Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0208 Acta Polytechnica 62(1):208–210, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague FROM QUARTIC ANHARMONIC OSCILLATOR TO DOUBLE WELL POTENTIAL Alexander V. Turbiner∗, Juan Carlos del Valle Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México-City, Mexico ∗ corresponding author: turbiner@nucleares.unam.mx Abstract. Quantum quartic single-well anharmonic oscillator Vao(x) = x2 + g2x4 and double-well anharmonic oscillator Vdw(x) = x2(1−gx)2 are essentially one-parametric, they depend on a combination (g2ℏ). Hence, these problems are reduced to study the potentials Vao = u2 + u4 and Vdw = u2(1 − u)2, respectively. It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction Ψao(u), obtained recently, see JPA 54 (2021) 295204 [1] and arXiv 2102.04623 [2], and then forming the function Ψdw(u) = Ψao(u)±Ψao(u−1) allows to get the highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues. Keywords: Anharmonic oscillator, double-well potential, perturbation theory, semiclassical expansion. 1. Introduction It is already known that for the one-dimensional quantum quartic single-well anharmonic oscillator Vao(x) = x2 + g2x4 and double-well anharmonic oscillator with potential Vdw(x) = x2(1 − gx)2 the (trans)series in g (which is the Perturbation Theory in powers of g (the Taylor expansion) in the former case Vao(x) supplemented by exponentially-small terms in g in the latter case Vdw(x)) and the semiclassical expansion in ℏ (the Taylor expansion for Vao(x) sup- plemented by the exponentially small terms in ℏ for Vdw(x)) for energies coincide [3]. This property plays crucially important role in our consideration. Both the quartic anharmonic oscillator V = x2 + g2x4 , (1) with a single harmonic well at x = 0 and the double- well potential V = x2(1 − gx)2 , (2) with two symmetric harmonic wells at x = 0 and x = 1/g, respectively, are two particular cases of the quartic polynomial potential V = x2 + agx3 + g2x4 , (3) where g is the coupling constant and a is a parameter. Interestingly, the potential (3) is symmetric for three particular values of the parameter a: a = 0 and a = ±2. All three potentials (1), (2), (3) belong to the family of potentials of the form V = 1 g2 Ṽ (gx) , for which there exists a remarkable property: the Schrödinger equation becomes one-parametric, both the Planck constant ℏ and the coupling constant g appear in the combination (ℏg2), see [2]. It can be immediately seen if instead of the coordinate x the so-called classical coordinate u = (g x) is introduced. This property implies that the action S in the path integral formalism becomes g-independent and the fac- tor 1ℏ in the exponent becomes 1 ℏg2 [4]. Formally, the potentials (1)-(2), which enter to the action, appear at g = 1, hence, in the form V = u2 + u4 , (4) V = u2(1 − u)2 , (5) respectively. Both potentials (4), (5) are symmetric with respect to u = 0 and u = 1/2, respectively. Namely, this form of the potentials will be used in this short Note. This Note is the extended version of a part of presentation in AAMP-18 given by the first author [5]. 2. Single-well potential In [1] for the potential (4) matching the small distances u → 0 expansion and the large distances u → ∞ expansion (in the form of semiclassical expansion) for the phase ϕ in the representation Ψ = P (u) e−ϕ(u) , of the wave function, where P is a polynomial, it was constructed the following function for the (2n + p)-excited state with quantum numbers (n, p), n = 0, 1, 2, . . . , p = 0, 1 : Ψ(n,p)(approximation) = upPn,p(u2) (B2 + u2) 1 4 ( B + √ B2 + u2 )2n+p+ 12 208 https://doi.org/10.14311/AP.2022.62.0208 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 62 no. 1/2022 From quartic anharmonic oscillator to double well potential -2 -1 1 2 u -1 -0.5 0.5 1 1.5 V(u) -2 -1 1 2 3 u -1 -0.5 0.5 1 1.5 V(u) Figure 1. Two lowest, normalized to one eigenfunc- tions of positive/negative parity: for single-well poten- tial (4), see (6) (top) and for double-well potential (5), see (9)(bottom). Potentials shown by black lines. × exp ( − A + (B2 + 3) u2/6 + u4/3 √ B2 + u2 + A B ) , (6) where Pn,p is some polynomial of degree n in u2 with positive roots. Here A = An,p, B = Bn,p are two pa- rameters of interpolation. These parameters (−A), B are slow-growing with quantum number n at fixed p taking, in particular, the values A0,0 = −0.6244 , B0,0 = 2.3667 , (7) A0,1 = −1.9289 , B0,1 = 2.5598 , (8) for the ground state and the first excited state, re- spectively. This remarkably simple function (6), see Figure 1 (top), provides 10-11 exact figures in energies for the first 100 eigenstates. Furthermore, the func- tion (6) deviates uniformly for u ∈ (−∞, +∞) from the exact function in ∼ 10−6. 3. Double-well potential: wavefunctions Following the prescription, usually assigned in folklore to E. M. Lifschitz – one of the authors of the famous Course on Theoretical Physics by L. D. Landau and E. M. Lifschitz – when a wavefunction for single well potential with minimum at u = 0 is known, Ψ(u), the wavefunction for double well potential with min- ima at u = 0, 1 can be written as Ψ(u) ± Ψ(u − 1). This prescription was already checked successfully for the double-well potential (2) in [6] for somehow sim- plified version of (6), based on matching the small distances u → 0 expansion and the large distances u → ∞ expansion for the phase ϕ but ignoring sub- tleties emerging in semiclassical expansion. Taking the wavefunction (6) one can construct Ψ(n,p)(approximation) = Pn,p(ũ2) (B2 + ũ2) 1 4 ( αB + √ B2 + ũ2 )2n+ 12 exp ( − A + (B2 + 3) ũ2/6 + ũ4/3 √ B2 + ũ2 + A B ) D(p) , (9) where p = 0, 1 and D(0) = cosh ( a0ũ + b0ũ3√ B2 + ũ2 ) , D(1) = sinh ( a1ũ + b1ũ3√ B2 + ũ2 ) . Here ũ = u − 1 2 , (10) α = 1 and A, B, a0,1, b0,1 are variational param- eters. If α = 0 as well as b0,1 = 0 the func- tion (9) is reduced to ones which were explored in [6], see Eqs.(10)-(11) therein. The polynomial Pn,p is found unambiguously after imposing the orthogonal- ity conditions of Ψ(n,p)(approximation) to Ψ (k,p) (approximation) at k = 0, 1, 2, . . . , (n − 1), here it is assumed that the polynomials Pk,p at k = 0, 1, 2, . . . , (n − 1) are found beforehand. 4. Double-well potential: Results In this section we present concrete results for energies of the ground state (0, 0) and of the first excited state (0, 1) obtained with the function (9) at p = 0, 1, re- spectively, see Figure 1 (bottom). The results are com- pared with the Lagrange-Mesh Method (LMM) [7]. 4.1. Ground State (0,0) The ground state energy for (5) obtained variationally using the function (9) at p = 0 and compared with LMM results [7], where all printed digits (in the second line) are correct, E(0,0)var = 0.932 517 518 401 , E (0,0) mesh = 0.932 517 518 372 . Note that ten decimal digits in E(0,0)var coincide with ones in E(0,0)mesh (after rounding). Variational parame- ters in (9) take values, A = 2.3237 , B = 3.2734 , a0 = 2.3839 , b0 = 0.0605 , cf. (7). Note that b0 takes a very small value. 209 Alexander V. Turbiner, Juan Carlos del Valle Acta Polytechnica 4.2. First Excited State (0,1) The first excited state energy for (5) obtained varia- tionally using the function (9) at p = 1 and compared with LMM results [7], where all printed digits (in the second line) are correct, E(0,1)var = 3.396 279 329 936 , E (0,1) mesh = 3.396 279 329 887 . Note that ten decimal digits in E(0,1)var coincide with ones in E(0,1)mesh. Variational parameters in (9) take values, A = −2.2957 , B = 3.6991 , a1 = 4.7096 , b1 = 0.0590 , cf. (8). Note that b1 takes a very small value similar to b0. 5. Conclusions It is presented the approximate expression (9) for the eigenfunctions in the double-well potential (5). In Non- Linearization procedure [8] it can be calculated the first correction (the first order deviation) to the func- tion (9). It can be shown that for any u ∈ (−∞, +∞) the functions (9) deviate uniformly from the exact eigenfunctions, beyond the sixth significant figure sim- ilarly to the function (6) for the single-well case. It increases the accuracy of the simplified function, pro- posed in [5] with α = 0 and b0,1 = 0, in the domain under the barrier u ∈ (0.25, 0.75) from 4 to 6 sig- nificant figures leaving the accuracy outside of this domain practically unchanged. Acknowledgements This work is partially supported by CONACyT grant A1- S-17364 and DGAPA grants IN113819, IN113022 (Mexico). AVT thanks the PASPA-UNAM program for support dur- ing his sabbatical leave. References [1] A. V. Turbiner, J. C. del Valle. Anharmonic oscillator: a solution. Journal of Physics A: Mathematical and Theoretical 54(29):295204, 2021. https://doi.org/10.1088/1751-8121/ac0733. [2] A. V. Turbiner, E. Shuryak. On connection between perturbation theory and semiclassical expansion in quantum mechanics, 2021. arXiv:2102.04623. [3] E. Shuryak, A. V. Turbiner. Transseries for the ground state density and generalized Bloch equation: Double- well potential case. Physical Review D 98:105007, 2018. https://doi.org/10.1103/PhysRevD.98.105007. [4] M. A. Escobar-Ruiz, E. Shuryak, A. V. Turbiner. Quantum and thermal fluctuations in quantum mechanics and field theories from a new version of semiclassical theory. Physical Review D 93:105039, 2016. https://doi.org/10.1103/PhysRevD.93.105039. [5] A. V. Turbiner, J. C. del Valle. Anharmonic oscillator: almost analytic solution, 2021. Talk presented by AVT at AAMP-18 (Sept.1-3), Prague, Czech Republic (September 1, 2021). [6] A. V. Turbiner. Double well potential: perturbation theory, tunneling, WKB (beyond instantons). International Journal of Modern Physics A 25(02n03):647–658, 2010. https://doi.org/10.1142/S0217751X10048937. [7] A. V. Turbiner, J. C. del Valle. Comment on: Uncommonly accurate energies for the general quartic oscillator. International Journal of Quantum Chemistry 121(19):e26766, 2021. https://doi.org/10.1002/qua.26766. [8] A. V. Turbiner. The eigenvalue spectrum in quantum mechanics and the nonlinearization procedure. Soviet Physics Uspekhi 27(9):668–694, 1984. English Translation, https://doi.org/10.1070/PU1984v027n09ABEH004155. 210 https://doi.org/10.1088/1751-8121/ac0733 http://arxiv.org/abs/2102.04623 https://doi.org/10.1103/PhysRevD.98.105007 https://doi.org/10.1103/PhysRevD.93.105039 https://doi.org/10.1142/S0217751X10048937 https://doi.org/10.1002/qua.26766 https://doi.org/10.1070/PU1984v027n09ABEH004155 Acta Polytechnica 62(1):208–210, 2022 1 Introduction 2 Single-well potential 3 Double-well potential: wavefunctions 4 Double-well potential: Results 4.1 Ground State (0,0) 4.2 First Excited State (0,1) 5 Conclusions Acknowledgements References