Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0003115 · American Journal of Physics

Tunnel splittings for one-dimensional potential wells revisited

Anupam Garg

Published May 1, 2000• 120 citations
arXivHTMLPDF

Abstract

The Wentzel–Kramers–Brillouin and instanton answers for the tunnel splitting of the ground state in a symmetric double-well potential are both reduced to an expression involving only functionals of the potential, without the need for solving any auxilliary problems. This formula is applied to simple model problems. The prefactor for the splitting in the textbook by Landau and Lifshitz is amended so as to apply to the ground and low lying excited states.

DOI
10.1119/1.19458

Related papers

Tunnel splittings for one dimensional potential wells revisited

Anupam Garg

The WKB and instanton answers for the tunnel splitting of the ground state in a symmetric double well potential are both reduced to an expression involving only the functionals of the potential, without the need for solving any auxilliary problems. This formula is applied to simple model problems. The prefactor for the splitting in the text book by Landau and Lifshitz is amended so as to apply to the ground and low lying excited states. Typeset using REVTEX 1 I.

•Jan 1, 2000•72 citations

Tunnel Splitting in Asymmetric Double Well Potentials: An Improved WKB Calculation

Seyyed M. H. Halataei, Anthony J. Leggett

We present an improved Wentzel-Kramers-Brillouin (WKB) calculation of tunnel splitting in one dimensional asymmetric double well potentials. We show that the tunnel splitting in general can have linear dependence on the bias energy, beside the well-known quadratic dependence. We demonstrate that the linear correction is greater than previously thought.

arXiv:1703.05758•Mar 15, 2017•6 citations

Escape from a one-dimensional metastable potential well revisited

Alex Edison, Anupam Garg

The Wentzel–Kramers–Brillouin and bounce (path integral) answers for the rate of escape from the ground state of a metastable potential well are both reduced to an expression involving only functionals of the potential, without the need for solving any auxiliary problems. The results are extended to low-lying excited states, and they are applied to the classic cubic parabola and other model potentials. A numerical calculation is done of the escape rate from the ground state of the cubic parabolic well for a variety of barrier heights, and is found to compare well with the WKB result.

American Journal of Physics•Aug 21, 2025•0 citations

Semiclassical formula for quantum tunneling in asymmetric double-well potentials

Gianluca Rastelli

Despite the fact that quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, I derive a succinct analytical formula based on the Wentzel-Kramers-Brillouin semiclassical approach. Two different examples of asymmetric potentials are discussed: the cases when the two localized levels are degenerate and when they are not degenerate. For the first case, I also discuss a time-dependent problem showing the quantum Zeno effect.

arXiv:1205.0366•Physical Review A•Jul 16, 2012•32 citations

The double-well potential in quantum mechanics: a simple, numerically exact formulation

Vedran Jelic, F. Marsiglio

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.

arXiv:1209.2521•European Journal of Physics•Sep 14, 2012•93 citations

Tunneling and energy splitting in an asymmetric double-well potential

Dae-Yup Song

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

•Aug 4, 2013•22 citations
The Wentzel–Kramers–Brillouin and instanton answers for the tunnel splitting of the ground state in a symmetric double-well potential are both reduced to an expression involving only functionals of the potential, without the need for solving any auxilliary problems. This formula is applied to simple model problems. The prefactor for the splitting in the textbook by Landau and Lifshitz is amended so as to apply to the ground and low lying excited states.
The Wentzel–Kramers–Brillouin and bounce (path integral) answers for the rate of escape from the ground state of a metastable potential well are both reduced to an expression involving only functionals of the potential, without the need for solving any auxiliary problems. The results are extended to low-lying excited states, and they are applied to the classic cubic parabola and other model potentials. A numerical calculation is done of the escape rate from the ground state of the cubic parabolic well for a variety of barrier heights, and is found to compare well with the WKB result.
Despite the fact that quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, I derive a succinct analytical formula based on the Wentzel-Kramers-Brillouin semiclassical approach. Two different examples of asymmetric potentials are discussed: the cases when the two localized levels are degenerate and when they are not degenerate. For the first case, I also discuss a time-dependent problem showing the quantum Zeno effect.
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.
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