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arXiv:quant-ph/9708036 · Journal of Physics A Mathematical and General

On the convergence of the WKB series for the angular momentum operator

Luca Salasnich, F. Sattin

Published Nov 7, 1997• 8 citations
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Abstract

In this paper we prove a recent conjecture about the convergence of the WKB series for the angular momentum operator. We demonstrate that the WKB algorithm for the angular momentum gives the exact quantization formula if all orders are summed. Finally, we discuss the supersymmetric semiclassical quantum mechanics (SWKB), which gives the correct quantization of the angular momentum at the leading order.

DOI
10.1088/0305-4470/30/21/028

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In this paper we prove a recent conjecture about the convergence of the WKB series for the angular momentum operator. We demonstrate that the WKB algorithm for the angular momentum gives the exact quantization formula if all orders are summed. Finally, we discuss the supersymmetric semiclassical quantum mechanics (SWKB), which gives the correct quantization of the angular momentum at the leading order.
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 perform a systematic WKB expansion to all orders for a one-dimensional system with potential . We are able to sum the series to the exact energy spectrum. Then we show that at any finite order the error of the WKB approximation measured in the natural units of the mean energy level spacing does not go to zero when the quantum number goes to infinity. Therefore we make the general conclusion that the semiclassical approximations fail to predict the individual energy levels within a vanishing fraction of the mean energy level spacing.
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.
Semiclassical angular momentum matrices are calculated using a new contour-integral formula for matrix-elements. WKB spherical harmonic functions are found to be exaclty orthonormal with the contour-integral inner-product. Matrix-elements obtained from these wave-functions are accurate to about 2 % and the matrices obey the commutation relations expected of quantum angular momentum operators. The semiclassical wave-functions are related to a superposition of allowed classical orbits and this illustrates the connection to the Feynman pathsummation formulation of quantum mechanics for electrons in a spherically symmetric potential.
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.