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arXiv:1107.4221v1 [quant-ph] 21 Jul 2011

Parameter-free ansatz for inferring ground state

wave functions of even potentials

S.P. Flego1, A. Plastino2,4,5, A.R. Plastino3,4

1Universidad Nacional de La Plata, Facultad de Ingenier´ ıa,´Area Departamental de

Ciencias B´ asicas, 1900 La Plata, Argentina

2Universidad Nacional de La Plata, Instituto de F´ ısica (IFLP-CCT-CONICET), C.C.

727, 1900 La Plata, Argentina

3CREG-Universidad Nacional de La Plata-CONICET, C.C. 727, 1900 La Plata,

Argentina

4Instituto Carlos I de Fisica Teorica y Computacional and Departamento de Fisica

Atomica, Molecular y Nuclear, Universidad de Granada, Granada, Spain

5Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca, Spain

E-mail: angeloplastino@gmail.com (corresponding author)

Abstract.

based on Fisher’s information measure (FIM) are intimately linked, which entails the

existence of a Legendre transform structure underlying the SE. In this comunication

we show that the existence of such an structure allows, via the virial theorem, for

the formulation of a parameter-free ground state’s SE-ansatz for a rather large family

of potentials. The parameter-free nature of the ansatz derives from the structural

information it incorporates through its Legendre properties.

Schr¨ odinger’s equation (SE) and the information-optimizing principle

PACS numbers: 05.45+b, 05.30-d

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Parameter-free ansatz for inferring ground state wave functions of even potentials2

1. Introduction

Few quantum-mechanical models admit of exact solutions. Approximations of diverse

type 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 pertinent

arsenal [1, 2]. We revisit here the subject in an information-theory context, via Fisher’s

information measure (FIM) with emphasis on i) its Legendre properties and ii) its

relation with the virial theorem.

Remark that the notion of using a small set of relevant expectation values so as to

describe the main properties of physical systems may be considered the leit-motiv

of statistical mechanics [3]. Developments based upon Jaynes’ maximum entropy

principle constitute a pillar of our present understanding of the discipline [4]. This

type of ideas has also been fruitfuly invoked for obtaining the probability distribution

associated to pure quantum states via Shannon’s entropy (see for instance, [5] and

references therein). In such a spirit, Fisher information, the local counterpart of

the global Shannon quantifier [6], first introduced for statistical estimation purposes

[6]. has been shown to be quite useful for the variational characterization of quantal

equations of motion [7]. In particular, it is well-known that a strong link exists

between 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 constrained

Fisher-minimization leads to a SE-like equation [6, 8, 10, 9, 11, 12, 13, 14]. In turn, this

guarantees the existence of intriguing relationships between various quantum quantities

reminiscent of the ones that characterize thermodynamics due to its Legendre-invariance

structure [8, 10]. Interestingly enough, SE-consequences such as the Hellmann-Feynman

and the Virial theorems can be re-interpreted in terms of thermodynamics’ Legendre

reciprocity relations [12, 11], a fact suggesting that a Legendre-transform structure

underlies the non-relativistic Schr¨ odinger equation.

energy-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, unrelated

to 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 was

applied to the quantum anharmonic oscillator. Exploring further interesting properties

of this “quantal-Legendre” structure will occupy us below. As a result, it will be seen

that, as a direct consequence of the Legendre-symmetry that underlies the connection

between Fisher’s measure and Schr¨ oedinger’s equation one immediately encounters an

elegant expression for an ansatz, in terms of quadratures, of the ground state (gs) wave

function of a rather wide category of potential functions.

As a consequence, the possible

2. Basic ideas

A special, and particularly useful FIM-expression (not the most general one) is to be

quoted. Let x be a stochastic variable and f(x) = ψ(x)2the probability density function

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Parameter-free ansatz for inferring ground state wave functions of even potentials3

(PDF) for this variable. Then I reads [6]

I =

?

f(x)

?∂ lnf(x)

∂x

?2

dx = 4

?

dx[∇ψ(x)]2;f = ψ2. (1)

Focus attention now a system that is specified by a set of M physical parameters µk.

We can write µk= ?Ak?, with Ak= Ak(x). The set of µk-values is to be regarded as our

prior knowledge (available empirical information). Again, the probability distribution

function (PDF) is called f(x). Then,

?Ak? =

?

dx Ak(x) f(x),k = 1,...,M.(2)

It can be shown (see [8, 9]) that the physically relevant PDF f(x) minimizes FIM

subject to the prior conditions and the normalization condition. Normalization entails

?dxf(x) = 1, and, consequently, our Fisher-based extremization problem becomes

δI − α

k=1

with (M +1) Lagrange multipliers λk(λ0= α). The reader is referred to Ref. [8] for the

details of how to go from (3) to a Schr¨ odinger’s equation (SE) that yields the desired

PDF in terms of the amplitude ψ(x). This SE is of the form

?

?

dx f(x) −

M

?

λk

?

dx Ak(x) f(x)

?

= 0, (3)

?

−1

2

∂2

∂x2+ U(x)

?

ψ =

α

8ψ,U(x) = −1

8

M

?

k=1

λkAk(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 which

the normalization-Lagrange multiplier (α/8) plays the role of an energy eigenvalue.

The λk are fixed by recourse to the available prior information. For one-dimensional

scenarios, ψ(x) is real [20] and

I =

?

ψ2

?∂ lnψ2

∂x

?2

dx = 4

??∂ψ

∂x

?2

dx = −4

?

ψ∂2

∂x2ψ dx (5)

so from (4) one finds a simple and convenient I−expression

M

?

I = α +

k=1

λk?Ak?.(6)

Legendre structure

The connection between the variational solutions f and thermodynamics was established

in Refs. [8] and [10] in the guise of typical Legendre reciprocity relations.

constitute thermodynamics’ essential formal ingredient [21] and were re-derived ` a la

Fisher in [8] by recasting (6) in a fashion that emphasizes the role of the relevant

independent variables,

These

I(?A1?,...,?AM?) = α +

M

?

k=1

λk?Ak?. (7)

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Parameter-free ansatz for inferring ground state wave functions of even potentials4

Obviously, the Legendre transform main goal is that of changing the identity of our

relevant independent variables. As for the normalization multiplier α, that plays the

role of an energy-eigenvalue in Eq. (4 ), we have

α(λ1,...,λM) = I −

M

?

k=1

λk?Ak?. (8)

After these preliminaries we straightforwardly encounter the three reciprocity relations

[8]

∂α

∂λk

= −?Ak? ;

∂I

∂?Ak?= λk;

∂I

∂λi

=

M

?

k

λk∂?Ak?

∂λi

, (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 L2, it admits of a series expansion in the

basis x, x2x3, etc. [22]. The Ak(x) themselves belong to L2as well, and can also be

series-expanded in similar fashion. This enables us to base our future considerations on

the assumption that the a priori knowledge refers to moments xkof the independent

variable, i.e., ?Ak?

moments ?xk?. Our “information” potential U thus reads

U(x) = −1

8

k

We will assume that the first M terms of the above series yield a satisfactory

representation ofU(x). Consequently, the Lagrange multipliers are identified with

U(x)’s series-expansion’s coefficients.

In a Schr¨ odinger-scenario the virial theorem states that [11]

=?xk?, and that one possesses information about M of these

?

λkxk. (10)

?∂2

∂x2

?

= −

?

x∂

∂xU(x)

?

=

1

8

M

?

k=1

k λk?xk? ,(11)

and thus, from (5) and (11) a useful, virial-related expression for Fisher’s information

measure can be arrived at [11]

I = −

M

?

k=1

k

2λk?xk?, (12)

I is explicit function of the M physical parameters ?xk?.

information provided by the virial theorem [12, 11].

Eq. (12) encodes the

Interestingly enough, the reciprocity relations (RR) (9) can be re-derived on a strictly

pure quantum mechanical basis [11], starting from the quantum Virial theorem [which

leads to Eq.(12) ] plus information provided by the quantum Hellmann-Feynman

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Parameter-free ansatz for inferring ground state wave functions of even potentials5

theorem.

dimensional Schr¨ oedinger equation [11]. Thus, with ?Ak? = ?xk?, our “new” reciprocity

relations are given by

This fact strongly suggests that a Legendre structure underlays the one-

∂α

∂λk

= −?xk? ;

∂I

∂?xk?= λk;

∂I

∂λi

=

M

?

k

λk∂?xk?

∂λi

, (13)

FIM expresses a relation between the independent variables or control variables (the

prior information) and I.Such information is encoded into the functional form

I = I(?x1?,...,?xM?). For later convenience, we will also denote such a relation or

encoding-process as {I,?xk?}.

involves both eigenvalues of the “information-Hamiltonian” and Lagrange multipliers.

Information is encoded in I via these Lagrange multipliers, i.e., α = α(λ1,...λM),

together with a bijection {I,?xk?}

In a

?

in [12], substituting the RR given by (13) in (12) one is led to a linear, partial differential

equations (PDE) for I,

We see that the Legendre transform FIM-structure

←→{α,λk}.

I,?xk?

?

- scenario, the λkare functions dependent on the ?xk?-values. As shown

λk =

∂I

∂?xk?

−→I = −

M

?

k=1

k

2?xk?

∂I

∂?xk?. (14)

and a complete solution is given by

I(?x1?,...,?xM?) =

M

?

k=1

Ck

????xk?

???

−2/k, (15)

where Ckare positive real numbers (integration constants). The I - domain is DI =

?

decreasing function of ?xk?, and as one expects from a “good” information measure

[6], I is a convex function. We may obtain λkfrom the reciprocity relations (13). For

?xk? > 0 one gets,

λk =

∂?xk?

and then, using (6), we obtain the α - normalization Lagrange multiplier.

discussion on how to obtain the reference quantities Cksee [15].

The general solution for the I - PDE does exist and its uniqueness has been demonstrated

via an analysis of the associated Cauchy problem [12]. Thus, Eq. (15) implies what

seems to be a kind of “universal” prescription, a linear PDE that any variationally (with

constraints) obtained FIM must necessarily comply with.

(?x1?,...,?xM?)/?xk? ∈ ℜo

?

. Eq. (15) states that for ?xk? > 0, I is a monotonically

∂I

= −

2

kCk ?xk?− (2+k)/k< 0 . (16)

For a

4. Present results

4.1. Inferring the PDF for even potentials

For even informational potentials good SWE-ansatz can be formulated via probability

distribution functions (PDF) that satisfy the virial theorem. The potentials are of the

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Parameter-free ansatz for inferring ground state wave functions of even potentials6

form

U(x) = −1

8

M

?

k=1

λ2kx2k, (17)

and the ansatz can be straightforwardly derived from (1) and (11). This constitutes

our main present result. The procedure is as follows. Begin with the Fisher measure I,

“virially” expressed as

I = − 4

?∂2

∂x2

?

= 4

?

x∂

∂xU(x)

?

−→I = −

?

M

?

k=1

k λ2kx2k

?

, (18)

which, in the Fisher-scenario, can obviouly be written as

?

dx f(x)

?∂ lnf(x)

∂x

?2

=−

?

dx f(x)

M

?

k=1

k λ2kx2k, (19)

or

?

dx f(x)

?∂ lnf(x)

∂x

?2

+

M

?

k=1

kλ2kx2k

= 0. (20)

We devise an ansatz fAthat by construction verifies (20). We merely require fulfillment

of

?2

k=1

Clearly, we inmediatly obtain,

?∂ lnfA(x)

∂x

+

M

?

k λ2kx2k= 0. (21)

?∂ lnfA(x)

∂x

?2

= −

M

?

k=1

k λ2kx2k, (22)

that leads to

fA(x) = exp

−

0. Eq. (23) provides us with a nice, rather general and

virially motivated ansatz. Is it good enough for dealing with the SWE?. We look for an

answer below.

?

dx

?

?

?

?−

M

?

k=1

k λ2kx2k

, (23)

where the minus sign in the exponential argument was chosen so as to enforce the

condition that f(x) −→

x→±∞

4.2. Harmonic oscillator (HO)

It is obligatory to start our investigation with reference to the harmonic oscillator. One

assumes that the prior Fisher-information is given by

1

2ω.

The pertinent FIM can now be obtained by using (15),

?x2? =

(24)

I = I(?x2?) = C2?x2?−1,

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Parameter-free ansatz for inferring ground state wave functions of even potentials7

which saturates the Cramer-Rao bound [6] when C2= 1,

I ?x2? = C2 = 1=⇒I = ?x2?−1.(25)

The pertinent Lagrange multiplier can be obtained by recourse to the reciprocity

relations (9) and (25),

λ2 =

∂I

∂?x2?= − ?x2?−2. (26)

The prior-knowledge (24) is encoded into the FIM (25), and the Lagrange multiplier λ2

(26),

I = ?x2?−1= 2ω;λ2 = − ?x2?−2= − 4ω2.(27)

and the α−value is gotten from (8),

α = I − λ2?x2? = 4 ω.

Our ansatz-PDF can be extracted from (23) as follows

(28)

f(x) = exp

?

−

?

dx

√

4ω2x2

?

= N exp

?

−ωx2?

, (29)

with,

?

f(x) dx = 1−→N =

?ω

π

?1/2

, (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 an

anharmonic oscillator of the form (we shall herefrom omit the subscript A)

?

−1

2

d2

dx2+

?a2kx2k

?

ψ(x) = Eψ(x) (31)

According to [13, 14], we can ascribe to (34) a Fisher measure and effect then the

following identifications:

α = 8E ,λ2k= −8 a2k,f(x) = ψ2(x). (32)

Accordingly, we get our ansatz by substituting into (23) the quantities given by (32).

ψ(x) = exp

−1

2

?

dx

?

?

?

?

M

?

k=1

8 ka2kx2k

, (33)

As an illustration of the procedure, we deal below with the quartic anharmonic oscillator.

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Parameter-free ansatz for inferring ground state wave functions of even potentials8

5.1. Quartic anharmonic oscillator

The Schr¨ odinger equation for a particle of unit mass in a quartic anharmonic potential

reads,

∂2

∂x2+

?

−1

2

1

2ω2x2+1

2λ x4

?

ψ = E ψ,(34)

where λ is the anharmonicity constant. Expression (33) takes the form

−1

2

Now, from an elemental integration, we obtain the desired eigenfunction

ψ(x) = exp

?

?

√4ω2x2+ 8λ x4dx

?

.

ψ(x) = N exp

ω3

6λ

1 −

?

1 +2λ

ω2x2

?3/2

, (35)

where N is the normalization constant.

When λ → 0 one re-obtains the Gaussian form,

lim

λ→0ψ(x) = ψHO = N exp

?

−ωx2?

√2λ

, (36)

and, when ω → 0 the pure anharmonic oscillator eigenfunction is given by,

lim

ω→0ψ(x) = ψPAO = N exp

?

−

3

|x|3

?

, (37)

Once we have at our dispossal the anzsatz gs-eigengenfunction, we obtain the

corresponding eigenvalues following one of the two procedures.

Schr¨ oedinger procedure:

E ≈ ?ψ|H|ψ? =

?

dx ψ(x)

?

−1

2

?1/2

∂2

∂x2+

1

2ω2x2+1

2λ x4

?

ψ(x) =

=

?

dx ψ(x)

ω

2

?

1 +2λ

ω2x2

+λ

ωx2

?

1 +2λ

ω2x2

?−1/2

−λ

2x4

ψ(x) . (38)

Fisher procedure:

From (6) and (12), with λ2= −4ω2, λ4= −4λ, we have

M

?

Evaluating the moments with the anzsatz function, we have

α = I −

k=1

λk

?

xk?

= −

M

?

k=1

?k

2+ 1

?

λk?xk? = 8 ω2?x2? + 12 λ ?x4? , (39)

?xp?A≈

?

dx xpf(x) =

?

dx xpψ2(x) (40)

and, accordingly,

E =α

8

≈ ω2?x2?A+3

2λ ?x4?A, (41)

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Parameter-free ansatz for inferring ground state wave functions of even potentials9

We determine E without passing first through a Schr¨ odinger equation, which is a nice

aspect of the present approach. The question for the suitability of our ansatz is answered

by 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 to

those one finds in the literature, obtained via a

numerical approach to the SE. These results, in

turn, are nicely reproduced by some interesting

theoretical approaches that, however, need to

introduce and adjust some empirical constants

[19]. Our ansatz-values, in the third column,

are obtained by a parameter-free procedure. The

fourth column displays the associated Cramer-

Rao bound, which is almost saturated in all

instances.

λEnum

0.50003749

0.50037435

0.50368684

0.53264275

0.69617582

1.22458704

2.49970877

5.31989436

EI ?x2?

1.000000015

1.000001477

1.000129847

1.000129847

1.046344179

1.099588057

1.123451126

1.130099216

0.0001

0.001

0.01

0.1

1

10

100

1000

0.50003749

0.50037444

0.50369509

0.53305374

0.70188134

1.25080186

2.57093830

5.48276171

6. Conclusions

The link Schr¨ oedinger equation - Fisher measure has been employed so as to infer, via the

pertinent reciprocity relations, a parameter-free ground state ansatz wave function for a rather

ample family of even potentials, of the form

U(x) =

?

a2kx2k,

(42)

in terms of the coefficients a2k. Its parameter-free character notwithstanding, our ansatz

provides good results, as evidenced by the examples here examined. It incorporates only the

knowledge of the virial theorem, via the Legendre-symmetry that underlies the connection

between Fisher’s measure and Schr¨ oedinger equation. One may again speak here of the power

of 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).

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Parameter-free ansatz for inferring ground state wave functions of even potentials 10

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