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–Schrödingertheory 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 P∞ n does [3]. energy corrections, E − E0 = n=1 an(g) , for different systems would be: The mathematician J.Ecalle´ provided a solid mathematical 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 consider 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 Schrödingerequation − ~ ∆Ψ + V (x)Ψ = EΨ 2m double-well potential as in [8], §36: through an ansatz of the form

i 2 2 φ(x) ~ d 1 2 √ 2 Ψ(x) = C(x)e ~ (1) H = − + x (1 − gx) (3) 2 dx2 2 Imposing some reasonable assumptions on φ(x)00, we obtain the approximate solution where g is a small parameter. By performing a change of variable x 7→ √x 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 √1 at the origin and one at g , and admits a reﬂection sym- ∗ √1 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- √ √ 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 √ √ 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 √ ~ 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- √ 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 √ Ψ0 √ (6) ~ 2 g 2 g 2 1 −1 Z 1−c ∆E ≈ ~ √ exp p(t)dt (15) where the formula holds for the wave function evaluated g eπ g at the reflection point p which satisfies ψ(p − x)=ψ(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 √1 Fixing 2 g as the integral base point, we find that the 2 − 1 energy splitting can be approximated as ∆E ∼ √ e 6g (16) πg 2 2 ∆E ≈ ~ C2 (8) g This result can be derived in a completely different framework, that of instanton mechanics (see [8] §36), 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 √ −x /2~ ψ0(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 overlapping√ 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 √1 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 ∞ ∞ 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) ≈ √ 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 ∈ 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. Z ∞ (n) X (n) n In what follows, we shall consider complex variables Πγ := qn(z)dz Πγ (~) = Πγ ~ (25) and functions instead of real-valued ones, since all con- γ n=0 sidered functions can be extended reasonably in a unique way. In fact, the mathematical foundations of resurgence We would expect Πγ (~) to be an analytical function of ~, makes extensive use of the notion of holomorphicity, in- thus making sense of Ψ as a wave function, which would trinsic to complex analysis. be a solution to the Schrödingerproblem. However it can be proven that the formal power series in eq. (25) has a (n) zero radius of convergence, with growth Πγ ∼ n!. A. Exact WKB In order to make sense of the WKB periods, we in- troduce the Borel and Laplace transforms, which are what will give rise to the instanton (e−1/x) and quasi- In the complex setting, the ansatz (1) is rewritten as zero mode (ln(1/x)) corrections . i Z z Ψ(z) = exp ψ(˜z, ~)d˜z (18) ~ B. Borel and Laplace transforms which, when plugged into the Schrödingerequation, leads to the Riccati equation The Borel and Laplace transforms will only be presented, without much further comment. Further discus- dψ sion can be found in [5]. ~ + ψ2 = E − V (z) (19) i dz The Borel transfrom acts on formal power series as P n follows. Given a formal power series S = n≥0 ang , we This equation does not admit a solution in terms of usual define its Borel transform by functions, but can be written down as a formal power ∞ series in . X an ~ BS(ξ) = ξn (26) n! ∞ n=0 X n ψ(z) = pn(z)~ (20) n=0 A remarkable aspect of the Borel transform is that it guarantees us that the Borel transform of our WKB peri- It can be proven that the odd terms of such a solution ods will define a holomorphic function in a neighborhood (n) will not contribute to the integral in eq. (18). Thus, and of 0, as long as the growth condition Πγ ∼ n! holds. By by imposing normalizing conditions, we obtain an exact means of analytic continuation, we can extend B(Πγ ) to solution to the Schrödingerequation in terms of formal a larger open set. power series The Laplace transform is a widely used tool in analysis and admits several definitions. We shall use a particu- Z z 1 i lar case that suits us due to our index choices and the Ψ(z) = p exp P (˜z, ~)d˜z (21) P (z) ~ complex setting we are working with. Let f(z) be an a.e. ∞ ∞ integrable function whose growth is dominated by eAz at X 2n X n P (z) = p2n(z)~ = qn(z)~ (22) infinity. Then, the Laplace transform of f is defined as n=0 n=0 Z teiϕ 1 −z/ξ q Lϕ(f)(ξ) = lim f(z)e dz (27) Notice that p0(z) = p(z) = 2m E − V (z) . ξ t→∞ 0 In virtue of equation (19), the meromorphic differential The Laplace and Borel can be viewed as inverse of each P (z)dz defined in equation (22) can be regarded as a other, due to the fact that any holomorphic function f on curve in the two dimensional Riemann surface Σ Az WKB C, whose growth is dominated by e at infinity, satisfies 2 w = 2m E − V (z) (23) B L0(f) (z) = L0 B(f) (z) = f(z) (28) seen as a section of T ∗Σ, where Σ is the coordinate man- Remarkably, this procedure can provide us with some ifold. We will assume Σ = 1 . Because of this, the interesting new results to sum divergent series whenever PC

3 Exact WKB and resurgence the previous growth condition is not met. In the context A. Quantization condition of exact WKB, we define the Borel sum of a period as When first studying the double potential it was ar- sϕ(Πγ )(~) = Lϕ B(f) (~) (29) gued that there should be a splitting of the energy and and say Πγ is summable if sϕ(Πγ ) is well-defined for a that the corresponding wave eigenfunctions should have small enough ~. well defined parity, the even wavefunction having lower The instanton corrections appear whenever sϕ(Πγ ) is energy. ill-defined and one needs to take lateral re-summations. The uniform WKB method allows us to derive a quan- In that case, they are related to the quantity tization condition for our problem. But in order to fully capture the essence of this, it is necessary to use the discϕ(Π) = lim sϕ+δ(Πγ ) − sϕ−δ(Πγ ) (30) δ→0 resurgent expansion of the parabolic cylinder functions, which can be calculated for the different regions of the complex plane as explained in §III.B. and becomes rele- IV. DOUBLE-WELL: REVISITED vant whenever ν∈ / N. According to the NIST function database [9], Let us reconsider the example of §2 from an exact per- √ spective. Whilst exact WKB provides the conceptual 2 2π 2 D (z) ∼ zν e−z /4F (z2) + e±iπν z−1−nez /4F (z2) framework to understand the WKB in order to actually ν 1 Γ(−ν) 2 carry out computations, we will resort to uniform WKB, (36) detailed in [3]. The uniform WKB studies the problem of with the double-well by making use of the parabolic cylinder functions ∞ ν ν−1 k 2 X Γ(k − 2 )Γ(k − 2 ) −2 F1(z ) = d2 z2 Γ( ν )Γ( 1−ν )k! z2 k=0 2 2 − 2 Dν (z) + Dν (z) = ν + 1/2 (31) dz 4 ∞ ν+1 ν+2 k 2 X Γ(k + 2 )Γ(k + 2 ) 2 rather than perturbating a plane wave, like the usual F2(z ) = Γ( ν+1 )Γ( ν+2 )k! z2 WKB method. Notice that parabolic cylinder functions k=0 2 2 when ν ∈ N are related to the solution of the nth energy state of the harmonic oscillator by which can be related to the hypergeometric function of the first kind through the reflection formula of the r r π 1 4 mω 2mω Gamma function: Γ(z)Γ(1 − z) = . ψn(x) = √ Dn x (32) sin(πz) π n! ~ ~ The parity condition can be translated in terms of the In the case of the double well, by performing the trans- value of the wavefunction or its derivative at the critical √ 0 1 formation y = gz, our ansatz for the wave function is point. Thus, an even function will fulfill Ψ 2 = 0 whilst an odd function will be characterized by Ψ 1 = 0. 1 u(y) 2 Ψ(y) = p Dν √ (33) Using the expansion (36), and performing some calcu- u0(y) g lations, we arrive at where ν is given by some global conditions which we 1 2e±iπ −ν will discuss in the next subsection. Plugging Ψ in the = (−1)(1+PΨ)/2 ξH(ν, g) (37) Schrödingerequation and making use of eq. (31) we ob- Γ(−ν) g tain the differential equation where PΨ is the parity eigenvalue of Ψ, and 2 √ 00 0 2 0 2 1 g 0 u u (u ) 2 2 g ν+ + u − = 2gE−y (1−y) 2 2 2 (u0)3/2 4 1 u0 (1/2) ξ = √ exp − (38) (34) πg 2g which can be solved recursively in terms of g, in the same 2 way we did with eq.(19). Although this expansion might F u (1/2) u2(1/2)ν+1/2 1 g seem more complicated to work with than (19), uniform H = WKB offers some advantages in front of exact WKB. If 2 u2(1/2) F2 we set n = ν + 1/2, the first terms of the energy expan- g sions are 1 × exp − u2(1/2) − u 2(1/2) (39) 1 19 2g 0 E(g) = n − g 3n2 + − g2n 17n2 + + ... (35) 4 4 with u0 the zeroth order term of eq.(34), which is given Notice that if we take ν = 0, the quantization condition by of the unperturbed oscillator, we obtain the same expan- Z y √ sion (5) which had been calculated through Rayleigh- 2 0 2 2 u0(u0) = 4V =⇒ u0(y) = 4 2V dy (40) Schrödingerperturbation theory. 0

4 Exact WKB and resurgence with V (y) = y2(1 − y)2. Evaluating for the double well, stated. Thus, the ﬁrst terms of the energy trans-series of we get the ground state are 2 1 1 1 −1 u = =⇒ ξ = √ e 6g (41) 0 2 3 πg 1 E(g) = − ξH + ξ2 H H(1) + (γ + σ )H2 2 0 0 0 ± 0 The reader will immediately realize that 2ξ = ∆E from h i §II. This follows from the parity dependence of (37). (1) − g 1 + 3ξH0 1 − ξH0(1 + γ + σ±) + ξH0 (45)

B. Energy trans-series since β = −1 as the ground state is even. The provided tools would allow us to continue expansions (44) and (45) algorithmically to any order, yielding an expansion of the Let us end this dissertation by giving the first terms form of (17), in agreement to the previous statements. of the trans-series of what has been our main topic of discussion: the ground state energy of the double well V. CONCLUSIONS potential. First, recall the Taylor expansion [9] The Schrödinger equation gives rise to non classi- 1 1 cal phenomena which can not be captured by standard ≈ −x + γx2 + π2 + 6γ2 x3 + O(x4) (42) means of perturbation theory, but their physical signifi- Γ(−x) 12 cance can be captured through instanton and quasi-zero where γ is the Euler-Mascheroni constant. mode corrections in the form of (17). The strategy will be to approximate the quantization Our study has led us to two different approaches on condition (37) around zero in terms of the instanton fac- how to calculate such generalized corrections. Exact P k 2 WKB presents itself as the more geometric approach to tor ξ by taking δν = k akξ . Take σ± = ln g ± iπ (i) di the problem and allows us to investigate the conceptual and H0 = H(0, g), H0 = dνi H(0, g). By (42), we get nature behind such phenomena. The paper [2] points out how this more conceptual understanding has allowed to X X 1 (i) P (σ )δνi+1 = (−1)(1+PΨ)/2 ξ H δνi (43) relate trans-series with problems in the theory of inte- i ± i! 0 i i grable systems. In contrast, uniform WKB can easily be used to derive where the Pi are polynomials in σ±, the first three being the precise coefficients of the trans-series expansion algorithmically for a given potential, as it has been done for P0(σ±) = −1 the double well potential. Geometrical and topological P1(σ±) = γ + σ± information can also be obtained through this method, π2 1 P (σ ) = − (γ + σ )2 as explained in [3]. 2 ± 12 2 ± Solving for δν, we get Acknowledgments δν = βξH + ξ2 H H(1) + (γ + σ )H2 + ... (44) 0 0 0 ± 0 I would like to thank Dr. Bartomeu Fiol, without whom this work would have not been possible and Dr. where β = −(−1)(1+PΨ)/2. All that remains is to replace Joan Carles Naranjo, for the mathematical perspective 1 n = 2 + δν into the expansion (35) with δν as previously he shed on this work.

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