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Precision Semiclassics: Resummed WKB from Variational Principles

Revista, Zen, PHYSICS, 10

Published Nov 28, 2025• 0 citations
Publisher

Abstract

The semiclassical approximation, particularly the Wentzel-Kramers-Brillouin (WKB) method, has been a cornerstone in quantum mechanics for understanding systems where quantum effects are present but allow for an approximate classical description. Despite its widespread success in providing intuitive insights into phenomena like tunneling and energy quantization, the standard WKB expansion often suffers from asymptotic divergence and limited accuracy, especially for lower quantum numbers or near turning points. This paper introduces a novel approach to enhance the precision and convergence of the WKB method by integrating variational principles with advanced resummation techniques. We propose a framework where the WKB expansion for key physical quantities is treated as an asymptotic series, and its coefficients are optimized through a variational procedure derived from the underlying quantum dynamics. Subsequently, these optimized series are subjected to resummation methods, such as generalized Borel transforms or Padé approximants, to extract highly accurate and convergent results. Our methodology aims to transcend the limitations of conventional WKB by systematically incorporating non-perturbative information and improving the analytic properties of the semiclassical series. We present the theoretical foundations of this "Resummed Variational WKB" approach, outlining its potential to yield unprecedented precision in calculating energy eigenvalues, tunneling rates, and other quantum observables, thereby bridging the gap between purely classical and fully quantum mechanical descriptions with enhanced quantitative fidelity.

DOI
10.5281/zenodo.17750317

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The semiclassical approximation, particularly the Wentzel-Kramers-Brillouin (WKB) method, has been a cornerstone in quantum mechanics for understanding systems where quantum effects are present but allow for an approximate classical description. Despite its widespread success in providing intuitive insights into phenomena like tunneling and energy quantization, the standard WKB expansion often suffers from asymptotic divergence and limited accuracy, especially for lower quantum numbers or near turning points. This paper introduces a novel approach to enhance the precision and convergence of the WKB method by integrating variational principles with advanced resummation techniques. We propose a framework where the WKB expansion for key physical quantities is treated as an asymptotic series, and its coefficients are optimized through a variational procedure derived from the underlying quantum dynamics. Subsequently, these optimized series are subjected to resummation methods, such as generalized Borel transforms or Padé approximants, to extract highly accurate and convergent results. Our methodology aims to transcend the limitations of conventional WKB by systematically incorporating non-perturbative information and improving the analytic properties of the semiclassical series. We present the theoretical foundations of this "Resummed Variational WKB" approach, outlining its potential to yield unprecedented precision in calculating energy eigenvalues, tunneling rates, and other quantum observables, thereby bridging the gap between purely classical and fully quantum mechanical descriptions with enhanced quantitative fidelity.
The accurate simulation and understanding of quantum dynamics are paramount across diverse fields, from quantum computing and material science to molecular biology. However, achieving high precision in complex quantum systems often poses significant computational challenges due to the exponential growth of Hilbert space. This paper introduces a novel hybrid theoretical framework for precision quantum dynamics, integrating variational methods, the Wentzel-Kramers-Brillouin (WKB) approximation, and perturbative corrections. The proposed approach leverages the strengths of each method: WKB provides an efficient semi-classical foundation and zeroth-order solutions for complex potentials, particularly effective in regimes where quantum effects are localized or well-approximated semi-classically. Variational methods, employing parameterized quantum circuits or classical function spaces, are then used to capture non-adiabatic effects, refine wavefunctions, and optimize system parameters by minimizing an energy functional or other relevant cost functions. Finally, perturbation theory is applied to systematically correct for residual interactions, small deviations from the WKB or variational solutions, or to account for higher-order couplings that enhance the overall precision. We elucidate the conceptual underpinnings, outline a comprehensive methodology for its implementation, and discuss the synergistic advantages of this hybrid scheme. Theoretical implications for enhancing accuracy, computational efficiency, and applicability to a broader class of quantum mechanical problems, including open quantum systems and time-dependent phenomena, are explored. This framework aims to significantly advance the capabilities for robust and precise characterization of quantum evolutionary processes, paving the way for more accurate predictions and design principles in quantum technologies.
The accurate description of dynamics in complex quantum and classical systems presents a formidable challenge, particularly when dealing with phenomena involving strong interactions, non-adiabatic transitions, or chaotic behavior. Traditional methods often struggle with either computational cost or insufficient accuracy in such regimes. This paper introduces and rigorously develops an Optimized Variational-Perturbation-WKB Integration (VPWKB) method designed to achieve high precision in the propagation of dynamics within these challenging systems. By synergistically combining the strengths of variational principles for robust global optimization, perturbation theory for handling intricate interactions and higher-order corrections, and the Wentzel-Kramers-Brillouin (WKB) approximation for efficient semiclassical trajectory generation, the VPWKB framework offers a powerful new approach. The method leverages variational techniques to determine optimal initial conditions or trial functions, subsequently refines the dynamics through a controlled perturbative expansion around a WKB-derived semiclassical path, and integrates these components into a coherent, self-consistent propagation scheme. This optimization significantly enhances the accuracy and stability of semiclassical calculations, enabling a more faithful representation of quantum effects and non-classical trajectories in complex landscapes. We discuss the theoretical underpinnings, computational advantages, and potential applications of VPWKB, demonstrating its capability to provide precision dynamics where conventional techniques falter.
We demonstrate that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the bound-state problem in a single well of one degree of freedom. While the WKB expansion formulas are basically the usual ones, in this representation they describe approximations that are uniform and nonsingular in the classically allowed region of phase space because no turning points appear there. The quantization of energy levels relies on a complex contour integral that tests the eigenfunction for analyticity. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a self-consistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions. All in all, the Bargmann representation appears to combine the advantages of a linear description and of a phase-space representation of the quantum state vectors.
This paper introduces a novel Self-Consistent WKB-Variational Theory designed to investigate the complex dynamics of non-equilibrium quantum fields. Traditional approaches often struggle with strongly correlated systems or those far from thermal equilibrium, necessitating methods that capture both semiclassical evolution and quantum fluctuations. Our proposed framework unifies the Wentzel-Kramers-Brillouin (WKB) approximation, which provides a semiclassical description of the background field, with a time-dependent variational principle applied to quantum fluctuations around this background. The core of the theory lies in the self-consistent determination of the classical background field dynamics, influenced by quantum backreaction, and the evolution of the quantum fluctuations. This intricate interplay allows for a comprehensive description of non-perturbative phenomena, particle production, and phase transitions in dynamic settings. We elaborate on the theoretical derivation, outlining the coupled equations of motion for the background field and the variational parameters that characterize the quantum state. The advantages of this approach include its ability to address strong coupling regimes and provide a physically intuitive understanding of the interplay between classical and quantum degrees of freedom. Potential applications span diverse areas, from early universe cosmology to heavy-ion collisions and quantum quenches in condensed matter systems.
The accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum-mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion. In the semiclassical limit of quantum mechanics such an expansion converges towards the result of the Jeffreys–Wentzel–Kramers–Brillouin method to all orders. The hypervirial and Hellmann–Feynman theorems facilitate the derivation of the semiclassical series from Rayleigh–Schrödinger perturbation theory.
The mutual consistency and structural interconnection between the Wentzel-Kramers-Brillouin (WKB) and Wigner-Kirkwood (WK) semiclassical approximations is investigated for nonrelativistic N-particle systems, with mutual scalar interactions and coupling to an external time-varying electromagnetic field. The generalized WK expansion of the propagator 〈x\ensuremath{\Vert}U(t,s)\ensuremath{\Vert}y〉 is obtained from a large-mass expansion of the higher-order WKB approximation. Two techniques are described for computing the WK coefficient functions. One relies on a large-mass expansion of classical paths and the transport representation of the WKB approximation; the other is recursive in nature. For time-independent Hamiltonians H the standard WK expansion of the heat kernel 〈x\ensuremath{\Vert}${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$\ensuremath{\Vert}y〉 is recovered.