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arXiv:chao-dyn/9603014 · Journal of Physics A Mathematical and General

WKB expansion for the angular momentum and the Kepler problem: from the torus quantization to the exact one

Marko Robnik, Luca Salasnich

Published Mar 7, 1997• 25 citations
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Abstract

We calculate the WKB series for the angular momentum and the non-relativistic three-dimesional Kepler problem. This is the first semiclassical treatment of the angular momentum for terms beyond the leading WKB approximation. We explain why the torus quantization (the leading WKB term) of the full problem is exact, even if the individual torus quantization of the angular momentum and of the radial Kepler problem separately is not exact. In this way we derive Langer's rule, calculate the first correction to the leading Langer's term and conjecture the form of all higher terms.

DOI
10.1088/0305-4470/30/5/032

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We calculate the WKB series for the angular momentum and the non-relativistic three-dimesional Kepler problem. This is the first semiclassical treatment of the angular momentum for terms beyond the leading WKB approximation. We explain why the torus quantization (the leading WKB term) of the full problem is exact, even if the individual torus quantization of the angular momentum and of the radial Kepler problem separately is not exact. In this way we derive Langer's rule, calculate the first correction to the leading Langer's term and conjecture the form of all higher terms.
We obtain the rigorous WKB expansion to all orders for the radial Kepler problem, using the residue calculus in evaluating the WKB quantization condition in terms of a complex contour integral in the complexified coordinate plane. The procedure yields the exact energy spectrum of this Schrödinger eigenvalue problem and thus resolves the controversies around the so-called "Langer correction". The problem is nontrivial also because there are only a few systems for which all orders of the WKB series can be calculated, yielding a convergent series whose sum is equal to the exact result, and thus sheds new light to similar and more difficult problems.
It is well-known that applications of the JWKB method to central-field problems in three dimensions require half-integral quantization of the angular momentum for their success. Thus, the square of the angular momentum must be represented by the term (𝓁 + 1 / 2 ) 2 ℏ 2 rather than 𝓁 (𝓁 + 1)ℏ 2 . This was first shown by Kramers in 1926 and has subsequently been discussed by several authors including, in particular, Langer (1937). While Kramers based his discussion on the ordinary radial variable r, Langer switched to the variable x defined by r = e x . In this new representation of the central-field problem, the expression (𝓁 + 1 / 2 ) 2 ℏ 2 emerges naturally. The ad hoc character of the Langer transformation has, however, often been emphasized. In the present communication, we choose a different entry to the problem. We keep the variable r and focus on physically equivalent forms of the radial Schrödinger equation in this variable. This leads to a smoother emergence of the (𝓁 + 1 / 2 ) 2 ℏ 2 term. Our analysis is carried out for a general dimension D . For a given D, there are D physically equivalent radial equations, corresponding to the subdimensions d = 1, 2, ..., D . We show that it is only the d = 2 equation that can be satisfactorily treated by the JWKB approximation. In the past, the focus was always on the d = 1 equation, and this was the reason behind the problems encountered by Kramers and Langer. As to the d = 2 equation, we finally show that this equation also is the most convenient starting point for determining the exact solutions of a central-field problem for general values of D and angular-momentum quantum number L .
The Wigner–Weyl quantum-to-classical correspondence rule is nonunique with respect to coordinate choice. This ambiguity can be exploited to improve the accuracy of semiclassical approximations. For instance, the well-known Langer modification was recently derived by applying a coordinate transformation to the radial Schrödinger equation prior to using the Wigner–Weyl rule—albeit only by presuming exact quantum solutions for all nonradial degrees of freedom [J. J. Morehead, J. Math. Phys. 36, 5431 (1995)]. In this paper, the full classical Hamiltonian is derived in all degrees of freedom, using a (hyper)spherical coordinate Wigner–Weyl correspondence with a Langer-like modification of polar angles. For central force Hamiltonians, the new result is radially equivalent to that of Langer, and to the standard Cartesian form. The new correspondence is superior with respect to all angular momentum operators however, in that the resultant semiclassical eigenvalues are exact—a desirable goal, evidently achieved here for the first time.
We perform a systematic WKB expansion to all orders for a one--dimensional system with potential V(x)=U0/cos⁡2(αx)V(x)=U_0/\cos^2{(αx)}V(x)=U0​/cos2(αx). We are able to sum the series to the exact energy spectrum. Then we show that any finite order WKB approximation fails to predict the individual energy levels within a vanishing fraction of the mean energy level spacing.