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Exact WKB and Resurgence in Quantum Mechanics

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Exact WKB and Resurgence in Quantum Mechanics Exact WKB and resurgence in Quantum Mechanics Enric Sol´e Farr´e∗ Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain. Advisor: Bartomeu Fiol N´u~nez Abstract: A review of non-perturbative effects in Quantum Mechanics is presented. We pay special attention to the exact and uniform WKB methods. The double well potential is considered as a motivating example due to its physical modeling relevance. The first terms of the ground-state energy trans-series for this potential are calculated as a direct application of the uniform WKB method. I. INTRODUCTION The WKB ansatz has proven to be remarkably useful in countless problems, although it only provides us with Quantum mechanics is one of the cornerstones of mod- an approximate solution to the problem and yields no ern physics. Despite its countless applications, very few further understanding on the topic. cases can be solved exactly by means of ordinary func- A more insightful view was developed in the 700s by tions. Instead, we need to turn to approximation tech- the physicists C. Bender, T. Wu, A. Voros and J. Zinn- niques. A widely used method is perturbation theory, Justin, known as exact WKB, which tries to unveil the known as Rayleigh{Schr¨odingertheory when we consider geometric background behind the WKB approximation time independent problems, on which we will focus. [1], [2]. An alternative approach to the problem is that of However, some subtleties may arise in perturbation uniform WKB, which tries to approximate the solution theory, such as non-convergence or undetectable effects. by perturbing the solutions of the harmonic oscillator Some illustrative examples of non-converging ground rather than plane waves, as the usual WKB approach P1 n does [3]. energy corrections, E − E0 = n=1 an(g) , for different systems would be: The mathematician J.Ecalle´ provided a solid mathe- matical basis for such developments in the early 800s, • Zeeman effect: a ∼ (−1)n(2n)! n referred as resurgence theory [4], [5], which relies heavily • Stark effect: an ∼ (2n)! on complex analysis techniques. In the next section, the need for this framework will where g denotes the coupling associated to the perturba- be illustrated through a simple example, followed by a tion. discussion of the main points of exact WKB and resur- This divergence problem, among others, challenges the gence. To conclude, there will be a review of the example intuitive idea that the physics from QM should be cap- through this new perspective. tured by analytic functions of the couplings. The issue becomes extremely relevant when we con- sider g = . This was studied in the early days of ~ II. DOUBLE-WELL: NON-PERTURBATIVE quantum mechanics, which led to the WKB (Wentzel, CORRECTIONS Kramers & Brillouin) approximation. The WKB method consists of approximately solv- 2 Let us first consider an introductory example, the ing the Schr¨odingerequation − ~ ∆Ψ + V (x)Ψ = EΨ 2m double-well potential as in [8], x36: through an ansatz of the form i 2 2 φ(x) ~ d 1 2 p 2 Ψ(x) = C(x)e ~ (1) H = − + x (1 − gx) (3) 2 dx2 2 Imposing some reasonable assumptions on φ(x)00, we ob- tain the approximate solution where g is a small parameter. By performing a change of variable x 7! px and multiplying the Hamiltonian by g, C i Z x g we obtain Ψ(x) ≈ p exp ± p(~x)d~x (2) p(x) ~ 2g2 d2 1 q gH = −~ + x2(1 − x)2 (4) where p(x) = 2mE − V (x) and C is the normaliza- 2 dx2 2 tion constant. Notice that Ψ(x) is not analytic in , and ~ Thus, identifying g~ as a redefined Planck's constant, we in fact presents an essential singularity for ~ = 0. 1 2 2 get the Hamiltonian for V (x) = 2 x (1 − x) times g. The potential associated to (3) has two minima, one p1 at the origin and one at g , and admits a reflection sym- ∗ p1 Electronic address: [email protected] metry R around the axis x = 2 g . Exact WKB and resurgence We shall focus on studying the ground state of this The second integral on the r.h.s can be solved exactly problem. We might try to find the ground state by ex- p p 1 Z c p x x2 − c2 c2 x2 − c2 + x panding from the unperturbed quantum harmonic oscil- t2 − c2dt = − + ln lator (g = 0). However, such perturbative expansion ~ x 2~ 2~ c fails for two separate reasons. First, the ground energy (11) expansion is not convergent, as aforementioned. The first Since the harmonic part dominates, it is reasonable to p p terms, up to fifth order in g, are assume c ≈ ~ and that c << x so that c=x is small. Taylor-expanding the r.h.s of eq. (11) we get p ~ 2 9 3 2 3 E0(~; g) = − ~ g − ~ g + O(g ) (5) Z c 2 2 2 1 p 2 2 x 1 1 2x 1 t − c dt ≈ − + + ln + O 2 Second, the expansion does not yield a splitting of the ~ x 2~ 4 2 ~ x ground energy, as required for 1-D potentials in Quantum (12) Mechanics (see [6] Prob. 2.42 for non-degeneracy in 1-D Thus, we are left with QM). Intuitively, any state of the unperturbed harmonic 1=4 1=2 4e 2 1 Z well should split into two different eigenfunctions with Ψ(x) ≈ C e−x =2 exp p(t)dt (13) c well defined parity and satisfying E− > E+, where the ~ ~ subscript denotes the parity with respect to R. Let us Comparing equations (13) and (10), we deduce that focus on the energy splitting, following the work of [7] . 1 −1 Z 1=2 The cornerstone of this approach is the Herring's for- p C ≈ 4 exp p(t)dt (14) mula (see [7] Appendix A), which characterizes the en- 4πe ~ c ergy splitting of an approximate solution of energy E 0 Substituting in eq. (8), and using the fact that the po- 1 1 tencial is symmetric around 1=2, we get ∆E = 2 2Ψ p Ψ0 p (6) ~ 2 g 2 g 2 1 −1 Z 1−c ∆E ≈ ~ p exp p(t)dt (15) where the formula holds for the wave function evaluated g eπ g at the reflection point p which satisfies (p − x)= (p + c x). Similarly to the WKB ansatz of eq. (2), we can where the integral of p(t) is the one associated to the approximate our wave function as Hamiltonian (4). Notice that eq.(15) depends on c, which Z x is also unknown to us. An elaborate computation allows C i us to evaluate this instanton correction [7]. Ψ(x) ≈ p exp ± p(t)dt (7) p(x) g Removing the extra factor g on eq.(4) and restoring natural units, the final result takes the form p1 Fixing 2 g as the integral base point, we find that the 2 − 1 energy splitting can be approximated as ∆E ∼ p e 6g (16) πg 2 2 ∆E ≈ ~ C2 (8) g This result can be derived in a completely different frame- work, that of instanton mechanics (see [8] x36), which is So we are just left with calculating the normalization how it was originally obtained. condition of Ψ. In order to estimate C we shall make use It is important to note that this is not in any case the of the Hamiltonian in (4) and calculate the normalization full answer. However, it should suffice to convince the constant of the ground state of the Hamiltonian reader that there is an energy splitting, and that it is of 2 d2 1 non-perturbative nature. H = −~ + x2(1 − x)2 (9) 2 dx2 2 by requiring that our wavefunction matches the ground state solution of the quantum oscillator in the overlap- III. EXACT WKB AND RESURGENCE ping region The reader should now feel confident that non pertur- 1 2 p −x =2~ 0(x) = 4 e (10) bative terms can be essential in capturing the physics in π~ QM problems. However, the question of what type of In the overlappingp region we can approximate the mo- non-perturbative effects should be considered arises. mentum by p(x) = x2 − c2 where c corresponds to the There is mathematical foundation (see [5]) that en- turning point V (c) = E and can be neglected when sub- sures us that considering perturbations in the form of p1 stituting it in p . Thus, we claim trans-series should be enough. Trans-series are the main object of study in resurgence, which illustrates that our 1=2 C 1 Z dependence on the coupling should be of the form Ψ(x) = p exp p(t)dt p(x) ~ x j−1 1 1 j Z 1=2 Z c X X X i − c k 1 C 1 1 p f(g) = c g e g ln ± (17) ≈ p exp p(t)dt + t2 − c2dt i;j;k g x ~ c ~ x i=0 j=0 k=0 2 Exact WKB and resurgence rather than just analytic functions (power series) of g, natural objects to study are the periods of P (z)dz along where ci;j;k; c are smooth functions of the generalized co- the cycles γ 2 H1(ΣWKB): ordinates. Z The first standard approach to obtain such trans- Πγ (~) := P (z)dz (24) series, which correspond to a more careful treatment of γ the WKB approach, is known as exact WKB. The basic idea, which we will just sketch, is that exact results for Similarly to P (z)dz,Πγ is defined as a formal power observables can be obtained from suitable continued ex- series in ~: pressions for the periods of an auxiliary Riemann surface.
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