Direct Fisher inference of the quartic oscillator's eigenvalues
aa r X i v : . [ qu a n t - ph ] J un Direct Fisher inference of the quartic oscillator’s eigenvalues
S.P. Flego , A. Plastino , and A.R. Plastino , Universidad Nacional de La Plata, Facultad de Ingenier´ıa,´Area Departamental de Ciencias B´asicas, 1900 La Plata, Argentina Universidad Nacional de La Plata,Instituto de F´ısica (IFLP-CCT-CONICET),C.C. 727, 1900 La Plata, Argentina CREG-Universidad Nacional de La Plata-CONICET,C.C. 727, 1900 La Plata, Argentina Instituto Carlos I de Fisica Teorica y Computacional andDepartamento de Fisica Atomica, Molecular y Nuclear,Universidad de Granada, Granada, Spain
Abstract
It is well known that a suggestive connection links Schr¨odinger’s equation (SE) and the information-optimizing principle based on Fisher’s information measure (FIM). It has been shown that thisentails the existence of a Legendre transform structure underlying the SE. Such a structure leadsto a first order partial differential equation (PDE) for the SE’s eigenvalues from which a com-plete solution for them can be obtained. As an application we deal with the quantum theory ofanharmonic oscillators, a long-standing problem that has received intense attention motivated byproblems in quantum field theory and molecular physics. By appeal to the Cramer Rao bound weare able to Fisher-infer the particular PDE-solution that yields the eigenvalues without explicitlysolving Schr¨odinger’s equation. Remarkably enough, and in contrast with standard variationalapproaches, our present procedure does not involve free fitting parameters.KEYWORDS: Information Theory, Fisher’s Information measure, Legendre transform, Quarticanharmonic oscillator.PACS: 05.45+b, 05.30-d . INTRODUCTION It is well-known that a strong link exists between Fisher’ information measure (FIM) I [1] andSchr¨odinger wave equation (SE) [2–8]. In a nutshell, this connection is based upon the factthat a constrained Fisher-minimization leads to a SE-like equation [1–8]. In turn, this impliesthe existence of intriguing relationships between various SE-facets, and Jaynes’s maximumentropy principle. In particular, basic SE-consequences such as the Hellmann-Feynman andthe Virial theorems can be re-interpreted in terms of a special kind of reciprocity relationsbetween relevant physical quantities, similar to the ones exhibited by the thermodynam-ics’ formalism via its Legendre-invariance property [5, 6]. This fact demonstrates that aLegendre-transform structure underlies the non-relativistic Schr¨odinger equation. As a con-sequence, the possible energy-eigenvalues are now seen to be constrained by such structurein a rather unsuspected way [5–8], a fact that allows one to obtain a first-order differentialequation, unrelated to Schroedinger’s equation [7, 8], that energy eigenvalues must neces-sarily satisfy. The predictive power of this new equation will be explored here.We will apply our formalism here to the quantum anharmonic oscillator, which is the paradig-matic testing-ground for new approaches to Schroedinger eigenvalue equation. Besides theirintrinsic conceptual and mathematical interest, anharmonic oscillators have received consid-erable attention over the years due to their practical relevance in connection with severalareas of physics, such as quantum field theory and molecular physics, among others. Inthis kind of systems, the most intense focus has been traditionally concentrated upon thequartic oscillator. General accounts containing illuminating references on this problem maybe found, for instance, in [9, 10]. Note that a perturbation series solution to this problem inpowers of the anharmonicity-parameter λ is divergent-asymptotic for all λ > . BASIC IDEAS Let x be a stochastic variable and f ( x, θ ) the probability density function (PDF) for thisvariable, which depends on the parameter θ . If an observer were to make a measurement of x and had to best infer θ from such measurement, calling the resulting estimate ˜ θ = ˜ θ ( x ),one might ask how well could θ be determined. Estimation theory [1] tells us that the bestpossible estimator ˜ θ ( x ), after a very large number of x -samples is examined, suffers a mean-square error ∆ x from θ obeying the rule I (∆ x ) = 1, where I is an information quantifiercalled the Fisher information measure (FIM), a non linear functional of the PDF that reads I = Z dx f ( x, θ ) ( ∂∂θ ln [ f ( x, θ )] ) . (1)Any other estimator must have a larger mean-square error (all estimators must be unbiased,i.e., satisfy h ˜ θ ( x ) i = θ ). Thus, FIM has a lower bound. No matter what the parameter θ might be, I has to obey I (∆ x ) ≥ , (2)the celebrated Cramer–Rao bound [1].In the case of physical Fisher applications the particular instance of translational familiesmerits special consideration. These are mono-parametric distribution families of the form f ( x, θ ) = f ( x − θ ) , known up to the shift parameter θ . All family members exhibit identicalshape. For such families we get I = Z f ( x ) ∂ ln f ( x ) ∂x ! dx. (3)Focus attention now a system that is specified by a set of M physical parameters µ k . Wecan write µ k = h A k i , with A k = A k ( x ) . The set of µ k -values is to be regarded as our priorknowledge. It represents our available empirical information. Let the pertinent probabilitydistribution function (PDF) be f ( x ). Then, h A k i = Z dx A k ( x ) f ( x ) , k = 1 , . . . , M. (4)In this context it can be shown (see for example [2, 3]) that the physically relevant PDF f ( x ) minimizes FIM subject to the prior conditions and the normalization condition. Nor-malization entails R dxf ( x ) = 1 , and, consequently, our Fisher-based extremization problem3dopts the appearance δ I − α Z dx f ( x ) − M X k =1 λ k Z dx A k ( x ) f ( x ) ! = 0 , (5)where we have introduced the ( M + 1) Lagrange multipliers λ k ( λ = α ). In Ref. [2] oncan find the details of how to go from (5) to a Schr¨odinger’s equation (SE) that yields thedesired PDF in terms of the amplitude ψ ( x ) defined by f ( x ) = ψ ( x ) . This SE is of theform " − ∂ ∂x + U ( x ) ψ = α ψ, U ( x ) = − M X k =1 λ k A k ( x ) , (6)and can be formally interpreted as the (real) Schr¨odinger equation (SE) for a particle ofunit mass (¯ h = 1) moving in the effective, “information-related pseudo-potential” U ( x ) [2]in which the normalization-Lagrange multiplier ( α/
8) plays the role of an energy eigenvalue.The λ k are fixed, of course, by recourse to the available prior information. In the case ofone-dimensional scenarios, ψ ( x ) is real [13] and I = Z ψ ∂ ln ψ ∂x ! dx = 4 Z ∂ψ∂x ! dx = − Z ψ ∂ ∂x ψ dx (7)so that using the SE (6) we obtain I = α + M X k =1 λ k h A k i . (8) Legendre structure
The connection between the variational solutions f and thermodynamics was established inRefs. [2] and [4] in the guise of reciprocity relations that express the Legendre-transformstructure of thermodynamics. They constitute its essential formal ingredient [14] and werere-derived `a la Fisher in [2] by recasting (8) in a fashion that emphasizes the role of therelevant independent variables, I ( h A i , . . . , h A M i ) = α + M X k =1 λ k h A k i . (9)Obviously, the Legendre transform main goal is that of changing the identity of our relevantvariables. As for I we have α ( λ , . . . , λ M ) = I − M X k =1 λ k h A k i , (10)4o that we encounter the three reciprocity relations (proved in [2]) ∂α∂λ k = −h A k i ; ∂I∂ h A k i = λ k ; ∂I∂λ i = M X k λ k ∂ h A k i ∂λ i , (11)the last one being a generalized Fisher-Euler theorem.
3. FISHER MEASURE AND QUANTUM MECHANICAL CONNECTION
The potential function U ( x ) belongs to L and thus admits of a series expansion in the basis x, x x , etc. [15]. The A k ( x ) themselves belong to L as well and can also be series-expandedin similar fashion. This enables us to base our future considerations on the assumption thatthe a priori knowledge refers to moments x k of the independent variable, i.e., h A k i = h x k i ,and that one possesses information about M of these moments h x k i . Our “information”potential U then reads U ( x ) = − X k λ k x k . (12) We will assume that the first M terms of the above series yield a satisfactory representationof U ( x ). Consequently, the Lagrange multipliers are identified with U(x)’s series-expansion’scoefficients.In this Schr¨odinger-scenario the virial theorem states that [5] * ∂ ∂x + = − * x ∂∂x U ( x ) + = 18 M X k =1 k λ k h x k i , (13)and thus, from (7) and (13) a useful, virial-related expression for Fisher’s information mea-sure can be arrived at [5] I = − M X k =1 k λ k h x k i , (14)Also, substituting the above I -expression into (8) and solving for α , we obtain α = − M X k =1 k ! λ k h x k i . (15) α ( I ) is explicit function of the M Lagrange multipliers - U ( x )’s series-expansion coefficients λ k (associated to the physical parameters h x k i ). Eqs. (14) and (15) encode the informationprovided by the virial theorem [5, 6]. 5 isher-Schr¨oedinger Legendre structure Interestingly enough, the reciprocity relations (RR) (11) can be re-derived on a strictly purequantum mechanical basis [5], starting from1. the quantum Virial theorem [which leads to Eqs. (14) and (15)] plus2. information provided by the quantum Hellmann-Feynman theorem.This fact indicates that a Legendre structure underlays the one-dimensional Schr¨oedingerequation [5]. Thus, with h A k i = h x k i , our “new” reciprocity relations are given by ∂α∂λ k = −h x k i ; ∂I∂ h x k i = λ k ; ∂I∂λ i = M X k λ k ∂ h x k i ∂λ i , (16)FIM expresses a relation between the independent variables or control variables (the prior in-formation) and I . Such information is encoded into the functional form I = I ( h x i , ..., h x M i ).For later convenience, we will also denote such a relation or encoding as { I, h x k i} . Wesee that the Legendre transform FIM-structure involves eigenvalues of the “information-Hamiltonian” and Lagrange multipliers. Information is encoded in I via these Lagrangemultipliers, i.e., α = α ( λ , ...λ M ) , together with a bijection { I, h x k i} ←→ { α, λ k } . Two scenarios
In a n I, h x k i o - scenario , the λ k are functions dependent on the h x k i -values. As shownin [6], substituting the RR given by (16) in (14) one is led to a linear, partial differentialequations (PDE) for I , λ k = ∂I∂ h x k i −→ I = − M X k =1 k h x k i ∂I∂ h x k i . (17)and a complete solution is given by I ( h x i , ..., h x M i ) = M X k =1 C k (cid:12)(cid:12)(cid:12) h x k i (cid:12)(cid:12)(cid:12) − /k , (18)where C k are positive real numbers (integration constants). The I - domain is D I = n ( h x i , ..., h x M i ) / h x k i ∈ ℜ o o . Eq. (18) states that for h x k i > I is a monotonicallydecreasing function of h x k i , and as one expects from a “good” information measure [1], I is6 convex function. We may obtain λ k from the reciprocity relations (16). For h x k i > λ k = ∂I∂ h x k i = − k C k h x k i − (2+ k ) /k < . (19)and then, using (8), we obtain the α - normalization Lagrange multiplier.The general solution for the I - PDE does exist and its uniqueness has been demonstratedvia an analysis of the associated Cauchy problem [6]. Thus, Eq. (18) implies what seems tobe a kind of “universal” prescription, a linear PDE that any variationally (with constraints)obtained FIM must necessarily comply with. In the { α, λ k } scenario , the h x k i are functions that depend on the λ k -values. As we showedin [7], an analog α -PDE exists. Substituting the RR given by (16) in (15) we are led to ∂α∂λ k = −h x k i −→ α = M X k =1 k ! λ k ∂α∂λ k . (20)and a complete solution is given by α ( λ , ..., λ M ) = M X k =1 D k | λ k | / (2+ k ) , (21)where the D k s are positive real numbers (integration constants). The α -domain is D α = { ( λ , · · · , λ M ) /λ k ∈ ℜ} = ℜ M . Also, Eq.(21) states that for λ k < α is a monotonicallydecreasing function of the λ k , and as one expect from the Legendre transform of I , we endup with a concave function. We may obtain the h x k i ’s from the reciprocity relations (16).For λ k < h x k i = − ∂α∂λ k = 2(2 + k ) D k | λ k | − k/ (2+ k ) > . (22)and then, using (8) one us able to build up I .The general solution for α - PDE exists. Uniqueness is, again, proved from an analysis ofthe associated Cauchy problem [7]. Thus, Eq. (18) implies once more a kind of “universal”prescription, a linear PDE that all SE-eigenvalues must necessarily comply with. The mathematical structure of the Legendre transform leads to a relation between the in-tegration constants C k and D k pertaining to the I and α expressions, respectively, given7y (18) and (21). In [7] we studied with some detail this relation. In our two scenarios, n I, h x k i o and { α, λ k } , we have [7] C k = k C k , D k = k + 22 ¯ D k , with ¯ D (2+ k ) k = ¯ C kk ≡ F k . (23)Consequently, expressions (18) and (21) take the form, I = M X k =1 k " F k |h x k i| /k , α = M X k =1 k + 22 [ F k | λ k | ] / (2+ k ) . (24)The reciprocity relations (19) and (22) can thus be economically summarized in the fashion F k = | λ k | k |h x k i| (2+ k ) . (25)
4. PRESENT RESULTSThe reference quantities F k The essential FIM feature is undoubtedly its being an estimation measure known to obeythe Cramer Rao (CR) bound of Eq. (2) [1]. Accordingly, since our partial differentialequation has multiple solutions, it is natural to follow Jaynes’s MaxEnt ideas and selectamongst them the one that optimizes the CR bound, that constitutes the informationaloperative constraint in Fisher’s instance. Of course, Jaynes needs to maximize the entropyinstead. We will also, without loss of generality, renormalize the reference quantities F k . Thisprocedure is convenient because it allows us to regard these quantities as statistical weightsthat optimize the CR-bound. In other words, our procedure entails that we extremize f ( F , · · · , F M ) = I (cid:16) h x i − h x i (cid:17) = M X k =1 k " F k |h x k i| /k (cid:16) h x i − h x i (cid:17) . (26)with the constraint φ ( F , · · · , F M ) = M X k =1 F /kk = 1 . (27)We are going to apply now the preceding considerations so as to obtain the eigenvalues ofthe quartic anharmonic oscillator. 8 uartic anharmonic oscillator The Schr¨odinger equation for a particle of unit mass in a quartic anharmonic potential reads, " − ∂ ∂x + 12 k x + 12 λ x ψ = E ψ. (28)where λ is the anharmonicity constant. According to [7, 8], we can ascribe to (28) a Fishermeasure and make then the following identifications: α = 8 E , λ = − k , λ = − λ. Accordingly, we have, in the { α, λ k } - scenario [Cf. (24)], α = 2 F / | λ | / + 3 F / | λ | / . (29)The functions f and φ defined by (26) and (27), respectively, can here be recast [using (25)]as f ( F , F ) = F + 2 F / F / | λ | − / | λ | / ,φ ( F , F ) = F + F / . After these preparatory moves we can recast our methodology in a convenient specializedfashion, suitable for the task at hand. We just face the simple two-equations system: ~ ∇ f ( F , F ) = µ ~ ∇ φ ( F , F ) φ ( F , F ) = 1 (30)where ~ ∇ ≡ ( ∂ F , ∂ F ). Straightforward solution of it yields F − / (1 − F ) − / (7 F −
3) = 3 | λ | / | λ | − / , F = (1 − F ) , (31)from which we obtain F and F . Substituting them into (29) we determine α and, ofcourse, the eigenvalue E = α/
8. Consider now our SE (28), taking k = 1 and a given valueof λ (0 . ≤ λ ≤ f ( F , F ) = I h x i exhibits, as afunction of its arguments, a unique “critical” point that satisfies (31). Using f = f critical ,that optimizes the CR-bound, we find a ground-state eigenvalue that is in good agreementthe literature. In this way, after properly dealing with (24), with the F k regarded as “FIMstatistical weights” that optimize the Cramer Rao inequalities, we determine α as a functionof the λ k without passing first through a Schr¨odinger equation , which is a notable aspect ofthe present approach. 9nterestingly enough, the Cramer-Rao inequality us equivalent to the quantum uncertaintyprinciple (see the Appendix for details and references). Thus, our methodology actuallyemploys Heisenberg’s celebrated principle to pick up just one solution among the severalones that our partial differential equation possesses. Table : Ground-state eigenvalues of the SE (28) for k = 1and several values of the anharmonicity constant λ . Thevalues of the second column correspond to those one findsin the literature, obtained via a numerical approach to theSE. These results, in turn, are nicely reproduced by someinteresting theoretical approaches that, however, need tointroduce and adjust some empirical constants [12]. Ourvalues, in the third column, are obtained by means thepresent theoretical, parameter-free procedure. The fourthcolumn displays the associated Cramer-Rao bound. λ E num E = α/ f = I h x i
5. CONCLUSIONS
On the basis of a variational principle based on Fisher’s information measure, free of ad-justable parameters, we have obtained the Schr¨odinger energy-eigenvalues for the funda-mental state of the quartic anharmonic oscillator (for several anharmoniticy-values). Ourtheoretical results, obtained without passing first through a Schr¨odinger equation, are in agood agreement with those of the literature. This constitutes an illustration of the power ofinformation-related tools in analyzing physical problems.Thus, we have in this communication introduced a new general technique for eigenvalue-problems of linear operators, whose use seems to constitute a promising venue, given theresults here displayed.
Acknowledgments-
This work was partially supported by the Projects FQM-2445 andFQM-207 of the Junta de Andalucia (Spain, EU).10
PPENDIX: CRAMER-RAO AND UNCERTAINTY PRINCIPLE
It is well known that the Cramer-Rao inequality may be regarded as an expression of Heisen-berg’s Uncertainty Principle (See, for instance, [1]). Remember that a precise statement ofthe position-momentum uncertainty principe reads [16](∆ x )(∆ p ) ≥ ¯ h or (∆ x ) (∆ p ) ≥ ¯ h , (32)where (∆ x ) = D ( x − h x i ) E = h x i − h x i (33)(∆ p ) = D ( p − h p i ) E = h p i − h p i . (34)In a one-dimensional configuration-space, if ψ is a normalizable real wave function, h p i = h− i ¯ h ∂∂x i = − i ¯ h Z ψ ∂∂x ψ dx = − i ¯ h Z ∂∂x ψ dx = 0 , (35) h p i = * − ¯ h ∂ ∂x + = − ¯ h Z ψ ∂ ∂x ψ dx . (36)Substituting (35) and (36) in (34) and using (7) leads to the above mentioned connectionbetween the uncertainty in momentum ∆ p and the Fisher’s measure I , i.e.,(∆ p ) = − ¯ h Z ψ ∂ ∂x ψ dx = ¯ h I . (37)If this relation is substituted into (32) we immediately arrive to the the CR-bound,(∆ x ) (∆ p ) ≥ ¯ h −→ I (∆ x ) ≥ . (38)Coming now back to the { α, λ k } -scenario, one easily ascertains that Eq. (26) can be givena clear “Heisenberg’s aspect” f ( F , · · · , F M ) = M X k =1 k F k | λ k | ] / (2+ k ) (cid:16) F / | λ | − / − F / | λ | − / (cid:17) . [1] B. R. Frieden, Science from Fisher Information: A Unification (Cambridge, University Press;Cambridge,2004).[2] B. R. Frieden, A. Plastino, A. R. Plastino, B. H. Soffer, Phys. Rev. E
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