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arXiv:0907.4485 · International Journal of Modern Physics A

DOUBLE WELL POTENTIAL: PERTURBATION THEORY, TUNNELING, WKB (BEYOND INSTANTONS)

Alexander V. Turbiner

Published Jan 30, 2010• 33 citations
arXivHTMLPDF

Abstract

A simple approximate solution for the quantum-mechanical quartic oscillator V = m2x2 + gx4 in the double-well regime m2 < 0 at arbitrary g ≥ 0 is presented. It is based on a combining of perturbation theory near true minima of the potential, semi-classical approximation at large distances and a description of tunneling under the barrier. It provides 9-10 significant digits in energies and gives for wavefunctions the relative deviation in real x-space less than ≲ 10-3.

DOI
10.1142/s0217751x10048937

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A simple approximate solution for the quantum-mechanical quartic oscillator V = m2x2 + gx4 in the double-well regime m2 < 0 at arbitrary g ≥ 0 is presented. It is based on a combining of perturbation theory near true minima of the potential, semi-classical approximation at large distances and a description of tunneling under the barrier. It provides 9-10 significant digits in energies and gives for wavefunctions the relative deviation in real x-space less than ≲ 10-3.
Extensions of the Bohr–Sommerfeld quantization formula to smooth double wells are established which preserve the formula’s generality and simplicity as well as its accuracy and the ease of use. Moreover, the proposed quantization equations for double wells, while being free from the limitations of the Wentzel–Kramers–Brillouin approximation which is restricted to deep energy levels alone, are equally well suited for determining higher energy levels lying in the vicinity of the barrier’s top as well as above the latter, and they have the correct connection to the conventional Bohr–Sommerfeld formula as the particle’s energy becomes large enough. As a particular application, a simple quantization equation for the pure quartic oscillator is derived which yields correct energy eigenvalues for all eigenstates of the oscillator including the ground one, with relative errors being all less than 1%. Conditions for the validity of the extended quantization equations are discussed and their numerical verification is made.
We analyse the accuracy of the approximate WKB quantization for the case of general one-dimensional quartic potential. In particular, we are interested in the validity of semiclassically predicted energy eigenvalues when approaching the limit E→∞, and in the accuracy of low lying energy levels below the potential barrier in the case of generally asymmetric double-well quartic potential. In the latter case, using the standard WKB quantization an unnatural localization of eigenstates due to the negligence of tunneling is implied and thus the validity of semiclassics is uncertain. In all computations the higher order corrections to the leading semiclassical approximation are included using the complex contour integration technique. We show that these corrections can improve accuracy of semiclassical approximation greatly by many orders of magnitude.
An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of �ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1
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.
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.
The generalized Bohr-Sommerfeld quantization condition up to the ninth-order approximation is given explicitly, in terms of complete elliptic integrals, for two kinds of quartic oscillators V(z)=v2{\mathit{v}}_{2}v2​z2{\mathit{z}}^{2}z2+v4{\mathit{v}}_{4}v4​z4{\mathit{z}}^{4}z4: (i) for the quartic double barrier, where v2{\mathit{v}}_{2}v2​>0 and ${\mathit{v}}_{4}$0, and the energy is below the potential maximum; and (ii) for the quartic double well, where ${\mathit{v}}_{2}$0 and v4{\mathit{v}}_{4}v4​>0. Resonance energies for some quartic double barriers and energy levels for some quartic double wells were evaluated by solving the quantization condition numerically. In some cases the results are very good.