[Paper Review] On WKB Series for the Radial Kepler Problem

Valery G. Romanovski, Marko Robnik|ArXiv.org|Mar 30, 2000

Quantum Mechanics and Non-Hermitian Physics3 references3 citations

TL;DR

This paper rigorously derives the full WKB series for the radial Kepler problem using residue calculus in the complexified coordinate plane, confirming that summing all orders yields the exact Coulomb energy spectrum. It resolves the long-standing controversy around the Langer correction by showing that higher-order WKB terms exactly compensate its ad hoc adjustment, proving the correction is unnecessary when the full series is summed.

ABSTRACT

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.

Motivation & Objective

• To resolve the controversy surrounding the Langer correction in the WKB quantization of the radial Kepler problem.
• To provide a rigorous derivation of the WKB series to all orders for the radial Kepler potential.
• To confirm the conjecture by Robnik and Salasnich (1997b) on the analytical form of all WKB terms in the Coulomb problem.
• To demonstrate that the full WKB series, when summed, yields the exact energy spectrum without any ad hoc adjustments.

Proposed method

• The Schrödinger equation for the radial Kepler problem is transformed into a complex phase equation using the WKB ansatz for the phase function σ(r).
• The quantization condition is formulated as a contour integral in the complex r-plane, requiring the total phase change to be 2πn_rħ.
• Residue calculus is applied to evaluate the contour integrals of the WKB phase derivatives dσ_k, focusing on poles at r=0 and r=∞.
• The recursion relations for σ_k are derived order-by-order from the differential equation σ′² + (ħ/i)σ′′ = 2(E - V(r)), with V(r) = L²/(2r²) - α/r.
• The structure of the rational functions σ′_k is analyzed using polynomial differential equation techniques, enabling identification of the general form of σ′_k.
• The vanishing of odd-order integrals ∫dσ_{2k+1} is proven via residue analysis, and the even-order integrals are computed exactly using residues at r=0 and r=∞.

Experimental results

Research questions

• RQ1 Can the full WKB series for the radial Kepler problem be derived rigorously to all orders using complex analysis?
• RQ2 Does the sum of the WKB series yield the exact energy spectrum of the Coulomb problem without the Langer correction?
• RQ3 What is the analytical structure of the WKB terms σ_k for the radial Kepler potential?
• RQ4 Why does the Langer correction, which replaces l(l+1) with (l+1/2)², yield the correct spectrum if it lacks physical justification?
• RQ5 Can residue calculus be systematically applied to compute WKB series for one-dimensional quantum potentials?

Key findings

• The WKB series for the radial Kepler problem converges exactly to the known Coulomb energy spectrum when all orders are summed.
• The residue calculus confirms the conjectured form of the even-order WKB terms: (ħ/i)^{2k} ∫γ dσ_{2k} = -2πħ (1/2 choose k) 2^{-2k} λ^{1-2k}, where λ = L/ħ.
• The odd-order integrals vanish: ∫γ dσ_{2k+1} = 0, which is essential for the exact cancellation of errors from the Langer correction.
• The residue at r=0 contributes to the even-order terms, while the residue at infinity vanishes, simplifying the computation.
• The exact energy spectrum is recovered as E = -α²m/(2ħ²n²), with n = n_r + l + 1, confirming the correctness of the full series.
• The Langer correction is shown to be an artifact of truncating the WKB series; its effect is exactly canceled by higher-order terms.