Abstract
A new method is proposed for determining level splitting Δ in a double-well 1D potential. Two “partner” functions (one symmetric Ψ+ and the other antisymmetric Ψ–) are determined. From these functions, potentials V+(x) and V–(x) and energies \(E_{ + }^{0}\) and \(E_{ - }^{1}\) corresponding to them are determined from the Schrödinger equation. A unique property of Ψ+ and Ψ– is identity \(E_{ + }^{0}\) = \(E_{ - }^{1}\), which makes it possible to determine Δ from the perturbation theory in parameter V+(x) – V–(x). For a double-well oscillator potential, the expression for the level splitting, which connects the instanton and single-well limits, is obtained. These results can be employed in the field theory, for which the possibility of obtaining instanton solutions from perturbation theory has been discussed more than once. A number of potentials are considered, for which the value of Δ can be determined without using the semiclassical approximation. Singular potentials of the funnel type are analyzed. The value of Δ determined in this study is compared with the results of numerical solution of the Schrödinger equation for the instanton potential.
REFERENCES
H. A. Kramers, Physica (Amsterdam, Neth.) 7, 284 (1940).
S. Chandrasekar, Rev. Mod. Phys. 15, 1 (1943).
W. Miller, J. Chem. Phys. 61, 1823 (1974).
A. I. Vainshtein, V. I. Zakharov, V. A. Novikov, and M. A. Shifman, Sov. Phys. Usp. 25, 195 (1982).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 3: Quantum Mechanics: Non-Relativistic Theory (Pergamon, New York, 1977; Fizmatlit, Moscow, 2001).
R. Dutt, A. Khare, and U. Sukhatme, Phys. Lett. B 181, 295 (1986).
J. W. Harald Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (World Sci., Singapore, 2012).
R. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).
M. Bernstein and L. S. Brown, Phys. Rev. Lett. 52, 1933 (1984).
P. Kumar, M. Ruiz-Altaba, and B. S. Thomas, Phys. Rev. Lett. 57, 2749 (1986).
Wai-Yee Keung, E. Kovacs, and U. P. Sukhatme, Phys. Rev. Lett. 60, 41 (1988).
A. V. Turbiner, Lett. Math. Phys. 74, 169 (2005). https://doi.org/10.1007/s11005-005-0012-z
A. V. Turbiner, Int. J. Mod. Phys. A 25, 647 (2010). https://doi.org/10.1142/S0217751X10048937
A. V. Turbiner and J. C. del Valle, Acta Polytech. 62, 208 (2022). https://doi.org/10.14311/AP.2022.62.0208
Yu. I. Bogdanov, N. A. Bogdanova, D. V. Fastovets, and V. F. Lukichev, JETP Lett. 114, 354 (2021).
A. M. Polyakov, Nucl. Phys. B 120, 429 (1977).
J. Zinn-Justin, Nucl. Phys. B 192, 125 (1981);
Nucl. Phys. B 218, 333 (1983).
J. Zinn-Justin and U. D. Jentschura, Ann. Phys. 313, 197 (2004);
Ann. Phys. 313, 269 (2004);
Phys. Lett. B 596, 138 (2004).
G. V. Dunne and M. Unsal, Phys. Rev. D 89, 105009 (2014).
M. A. Escobar-Ruiz, E. Shuryak, and A. V. Turbiner, Phys. Rev. D 92, 025046 (2015);
Phys. Rev. D 92, 089902(E) (2015).
E. Shuryak and A. V. Turbiner, Phys. Rev. D 98, 105007 (2018).
A. M. Dyugaev and P. D. Grigoriev, JETP Lett. 112, 101 (2020).
A. V. Turbiner, JETP Lett. 30, 352 (1979).
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Nauka, Moscow, 1971; Academic, New York, 1980).
I. V. Andreev, Chromodynamics and Hard Processes at High Energies (Nauka, Moscow, 1981) [in Russian].
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics (Pergamon, Oxford, 1980; Fizmatlit, Moscow, 2005).
Funding
The work of A.M.D. was fulfilled under State assignment no. 0033-2019-0001 “Development of the theory of condensed state of matter.” The work of P.D.G. was supported by the Foundation for Development of Theoretical Physics and Mathematics “Basis” and the program of strategic academic leadership “Priority-2030” (grant no. K2-2022-025 for the National Research Technological University “MISiS”).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Translated by N. Wadhwa
Rights and permissions
About this article
Cite this article
Dyugaev, A.M., Grigoriev, P.D. Modeling of Double-Well Potentials for the Schrödinger Equation. J. Exp. Theor. Phys. 137, 17–22 (2023). https://doi.org/10.1134/S1063776123070014
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063776123070014