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14Classical and quantum behavior of the harmonic and the quartic oscillators

David Brizuela

Fisika Teorikoa eta Zientziaren Historia Saila, UPV/EHU, 644 P.K., 48080 Bilbao, Spain andInstitut fur Theoretische Physik, Universitat zu Koln, Zulpicher Strae 77, 50937 Koln, Germany

In a previous paper a formalism to analyze the dynamical evolution of classical and quantumprobability distributions in terms of their moments was presented. Here the application of thisformalism to the system of a particle moving on a potential is considered in order to derive physicalimplications about the classical limit of a quantum system. The complete set of harmonic potentialsis considered, which includes the particle under a uniform force, as well as the harmonic and theinverse harmonic oscillators. In addition, as an example of anharmonic system, the pure quarticoscillator is analyzed. Classical and quantum moments corresponding to stationary states of thesesystems are analytically obtained without solving any differential equation. Finally, dynamical statesare also considered in order to study the differences between their classical and quantum evolution.

PACS numbers: 03.65.-w, 03.65.Sq, 98.80.Qc

I. INTRODUCTION

Even if the foundations of the theory of quantum me-chanics are very well settled, there are still open questionsabout its classical limit and the interaction between clas-sical and quantum degrees of freedom. In fact, there arehybrid theories which take into account classical as wellas quantum degrees of freedom (see for instance [17]),but will not be considered here. Concerning the classicallimit of quantum mechanics, in Ref. [8] the idea that sucha limit should be an ensemble of classical orbits was pro-posed. This classical ensemble should be described bya classical probability distribution on phase space and,thus, its evolution would be given by the Liouville equa-tion. It is not possible to compare directly classical andquantum probability distributions since they are definedon different spaces. Therefore, a very convenient wayto perform such a comparison is by decomposing bothprobability distributions in terms of its infinite set of mo-ments. These moments are the observable quantities andone could directly relate (and experimentally measure)their classical and quantum values.The formalism to analyze the evolution of these mo-

ments was first developed in [9] for the Hamiltonian ofa particle on a potential. A formalism similar to thisone, but with a different ordering of the basic variables,was presented in [10, 11] on a canonical framework andfor generic Hamiltonians. Let us comment that this lat-ter formalism has found several applications in the con-text of quantum cosmology [12]. For example, isotropicmodels with a cosmological constant have been analyzed[13, 14]. Bounce scenarios have also been studied withinthe framework of loop quantum cosmology [15]. In ad-dition, the problem of time has also been considered in[16, 17]. Remarkably this framework is also useful whenthe dynamics is generated by a Hamiltonian constraint,as opposed to a Hamiltonian function [18].

Electronic address: david.brizuela@ehu.es

Recently the classical counterpart of the formalism de-veloped in [10, 11] was presented [19]. In this referenceit was argued that the quantum effects have two differ-ent origins. On the one hand, distributional effects aredue to the fact that, because of the Heisenberg uncer-tainty principle, one needs to consider an extensive (asopposed to a Dirac delta) distribution with nonvanishingmoments. These effects are also present in the evolutionof a classical ensemble and, for instance, they genericallyprevent the centroid of the distribution (the expectationvalue of the position and momentum) from following aclassical trajectory on the phase space. On the otherhand, noncommutativity or purely quantum effects ap-pear as explicit ~ terms in the quantum equations of mo-tion and have no classical counterpart. In the presentpaper, this formalism for the evolution of classical andquantum probability distributions will be applied to thecase of a particle moving on a potential with the particu-lar aim of measuring the relative relevance of each of thementioned effects.

The analysis will be made in two parts. On the onehand, the systems with a harmonic Hamiltonian will beconsidered, that is, those that are at most quadratic onthe basic variables. This includes the system of a par-ticle under a uniform force (which trivially includes alsothe free particle case), the harmonic oscillator, and theinverse harmonic oscillator. One of the properties of thiskind of Hamiltonians is that there is no purely quantumeffect and, thus, they generate the same dynamics in thequantum and in the classical (distributional) cases. Inaddition, the equations of motion generated by this har-monic Hamiltonians are much simpler than in the generalcase, so it will be possible to obtain analytically the ex-plicit form of their moments corresponding to stationaryas well as to dynamical states.

On the other hand, due to the complexity of the anhar-monic case, a concrete particular example must be ana-lyzed. In our case, between the large set of anharmonicsystems, we have chosen the pure quartic potential inorder to study both its stationary and dynamical stateswith this formalism. As simple as it might seem, the

2quartic harmonic oscillator can not be solved analyticallyand one usually resorts either to numerical or analyticalmethods of approximation. Nonetheless, from a pertur-bative perspective the model of the quartic oscillator cor-responds to a singular perturbation problem due to thefact that in the limit of a vanishing coupling constant,several physical quantities diverge [20, 21]. Hence, evenif it has been studied during decades and, for instance, itsenergy eigenvalues are well known from numerical com-putations [22, 23], this model is still considered of interestin different context and new approximation techniquesare being developed to treat it, see e.g. [24, 25].The rest of the paper is organized as follows. In Sec.

II a summary of the formalism presented in Ref. [19] isgiven. Section III presents the equations of motion fora Hamiltonian of a particle on a potential. In Sec. IVthe harmonic cases are analyzed. Section V deals withthe anharmonic example of the pure quartic oscillator.Finally, Sec. VI summarizes the main results and detailsthe conclusions of the paper.

II. GENERAL FORMALISM

Given a quantum system with one degree of freedomdescribed by the basic conjugate variables (q, p), it is pos-sible to define the quantum moments as follows:

Ga,b := (p p)a (q q)bWeyl. (1)In this equation p := p and q := q have been defined,and Weyl (totally symmetric) ordering has been chosen.The sum between the two indices of a given moment,(a+ b), will be referred as its order.The evolution equations for these moments are given

by the following effective Hamiltonian, which is definedas the expectation value of the Hamiltonian operator H,and it is Taylor expanded around the position of its cen-troid (q, p):

HQ(q, p,Ga,b) := H(q, p)Weyl

= H(q q + q, p p+ p)Weyl

=a=0

b=0

1

a!b!

a+bH

paqbGa,b

= H(q, p) +

a+b2

1

a!b!

a+bH

paqbGa,b.(2)

The Hamiltonian H(q, p) is the function obtained by re-

placing in the Hamiltonian operator H(q, p) every oper-ator by its expectation value.The equations of motion for the expectation values (q,

p) and for the infinite set of moments Ga,b are directlyobtained by computing the Poisson brackets between thedifferent variables with the Hamiltonian (2). In partic-ular, it is easy to show that Poisson brackets betweenexpectation values and moments vanish. Furthermore, aclosed formula is known for the Poisson bracket between

any two moments [11, 14]. In this way an infinite sys-tem of ordinary differential equations is obtained, whichis completely equivalent to the Schrodinger flow of states.In the general case, as will be shown below, in order toperform the resolution of this system, it is necessary tointroduce a cutoff Nmax and drop all moments of an orderhigher than Nmax.The classical counterpart of this formalism is obtained

by assuming a classical ensemble described by a prob-ability distribution function (q, p, t) on a phase spacecoordinatized by (q, p). As it is well known, the evolu-tion equation of such a distribution is given by the Li-ouville equation. Following the same procedure as in thequantum case, making use of the probability distribution(q, p, t), one can define a classical expectation value op-eration on the phase space:

f(q, p)c :=

dqdpf(q, p)(q, p, t), (3)

where the integration extends to the whole domain of theprobability distribution. With this operation at hand,the classical moments can be defined as

Ca,b := (p p)a(q q)bc, (4)q and p being the position of the centroid of the distri-bution, that is, q := qc and p := pc. Note that inthis classical case, everything commutes and, thus, theordering in the definition of the moments is indifferent.As in the quantum case, the effective Hamiltonian thatencodes the dynamical information of these variables isconstructed by computing the expectation value of theHamiltonian and expanding it around the position of thecentroid. In this way, one obtains the classical effectiveHamiltonian:

HC(q, p, Ca,b) := H(q, p)c (5)

= H(q, p) +

a+b2

1

a!b!

a+bH(q, p)

paqbCa,b.

The equations of motion for the classical moments andexpectation values (q, p) are then obtained by comput-ing their Poisson brackets with this Hamiltonian. Theinfinite system of equations of motion that is obtainedby this procedure is then completely equivalent to theevolution given by the Liouville equation.The evolution equations obeyed by the classical mo-

ments are the same as the ones fulfilled by their quantumcounterparts with the particularization ~ = 0. These ~factors only appear when computing the Poisson brack-ets between two moments due to the noncommutativityof the basic operators q and p.In this formalism it is very clear that the classical limit,

understood as ~ 0, of a quantum theory is not a uniquetrajectory on the phase space, but an ensemble of clas-sical trajectories described by a probability distribution or its corresponding moments Ca,b. In this way, thequantum effects have two different origins. On the one

3hand, distributional effects are due to the fact that mo-ments can not be vanishing (due to the Heisenberg uncer-tainty relation) and generically the centroid of a distribu-tion (q, p) does not follow a classical point trajectory onphase space. (The classical orbit obtained with an initialDirac delta distribution, for which all moments vanish,will be referred as classical point trajectory.) These dis-tributional effects are also present in a classical setting.On the other hand, there are noncommutativity or purelyquantum effects, which appear as explicit ~ factors in thequantum equations of motion. These latter effects aredue to the noncommutativity of the basic operators andhave no classical counterpart.The evolution of the classical and quantum moments

differ for a generic Hamiltonian due to the commented~ terms. Nevertheless the harmonic Hamiltonians, de-fined as those that are at most quadratic in the basicvariables, have very special properties and, in particular,they generate exactly the same evolution in the classicaland quantum frameworks. In this paper the Hamiltonianof a particle on a potential will be studied and, due tothese special properties of the harmonic Hamiltonians,the analysis will be separated between the harmonic andthe anharmonic case. All possible harmonic systems willbe studied but, regarding the anharmonic sector, whichis much more involved, only a particular example will beworked out: the pure quartic oscillator.Once the equations of motion are obtained, the only

information left to obtain a dynamical state are the ini-tial conditions. Nonetheless, the stationary states playa fundamental role in quantum mechanics. In this set-ting, moments corresponding to a stationary state can beobtained as fixed points of the dynamical system underconsideration; that is, by dropping all time derivatives onthe equations of motion for (q, p,Ga,b) and solving the re-maining algebraic system. This system of algebraic equa-tions, as will be made explicit below, is sometimes incom-plete and thus it is not possible to fix the values of allvariables (q, p,Ga,b) of a stationary state by this method.Nonetheless, as shown in [2628], another condition forthe stationary states can be derived as a recursive relationbetween moments of the form G0,n, by making use of thefact that these states are eigenstates of the Hamiltonianoperator (H = E). For the kind of Hamiltonians thatwill be treated in this paper, corresponding to mechani-cal systems of a particle on a potential H = p2/2+ V (q)with potentials of the form V (q) = qm and vanishing ex-pectation value q in its stationary state, this recursiverelation can be written in the following way (see [19] formore details):

(2k +m+ 2)G0,k+m = 2E(k + 1)G0,k

+~2

4(k + 1)k(k 1)G0,k2. (6)

In consequence, whenever moments up to order G0,m areknown, the higher-order fluctuations of the position canbe obtained directly. Classical stationary moments obeythis very same equation dropping the last term.

In order to finalize the summary of previous works, letus comment that the moments corresponding to a validprobability distribution (wave function) are not free andobey several inequalities. The most simple examples arethe non-negativity of moments with two even indices,

G2n,2m 0 , for n,m N, (7)

and the Heisenberg uncertainty principle,

(G1,1)2 G2,0G0,2 ~2

4. (8)

As always, inequalities for classical moments are obtainedfrom the ones of the quantum moments by taking ~ = 0.In Ref. [19] several inequalities for high-order momentswere obtained. These inequalities will be used below toconstrain the values of certain moments of stationarystates as well as to monitor the validity of the numer-ical resolution of dynamical states.

III. PARTICLE ON A POTENTIAL

For definiteness, in order to check the interpretationand applicability of the formalism for classical and quan-tum moments summarized in previous section, here theHamiltonian for a particle moving on a potential will beassumed,

H =p2

2+ V (q). (9)

Let us define the dynamics for the quantum expecta-tion values and moments. The effective quantum Hamil-tonian is given by

HQ =p2

2+ V (q) +

1

2G2,0 +

n=2

1

n!

dnV (q)

dqnG0,n. (10)

From there, it is straightforward to obtain the equa-tions of motion for the centroid of the distribution:

dq

dt= p, (11)

dp

dt= V (q)

n=2

1

n!

dn+1V (q)

dqn+1G0,n. (12)

Note that the evolution equation of the position q is notmodified by the moments. On the contrary, the equationof motion for its conjugate momentum p does receivecorrections due to the presence of the moments G0,a inthe right-hand side of Eq. (12). It is straightforward tosee that the Hamiltonian HC , which would describe theevolution of a classical distribution on the phase space,it is obtained by replacing the quantum moments Ga,b

by the classical ones Ca,b in Eq. (10). The centroid ofthat classical distribution will follow the evolution givenby (11-12), replacing again Ga,b by Ca,b.

4It is enlightening to combine last two equations in orderto obtain the corrected Newton equation,

d2q

dt2= V (q)

n=3

1

n!

dnV (q)

dqnG0,n1. (13)

The moment terms that appear in this modified equationare sometimes referred as the quantum contributions tothe Newton equations. Nevertheless, we see from ouranalysis that the equations of motion for a centroid of a

classical distribution in the phase space characterized bymoments Ca,b will obey this very same equation. There-fore this equation must be understood as the fact thatthe centroid of a distribution does not follow a classicalpoint trajectory.

Taking the Poisson brackets between moments Ga,b

and the Hamiltonian (10), and separating the terms withan explicit dependence on ~, the equations of motion forthe quantum moments Ga,b can be written as

dGa,b

dt= bGa+1,b1+a

n=2

V (n)(q)

(n 1)![G0,n1Ga1,b Ga1,b+n1]

n=3

Mk=1

V (n)(q)

(n 2k 1)!(

a2k + 1

)(~

2

4

)kGa2k1,b+n2k1, (14)

with M being the integer part of [Min(a, n) 1]/2. Theevolution equation for the classical moments can be for-mally obtained from last equation by replacing all Ga,b

by Ca,b moments and imposing ~ = 0, that is, removingall terms that appear in the second line:

dCa,b

dt= b Ca+1,b1 (15)

+ a

n=2

V (n)(q)

(n 1)! [C0,n1Ca1,b Ca1,b+n1].

In summary, Eqs. (11) and (12), in combination with(14), form an infinite closed system of ordinary differen-tial equations that describes the quantum dynamics of aparticle on a potential V (q) and are completely equiva-lent to the Schrodinger flow of quantum states (or theHeisenberg flow of quantum operators). On the otherhand, the infinite system composed by Eqs. (11), (12)[replacing G0,n terms by C0,n], and (15) describes theclassical evolution of a probability distribution on thephase space, which is equivalent to the Liouville equa-tion.As can be seen in these equations of motion, for a

generic potential V (q), all orders couple. Hence, in orderto make these equations useful for a practical purpose, itis necessary to introduce a cutoff by hand, and assumeGa,b to be vanishing for all a+b > Nmax, Nmax being themaximum order to be considered. In order to impose thiscutoff, due to the special properties of the Poisson brack-ets between two moments, care is needed (see [19] fora more detailed discussion). In order to truncate prop-erly the system at an order Nmax, taking into accountall contributions up to this order, it is straightforward tosee that the upper limit of the summation in Eq. (12)should be taken as Nmax. Regarding the equation for themoments (14), the sum of the first line should clearly go

up to (Nmax+1) for the quadratic term in moments, butonly up to (Nmax a b 2) for the second linear term.The summations in the second line of that equation aremore involved and should be replaced by

n=3

Mk=1

nmaxn=3

Mk=kmin

, (16)

with nmax = Nmax + a b and, for every fixed n, kminthe maximum between 1 and (a+b+nNmax2)/4).For the classical equations, the same limits as in theircorresponding quantum equations should be imposed.The validity of this cutoff should be proved a posteriori

by solving the equations of motion with different cutoffsand checking that the solution converges with the cutofforder.If an integer Nmax exists, for which V

(n)(q) vanishesfor all values n > Nmax, the infinite sums on the right-hand side of Eq. (12) will become finite. Regarding thequantum Ga,b (14) and the classical moments Ca,b (15)of order a + b, the highest order that appears in theircorresponding equations of motion is of order (a + b +Nmax 2). Therefore, only in the case that Nmax 2the introduction of a cutoff will not be necessary. Thisis in fact the case of a harmonic Hamiltonian, which willbe analyzed in the next section.

IV. HARMONIC POTENTIALS: V (q) = 0

The harmonic Hamiltonians H(q, p) are defined asthose for which all derivatives with respect to the ba-sic variables (q, p) higher than second order vanish. Inthe case of a Hamiltonian of a particle on a potential (9),this happens when V (q) =: 2 is a constant.

5This kind of Hamiltonians has very special properties,which were analyzed in Ref. [19]. Let us briefly summa-rize its main properties. First, for this kind of Hamil-tonians, equations at every order decouple from the restof the orders. Second, equations of motion of expecta-tion values (q, p) do not get any correction from momentterms and thus there is no back-reaction. Hence, thecentroid of the distribution follows a classical point tra-jectory. In addition, given the same initial data, classicaland quantum moments have exactly the same evolutionsince no ~ term appears in the equations of motion. Aswill be shown in this section, classical and quantum sta-tionary states differ because the equations of motion donot provide the complete information to fix the value ofall moments and thus recursive relation (6) will have tobe used.Due to the mentioned properties, most of the analysis

of this section applies equally to classical as well as toquantum moments. Thus the whole analysis will be per-formed for quantum moments and emphasis will be madein the particular points where the situation is different forclassical moments.The expectation value of a Hamiltonian of a particle

on a potential V (q), such that V (q) = 2 is a constantvalue, can be written in the following way in terms ofexpectation values and moments:

HQ =p2

2+2

2q2 +

1

2G2,0 +

2

2G0,2. (17)

This is, as explained in previous section, the effectivequantum Hamiltonian that can be used to obtain theequations of motion. In particular, the equations of mo-tion for the expectation values q and p reduce to theirusual form,

dq

dt= p, (18)

dp

dt= V (q). (19)

Here it can be seen that, as already commented above,there is no back-reaction of moments in the equations forthe centroid, in such a way that the centroid follows aclassical phase space orbit.The equations for the moments (14) reduce to,

dGa,b

dt= bGa+1,b1 a2Ga1,b+1. (20)

The classical moments Ca,b fulfill this very same equa-tion, replacing all quantum moments Ga,b by their clas-sical counterparts Ca,b, as can be readily checked from(15).As it is well known, it is not necessary to solve Eqs.

(1819) explicitly to obtain the phase-space orbit that isfollowed by the centroid. It is sufficient to divide bothequations to remove the dependence on time and inte-grate the resulting equation. This procedure leads to theimplicit solution,

Ecentroid = p2/2 + V (q), (21)

Ecentroid being the integration constant that parametrizesdifferent orbits, which can obviously be interpreted as theenergy of the centroid. Note that this Ecentroid energyis not the expectation value of the Hamiltonian HQ. Inparticular, since HQ (and for the classical treatment HC)is also a constant of motion, the difference between both,leads to another conserved quantity in terms of second-order moments: G2,0+2G0,2 (and C2,0+2C0,2 for theclassical moments).

The first derivative of the potential V (q) only ap-pears in the evolution equation for the momentum p (19)and, certainly, the phase-space orbit followed by the cen-troid (21) depends on the precise form of the potential.Nonetheless, note that the equations of the moments (20)only depend on the second derivative of the potential 2.Therefore, in order to fully analyze the evolution of themoments, the study will be split in the two possible andphysically different cases: 2 = 0 and 2 6= 0. The for-mer describes a particle moving under a uniform force,whereas the latter corresponds to the harmonic (2 > 0)and the inverse harmonic (2 < 0) oscillators.

A. Particle under a uniform force: V (q) = 2 = 0

In this subsection the generic linear potential V = q+V0 will be analyzed. Without loss of generality, V0 willbe chosen to be vanishing. This potential represents aparticle under a constant force. The case of a free particle( = 0) will also be included in the analysis.

As explained above, in this case all orders decouple andthe centroid of the distribution follows a classical pointtrajectory in phase space: q + p2/2 = Ecentroid, withEcentroid a constant value. Since the full HamiltonianHQ is also a constant of motion, it is obvious then thatthe moment G2,0 is also constant during the evolution.In fact, looking at the equations of motion (20), it isimmediate to see that the fluctuations of the momentumat all orders Ga,0 are constants of motion.

Let us first analyze the stationary states, that is, thefixed points of the dynamical system. Dropping alltime derivatives in the system of equations (1820), it isstraightforward to see that only the free particle ( = 0)case allows for stationary solutions that would be givenby p = 0 (particle at rest) and all momentsGa,b vanishingfor all a 1 and b 0. The position q and its fluctua-tions at all orders G0,b could, in principle, take any value.That is, the particle can be anywhere and with an un-bounded uncertainty in its position. Nonetheless, even ifthis choice of moments is valid for the classical case, it isnot for the quantum case since it violates the Heisenberguncertainty relation (8). Therefore as it is well known,and contrary to the classical case, no stationary state canbe constructed for the free quantum particle.

The analytical solution for a dynamical state can be

6found explicitly for the evolution of all moments,

Ga,b(t) =b

n=0

(bn

)(t t0)bnGa+bn,n0 , (22)

for initial data Ga,b0 := Ga,b(t0). The evolution of the mo-

ments is independent of the value of , thus this solutionis valid both for the case of the free particle and the parti-cle under a uniform force. As can be seen, each momentis given by a linear combination of the initial value ofthe moments of its corresponding order with polynomialcoefficients on the time parameter. The state spreadsaway from its initial configuration and, for large times,the moments Ga,b increase as tb. The initial conditionsof this state are still free. For instance, it is possible tochoose an initial state of minimum uncertainty but, evenso, all moments, except the constants of motion Ga,0, willincrease with time.

B. Harmonic and inverse harmonic oscillators:

V (q) = 2 6= 0

It is well known that any potential of the form V =2 q

2 + q + V0 can be taken to the form V =2

2 q2 by a

shift of the variable q = q + 2 and a redefinition of the

value of the potential at its minimum (V0 =2

22 ), which

does not have any physical meaning. If 2 is positive,this is the potential of a harmonic oscillator, a ubiqui-tous system in all branches of physics. Since the equa-tions of motion for expectation values (1819) do not getany backreaction by moments, their solutions are oscilla-tory functions and they follow an elliptical orbit in phasespace. On the other hand, the case 2 < 0 correspondsto the inverted harmonic oscillator. This system can beviewed as an oscillator with imaginary frequency. Thesolution for the expectation values (q, p) are hyperbolicfunctions and they follow hyperbolas in phase space. Inthe rest of this subsection the behavior of the momentswill be considered for both systems.Let us first analyze the stationary states. Equaling

to zero the right-hand side of the equations of motion(1820), the equilibrium point p = 0 = q for the expecta-tion values, as well as the recursive relation bGa+1,b1 =a2Ga1,b+1 for the moments are obtained. The solu-tion to this recursive relation is given by the followingcondition for moments with both indices even numbers,

G2a,2b =2a! 2b!

(2(a+ b))!

(a+ b)!

a! b!2aG0,2(a+b), (23)

whereas the rest of the moments must vanish. If the signof 2 was negative, that would impose some momentswith even indices to be negative. This is not acceptablesince all moments of the form G2a,2b are non-negative byconstruction (7). Thus, from here it is immediately con-cluded that the inverse oscillator can not have stationarystates.

As can be appreciated in the last relation (23), evenif the information concerning the stationary state con-tained in the equations of motion has been exhausted,there is still one freedom left at each order. This free-dom is represented in this equation by the high-orderfluctuations of the position G0,n.In order to fix the moments G0,n, the recursive rela-

tion (6) can be made use of. For the potential underconsideration, that relation reads

2 (k+2)G0,k+2 = 2(k+1)EG0,k+~2

4(k+1)k(k1)G0,k2.

(24)This last equation allows us to compute all G0,n mo-ments as function of the energy at the stationary pointE = (G2,0 + 2G0,2)/2 = G2,0 and Planck constant ~.Taking the limit ~ 0, the (two point) recursive rela-tion obeyed by classical moments is obtained, which canbe easily solved. Combining this solution with (23), theclassical moments corresponding to a stationary situa-tion of the harmonic oscillator can be written in a closedform. Those with two even indices read

C2a,2b =(2a)!(2b)!

a!b!(a+ b)!

Ea+b

2a+b2b, (25)

and the rest are vanishing.The quantum case is a little bit more involved. The

second-order moments G2,0 and G0,2 have the same formas their classical counterparts in terms of the energy Eand the frequency (25). But higher-order momentswill take corrections as a power series in the parameter ~2

when solving the recursive relation (24). Here we give theexplicit expression of all the fluctuations of the positionG0,n up to order ten:

G0,2 =E

2,

G0,4 =3

2

(E

2

)2+

3

8

(~

)2,

G0,6 =5

2

(E

2

)3+

25

8

(E

2

)(~

)2,

G0,8 =35

8

(E

2

)4+245

16

(E

2

)2 (~

)2+

315

128

(~

)4,

G0,10 =63

8

(E

2

)5+

945

16

(E

2

)3(~

)2+

5607

128

(E

2

)(~

)4.

The rest of the nonvanishing moments are proportionalto these and can be obtained by using the solution (23).Note that a quantum moment Ga,b is equal to its classicalcounterpart (25) plus certain corrections that are givenas an even power series in ~. This power series goes from~2 up to ~2n, n being the integer part of (a+ b)/4.The only information that is left here is the exact form

of the energy spectrum: E = ~(n + 1/2). This is theonly input needed in order to obtain the complete re-alization of the system. In fact, one could obtain all

7the moments corresponding to the ground state by as-suming that it is an unsqueezed state with minimumuncertainty that saturates the Heisenberg relation (8),G2,0G0,2 = ~2/4, which implies Eground = ~/2. In ad-dition note that, as expected, for this ground state theexpression of the quantum moments reduces to the mo-ments corresponding to a Gaussian probability distribu-tion with width

~/. [The explicit expression for the

moments of a Gaussian state is given below (42).]Regarding the dynamical states, it is easy to solve the

equations of motion (1820). The solution for the mo-ments Ga,b can be written as a linear combination offunctions of the form eit. For moments of even or-ders, a + b = 2n, takes even values: = 0, 2, . . . , 2n;whereas for those of odd orders, a+ b = 2n+ 1, it takesodd values: = 1, 3, . . . , 2n + 1. Thus, the dynamicalbehavior of the harmonic oscillator (2 > 0) and the in-verse oscillator (2 < 0) is completely different. For theoscillatory case (2 > 0), all moments Ga,b are boundedand they are oscillating functions. On the contrary, themoments corresponding to the inverse oscillator are ex-ponentially growing and decreasing functions of time.

V. THE ANHARMONIC CASE: THE PURE

QUARTIC OSCILLATOR

The potential of the pure quartic oscillator is given by

V (q) = q4, (26)

which leads to an effective Hamiltonian of the form

HQ =p2

2+ q4+

1

2G2,0 + 6q2G0,2 + 4qG0,3 + G0,4.

(27)From this Hamiltonian it is easy to get the equations ofmotion for the expectation values,

dq

dt= p, (28)

dp

dt= 4(q3 + 3qG0,2 +G0,3), (29)

and for the moments

dGa,b

dt= bGa+1,b1 + 4 a [3 q G0,2 +G0,3]Ga1,b

4 a [3 q2Ga1,b+1 + 3 q Ga1,b+2 +Ga1,b+3]+ a ~2 (a 2) (a 1) [q Ga3,b +Ga3,b+1].(30)

As can be seen, in this case all orders couple. Morespecifically, in the equation for a moment Ga,b there ap-pear moments of order two, three and of all orders fromO(a+ b 3) to O(a+ b+ 2).The centroid of a classical distribution will follow the

same equations (28) and (29), replacing moments Ga,b

by their classical counterparts,

dq

dt= p, (31)

dp

dt= 4(q3 + 3qC0,2 + C0,3), (32)

whereas the evolution of the classical moments will begiven by

dCa,b

dt= b Ca+1,b1 + 4 a [3 q C0,2 + C0,3]Ca1,b (33)

4 a [3 q2 Ca1,b+1 + 3 q Ca1,b+2 + Ca1,b+3] .The explicit order coupling differs a little bit from thequantum case, since in this equation there are only mo-ments of order two, three and of all orders betweenO(a+ b 1) and O(a+ b+ 2).

A. Stationary states

In order to obtain the stationary states of the purequartic oscillator, the infinite set of algebraic equationsobtained by equaling to zero the right-hand side of Eqs.(2830) must be solved. Furthermore, recursive relation(6) must also be obeyed. In this particular case, thatrelation takes the following form:

2(a+3)G0,a+4 = 2E(a+1)G0,a+~2

4(a+1)a(a1)G0,a2,

(34)with the energy given by the numerical value of the ex-pectation value of the Hamiltonian,

E = HQ. (35)

In practice, due to the coupling of the system, it is nec-essary to introduce a cutoff in order to get a finite systemand be able to solve it. In our case different cutoffs havebeen considered (specifically Nmax= 15, 20, 25, and 30)and the mentioned system of equations, in combinationwith relation (34) and the definition of the energy (35),has been analytically solved. The idea behind perform-ing this computation for several cutoffs is to study theconvergence of the solution, that is, to check whether thesolution for the moments does not change when consid-ering higher-order cutoffs.In principle, there are two different solutions: one that

corresponds to the classical stationary configuration (andthus its equilibrium position is at the origin q = 0) andanother, for which the position must not be vanishing[note that this is possible due to the moment terms thatappear in the Hamilton equation (29) ] and does not havea classical point counterpart. Nevertheless, for this lat-ter case, the solution for some moments with both evenindices turns out to be negative, which makes this solu-tion invalid. Therefore, and as one would expect fromsymmetry considerations, the expectation values of anystationary state of the quartic oscillator corresponds tothe origin of the phase space (p = 0 = q). Furthermore,it can be seen that all its corresponding moments Ga,b arevanishing in case any of the indices a or b is an odd num-ber. The remaining moments can be written in terms ofthe energy E and the fluctuation of the position G0,2, orany other chosen moment. That is, there is not enough

8information in our system of equations to fix all momentsand one of them is free.Regarding the convergence of the solution, comparing

the solution obtained with the cutoff Nmax = 30 withthe one corresponding to Nmax = 15, we see that the ex-pression of all moments coincides up to order 8, whereasthe solution with Nmax = 30 and Nmax = 20 give thesame expression for all moments up to order 12. Finally,solutions that correspond to Nmax = 30 and Nmax = 25coincide up to order 14. From here the existence of aclear convergence of the solution with the cutoff order isconcluded. Nevertheless, this convergence seems to beslower with higher orders. Here the explicit expressionsfor all nonvanishing moments up to sixth order is pro-vided:

G2,0 =4

3E,

G4,0 =2

7

(8E2 + 15~2G0,2

),

G2,2 =1

5

(4EG0,2 + ~2

),

G0,4 =1

3E,

G6,0 =10

77

(32E3 + 228E~2G0,2 + 21~4

),

G4,2 =2

45E(24EG0,2 + 41~2

),

G2,4 =4

21E2 +

6

7~2G0,2,

G0,6 =3

20

(4EG0,2 + ~2

). (36)

The classical moments Ca,b, as always, take the samevalues as their quantum counterparts with the particu-larization ~ = 0. In these expressions the singular behav-ior of the limit 0 is made explicit as the divergenceof several moments. This fact does not allow to performregular perturbative treatments of this system.In summary, after imposing the stationarity condition

on Eqs. (2830) and using the definition of the energy(35) in combination with the recursive relation (34), theonly information left in order to characterize completelyany stationary state of the pure quartic oscillator is theenergy E and the fluctuation of the position G0,2.In addition to these equations already mentioned, there

is still some information more than we can get by makinguse of the inequalities obtained in Ref. [19]. In the follow-ing, use will be made of those relations to constrain thevalues of G0,2 and the energy E. For instance, Heisen-berg uncertainty principle (8) provides a lower bound forthe product between E and G0,2:

3~2

16 EG0,2. (37)

Higher-order inequalities give more complicated rela-tions, which must be fulfilled by the energy E and thefluctuation of the position G0,2 of any stationary state ofthis system.

For the particular case of the ground state a reason-able assumption is that, as happens for the harmonicoscillator, it saturates the above relation. This wouldgive G0,2ground = 3~

2/(16Eground) and let the energy ofthe ground state Eground as the only unknown physicalquantity in (36). Introducing then these expressions ofthe moments of the ground state in terms of Eground inthe higher-order inequalities, an upper and lower boundfor the energy is obtained. By considering inequalitiesthat only contain moments up to fourth-order yields thefollowing result:

3

4

(45

68

)1/3 Eground

(~4)1/3 1

4

(85

4

)1/3, (38)

or, in decimal notation,

0.654 Eground(~4)1/3

0.692, (39)

which already provides a good constraint on the energy.Furthermore, all inequalities that contain moments upto order six reduce to the following tighter interval ofvalidity for the energy:

3

4

(45

68

)1/3 Eground

(~4)1/3 9

4

(3

116

)1/3, (40)

or, writing these fractions as decimal numbers,

0.654 Eground(~4)1/3

0.665. (41)

This gives a very tight constraint on the energy of thisbound state. Nevertheless, the exact (numerically com-puted) energy of this state is available in the literature(see e.g. [22, 23]): Eground = 0.670039(~

4)1/3 [29]. Thisnumerical value is very close but outside the derived in-terval. Therefore, we can conclude that, even if the sat-uration of the Heisenberg uncertainty is a reasonable as-sumption for the ground state that provides a good es-timation of the ground energy, this assumption is notsatisfied and the uncertainty relation is not completelysaturated for the present model.This analysis shows the practical relevance of the in-

equalities that were derived in Ref. [19] as a complemen-tary method to extract physical information from thesystem. Certainly the inequalities will not give exact re-lations between different quantities, but intervals of va-lidity can be extracted from them. Finally, let us stressthe importance of considering higher-order inequalities.Note that the interval derived from fourth-order inequal-ities (39) does indeed allow the exact (numerical) value ofthe ground energy, and thus in principle permits the satu-ration of the uncertainty relation. Therefore, in this par-ticular example inequalities up to fourth order allowed aproperty of the system, which is forbidden by the strongercondition derived from higher-order ones.

9B. Dynamical states

The classical point trajectory of the pure quartic oscil-lator, that is, the solution to Eqs. (28-29) neglecting allmoments, can only be written in terms on hypergeomet-ric functions. Nevertheless, the orbits on the phase spaceare easily obtained by the conservation of the classical

energy: Eclass =p2

2 + q4. Contrary to the harmonic

oscillator, the period depends on the energy Eclass of theorbit, and it is not a constant for different orbits. For lat-ter use, note that the maximum (classical) value of theposition and the momentum can be directly related to theenergy as q4max = Eclass/ and p

2max = 2Eclass. In order

to compare different solutions, below we will also makeuse of a (squared) Euclidean distance on the phase space(p2 + q2). The maximum distance from the origin of agiven orbit is reached at ((p2+q2)max = 2Eclass+1/(8)).We are interested on analyzing the quantum and clas-

sical evolution of a distribution that, respectively, fol-lows Eqs. (28-30) and (31-33). Nonetheless, due to thecomplicated form of these evolution equations, the pos-sibility of getting an analytical solution seems unlikely.Hence, in order to analyze the dynamics of the system,it is necessary to resort to numerical methods. Here acomment about notation is in order. When the meaningis not clear from the context, we will sometimes denote asqq(t) the solution of the quantum distributional system(28-30), qc(t) the solution of the classical distributionalsystem (31-33), and finally qclass(t) the solution corre-sponding to the classical point trajectory, that is, thesolution to Eqs. (28-29) dropping all moments. The verysame notation will be used for the different solutions ofthe momentum p(t).For a numerical resolution of the system, two choices

have to be made. On the one hand, for practical reasons,a cutoff Nmax has to be considered in order to truncatethe infinite system. On the other hand, it will be neces-sary to choose initial conditions for the state to be ana-lyzed.Regarding the truncation of the system, the dynamical

equations for different values of the cutoff will be consid-ered. More precisely, both the quantum system (28-30)and the classical distributional system (31-33) for everyorder up to tenth order will be solved. In this way, it willbe possible to check the convergence of the solution withthe consideredNmax, as well as study differences betweenthe classical and quantum moments.Concerning the initial conditions, since the movement

of the system is oscillatory around the equilibrium pointq = 0, a vanishing value for the initial expectation ofthe position q(0) = 0 will be considered without lossof generality. For the expectation of the momentum p,in order to check the dependence of the properties ofthe system with the energy, we will make evolutions forseveral values, namely p(0) = 10, 102, and 103. Notethat the initial classical (point) energy (p(0)2/2) will notbe conserved through evolution; instead, the completeHamiltonian (27) will be constant. Nevertheless, due to

the correspondence principle, the larger the classical en-ergy, a somehow more classical behavior is expected tobe found. This can already be inferred from the equa-tions themselves: in the case that moments are negligiblewith respect to expectation values q and p, the centroidwill approximately follow a classical point orbit on phasespace.As for the initial values of the fluctuations and higher-

order moments, a peaked state given by a Gaussian ofwidth

~ will be chosen. Its corresponding moments

Ga,b are vanishing if any of the indices a or b are odd.The only nonvanishing moments take the following values[14]:

G2a,2b = ~a+b2a! 2b!

22(a+b) a! b!. (42)

Therefore, initially the fluctuation of the position and ofthe momentum are G0,2 = G2,0 = ~2/2. In principlethe initial conditions for the classical pair q and p shouldbe chosen large in comparison with their fluctuations, sothat we can be safely say that we are in a semiclassicalregion where this method is supposed to provide trustableresults. Nonetheless, in this case the system oscillatesaround q = 0 and in the turning points the momentumvanishes p = 0. Thus, for this case the condition of q andp being much larger than their corresponding fluctuationscan not be a good measure of semiclassicality. We willcheck if, as already mentioned above, the classical (point)energy of the system does play such a role.Given this setting, we will be interested in analyzing

several aspects of the system. i/ The validity of thismethod based on the decomposition of the classical andquantum probability distributions in terms of moments.In particular the convergence of the system with the trun-cation order Nmax as well as other control methods, likethe conservation of the full Hamiltonian, will be ana-lyzed. ii/ The dynamical behavior of the moments. iii/The deviation, due to quantum effects, from the classicaltrajectory on the phase space. iv/ The relative relevanceof the two different quantum effects that have been dis-cussed in Sec. II: the distributional ones and the non-commutativity or purely quantum ones. v/ The validityof the correspondence principle. That is, do systems witha larger energy have somehow a more classical behaviorthan those with lower energy?Regarding the first two question (i/ and ii/) all re-

sults that will be commented for the quantum momentsapply also to the classical ones. Furthermore, except forthe last issue (v/) about the correspondence principle,the qualitative behavior of the system is the same for allconsidered values of initial momentum p(0). Hence, theresults regarding the first four points (i/ to iv/) will bepresented for the particular case of p(0) = 10 and, fi-nally, the last point (v/) will be discussed by comparingresults obtained for different initial values of the classicalenergy. In all numerical simulations = 1 and ~ = 102

have been considered.i/ The natural tendency of the both quantum and clas-

10

sical moments is to increase with time, since the dynam-ical states are deformed through evolution. This formal-ism is best suited for peaked states so, when higher-ordermoments become important, it is expected not to givetrustable results. Numerically this is seen in the factthat, after several periods, the system becomes unstableand thus the results are no longer trustable.

In order to check the validity of our results we haveseveral indicators at hand: numerical convergence of thesolution, conservation of the constants of motion (in thiscase the full Hamiltonian), convergence of the resultswith the order of the cutoff, and fulfillment of the inequal-ities derived in [19]. The numerical convergence has beenchecked by the usual method: by computing several so-lutions with an increasing precision and confirming thatthe difference between them and the most precise onetends to zero. The full Hamiltonian has also been ver-ified to be conserved during the evolutions presented inthis paper.

For the analysis of the convergence of the system withthe truncation order, we define the squared Euclideandistance between points on the phase space as n(t) :=[qn(t)qn1(t)]2+[pn(t)pn1(t)]2, with qn(t) and pn(t)being the solution of the system truncated at nth order.In particular q1(t) = qclass(t) and p1(t) = pclass(t) corre-spond to classical point orbits. This will serve as a mea-sure of the departure of the solution at every order fromthe previous order. In Fig. 1 the distance n betweenconsecutive solutions is drawn in a logarithmic scale forn = 2, . . . , 8 for the first three periods of the evolution.From this plot it is clear that, at least during a few pe-riods, the convergence is very fast with the truncationorder. We have not included the 9 and 10, since theirvalue is lower than 1016, the estimated numerical er-ror for the solutions shown in this plot, during the firstthree periods. It is interesting to note that, whereas therest of the n have a more complicated structure, 2(shown by the thickest black line in Fig. 1) follows aperiodic pattern with a local minimum every quarter ofa period. These points of minimum deviation from theclassical orbit correspond to points with maximum mo-mentum (q = 0) and to turning points (p = 0).

Remarkably we have found that the inequalities arethe first indicator to signalize the wrong behavior of thesystem. In the particular case with p(0) = 10, the tenth-order solution obeys all inequalities that contain only mo-ments up to fourth order during more than five periods.But some of the inequalities that contain moments ofsixth order are violated soon after the fourth cycle. Fi-nally, some inequalities with eighth order moments areviolated after around 2.53 cycles. In fact, it is expectedthat the values obtained for higher-order moments be lessaccurate than those for lower-order ones due to the trun-cation of the system. As already commented above, inthe evolution equation of a moment Ga,b, there appearmoments from orderO(a+b3) to O(a+b+2) [only fromO(a + b 1) to O(a + b+ 2) for the classical moments].Thus, when we perform the truncation, let us say, at or-

1 2 3T

-15

-10

-5

log10HDnL

FIG. 1: The squared Euclidean distance on phase spacebetween orbits corresponding to consecutive orders n :=[qn(t) qn1(t)]2 + [pn(t) pn1(t)]2 is shown in a logarith-mic plot for n = 2, . . . , 8. The distance between the second-order and the classical point trajectory (2) corresponds tothe black (thickest) line. For the distance correspondingto higher orders (n), the following colors have been used:brown (n = 3), green (n = 4), red (n = 5), blue (n = 6),purple (n = 7), and orange (n = 8); the thickness of the linesbeing decreasing with the order. The estimated numericalerror of these solutions is around 1016, thus higher ordersare almost numerical error during the first two periods. Notethat, at any time, we get a very rapid and strong convergencewith the considered order.

der Nmax, we remove several terms from the equations ofmotion for moments of order (Nmax 1) and (Nmax 2),whereas evolution equations for lower-order moments areconsidered in a complete form. Therefore, moments of or-der (Nmax 1) and (Nmax 2) suffer the presence of thecutoff directly. On the other hand, lower-order momentsonly feel the presence of the cutoff indirectly, due to thecoupling of the equations.

In summary, after the analysis explained in the last fewparagraphs, it is quite safe to assert that the results de-rived during the first 2.5 cycles are completely trustable[for p(0) = 10]. As can be seen, in most of the plots onlytwo periods are shown.

ii/ The fluctuations and higher-order moments are os-cillatory functions that evolve increasing their amplitude.In Fig. 2 the evolution of some moments, as well as ofthe expectation value of the position q, is shown as anexample. Note that, for illustrational purposes, the mo-ments have been multiplied by different factors and theposition is divided by its (classical) maximum value qmax.Interestingly, moments G0,2 and G0,4 are almost vanish-ing at turning points, when the position takes its maxi-mum value, and have a maximum soon after q crosses itsorigin.

iii/ and iv/ In order to analyze the deviation of thequantum and classical distributional trajectories fromtheir corresponding classical point orbit, two operators,

11

1 2T

-1.0

-0.5

0.5

1.0

qqmax, Ga,b

FIG. 2: In this figure the evolution of the position q overits maximum value qmax (black continuous thick line) withrespect to to the time (measured in terms of the period T )is shown. The rest of the lines correspond to some momentsGa,b rescaled by a factor for illustrational purposes. Moreprecisely, the red (long-dashed) line corresponds to 50G0,2,the green (dot-dashed) line to 103G0,4, the blue (dotted) lineto 10G1,1, and the gray (continuous thin) line to 102G2,1.The behavior of the moments is oscillatory, with an increasingamplitude.

1 and 2, are defined as follows

1q(t) := qc(t) qclass(t), (43)2q(t) := qq(t) qc(t). (44)

The same definitions apply for 1p and 2p. These op-erators are a measure of the two quantum effects thatwere defined in [19] and have been discussed in Sec. IIof the present paper. On the one hand, the operator1 will contain the strength of the distributional effects.On the other hand, 2 will encode the intensity of purelyquantum effects, whose origin is due to the ~ factors thatappear explicitly in the quantum equations of motion. Inour numerical analysis qc(t) and qq(t) will be consideredto be the solutions to the corresponding truncated sys-tem at order 10. Finally, the complete departure fromthe classical orbit will be given by the sum of both dif-ferences:

q = 1q + 2q = qq qclass. (45)Figure 3 shows the evolution of the system as well as

the differences given by the operators 1 and 2 actingon different variables in terms of time. (Note that thesedifferences are multiplied by certain enhancement factorsfor illustrational purposes.) More precisely, in the upperplot of the mentioned figure the evolution of the positiondivided by its (classical) maximum q/qmax, as well asthe differences 1q and 2q, are shown. The middle plotrepresents the evolution of p/pmax with its corresponding1p and 2p. Finally, in the lower graphic the squaredEuclidean distance from the origin of the phase spaceis plotted, as well as the deviations (1p

2 + 1q2) and

(2p2 + 2q

2) [30]. This distance has been divided by

FIG. 3: In these plots the evolution of q, p, and (q2 + p2) (di-vided by their maximum values) is shown in combination withthe operators 1 and 2 acting on them. The black (thinnest)line represents the evolutions of the quantity we are consider-ing, for instance in the upper plot q/qmax, the blue (thickest)line represents the distributional effects, in the mentioned plot1q, whereas the red line stands for the purely quantum ef-fects, in the considered graphic 2q.

its maximum classical value which, as commented above,can be easily related to the initial conditions as (p2 +q2)max = p(0)

2 + 1/(8).Looking at the enhancement factors that have been in-

troduced for the differences 1 and 2 so that objects thathave been plotted appear approximately with the sameorder of magnitude, it is straightforward to see that for

12

all quantities the departure from the classical point tra-jectory is mainly due to the distributional effects mea-sured by 1. In particular, during the two cycles that areshown, the absolute maximum departure from the clas-sical trajectory is of the order of q 1q 5 103 forthe position and p 1p 3102 for the momentum.Combining this result, it is direct to obtain the maxi-mum departure as measured by the squared Euclideandistance on the phase space: q2 + p2 103; whichalso can be obtained from the lower plot of Fig. 3.

As already commented, and as one of the main resultsof this paper, in this model we have shown that the distri-butional effects are much more relevant than the purelyquantum ones. Let us analyze its relative importance:from the values that can be seen in Fig. 3 we have that2q/1q 2p/1p 104. This ratio happens to be ofthe order of ~2, which is a measure of the purely quan-tum effects in the equations of motion. Nevertheless, aswe will be shown below when considering initial condi-tions of higher energy, this is not generic. In fact, thisis a property of the nonlinearity of the equations: theeffects of a term of order ~2 on the equations of motionare not necessarily of the same order in the solution.

Finally, it is of interest to analyze the time evolution ofthe terms 1q and 2q. Note that both are periodic func-tions, with approximately the same period as the classicalsystem T , with an amplitude that increases with time. Infact, 1q and 2q follow the same pattern, that is, theyhave qualitatively the same form, but with a phase dif-ference of T/4 so that when one of them is at a maximum(or at a minimum) the other one is around zero. In thecase of 1p and 2p, they are also periodic functions withperiod T , follow the same pattern and T/4 dephased.The main difference between the pattern followed by 1qand 2q with respect to the one followed by 1p and 2pis that, whereas the formers have just a critical pointbetween consecutive changes of sign, the latters oscillatetwice (producing three critical points) between two oftheir zeros.

The net result of all commented effects on the phase-space orbits can be seen in the lower plot of Fig. 3.Minimum departure from classical orbit occurs at turn-ing points and when q crosses the origin. In this plotit is possible to see again that 1 and 2 follow qual-itatively the same pattern but, interestingly, they are(almost) not dephased; the phase differences in positionand in momentum compensate each other. In a more de-tailed level, it is possible to observe that critical pointsof (1q

2 + 1p2) and (2q

2 + 2p2) do not exactly coin-

cide in time: there is a slight delay between them. Inaddition, from these plots it can also be inferred thatthe orbit followed by the expectation values of quantumstates does not coincide at any point with its classicalcounterpart, since there is no time when all correctionsvanish: 1q = 2q = 1p = 2p = 0.

v/ Finally, regarding the correspondence principle, it isnecessary to relate the results commented above for thecase p(0) = 10, with results obtained for larger values

FIG. 4: In this figure the initial value of the momentum isp(t0) = 100. Note that enhancement factors, by which differ-ences between solutions are multiplied, differ from the previ-ous case.

of the initial condition of the momentum. In particular,Fig. 4 shows the plot equivalent to the last graphic of Fig.3 for the initial value p(0) = 100. As already commentedabove, the qualitative behavior of the system does notchange. Nonetheless, there are significant modificationsin quantitative aspects that lead us to conclude that thebehavior is more classical.First of all we notice that the larger the value of p(0),

the longer (in terms of its period) the system stays stable.This is due to the fact that the corrections due to the mo-ments are relatively smaller and take longer to move thesystem significantly from its classical trajectory. Moreprecisely, as can be seen in Fig. 4, the system has to beevolved during six cycles so that the departure from theclassical trajectory, dominated by distributional effects,(1q

2 + 1p2) is of the same order of magnitude as the

one obtained for the previous (p(0) = 10) case with justtwo cycles.In addition, as another important result of this paper,

we note that the relative importance between the twoquantum effects, which can be measured by the quantity

:= (2q2 + 2p

2)/(1q2 + 1p

2), (46)

is smaller the larger the energy of the system. Thatis, from Fig. 4, we get 1011 for the case withlarger energy (p(0) = 100), whereas 108 for theprevious less-energetic case with p(0) = 10. The casep(0) = 1 has also been checked for which, after a littlebit more than half a cycle, the following values are mea-sured: (1q

2 + 1p2) 103 and (2q2 + 2p2) 108.

These results give 105 for the case p(0) = 1. Thisresult shows that the quantity defines a semiclassicalbehavior of a system when its value is small. Nonetheless,when tends to zero there are still distributional effectspresent. This shows that, as commented in the introduc-tion, the classical limit of a quantum state is an ensembleof classical trajectories described, in this context, by itscorresponding classical moments.

13

VI. CONCLUSIONS

In this paper the formalism presented in Ref. [19],to analyze the evolution of classical and quantum prob-ability distributions, has been applied to the system ofa particle on a potential. Due to the kinetic term, theHamiltonian of this system is quadratic in the momen-tum, and its dependence on the position is completelyencoded in the potential. The special properties of theharmonic Hamiltonians, which are defined as those thatare at most quadratic on the basic variables, makes themmuch easier to be analyzed. Thus, the study has beendivided in two different sectors. On the one hand, thecomplete set of harmonic Hamiltonians has been stud-ied; and, on the other hand, for the anharmonic case aninteresting example has been chosen: the pure quarticoscillator.By choosing different functional forms of the potential,

three physically different harmonic Hamiltonians can beconstructed. First, the system of a particle moving un-der a uniform force, which also includes the free particlewhen the value of this force is considered to be zero. Sec-ond, the harmonic oscillator with a constant frequency .And finally the inverse harmonic oscillator, which can beunderstood as a harmonic oscillator with imaginary fre-quency. For all of them the moments corresponding totheir stationary and dynamical states have been explic-itly obtained. In this framework the stationary statescorrespond to fix points of the dynamical system, whichis composed by the infinite set of equations of motionsfor expectation values and moments. Therefore, in orderto find these stationary moments, the algebraic systemobtained by dropping all time derivatives must be solved.With this procedure, and contrary to the usual treatmentof considering the time-independent Schrodinger equa-tion, the stationary moments can be obtained withoutsolving any differential equation.

More precisely, regarding the particle under a uniformforce, it has been shown that even if the classical (distri-butional) case accepts a stationary state where the par-ticle is at rest at any position and with arbitrary value ofits corresponding (high-order) fluctuations, such a stateis forbidden in the quantum system by the Heisenberguncertainty principle. For the harmonic oscillator, themoments corresponding to any stationary state have beenobtained in terms of the frequency of the oscillator andthe energy of the state. These relations are valid for anystationary state. The only ingredient that is not derivedby the present formalism, and thus one needs to includeby hand, are the eigenvalues of the energy. Finally, it hasbeen proven that the inverse harmonic oscillator can nothave stationary states.

Concerning the pure quartic oscillator, the momentscorresponding to any stationary state have been derivedby making use of the above technique. In this case, thesystem of equations is not complete and thus it does notfix the whole set of moments. Hence, apart from the en-ergy of the state, the fluctuation of the position has been

left as a free parameter. Furthermore, in order to con-straint the values of these two parameters, use has beenmade of the high-order inequalities which were derived in[19]. For the particular case of the ground state, a rea-sonable assumption is that the Heisenberg uncertaintyrelation is saturated. This leads to a tight interval forthe value of the ground energy (41). It turns out that theexact (numerically computed) value of this energy is notcontained in this interval, but it is quite close. Thereforeone can assert that, even if the exact saturation of theHeisenberg uncertainty relation provides a good approxi-mation for the ground state of the pure quartic oscillator,it is not exactly obeyed.

The above analysis shows the practical relevance of theinequalities that were derived in [19] as a complementarymethod to extract physical information from the system.In particular, high-order inequalities are of relevance be-cause the conditions they provide are stronger than theones obtained from lower-order inequalities.

Finally, a numerical computation of the dynamicalstates corresponding to the pure quartic oscillator hasbeen performed. To that end, a Gaussian in the positionhas been assumed as the initial state. In this setting, anumber of interesting results have been obtained.

First, the validity of the method has been analyzed.The present formalism is valid as long as the high-ordermoments that one drops with the cutoff are small. Thenatural tendency of the moments in this system is tooscillate with a growing amplitude and thus, from certainpoint on, this method will not give trustable results. Inorder to find the region of validity of the method, onthe one hand, different cutoffs have been considered andthe convergence of the solution with the cutoff order hasbeen studied. On the other hand, the conservation of theHamiltonian, as well as the fulfillment of the high-orderinequalities mentioned above, has been monitored duringthe evolution. With these control methods at hand, onecan estimate when (after how many cycles) the formalismis not valid anymore. In particular, this validity timeincreases with the value of the initial classical energy.

Second, the departure of the centroid from its classicalpoint trajectory has been analyzed, as well as the relativerelevance of the two different quantum effects: the dis-tributional and the purely quantum effects. It has beenshown that, as one would expect, the former ones, whichare also present in the evolution of a classical probabil-ity distribution, are much more relevant than the latterones. Nonetheless, the strength of the purely quantum ef-fects in the equations of motion is of order ~2. Therefore,a change in the numerical value of the Planck constantwould tune the relative relevance of these effects.

Finally, the correspondence principle has also been ver-ified in the sense that the larger the classical initial valueof the energy is chosen, the smaller purely quantum ef-fects are measured. In particular, the smallness of thequantity , as defined in (46), gives a precise notion ofsemiclassicality. In fact the vanishing of would definea complete classical (distributional) behavior of the sys-

14

tem. Let us stress the fact that this classical behavioris distributional. In other words, and as commented al-ready throughout the paper, the classical limit of a quan-tum state is not a unique orbit on the phase space but,instead, an ensemble of classical trajectories which aredescribed by a probability distribution or, in the contextof the present formalism, by its classical moments.

Acknowledgments

The author thanks Carlos Barcelo, Raul Carballo-Rubio, Inaki Garay, Claus Kiefer, Manuel Kramer, and

Hannes Schenck for discussions and comments. Specialthanks to Martin Bojowald for interesting comments ona previous version of this manuscript. Financial supportfrom the Alexander von Humboldt Foundation through apostdoctoral fellowship is gratefully acknowledged. Thiswork is supported in part by Projects IT592-13 of theBasque Government and FIS2012-34379 of the SpanishMinistry of Economy and Competitiveness.

[1] W. Boucher and J. H . Traschen, Phys. Rev. D 37, 3522(1988).

[2] A. Anderson, Phys. Rev. Lett. 74, 621 (1995).[3] T. N. Sherry and E. C. G. Sudarshan, Phys. Rev. D 18,

4580 (1978); Phys. Rev. D 20, 857 (1979).[4] A. Peres and D. R. Terno, Phys. Rev. A 63, 022101

(2001).[5] H. T. Elze, Phys. Rev. A 85, 052109 (2012).[6] A. J. K. Chua, M. J. W. Hall, and C. M. Savage, Phys.

Rev. A 85, 022110 (2012).[7] C. Barcelo, R. Carballo-Rubio, L. J. Garay, and R.

Gomez-Escalante, Phys. Rev. A 86, 042120 (2012).[8] L. E. Ballentine, Y. Yang, and J. P. Zibin, Phys. Rev. A

50, 2854 (1994).[9] L. E. Ballentine and S. M. McRae, Phys. Rev. A 58, 1799

(1998).[10] M. Bojowald and A. Skirzewski, Rev. Math. Phys. 18,

713 (2006).[11] M. Bojowald, B. Sandhofer, A. Skirzewski, and A.

Tsobanjan, Rev. Math. Phys. 21, 111 (2009).[12] M. Bojowald, Class. Quantum Grav. 29, 213001 (2012).[13] M. Bojowald and R. Tavakol, Phys. Rev. D 78, 023515

(2008).[14] M. Bojowald, D. Brizuela, H. H. Hernandez, M. J. Koop,

and H. A. Morales-Tecotl, Phys. Rev. D 84, 043514(2011).

[15] M. Bojowald, Phys. Rev. D 75, 081301 (2007); Phys.Rev. D 75, 123512 (2007).

[16] M. Bojowald, P. A. Hohn, and A. Tsobanjan, Class.Quantum Grav. 28, 035005 (2011); Phys. Rev. D 83,

125023 (2011).[17] P. A. Hohn, E. Kubalova, and A. Tsobanjan, Phys. Rev.

D 86, 065014 (2012).[18] M. Bojowald and A. Tsobanjan, Phys. Rev. D 80, 125008

(2009); Class. Quantum Grav. 27, 145004 (2010).[19] D. Brizuela, Phys. Rev. D 90, 085027 (2014).[20] C. M. Bender and T. T. Wu, Phys. Rev. 184, 1231

(1969); Phys. Rev. Lett. 27, 461 (1971); Phys. Rev. D7,1620 (1973).

[21] B. Simon and A. Dicke, Ann. Phys. 58, 76 (1970).[22] C. M. Bender, K. Olaussen, and P. S. Wang, Phys. Rev.

D 16, 1740 (1977).[23] A. Voros, J. Phys. A: Math. Gen. 27, 4653 (1994).[24] E. Z. Liverts , V. B. Mandelzweig, and F. Tabakin, J.

Math. Phys. 47, 062109 (2006).[25] A. C. O. Oliveira, A. R. Bosco de Magalhaes, and J. G.

Peixoto de Faria, Physica A 391, 5082 (2012).[26] K. Banerjee, Phys. Lett. A 63, 223 (1977).[27] J. L. Richardson and R. Blankenbecler, Phys. Rev. D 19,

496 (1979).[28] J. B. Bronzan and R. L. Sugar, Phys. Rev. D 23, 1806

(1981).[29] Note that in the mentioned references the considered ki-

netic term in the Hamiltonian is chosen to be p2, insteadof p2/2. Therefore a factor 3

4 must be introduced to re-

late the energy eigenvalues given in those references withthe one obtained by the notation of the present paper.

[30] All objects of the form x2 must be understood as (x)2.