Abstract
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.
Highlights
Time-independent Schrodinger equation (TISE) is an eigenvalue problem in the form: H^ jφi 1⁄4 Ejφi ) À2mħ2 ∇2 þ UðrÞ!φn 1⁄4 Enφ (1)where the terms are in the usual meanings, namely, ∇2, the Laplacian operator; H^, Hamiltonian operator; m, mass; ħ, Planck’s constant divided by 2π; φ, wave function; E, total energy; and U(r), function of potential energy [1–7]
The Quantum harmonic oscillator (QHO) is a very good approximation in solving systems of diatomic molecules vibrating under the spring constant [1, 2, 5] and finds various modern physics applications such as in [8–10] as stated in a famous quotation: “the career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction by Sidney Coleman” [10, 11]
We study here one dimensional and non-frictional, that is, undamped case, and present its solution by the two following conventional semiclassical approximation methods: (i) the JWKB method [1–7, 14] and (ii) the MAF method [3, 18–23]
Summary
Time-independent Schrodinger equation (TISE) is an eigenvalue problem in the form: H^ jφi 1⁄4 Ejφi ) À2mħ2 ∇2 þ UðrÞ!φn 1⁄4 Enφ (1). JWKB method is known to give exact eigenenergies for the QHO, but eigenfunctions fail at and around the classical turning points (CTPs) where f 1⁄4 0 (or, equivalently, En 1⁄4 UðrÞ) in (3) as typical to the JWKB method [1–7, 14] These discontinuities prevent us from using continuity at the boundaries by equating the JWKB solutions of two neighboring regions directly at the CTPs to find the eigenenergy-dependent coefficients in the general JWKB eigenfunctions (wave functions). As to the MAF method, it does not exhibit discontinuities at the CTPs, though highly oscillating behavior of the Airy functions requires careful handling in finding their zeros and the parity treatment used in the JWKB solution seems straightforward to be applicable to the MAF solution of the QHO [3, 19–22] It was originally suggested in 1931 by Langer in [25], finding zeros of highly oscillatory Airy functions became practical as the advances in computational software and the MAF method became widespread by the 1990s [3, 18–23]. We will discuss the treatment of parity matching and asymptotic matching in solving the QHO by these semiclassical methods
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