ( $n=0$ ) state, apart from a tiny shift of its zeros that remains constant for large $n$ .We demonstrate that the ($s$-wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cutoff radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wave functions, regularized near the classical turning point by Langer's generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest ($n=0$) state, apart from a tiny shift of its zeros that remains constant for large $n$.

arXiv:1103.1737Physical Review AJun 13, 20112 citations

Simple rule for normalizing WKB-type wave functions

Harvey C. Woodsum

American Journal of PhysicsJul 1, 19780 citations

A semiclassical analysis of the Efimov energy spectrum

Rajat K. Bhaduri, M. Brack, M. V. N. Murthy

We demonstrate that the (s-wave) geometric spectrum of the shallow Efimov energy levels is generated by the radial motion of a primitive periodic orbit of the corresponding classical system and its harmonics. The action of the primitive orbit is given by the logarithm of the energy. This is shown to be consistent with the leading term of an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized at the classical turning point using Langer’s connection formula, are practically indistinguishable from the exact wave functions even for the lowest (n = 0) state, apart from a tiny shift of its zeros that remains constant for large n.We demonstrate that the (s-wave) geometric spectrum of the shallow Efimov energy levels is generated by the radial motion of a primitive periodic orbit of the corresponding classical system and its harmonics. The action of the primitive orbit is given by the logarithm of the energy. This is shown to be consistent with the leading term of an inverse-squared radial potential with a lower cut-off radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wavefunctions, regularized at the classical turning point using Langer’s connection formula, are practically indistinguishable from the exact wave functions even for the lowest (n = 0) state, apart from a tiny shift of its zeros that remains constant for large n.

Mar 9, 20110 citations

We demonstrate that the (

s

-wave) geometric spectrum of the Efimov energy levels in the unitary limit is generated by the radial motion of a primitive periodic orbit (and its harmonics) of the corresponding classical system. The action of the primitive orbit depends logarithmically on the energy. It is shown to be consistent with an inverse-squared radial potential with a lower cutoff radius. The lowest-order WKB quantization, including the Langer correction, is shown to reproduce the geometric scaling of the energy spectrum. The (WKB) mean-squared radii of the Efimov states scale geometrically like the inverse of their energies. The WKB wave functions, regularized near the classical turning point by Langer's generalized connection formula, are practically indistinguishable from the exact wave functions even for the lowest (

n=0

) state, apart from a tiny shift of its zeros that remains constant for large

n