Convergent WKB series

First, two conditions are specified for the lowest order Wentzel-Kramers-Brillouin quantization rule to yield exact results. These rules are related to the periodic orbit decomposition of the quantum density of states. This approach is then applied to supersymmetric quantum mechanics. It leads to a new derivation of the result that shapeinvariant potentials give exact results when the classical action is calculated with the square of the super potential, but without the Maslov index or the Langer correction. PACS Nos.: 03.65.Sq, 12.60.Jv Résumé : Nous précisons d'abord deux conditions pour que la quantification WKB à l'ordre le plus bas donne des résultats exacts. Ces règles sont reliées à la décomposition en orbite périodiques de la densité quantique d'états. Cette approche est alors appliquée à la mécanique quantique super-symétrique. Cela donne une nouvelle dérivation du résultat que les potentiels invariants de forme donnent des résultats exacts lorsque l'action classique est calculée avec le carré du super-potentiel, mais sans indice de Maslov ou correction de Langer.[Traduit par la Rédaction]

show abstract

“…The (h-independent) constant η may be determined by using (11), and applying the condition given by (6) for E = 0. We then obtain…”

Section: Connection To Periodic Orbit Theorymentioning

confidence: 99%

When is the lowest order WKB quantization exact?

Bhaduri¹,

Sprung²,

Suzuki³

2006

Can. J. Phys.

View full text Add to dashboard Cite

show abstract

“…As already proved in [19,22], all the terms I 2n (E), n ≥ 1 of the isotonic potential are constant and once summed, the WKB series leads to the exact quantisation condition for this potential. To prove that this family is the most general, we use the result of appendix C, which shows that requiring I 2 (E) to be constant implies that S(X) corresponds to the function of the isotonic oscillator.…”

Section: Discussionmentioning

confidence: 76%

“…The second and fourth corrections become energy-independent in this limit. It can be shown by calculating the entire WKB series that all corrections are energy-independent and that their summation leads to the exact quantisation condition [19,22]. We can check that the two corrective terms I 2 and I 4 we have obtained are the coefficients of the Taylor expansion for the non-integral Maslov index (see [24]) of the exact quantisation condition in β = 0.…”

Section: Wkb Quantisation Condition To Fourth Ordermentioning

confidence: 99%

“…Finally, the requirement |S(X)| < 1 leads to the conclusion that I 2 has to be negative. Let us write I 2 = − 2 β 2 /(8ω) for some parameter β ≥ 0, (C7) simplifies to S(X) = βX 1 + β 2 X 2 (C8) which is nothing but the S function of an isotonic potential with scaling parameter β (see (22), (26) and (18)). Repeating what has been done in the previous example for the function I 2 (E) chosen above yields the following S function This function is readily odd and ∀X ∈ R, |S(X)| < 1.…”

Section: Appendix B: Spectrum Of the Split-harmonic Potentialmentioning

confidence: 99%

See 1 more Smart Citation

On the quantum spectrum of isochronous potentials

Dorignac¹

2005

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

In this paper, the quantum spectrum of isochronous potentials is investigated. Given that the frequency of the classical motion in such potentials is energy-independent, it is natural to expect their quantum spectra to be equispaced. However, as it has already been shown in some specific examples, this property is not always true. To gain some general insight into this problem, a WKB analysis of the spectrum, valid for any analytic potential, is performed and the first semiclassical corrections to its regular spacing are calculated. We illustrate the results on the two-parameter family of isochronous potentials derived in [1], which includes the harmonic oscillator, the asymmetric parabolic well, the radial harmonic oscillator and Urabe's potential as special limiting cases. In addition, some new analytical expressions for families of isochronous potentials and their corresponding spectra are derived by means of the above-mentioned method.

show abstract

“…There is a class of potentials called translationally shape invariant potentials which are exactly solvable. For TSIP the exact quantization rule can be written as [7][8][9][10][11][12]28,29 (9) with .…”

Section: Modmentioning

confidence: 99%

Langer Modification in WKB Quantization for Translationally Shape Invariant Potentials

Sun¹

2012

Bulletin of the Korean Chemical Society

View full text Add to dashboard Cite

When the Langer modification is applied to Coulomb potential, the standard WKB quantization yields an exact energy spectrum for the potential. This Langer modification has been known to be related to the centrifugal term appearing in Coulomb potential. But we find that a similar modification exists for all translationally shape invariant potentials without referring to the centrifugal term. The characteristic shape of the potentials accounts for the generalized version of Langer modification that makes the WKB quantization valid for all translationally shape invariant potentials.

show abstract

Convergent WKB series

Cited by 14 publications

References 15 publications

When is the lowest order WKB quantization exact?

When is the lowest order WKB quantization exact?

On the quantum spectrum of isochronous potentials

Langer Modification in WKB Quantization for Translationally Shape Invariant Potentials

Contact Info

Product

Resources

About