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arXiv:2502.15063 · AIP Advances

The modified airy function approximation applied to the double-well potential

N. Wine, J. Achtymichuk, F. Marsiglio

Published Mar 1, 2025• 6 citations

Abstract

The single harmonic oscillator and double-well potentials are important systems in quantum mechanics. The single harmonic oscillator is the paradigm in physics and is taught in nearly all beginner undergraduate classes, while the double-well potential illustrates the two important principles of quantum tunneling and linear superposition. While exact analytical solutions of the Schrödinger equation exist for both of these potentials, they are also employed to benchmark the use of approximate techniques, which may be the only recourse for more complicated potentials. In this paper, we review the Wentzel–Kramers–Brillouin (WKB) approximation for both these potentials. While this approximation is known for its accurate energies, we will instead emphasize how poor the WKB wave functions are. The inaccuracy of the WKB wave functions will then motivate us to adopt the lesser-known Modified Airy Function (MAF) approximation, which alleviates the deficiencies of the WKB wave functions. We will review the MAF solution to the simple harmonic oscillator potential and then apply the MAF to the double-well potential. We find accurate eigenvalues and, more importantly, very accurate wave functions. We conclude with the suggestion that an introduction to the MAF should be included in undergraduate courses to complement the WKB.

DOI: 10.1063/5.0241523

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Quantum harmonic oscillator (QHO) involves square law potential (x2) in the Schrodinger equation and is a fundamental problem in quantum mechanics. It can be solved by various conventional methods such as (i) analytical methods where Hermite polynomials are involved, (ii) algebraic methods where ladder operators are involved, and (iii) approximation methods where perturbation, variational, semiclassical, etc. techniques are involved. Here we present the general outcomes of the two conventional semiclassical approximation methods: the JWKB method (named after Jeffreys, Wentzel, Kramers, and Brillouin) and the MAF method (abbreviated for “modified Airy functions”) to solve the QHO in a very good precision. Although JWKB is an approximation method, it interestingly gives the exact solution for the QHO except for the classical turning points (CTPs) where it diverges as typical to the JWKB. As the MAF method, it enables very approximate wave functions to be written in terms of Airy functions without any discontinuity in the entire domain, though, it needs careful treatment since Airy functions exhibit too much oscillatory behavior. Here, we make use of the parity conditions of the QHO to find the exact JWKB and approximate MAF solutions of the QHO within the capability of these methods.

The double-well potential is arguably one of the most important potentials in quantum mechanics, because the solution contains the notion of a state as a linear superposition of 'classical' states, a concept which has become very important in quantum information theory. It is therefore desirable to have solutions to simple double-well potentials that are accessible to the undergraduate student. We describe a method for obtaining the numerically exact eigenenergies and eigenstates for such a model, along with the energies obtained through the Wentzel–Kramers–Brillouin (WKB) approximation. The exact solution is accessible with elementary mathematics, though numerical solutions are required. We also find that the WKB approximation is remarkably accurate, not just for the ground state, but also for the excited states.

We apply a many-body Wentzel-Kramers-Brillouin (WKB) approach to determine the leading quantum corrections to the semiclassical dynamics of the Josephson model, describing interacting bosons able to tunnel between two localized states. The semiclassical dynamics is known to divide between regular oscillations and self-trapped oscillations where the sign of the imbalance remains fixed. In both cases, the WKB wave functions are matched to Airy functions, yielding a modified Bohr-Sommerfeld quantization condition. At the critical energy dividing normal and self-trapped oscillations, the WKB wave functions should instead be matched to parabolic cylinder functions, leading to a quantization formula that is not just the Bohr-Sommerfeld formula, and recovering the known logarithmic quantum break times at this energy. This work thus provides another illustration of the usefulness of the WKB approach in certain many-body problems.

We present an approach to obtain eigenfunctions and eigenenergies for abruptly varying potentials in the framework of the Wentzel–Kramers–Brillouin (WKB) approximation. To illustrate it, two examples of the potentials are studied. The first one is the combination of a step barrier and a harmonic oscillator potential, and the second one consists of a step barrier and a linear potential. The formulation of a WKB quantization rule is proposed. Our approach shows that WKB energies and those from numerical calculation are in good agreement. According to matching conditions used, WKB wavefunctions in this present work are violated at only one classical turning point, but they behave well at another point where the potentials are discontinuous.