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Higher corrections to the WKB approximation

N. G. van Kampen

PhysicaJan 1, 196722 citations

Tunneling near the peaks of potential barriers: Consequences of higher-order Wentzel-Kramers-Brillouin corrections

Clifford M. Will, James Williams Guinn

We derive the quantum-mechanical tunneling transmission coefficient for energies near the peak of a general potential barrier, to fourth order in the Wentzel-Kramers-Brillouin (WKB) approximation, using a method based on an eighth-order Taylor expansion of the potential near the peak. The result agrees with contour-integral formulations of WKB tunneling. For the P\"oschl-Teller potential the WKB series converges rapidly to the known exact transmission coefficient. Higher-order WKB corrections may be important in attempts to determine the internuclear potential by inversion using low-energy fusion cross-section data.We derive the quantum-mechanical tunneling transmission coefficient for energies near the peak of a general potential barrier, to fourth order in the Wentzel-Kramers-Brillouin (WKB) approximation, using a method based on an eighth-order Taylor expansion of the potential near the peak. The result agrees with contour-integral formulations of WKB tunneling. For the P\"oschl-Teller potential the WKB series converges rapidly to the known exact transmission coefficient. Higher-order WKB corrections may be important in attempts to determine the internuclear potential by inversion using low-energy fusion cross-section data.

Physical Review AMay 1, 198823 citations

WKB Expansion for arbitrary Hamiltonians

Florentin Langouche, D. Roekaerts, E. Tirapegui

Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fieldsAug 1, 198111 citations

Accuracy of the WKB Approximation: The Case of General Quartic Potential

Marko Vranivcar, Marko Robnik

We analyse the accuracy of the approximate WKB quantization for the case of general one-dimensional quartic potential. In particular, we are interested in the validity of semiclassically predicted energy eigenvalues when approaching the limit E→∞, and in the accuracy of low lying energy levels below the potential barrier in the case of generally asymmetric double-well quartic potential. In the latter case, using the standard WKB quantization an unnatural localization of eigenstates due to the negligence of tunneling is implied and thus the validity of semiclassics is uncertain. In all computations the higher order corrections to the leading semiclassical approximation are included using the complex contour integration technique. We show that these corrections can improve accuracy of semiclassical approximation greatly by many orders of magnitude.We analyse the accuracy of the approximate WKB quantization for the case of general one-dimensional quartic potential. In particular, we are interested in the validity of semiclassically predicted energy eigenvalues when approaching the limit E→∞, and in the accuracy of low lying energy levels below the potential barrier in the case of generally asymmetric double-well quartic potential. In the latter case, using the standard WKB quantization an unnatural localization of eigenstates due to the negligence of tunneling is implied and thus the validity of semiclassics is uncertain. In all computations the higher order corrections to the leading semiclassical approximation are included using the complex contour integration technique. We show that these corrections can improve accuracy of semiclassical approximation greatly by many orders of magnitude.

arXiv:nlin/0003051Progress of Theoretical Physics SupplementJan 1, 20008 citations

The WKB Approximation Without Divergences

Aug 16, 20160 citations

Structural connections between two semiclassical approximations: The WKB and Wigner-Kirkwood approximations

T. A. Osborn, F. H. Molzahn

The mutual consistency and structural interconnection between the Wentzel-Kramers-Brillouin (WKB) and Wigner-Kirkwood (WK) semiclassical approximations is investigated for nonrelativistic N-particle systems, with mutual scalar interactions and coupling to an external time-varying electromagnetic field. The generalized WK expansion of the propagator 〈x\ensuremath{\Vert}U(t,s)\ensuremath{\Vert}y〉 is obtained from a large-mass expansion of the higher-order WKB approximation. Two techniques are described for computing the WK coefficient functions. One relies on a large-mass expansion of classical paths and the transport representation of the WKB approximation; the other is recursive in nature. For time-independent Hamiltonians H the standard WK expansion of the heat kernel 〈x\ensuremath{\Vert}${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$\ensuremath{\Vert}y〉 is recovered.The mutual consistency and structural interconnection between the Wentzel-Kramers-Brillouin (WKB) and Wigner-Kirkwood (WK) semiclassical approximations is investigated for nonrelativistic N-particle systems, with mutual scalar interactions and coupling to an external time-varying electromagnetic field. The generalized WK expansion of the propagator 〈x\ensuremath{\Vert}U(t,s)\ensuremath{\Vert}y〉 is obtained from a large-mass expansion of the higher-order WKB approximation. Two techniques are described for computing the WK coefficient functions. One relies on a large-mass expansion of classical paths and the transport representation of the WKB approximation; the other is recursive in nature. For time-independent Hamiltonians H the standard WK expansion of the heat kernel 〈x\ensuremath{\Vert}${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$\ensuremath{\Vert}y〉 is recovered.

Physical Review ASep 1, 198620 citations

WKB expansions with higher order corrections: The general case

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