We study the WKB method away from the short-wave limit. Incorporating the correct phase loss due to reflection at a classical turning point, in place of the usual choice $\ensuremath{\pi}/2$, can greatly improve the accuracy of the WKB wave function in the classically allowed region. For a repulsive ${1/x}^{2}$ potential this leads to far more accurate WKB wave functions that the usual Langer modification of the potential.We study the WKB method away from the short-wave limit. Incorporating the correct phase loss due to reflection at a classical turning point, in place of the usual choice $\ensuremath{\pi}/2$, can greatly improve the accuracy of the WKB wave function in the classically allowed region. For a repulsive ${1/x}^{2}$ potential this leads to far more accurate WKB wave functions that the usual Langer modification of the potential.

Physical Review LettersJun 24, 199669 citations

Classical limit

Carlo Rovelli C, Francesca Vidotto

Cambridge University Press eBooksNov 13, 20140 citations

Nonintegral Maslov indices

Hanno Friedrich, Johannes Trost

The phase loss of a wave reflected by a smooth potential generally varies continuously from \ensuremath{\pi} in the long-wave limit to \ensuremath{\pi}/2 in the limit of short waves. Incorporating the corresponding nonintegral multiples of \ensuremath{\pi}/2 as nonintegral Maslov indices in the formulation of the WKB approximation leads to a substantial improvement of accuracy when the conditions for applicability of the WKB method are violated only near the classical turning points. We demonstrate the efficacy of using nonintegral Maslov indices for a Woods-Saxon potential and a repulsive 1/ ${\mathit{x}}^{2}$ potential. The nonintegral Maslov index for a given 1/ ${\mathit{x}}^{2}$ potential yields far more accurate wave functions than the conventional Langer modification of the potential in conjunction with phase loss \ensuremath{\pi}/2. The energy spectrum of the radial harmonic oscillator (including the centrifugal potential), which is reproduced exactly by the standard WKB method with the Langer modification, is also reproduced exactly without the Langer modification when the nonintegral Maslov index is used. We suggest a method for approximately calculating the nonintegral Maslov index near the long-wave limit from the decaying WKB wave function in the classically forbidden region. \textcopyright{} 1996 The American Physical Society.The phase loss of a wave reflected by a smooth potential generally varies continuously from \ensuremath{\pi} in the long-wave limit to \ensuremath{\pi}/2 in the limit of short waves. Incorporating the corresponding nonintegral multiples of \ensuremath{\pi}/2 as nonintegral Maslov indices in the formulation of the WKB approximation leads to a substantial improvement of accuracy when the conditions for applicability of the WKB method are violated only near the classical turning points. We demonstrate the efficacy of using nonintegral Maslov indices for a Woods-Saxon potential and a repulsive 1/${\mathit{x}}^{2}$ potential. The nonintegral Maslov index for a given 1/${\mathit{x}}^{2}$ potential yields far more accurate wave functions than the conventional Langer modification of the potential in conjunction with phase loss \ensuremath{\pi}/2. The energy spectrum of the radial harmonic oscillator (including the centrifugal potential), which is reproduced exactly by the standard WKB method with the Langer modification, is also reproduced exactly without the Langer modification when the nonintegral Maslov index is used. We suggest a method for approximately calculating the nonintegral Maslov index near the long-wave limit from the decaying WKB wave function in the classically forbidden region. \textcopyright{} 1996 The American Physical Society.

Physical Review AAug 1, 199671 citations

The Classical Limit

WORLD SCIENTIFIC eBooksSep 8, 20140 citations

{1/x}^{2}

potential this leads to far more accurate WKB wave functions that the usual Langer modification of the potential.

{\mathit{x}}^{2}

potential. The nonintegral Maslov index for a given 1/

{\mathit{x}}^{2}

potential yields far more accurate wave functions than the conventional Langer modification of the potential in conjunction with phase loss \ensuremath{\pi}/2. The energy spectrum of the radial harmonic oscillator (including the centrifugal potential), which is reproduced exactly by the standard WKB method with the Langer modification, is also reproduced exactly without the Langer modification when the nonintegral Maslov index is used. We suggest a method for approximately calculating the nonintegral Maslov index near the long-wave limit from the decaying WKB wave function in the classically forbidden region. \textcopyright{} 1996 The American Physical Society.