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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: [email protected]
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.