Parameter-free ansatz for inferring ground state wave functions of even potentials
aa r X i v : . [ qu a n t - ph ] J u l Parameter-free ansatz for inferring ground statewave functions of even potentials
S.P. Flego , A. Plastino , , , A.R. Plastino , Universidad Nacional de La Plata, Facultad de Ingenier´ıa, ´Area Departamental deCiencias 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 and Departamento de FisicaAtomica, Molecular y Nuclear, Universidad de Granada, Granada, Spain Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca, SpainE-mail: [email protected] (corresponding author)
Abstract.
Schr¨odinger’s equation (SE) and the information-optimizing principlebased on Fisher’s information measure (FIM) are intimately linked, which entails theexistence of a Legendre transform structure underlying the SE. In this comunicationwe show that the existence of such an structure allows, via the virial theorem, forthe formulation of a parameter-free ground state’s SE-ansatz for a rather large familyof potentials. The parameter-free nature of the ansatz derives from the structuralinformation it incorporates through its Legendre properties.PACS numbers: 05.45+b, 05.30-d arameter-free ansatz for inferring ground state wave functions of even potentials
1. Introduction
Few quantum-mechanical models admit of exact solutions. Approximations of diversetype constitute the hard-core of the armory at the disposal of the quantum-practitioner.Since the 60’s, hypervirial theorems have been gainfully incorporated to the pertinentarsenal [1, 2]. We revisit here the subject in an information-theory context, via Fisher’sinformation measure (FIM) with emphasis on i) its Legendre properties and ii) itsrelation with the virial theorem.Remark that the notion of using a small set of relevant expectation values so as todescribe the main properties of physical systems may be considered the leit-motivof statistical mechanics [3]. Developments based upon Jaynes’ maximum entropyprinciple constitute a pillar of our present understanding of the discipline [4]. Thistype of ideas has also been fruitfuly invoked for obtaining the probability distributionassociated to pure quantum states via Shannon’s entropy (see for instance, [5] andreferences therein). In such a spirit, Fisher information, the local counterpart ofthe global Shannon quantifier [6], first introduced for statistical estimation purposes[6]. has been shown to be quite useful for the variational characterization of quantalequations of motion [7]. In particular, it is well-known that a strong link existsbetween Fisher’s information measure (FIM) I and Schr¨odinger’s wave equation (SE)[8, 10, 9, 11, 12, 13, 14]. Such connection is based upon the fact that a constrainedFisher-minimization leads to a SE-like equation [6, 8, 10, 9, 11, 12, 13, 14]. In turn, thisguarantees the existence of intriguing relationships between various quantum quantitiesreminiscent of the ones that characterize thermodynamics due to its Legendre-invariancestructure [8, 10]. Interestingly enough, SE-consequences such as the Hellmann-Feynmanand the Virial theorems can be re-interpreted in terms of thermodynamics’ Legendrereciprocity relations [12, 11], a fact suggesting that a Legendre-transform structureunderlies the non-relativistic Schr¨odinger equation. As a consequence, the possibleenergy-eigenvalues become constrained by such structure in a rather unsuspected way[11, 12, 13, 14], which allows one to obtain a first-order differential equation, unrelatedto Schr¨oedinger’s equation [13, 14], that energy eigenvalues must necessarily satisfy.The predictive power of that equation was explored in [15], where the formalism wasapplied to the quantum anharmonic oscillator. Exploring further interesting propertiesof this “quantal-Legendre” structure will occupy us below. As a result, it will be seenthat, as a direct consequence of the Legendre-symmetry that underlies the connectionbetween Fisher’s measure and Schr¨oedinger’s equation one immediately encounters anelegant expression for an ansatz, in terms of quadratures, of the ground state (gs) wavefunction of a rather wide category of potential functions.
2. Basic ideas
A special, and particularly useful FIM-expression (not the most general one) is to bequoted. Let x be a stochastic variable and f ( x ) = ψ ( x ) the probability density function arameter-free ansatz for inferring ground state wave functions of even potentials I reads [6] I = Z f ( x ) ∂ ln f ( x ) ∂x ! dx = 4 Z dx [ ∇ ψ ( x )] ; f = ψ . (1)Focus attention now a system that is specified by a set of M physical parameters µ k .We can write µ k = h A k i , with A k = A k ( x ) . The set of µ k -values is to be regarded as ourprior knowledge (available empirical information). Again, the probability distributionfunction (PDF) is called f ( x ). Then, h A k i = Z dx A k ( x ) f ( x ) , k = 1 , . . . , M. (2)It can be shown (see [8, 9]) that the physically relevant PDF f ( x ) minimizes FIMsubject to the prior conditions and the normalization condition. Normalization entails R dxf ( x ) = 1 , and, consequently, our Fisher-based extremization problem becomes δ I − α Z dx f ( x ) − M X k =1 λ k Z dx A k ( x ) f ( x ) ! = 0 , (3)with ( M + 1) Lagrange multipliers λ k ( λ = α ). The reader is referred to Ref. [8] for thedetails of how to go from (3) to a Schr¨odinger’s equation (SE) that yields the desiredPDF in terms of the amplitude ψ ( x ). This SE is of the form " − ∂ ∂x + U ( x ) ψ = α ψ, U ( x ) = − M X k =1 λ k A k ( x ) , (4)and is to be interpreted as the (real) Schr¨odinger equation (SE) for a particle of unit mass(¯ h = 1) moving in the effective, “information-related pseudo-potential” U ( x ) [8] in whichthe normalization-Lagrange multiplier ( α/
8) plays the role of an energy eigenvalue.The λ k are fixed by recourse to the available prior information. For one-dimensionalscenarios, ψ ( x ) is real [20] and I = Z ψ ∂ ln ψ ∂x ! dx = 4 Z ∂ψ∂x ! dx = − Z ψ ∂ ∂x ψ dx (5)so from (4) one finds a simple and convenient I − expression I = α + M X k =1 λ k h A k i . (6) Legendre structure
The connection between the variational solutions f and thermodynamics was establishedin Refs. [8] and [10] in the guise of typical Legendre reciprocity relations. Theseconstitute thermodynamics’ essential formal ingredient [21] and were re-derived `a laFisher in [8] by recasting (6) in a fashion that emphasizes the role of the relevantindependent variables, I ( h A i , . . . , h A M i ) = α + M X k =1 λ k h A k i . (7) arameter-free ansatz for inferring ground state wave functions of even potentials α , that plays therole of an energy-eigenvalue in Eq. (4 ), we have α ( λ , . . . , λ M ) = I − M X k =1 λ k h A k i . (8)After these preliminaries we straightforwardly encounter the three reciprocity relations[8] ∂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 , (9)the last one being a generalized Fisher-Euler theorem.
3. Fisher measure and quantum mechanical connection
Since the potential function U ( x ) belongs to L , it admits of a series expansion in thebasis x, x x , etc. [22]. The A k ( x ) themselves belong to L as well, and can also beseries-expanded in similar fashion. This enables us to base our future considerations onthe assumption that the a priori knowledge refers to moments x k of the independentvariable, i.e., h A k i = h x k i , and that one possesses information about M of thesemoments h x k i . Our “information” potential U thus reads U ( x ) = − X k λ k x k . (10) We will assume that the first M terms of the above series yield a satisfactoryrepresentation of U ( x ). Consequently, the Lagrange multipliers are identified withU(x)’s series-expansion’s coefficients.In a Schr¨odinger-scenario the virial theorem states that [11] * ∂ ∂x + = − * x ∂∂x U ( x ) + = 18 M X k =1 k λ k h x k i , (11)and thus, from (5) and (11) a useful, virial-related expression for Fisher’s informationmeasure can be arrived at [11] I = − M X k =1 k λ k h x k i , (12) I is explicit function of the M physical parameters h x k i . Eq. (12) encodes theinformation provided by the virial theorem [12, 11].Interestingly enough, the reciprocity relations (RR) (9) can be re-derived on a strictlypure quantum mechanical basis [11], starting from the quantum Virial theorem [whichleads to Eq. (12) ] plus information provided by the quantum Hellmann-Feynman arameter-free ansatz for inferring ground state wave functions of even potentials h A k i = h x k i , our “new” reciprocityrelations 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 , (13)FIM expresses a relation between the independent variables or control variables (theprior information) 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 orencoding-process as { I, h x k i} . We see that the Legendre transform FIM-structureinvolves both eigenvalues of the “information-Hamiltonian” and Lagrange multipliers.Information is encoded in I via these Lagrange multipliers, i.e., α = α ( λ , ...λ M ) , together with a bijection { I, h x k i} ←→ { α, λ k } . 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 [12], substituting the RR given by (13) in (12) 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 . (14)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 , (15)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. (15) 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[6], I is a convex function. We may obtain λ k from the reciprocity relations (13). For h x k i > λ k = ∂I∂ h x k i = − k C k h x k i − (2+ k ) /k < . (16)and then, using (6), we obtain the α - normalization Lagrange multiplier. For adiscussion on how to obtain the reference quantities C k see [15].The general solution for the I - PDE does exist and its uniqueness has been demonstratedvia an analysis of the associated Cauchy problem [12]. Thus, Eq. (15) implies whatseems to be a kind of “universal” prescription, a linear PDE that any variationally (withconstraints) obtained FIM must necessarily comply with.
4. Present results
For even informational potentials good SWE-ansatz can be formulated via probabilitydistribution functions (PDF) that satisfy the virial theorem. The potentials are of the arameter-free ansatz for inferring ground state wave functions of even potentials U ( x ) = − M X k =1 λ k x k , (17)and the ansatz can be straightforwardly derived from (1) and (11). This constitutesour main present result. The procedure is as follows. Begin with the Fisher measure I ,“virially” expressed as I = − * ∂ ∂x + = 4 * x ∂∂x U ( x ) + −→ I = − * M X k =1 k λ k x k + , (18)which, in the Fisher-scenario, can obviouly be written as Z dx f ( x ) ∂ ln f ( x ) ∂x ! = − Z dx f ( x ) M X k =1 k λ k x k , (19)or Z dx f ( x ) ∂ ln f ( x ) ∂x ! + M X k =1 k λ k x k = 0 . (20)We devise an ansatz f A that by construction verifies (20). We merely require fulfillmentof ∂ ln f A ( x ) ∂x ! + M X k =1 k λ k x k = 0 . (21)Clearly, we inmediatly obtain, ∂ ln f A ( x ) ∂x ! = − M X k =1 k λ k x k , (22)that leads to f A ( x ) = exp − Z dx vuut − M X k =1 k λ k x k , (23)where the minus sign in the exponential argument was chosen so as to enforce thecondition that f ( x ) x →±∞ −→
0. Eq. (23) provides us with a nice, rather general andvirially motivated ansatz. Is it good enough for dealing with the SWE?. We look for ananswer below.
It is obligatory to start our investigation with reference to the harmonic oscillator. Oneassumes that the prior Fisher-information is given by h x i = 12 ω . (24)The pertinent FIM can now be obtained by using (15), I = I ( h x i ) = C h x i − , arameter-free ansatz for inferring ground state wave functions of even potentials C = 1, I h x i = C = 1 = ⇒ I = h x i − . (25)The pertinent Lagrange multiplier can be obtained by recourse to the reciprocityrelations (9) and (25), λ = ∂I∂ h x i = − h x i − . (26)The prior-knowledge (24) is encoded into the FIM (25), and the Lagrange multiplier λ (26), I = h x i − = 2 ω ; λ = − h x i − = − ω . (27)and the α − value is gotten from (8), α = I − λ h x i = 4 ω. (28)Our ansatz-PDF can be extracted from (23) as follows f ( x ) = exp (cid:26) − Z dx √ ω x (cid:27) = N exp n − ωx o , (29)with, Z f ( x ) dx = 1 −→ N = (cid:18) ωπ (cid:19) / , (30)the exact result.
5. Ground state eigenfunction of the general,even-anharmonic oscillator
We outline here the methodology for constructing the ground state ansatz for ananharmonic oscillator of the form (we shall herefrom omit the subscript A) " − d dx + X a k x k ψ ( x ) = Eψ ( x ) (31)According to [13, 14], we can ascribe to (34) a Fisher measure and effect then thefollowing identifications: α = 8 E , λ k = − a k , f ( x ) = ψ ( x ) . (32)Accordingly, we get our ansatz by substituting into (23) the quantities given by (32). ψ ( x ) = exp − Z dx vuut M X k =1 k a k x k , (33)As an illustration of the procedure, we deal below with the quartic anharmonic oscillator. arameter-free ansatz for inferring ground state wave functions of even potentials The Schr¨odinger equation for a particle of unit mass in a quartic anharmonic potentialreads, " − ∂ ∂x + 12 ω x + 12 λ x ψ = E ψ, (34)where λ is the anharmonicity constant. Expression (33) takes the form ψ ( x ) = exp (cid:26) − Z √ ω x + 8 λ x dx (cid:27) . Now, from an elemental integration, we obtain the desired eigenfunction ψ ( x ) = N exp ω λ − λω x ! / , (35)where N is the normalization constant.When λ → λ → ψ ( x ) = ψ HO = N exp (cid:16) − ωx (cid:17) , (36)and, when ω → ω → ψ ( x ) = ψ P AO = N exp − √ λ | x | ! , (37)Once we have at our dispossal the anzsatz gs-eigengenfunction, we obtain thecorresponding eigenvalues following one of the two procedures. Schr¨oedinger procedure: E ≈ h ψ | H | ψ i = Z dx ψ ( x ) " − ∂ ∂x + 12 ω x + 12 λ x ψ ( x ) == Z dx ψ ( x ) ω λω x ! / + λω x λω x ! − / − λ x ψ ( x ) . (38) Fisher procedure:
From (6) and (12), with λ = − ω , λ = − λ , we have α = I − M X k =1 λ k D x k E = − M X k =1 k ! λ k h x k i = 8 ω h x i + 12 λ h x i , (39)Evaluating the moments with the anzsatz function, we have h x p i A ≈ Z dx x p f ( x ) = Z dx x p ψ ( x ) (40)and, accordingly, E = α ≈ ω h x i A + 32 λ h x i A , (41) arameter-free ansatz for inferring ground state wave functions of even potentials E without passing first through a Schr¨odinger equation , which is a niceaspect of the present approach. The question for the suitability of our ansatz is answeredby looking at the Table below. Table :SE-ground-state eigenvalues (34) for ω = 1 and several values of the anharmonicity-constant λ .The values of the second column correspond tothose one finds in the literature, obtained via anumerical approach to the SE. These results, inturn, are nicely reproduced by some interestingtheoretical approaches that, however, need tointroduce and adjust some empirical constants[19]. Our ansatz-values, in the third column,are obtained by a parameter-free procedure. Thefourth column displays the associated Cramer-Rao bound, which is almost saturated in allinstances. λ E num
E I h x i
6. Conclusions
The link Schr¨oedinger equation - Fisher measure has been employed so as to infer, via thepertinent reciprocity relations, a parameter-free ground state ansatz wave function for a ratherample family of even potentials, of the form U ( x ) = X a k x k , (42)in terms of the coefficients a k . Its parameter-free character notwithstanding, our ansatzprovides good results, as evidenced by the examples here examined. It incorporates only theknowledge of the virial theorem, via the Legendre-symmetry that underlies the connectionbetween Fisher’s measure and Schr¨oedinger equation. One may again speak here of the powerof symmetry considerations. in devising physical treatments. Acknowledgments-
This work was partially supported by the Projects FQM-2445 and FQM-207 of the Junta de Andalucia (Spain, EU). [1] F. M. Fernandez and E. A. Castro,
Hypervirial theorems (Springer-Verlag, Berlin, 1987).[2] A. R. Plastino, M. Casas, A. Plastino, A. Puente, Phys. Rev. A (1995) 2601.[3] L. Brillouin, Science and information theory (Academic, New york, 1956).[4] A. Katz,
Principles of statistical mechanics (Freeman, San Francisco, 1967).[5] 183- A. R. Plastino, A. Plastino, Phys. Lett. A (1993) 446; M.Casas, F. Garcias, A. Plastino,Ll. Serra, Physica A (1995) 376.[6] B. R. Frieden,
Science from Fisher Information: A Unification (Cambridge, University Press;Cambridge,2004). arameter-free ansatz for inferring ground state wave functions of even potentials [7] Lett. Math. Phys. , 243 (2000).[8] B. R. Frieden, A. Plastino, A. R. Plastino, B. H. Soffer, Phys. Rev. E
60 1999 (1999) 48.[9] M. Reginatto, Phys. Rev. E (1998)1775.[10] S. P. Flego, B. R. Frieden, A. Plastino, A. R. Plastino, B. H. Soffer, Phys. Rev. E (2003)016105.[11] S. P. Flego, A. Plastino, A. R. Plastino, Physica A (2011) 2276.[12] S. P. Flego, A. Plastino, A. R. Plastino: Inferring an optimal Fisher Measure . Physica A (2011).In press.[13] S. P. Flego, A. Plastino, A. R. Plastino, Cond-mat-stat mech, arXiv:1105.5054v1 [quant-ph] (2011).[14] S. P. Flego, A. Plastino, A. R. Plastino, Cond-mat-stat mech, arXiv:1101.4661v1 [cond-mat.stat-mech] (2011).[15] S. P. Flego, A. Plastino, A. R. Plastino, Cond-mat-stat mech, arXiv:1106.2078v2 [quant-ph] (2011).[16] F.T. Hioe and E.W. Montroll, J. Math. Phys. (1975)[17] K. Banerjee, S.P.Bhatnagar, V. Choudhry, and S.S.Kanwall, Proc.R.Soc.Lond. A (1978) 575.[18] C.M. Bender and T.T. Wu, Phys Rev. (1969) 1231.[19] K. Banerjee. Proc.R.Soc.Lond. A. (1978) 265.[20] R. P. Feynman, Phys. Rev. (1939) 340.[21] A. Desloge , Thermal Physics (Holt, Rinehart and Winston, New York, 1968).[22] W. Greiner and B. M¨uller,
Quantum mechanics. An Introduction. (Springer, Berlin, 1988).[23] P. M. Mathews and K. Venkatesan,