Aspects of perturbation theory in quantum mechanics: The BenderWu Mathematica ^® package

Abstract

We discuss a general setup which allows the study of the perturbation theory of an arbitrary, locally harmonic 1D quantum mechanical potential as well as its multi-variable (many-body) generalization. The latter may form a prototype for regularized quantum field theory. We first generalize the method of Bender–Wu,and derive exact recursion relations which allow the determination of the perturbative wave-function and energy corrections to an arbitrary order, at least in principle. For 1D systems, we implement these equations in an easy to use Mathematica ^® package we call BenderWu. Our package enables quick home-computer computation of high orders of perturbation theory (about 100 orders in 10–30 s, and 250 orders in 1–2 h) and enables practical study of a large class of problems in Quantum Mechanics. We have two hopes concerning the BenderWu package. One is that due to resurgence, large amount of non-perturbative information, such as non-perturbative energies and wave-functions (e.g. WKB wave functions), can in principle be extracted from the perturbative data. We also hope that the package may be used as a teaching tool, providing an effective bridge between perturbation theory and non-perturbative physics in textbooks. Finally, we show that for the multi-variable case, the recursion relation acquires a geometric character, and has a structure which allows parallelization to computer clusters.

Program summary

Program Title: BenderWu

Program Files doi: http://dx.doi.org/10.17632/vpg2zsbryc.1

Licensing provisions: CC By 4.0

Programming language: Wolfram Mathematica

Nature of problem: In 1D quantum mechanics, a perturbative expansions are known to generically be divergent. An analysis of such problems was so far limited to a case-by-case basis. The Mathematica package presented here allows a quick computation and analysis of all such 1D quantum mechanical problems.

Solution method: The program uses a general recursive relation, inspired by the works of Bender and Wu [1], which allows quick computation of the perturbative data.

References

• Carl M. Bender, Tai Tsun Wu, Phys. Rev. 184 (1969) 1231–1260

* Items marked with an asterisk are only required for new versions of programs previously published in the CPC Program Library.

Introduction

Time-independent perturbation theory in quantum mechanics, developed by Erwin Schrödinger, is almost as old as quantum mechanics itself. Published in 1926 [1], the same year as the Schrödinger equation [2], it is a standard topic of any textbook of quantum mechanics. However the method is rooted in wave-mechanics and dates back to Lord Rayleigh and his 1877 textbook on The Theory of Sound [3]. For this reason the theory is often referred to as  The Rayleigh–Schrö dinger perturbation theory.

Little did these great minds know that had they been able to compute large orders of perturbation theory, for most systems they would have come to a surprising revelation: the radius of convergence of the perturbation theory is zero. The reason for this is a factorial growth of the coefficient of the expansion parameter, which we refer to here as “the coupling” and denote by $g$.

Rather than being a curse, it is now becoming increasingly clear that the perturbative expansion is intimately tied up with non-perturbative physics. The analysis and study of such series, functions and theories goes under the name of Resurgence theory. Although the resurgence idea is old [4], it has been the subject of recent rapid development, from one-dimensional integrals [[5], [6], [7]], through quantum mechanics [[8], [9], [10], [11], [12], [13]] to quantum field theory [[14], [15]] and string theory, see e.g. [[16], [17], [18], [19]].

The resurgence structures are mostly studied in Quantum Mechanics. However, to our knowledge, all publications to date deal with case-by-case examples, and no practical general procedure of studying large orders of perturbation theory was published. In this work we adapt the method, developed originally by C. M. Bender and T. T. Wu [20] for the anharmonic oscillator with quartic term in the potential, to a perturbative expansion of an arbitrary locally-harmonic potential around one of its harmonic minima.¹ In addition we develop a workable Mathematica ^® computer code which can easily compute many orders of perturbation theory (about $\sim 100$ orders in $\sim$10–30 s, and $\sim$250 orders in 1–2 h on a modern home computer). The computation is done symbolically by default, so the result is an exact result in perturbation theory without any numerical error. Furthermore, the potential can be allowed to depend on arbitrary symbolic variables, allowing the study of the parametric dependence of the perturbation theory.

To be more specific, the code presented here allows a perturbative treatment of a one-dimensional systems with a Hamiltonian in the coordinate representation given by

$$H=-\frac{{ħ}^2}{2m}\frac{{\partial }^2}{\partial X^2}+\mathcal{V}\left(X\right),$$

where $\mathcal{V}\left(X\right)$ is an arbitrary non-singular potential. Such a potential is typically characterized by some length scale $a$ characterizing its spatial variation, and a frequency scale of the harmonic motion around one of its minima.² On dimensional grounds we can write in general

$$\mathcal{V}\left(x\right)=m{\omega }^2{a}^{2}v(X∕a),$$

where $v(X∕a)$ is an arbitrary (dimensionless) function defining a non-trivial potential. The perturbation theory is defined by an expansion of the potential $\mathcal{V}\left(X\right)=m{\omega }^2{a}^{2}v(X∕a)$ around one of its local minima, which coincide with the minima of $v(X∕a)$. Without loss of generality, we will take that one such minimum is at $x=0$, and construct the perturbation theory around it. A dimensionless combination of parameters is given by³ $g=\sqrt{\frac{ħ}{m\omega a^2}}$, which is made small when either $\omega ,a$ or $m$ is made large. A time-independent perturbation series is therefore an expansion of the wave-function and energy in this small coupling, by an expansion of the potential $v(X∕a)$ in powers of $X∕a$, treating non-quadratic terms as a perturbation of a Harmonic oscillator.

To make this more explicit, let us go to a dimensionless spatial coordinate⁴ $x=X∕\left(ag\right)$. Then the Hamiltonian becomes

$$H=\frac{{ħ}^2}{m{a}^2{g}^2}\left[-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}+\frac{v\left(xg\right)}{g^2}\right].$$

Finally, we define a reduced Hamiltonian given by

$$h=\frac{m{a}^2{g}^2}{{ħ}^2}H=-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}+\frac{v\left(xg\right)}{g^2}.$$

We wish to construct the perturbative solution to the Schrödinger equation by expanding the potential $v\left(xg\right)$ for small $g$ around one of its minima, taken to be at $x=0$,

$$\frac{v\left(xg\right)}{g^2}=\frac{v\left(0\right)}{g^2}+\frac{1}{2}{v}^{\prime \prime }\left(0\right)x^2+\frac{g}{3!}{v}^{\prime \prime \prime }\left(0\right)x^3+\cdots .$$

It is clear that the higher terms are made arbitrarily small by taking $g$ arbitrarily small. We can therefore treat all terms, except the quadratic term, as a perturbation, and solve the reduced Schrödinger equation

$$h\psi \left(x\right)=ϵ\psi \left(x\right)$$

for $\psi \left(x\right)$ and $ϵ$ as a power series

$$\psi \left(x\right)={\psi }_0\left(x\right)+{\psi }_1\left(x\right)g+{\psi }_2\left(x\right)g^2+\cdots$$

and⁵

$$ϵ={ϵ}_0+{ϵ}_{1}g+{ϵ}_2{g}^2+\cdots .$$

Note however that ${ϵ}_{2n+1}=0$ (see 2.3) for the potential at hand. To do this we will apply the Bender–Wu method, originally developed for an anharmonic oscillator with the $g{x}^4$ perturbation [20]. As we shall see, however, it is possible to generalize this method to an arbitrary potential, as long as its minima are harmonic and non-singular.

We will also be interested in a more general problem with the reduced Hamiltonian

$$h=-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}+V\left(x\right)$$

where $V\left(x\right)$ is given by

$$V\left(x\right)=\frac{v_0\left(xg\right)}{g^2}+v_2\left(xg\right).$$

The term $v_2\left(x\right)$ is clearly suppressed by a power of $g^2$ compared to the first term, and is important in the definition of supersymmetric quantum mechanics, as well as some quasi exactly solvable models which were an initial motivation for this work. Studies of these problems is the topic of a separate publication [21]. Note however that such potentials cannot be classical in nature, and must come from some quantum effects.

Although we will not implement it in the Mathematica ^® package which accompanies this work, we also give BenderWu recursion relations for the potential of the form

$$V\left(x\right)=\frac{1}{g^2}\sum _{m=0}^{\mathcal{N}}{g}^m{v}_m\left(xg\right)$$

for an arbitrary integer $\mathcal{N}$. This potential is of some interest in the literature (see for example an elegant book [22]).

Finally we also note the multi-variable (many-body) Schrödinger problem

$$\left(-\sum _{i=1}^N\frac{1}{2}{\partial }_i^2+\frac{1}{g^2}v(g{x}_1,g{x}_2,\dots ,g{x}_N)\right)\psi =ϵ\psi ,$$

is possible to treat by our methods in principle. This problem, although a natural generalization of the one-variable problem, is not pertinent to the rest of the paper. Nevertheless, given the importance of such a problem for higher dimensional and multi-particles QM problems, as well as its extensions to quantum field theory,⁶ we felt compelled to derive the recursion relations for (1.12) and explain how they are solved, at least in principle. We also restrict the discussion to non-degenerate perturbation theory only, leaving the study of general degenerate case for the future. The efficacy of this approach in the multi-dimensional problem above is also left for future work.

The paper is organized as follows: In Section 2 we generalize the method of Bender–Wu to the arbitrary potential in 1D quantum mechanics. In Section 3, we sketch how the discussion generalizes to the multi-variable quantum mechanics, and briefly discuss the geometric structure which arises there. Section 4 is dedicated to the explanation of the BenderWu Mathematica ^® package. The content of this section is largely independent of the rest of the paper, and the user interested in learning how to run and use the package is invited to go there immediately. We conclude in Section 5.

We dedicate this small section to briefly comment on the literature available and the application of the Bender–Wu method. Following the original publication the vast application of the method was used to analyze the anharmonic oscillator. A notable exception is [23], which attempted to generalize the method to a Mathieu potential, but encountered a numerical instability of the recurrence relations for level numbers $\nu >1$. Ref. [24] applied the method to the supersymmetric version of the double-well potential. In [25] the method was applied to a PT symmetric potential $x^2+ig{x}^3$, while in [26] the complex Hénon–Heiles potential was analyzed. All of these results, save the last one,⁷ are easily reproduced by the BenderWu function of our package, within seconds.