Some consequences of GUP induced ultraviolet wavevector cutoff in one-dimensional Quantum Mechanics

0.34 MB·English

by  K. Sailer

Some consequences of GUP induced ultraviolet wavevector cutoff in one-dimensional
Quantum Mechanics
K. Sailer,1 Z. P´eli,1 and S. Nagy1,2
1Department of Theoretical Physics, University of Debrecen, P.O. Box 5, H-4010 Debrecen, Hungary
2MTA-DE Particle Physics Research Group, P.O.Box 51, H-4001 Debrecen, Hungary
(Dated: January 30, 2013)
A projection method is proposed to treat the one-dimensional Schr¨odinger equation for a single
particlewhentheGeneralized UncertaintyPrinciple(GUP)generatesanultraviolet (UV)wavevec-
torcutoff. Theexistence of auniquecoordinate representation called thenaiveoneis derived from
theone-parameterfamily ofdiscrete coordinaterepresentations. Inthisbandlimited QuantumMe-
chanics a continuous potential is reconstructed from discrete sampled values observed by means of
3
a particle in maximally localized states. It is shown that bandlimitation modifies the speed of the
1
center and the spreading time of a Gaussian wavepacket moving in free space. Indication is found
0
that GUPaccompanied bybandlimitation may cause departuresof thelow-lying energy levels of a
2
particle ina boxfrom those in ordinary QuantumMechanics much less suppressedthan commonly

n thought when GUP without bandlimitation is in work.
a
J PACSnumbers: 04.60.Bc

9
2
I. INTRODUCTION continuously shifted to each other. This is a sign that

] positions can only be observedwith a maximalprecision
h
p There are several theoretical indications that Quan- ∆xmin ≈ a, while momenta can be measured with arbi-
- tum Gravity may have consequences in the behaviour of trary accuracy [9, 15–21]. Although there exist formal
h low-energy quantum systems [1–4]. The corresponding coordinate eigenstates even in that case, but – as op-
at effective Quantum Mechanics is believed to be based on posed to ordinary Quantum Mechanics, – they cannot
m theGeneralizedUncertaintyPrinciple(GUP)[5–11],var- be approximated now by a sequence of physical states
[ ious modifications of Heisenberg’s Uncertainty Principle with uncertainty in position decreasing to zero [15]. A
(HUP). Recently even a proposal has been put forward so-called naive coordinate representation can be built
1 toprobeexperimentallythedeparturefromHUPinadi- up representing the operator algebra of [xˆ,kˆx] = i~ as
v
rectmanner[12]. AmongthevariousrealizationsofGUP xˆ x and kˆ i∂ on square-integrable functions
3 x x
1 there is a class when the deformation of the commuta- of⇒the coordinat⇒e x− R, referred to below as coordi-
∈
9 torrelationforthe operatorsofthe coordinatexˆ andthe nate wavefunctions. Then the canonical momentum sat-
6 canonical momentum pˆ depends only on the canonical isfying the relation in Eq. (1) can be represented as
x
. momentum, pˆ α−1F−1( iα~∂ )whereF−1 standsfortheinverse
1 x x
30 [xˆ,pˆx]=i~f(α|pˆx|) (1) toofrt⇒shoefftuhnectwioanv−eFve(cut)or=an0dutfh(d|euu′′c|)anaonndicraellamteosmtehnetuompevraia-
1 withthedeformationfunctionf(u),whereu=αpx and kˆ = (α~)−1F(αpˆ ). LeRt us note that the commutator
: α= (ℓ /~)withthe Plancklen|g|thℓ 1.616 10−35 x x
v O P P ≈ × in the left-hand side of Eq. (1) is invariant under the
i misasmallparameter. Inthe presentpaperweshallre- reflection x x, p p so that the deformation
X x x
strict ourselves to deformation functions for which there → − → −
function should be an even function of p which implies
r exists a minimal wavelength, i.e. a maximal magnitude x
a thatbothfunctionsF(u)anditsinverseareeven,aswell.
K of the wavevector but the canonical momentum can
Furthermore, f(0)=1 has to be required in order to re-
take arbitrary large values (as opposed to the cases dis-
cover HUP in the limit α 0. Here we shall restrict
cussede.g. in[13]). Thenphysicalstatesarerestrictedto →
ourselves to particular choices of the deformation func-
those of finite band width, i.e. the wavevector operator
tion which are monotonically increasing with raising u
kˆx can only take values in the interval [ K,K]. and for which the limits F( )= α~K remain fini|te.|
−
The mathematical structure of Quantum Mechanics ±∞ ±
There are known a few explicit cases of such deforma-
with finite band width is rather delicate [14], even in tion functions, e.g. f =1+α2p2 with F(u)= arc tan u
x
the one-dimensional case. Namely, while both opera- [15, 22], f = exp(α2p2) with F(u) = (√π/2) erf u [20],
tors of the wavevector kˆx and the canonical momentum and f = exp(αpx )xwith F(u) = (1 e−|u|) sign u
pˆx are self-adjoint, the coordinate operator xˆ cannot be used after Taylor|ex|pansion in [23, 24] an−d all implying
self-adjoint, but only Hermitian symmetric in order to ~K 1/α and a α~ ℓ .
P
satisfy the deformed commutator relation (1). It has, ≈ ≈ ≈
however, a one-parameter family of self-adjoint exten- Coordinate space turns out to exhibit features of dis-
sions with eigenvalues determining equidistant grids on creteness [25] and continuity at the same time like infor-
the coordinate axis, defining a minimal length scale a, mation does [26–32]. The main idea is that space can
while the grids belonging to the various extensions are be thought of a differentiable manifold, but the physical
2
degreesoffreedomcannotfillitinarbitrarilydenseman- perturbation theory.
ner. Ithasalsobeenconjecturedthatdegreesoffreedom
Itisappropriatetomakethefollowingremarks: (i)As
corresponding to structures smaller than the resolvable
to our viewpoint of removing the UV components of the
Planck scale turn into internal degrees of freedom [33–
wavefunctions, it is rather a naive approach to estimate
36]. Although the coordinate wavefunctions introduced
the additional effects due to the existence of the finite
in the manner described above do not have the simple
UV cutoff. Our viewpoint would be exact when the co-
probabilisticmeaning,yetprovideusefultoolstocharac-
ordinate space were discrete, but it is not although the
terize the quantum states of the particle which can then
self-adjoint extensions of the coordinate operator have
be analyzed e.g. in terms of maximally localized states
discreteeigenvaluesformingagridonthecoordinateaxis.
[16–19]. Below we shall discuss the justification of that
However,there exists a one-parameterfamily ofsuchex-
naivecoordinaterepresentationforone-dimensionalban-
tensionsandthatofthecorrespondinggridswhichcanbe
dlimited Quantum Mechanics in more detail.
transformedintoeachotherbycontinuousshifts. Thisin-
In the present paper we shall concentrate on the so- dicates that quantum fluctuations of wavelength smaller
lution of the Schr¨odinger equation for the coordinate than the minimal length scale a cannot be probably ex-
wavefunctions in the case when GUP implies finite band cluded completely, but rather should have been treated
width. It is well-known that GUP directly affects the by more sophisticated methods, like e.g. renomalization
Hamiltonian through the modification of the canoni- groupmethods. Fortunately,intheone-dimensionalcase
cal momentum and that of the kinetic energy operator the solutions of the bandlimited Schr¨odingerequation in
(pˆ2/2m) (~2kˆ2/2m) which can be expanded – when thenaivecoordinaterepresentationusedbyusreflectin-
x − x herently the physical equivalence of any of those grids.
low-energystatesare considered–in powersofthe small
Therefore, our projection technique can give a reliable
parameter α. This pure GUP effect has been treated in
order-of-magnitudeestimateoftheimportanceofthead-
the framework of the perturbation expansion using the
ditional effect of the UV cutoff as compared to the pure
naive coordinate representation and discussed in detail
GUP effect. (ii) As to our choice of the model, the par-
for various quantum systems (see Refs. [22, 24, 37–49]
ticle in a box, it is rather a toy model. The pure GUP
withoutthequestofcompleteness),amongothersforthe
effectonthe low-lyingstationarystateshasalreadybeen
particleinabox[23,24,42,43,46,50,51]. Treatmentsin
discussedandnoticedthatthemodelwithpreciselygiven
the Bargmann-Fock representation [52–54], and various
box size L is ill-defined in the sense that a change of the
path-integralformulations[55–57]havebeenworkedout.
box size of the order ℓ , i.e. that of the maximal ac-
Here we shall take the viewpoint that the bandlimited P
curacy ∆x of the position determination causes an
Quantum Mechanics is an effective theory in the frame- min
energy shift of the order (ℓ /L) as compared to the
work of which no quantum fluctuations of wavelength P
O
pure GUP effect of the order (ℓ /L)2 [58]. In our
smaller than those of the order of the minimal length P
O
approach the Hamiltonian operates on the subspace of
are possible, i.e. the coordinate wavefunctions should
(cid:0) (cid:1)
states with finite band width and transforms the origi-
not contain Fourier components with wavevectors out-
nally local potential effectively into a nonlocal one. Pro-
side of the finite band with k [ K,K]. In order to
x
∈ − jectingouttheUVcomponentsofthepotentialresultsin
built in this restrictioninto the Schr¨odingerequation we
a kind of smearing out the edges of the box in regions of
propose to use Hamiltonians operating on the subspace
the size of (∆x ). Our purpose is to determine the
of the square-integrable coordinate wavefunctions with min
O
additional shift of the low-lying energy levels caused by
finite band width. To ensure this we introduce the pro-
jector Πˆ onto that subspace and restrict the solutions of the existence of the finite UV cutoff. We shall see that
this turns out to be of the order (ℓ /L), being much
the Schr¨odingerequationto the bandlimited subspace P
O
H more significant than the pure GUP effect, so that in
of the Hilbert space. We shall also show that the above
somesensethe resultof[58]willbe recovered. Neverthe-
mentioned naive coordinate representation of the ban-
less,ourapproachmaybe ahintthatthis kindofenergy
dlimitedHilbertspace uniquelyexistsandsolutionsof
H shift mightbe the true effect, whenthe physically realis-
the bandlimited Schr¨odinger equation automatically re-
ticboxwithsmearedoutwallsisconsideredandmodeled
flectthesymmetrythattheformulationsofthetheoryon
by performing the projection which determines the op-
any of the equidistant spatial grids of spacing a exhibit
eration of the Hamiltonian on the bandlimited Hilbert
the same physical content. Moreover, we shall discuss
space .
the reconstruction of a unique continuous bandlimited
H
potential from sampled values taken on such grids by Our paper is constructed as follows. In Sect. II the
meansofmaximallylocalizedstates. Thebandlimitation projectionmethod is introduced and the integralkernels
will be shown to broaden the peaks and smear out the for the various projectedoperatorsdetermined. For one-
sudden jumps of the microscopic potential over a region dimensional bandlimited Quantum Mechanics a justifi-
of the Planck scale. Below we shall apply our projec- cation of the naive coordinate representation is given in
tion method to determine the free motion of a Gaussian Sect. III. A method is given in Sect. IV which enables
wavepacket as well as the energy shifts of the low-lying one to reconstruct a bandlimited continuous potential
stationary states of a particle in a box. The stationary from sampled values obtained by means of a particle in
problem shall be treated in the framework of first-order maximallylocalizedstate. ThefreemotionofaGaussian
3
wavepacketisthendiscussedinSect. Vintheframework be projected onto that subspace by an appropriate pro-
of bandlimited Quantum Mechanics. In Sect. VI the de- jector Πˆ, Oˆ ΠˆOˆΠˆ. The projector should cut off
⇒
termination of the shifts of the low-lyingenergy levels of the UV components of any square integrable function
a particle in a potential is formulated in the framework f(x) L2( , ), i.e. for its kernel Π(x,y) the rela-
∈ −∞ ∞
of the first-order perturbation theory. The problem of a tion
particle in a box is considered in Sect. VII as the limit-
∞
ingcaseofaparticleinasquare-wellpotentialtakingthe (Πf)(x)= dyΠ(x,y)f(y)
limit of infinite depth. After setting some notations in Z−∞
Subsect. VIIA and recovering the well-known result for = K dkxeikxxf˜˜(k ) L2 ( , ) (4)
thepureGUPeffectinSubsect. VIIB,inSubsect. VIIC 2π x ∈ K −∞ ∞
theadditionalenergyshiftsofthelow-lyinglevelscaused ZK
by the existence of the finite UV cutoff are shown to be should hold that implies
dominant . Finally, the results are summarized in Sect.
K dk sin[K(x y)]
VIII. SeveraltechnicaldetailsaregivenintheAppendix. Π(x,y)= xeikx(x−y) = − . (5)
App. A reminds the reader on some mathematics rele- 2π π(x y)
Z−K −
vant for the self-adjoint extensionof the Hermitian sym-
metriccoordinateoperator. InApp. Bthewavefunctions It is straightforward to show that Πˆ is a projector sat-
oftheunperturbedsystem,i.e. thoseforaparticleinthe isfying Πˆ2 = Πˆ. When K the operation of any
square-wellpotentialintheframeworkofusualQuantum Hermitiansymmetricoperato→rOˆ∞onanystate ψ canbe
| i
Mechanics are derived. The operation of the GUP mod- represented as
ified kinetic energy operator on exponential functions is
determined in App. C. The details of the evaluation of ∞ dyO(x,y)ψ(y)= ∞ dkxψ˜˜(k )O˜˜ eikxx (6)
2π x kx
theadditionalenergyshiftsofthelow-lyingenergylevels Z−∞ Z−∞
foraparticleinaboxarepresentedinAppendicesDand
where the kernel O(x,y) and the formal differential op-
E. InApp. Fmaximallylocalizedstatesofaparticleare
constructed. Finally, in App. G the bandlimited poten- erators Ox and O˜˜kx are related as
tliikael preoctoennsttiarulcistepdrefsreonmtesda.mpled values of a Dirac-delta O(x,y)=O δ(x y)= ∞ dkxe−ikxyO˜˜ eikxx. (7)
x − 2π kx
Z−∞
Hermitian symmetry implies O(x,y) = [O(y,x)]∗ and
II. PROJECTORS ONTO THE SUBSPACE OF
WAVEFUNCTIONS WITH FINITE BAND O˜˜kx = [O˜˜−kx]∗. A few examples are summarized in the
WIDTH table:
Letψ(x), ψ˜(px)andψ˜˜(kx)bethewavefunctionsofthe O O˜˜
state ψ in the naive coordinate, canonical momentum x kx
| i
and wavevectorrepresentations, respectively. The scalar
product of arbitrarystates |ψi and |φi can be written as xn (n∈N) 12[(−i∂k→x)n+(i∂k←x)n]
φψ = ∞ dxφ∗(x)ψ(x)= K dkxφ˜˜∗(k )ψ˜˜(k ) (−i∂x)n (n∈N) kxn
x x
h | i 2π
Z−∞ Z−K
∞ dp |−i∂x|=p(i∂x)2 |kx|
= x φ˜∗(p )ψ˜(p ) (2)
2π~f(αp ) x x
Z−∞ | x|
It is straightforwardto show that the symmetrized form
and is kept invariant under the transformation
of O˜˜ for O =xn with partial derivatives ∂→ and ∂←
ψ(x)= K dkxeikxxψ˜˜(k ), actinkgxto thexright and left, respectively, is inkaxgreemeknxt
2π x with Heisenberg’s commutation relation [xˆ,kˆ ]=i.
Z−K x
∞ In Quantum Mechanics with finite band width K the
ψ˜˜(kx)= dxe−ikxxψ(x). (3) kernel O(x,y) should be projected as
Z−∞
∞ ∞
An arbitrary square integrable function ψ(x) (ΠOΠ)(x,y)= dz duΠ(x,z)O(z,u)Π(u,y)
L2(−∞,∞) contains ultraviolet (UV) Fourier compo∈- Z−∞ Z−∞
tnheentssowluittihon|skxo|f>quKan,tausmwmelel.chIanniocradleerigteonveanlsuuereeqthuaat- = K d2kπxe−ikxyO˜˜kxeikxx (8)
tions, as well as that of the Schr¨odinger equation belong Z−K
to the subspace L2 ( , ) of wavefunctions with fi- while Hermitian symmetry is preserved,
K −∞ ∞
nite bandwidth, anyoperatorOˆ ofanobservableshould [(ΠOΠ)(y,x)]∗ = (ΠOΠ)(x,y). Now, one easily
4
finds the kernel of ΠˆkˆnΠˆ, One might accept the usage of the naive coordinate
x
representationwithsomereservationbecausethe coordi-
K dk
(ΠknΠ)(x,y)= xeikx(x−y)kn nate operator xˆ is not self-adjoint (see App. A). There-
x 2π x
Z−K fore, no coordinate eigenstates exist in the physical do-
=( i∂x)nΠ(x,y), (9) main and the introduction of the coordinate represen-
−
and that of ΠˆxˆnΠˆ, tation becomes questionable. There exists, however, a
one-parameter family of the self-adjoint extensions xˆ
θ
(ΠxnΠ)(x,y) of the coordinate operator, labeled by the parameter
K dk 1 θ [0,a). Any ofthese extensions for fixedθ exhibits an
=Z−K 2πxe−ikxy2[(−i∂k→x)n+(i∂k←x)n]eikxx dorl∈itmhoitneodrmHaillbceormt sppleatceeset.oIfneitgheenvweacvtoevrsec|xtoθnririenptrheesebnatna--
= 1(xn+yn)Π(x,y) (10) tion the correspondingHeigenfunctions k xθ =ψ˜˜θ (k )
2 h x| ni xn x
with k [ K,K] are given by Eq. (A10) in App. A,
which imply for the kernels of the functions f(kˆx) and whereaxs∈the−correspondingdiscretenondegenerateeigen-
V(xˆ) of operators kˆx and xˆ, respectively, valuesxθn =an+θ formagridwithspacingaintheone-
dimensionalspace. Therefore,anyvector ψ canbe
(Πf(k )Π)(x,y)=f( i∂ )Π(x,y),
x x | i∈H
− represented as a linear superposition of the base vectors
1
(ΠV(x)Π)(x,y)= [V(x)+V(y)]Π(x,y). (11) xθ for any given θ, ψ = √a ∞ xθ ψ xθ ,
2 | ni ∈ H | i n=−∞h n| i| ni
sothatdiscretecoordinaterepresentations ofthenor-
θ
Making use of the projector Πˆ introduced above, the malizedvectors ψ viathevectoPrs ψθR= xθ ψ
time-dependent and the stationary Schr¨odinger equa- ℓ2 arise (the nor|mial∈izaHtion implies a ∞{ n ψhθn2|=i}1∈).
n=−∞| n|
tions can be written as ThentheoperatorsOˆ overthebandlimitedHilbertspace
P
i~∂ ψ =ΠˆHˆΠˆ ψ , (12) should be represented by the countably infinite di-
ΠˆHˆΠˆt|ψi=E ψ ,| i (13) Hmensional matrices Onθn′ = hxθn|Oˆ|xθn′i. Thus a one-
| i | i parameter family of discrete coordinate representations
respectively,intermsoftheprojectedHamiltonianΠˆHˆΠˆ is available.
θ
R
whenQuantumMechanicswithfinite bandwidthiscon- Using the wavevector representation, one realizes im-
sidered. The initial condition for the time-dependent mediatelythatthetransformationfromadiscretecoordi-
Schr¨odingerequation(12)shouldbe a physicalstate,i.e. naterepresentation θ toanotherone θ′ belongstothe
anlisaonboabnvdioluimsliyteedn.suTrehsethuasatgteheofsttahteewpriloljnecottecdonHtaaminiltthoe- U(1) group becauseRψ˜˜xθn′(kx) = eikx(θ′R−θ)ψ˜˜xθn(kx). Such
a transformation means a shift of the spatial grid from
UV Fourier components. The rules in Eq. (11) for pro- xθ =na+θ to xθ′ =na+θ′ onwhich one describes
jecting functions of operators enable one to make both { n } { n }
one-dimensionalbandlimited Quantum Mechanics. Nev-
the kinetice energy and the potential energy piece of the
ertheless, there should be a distinction of Quantum Me-
projected Hamiltonian explicit in the naive coordinate
chanicsdiscretizedonagridandthecasediscussedhere.
representation.
In the latter case space reveals discrete and continuous
features at the same time, similarly as information does
[26–36]. Namely, the physical content of any of the dis-
III. COORDINATE REPRESENTATIONS
crete coordinate representations should be identical,
θ
R
i.e. the U(1) group of the transformations among them
In this section we show that the naive coordinate rep-
should be a symmetry. Thus, physics contained in the
resentation for one-dimensional bandlimited Quantum
scalar products
Mechanics formulated by means of the projection tech-
nique proposed in Sect. II is equivalent with any of the ∞
discrete coordinate representations based on the com- φψ =a φθ∗ψθ (14)
h | i n n
plete orthonormalsetsofeigenvectorsofthe variousself- n=−∞
X
adjointextensions ofthe coordinateoperator. Moreover,
the time-dependent Schr¨odinger equation (12) and the of arbitrary vectors ψ , φ should be independent
| i | i ∈H
stationary one, Eq. (13) in the naive coordinate repre- oftheparticularchoiceofthe representation θ. Inthat
R
sentation as well as their solutions reflect inherently the respectit is importantto underline that the Schr¨odinger
U(1) symmetry which reveals itself in the unique physi- equations obtained by using the projection method are
cal content of the formulations of the theory in terms of givenin their abstractforms in Eqs. (12) and (13) with-
the various discrete coordinate representations. There- out referring to any representation, and their solutions
fore,the so-callednaive coordinaterepresentationis cor- ψ are authomatically in the bandlimited Hilbert space,
| i
rect for one-dimensional bandlimited Quantum Mechan- ψ .
| i∈H
ics, although the coordinate wavefunction looses its di- Now we show that to each vector ψ , i.e. to each
rect probability meaning as opposed to ordinary Quan- vector ψθ = xθ ψ of the represe|ntia∈tioHn one can
{ n h n| i} Rθ
tum Mechanics. associatea single coordinatewavefunctionψ(x), and the
5
latter is independent of the choice of the representation ψ is unique, independent of the representa-
| i ∈ H
used for its construction. tion usedtoitsconstruction. Ontheotherhand
θ θ
R R
italsomeansthattakingsampledvaluesonvarious
1. To any position x R belongs exactly a single spatiallyshiftedgrids xθ andinsertingthoseinto
eigenvalue xθ′ = an∈′+θ′ = x of a particular self- { n}
n′ Eq. (16)oneobtainsfinallythe samewavefunction
adjoint extension xˆθ′ of the coordinate operator of the continuous variable x. Similar arguments
(see the discussion at the end of App. A). This lead to the unique kernel O(x,y) associated to the
enables one to construct a wavefunction of contin- operator Oˆ mapping into itself.
uous variable, H
Thusonecanconcludethatthesolutionsofthebandlim-
∞
ψ(x)= xθ′ =xψ = xθ′ =xxθ ψθ (15) itedSchr¨odingerequationsforonespatialdimension,be-
h n′ | i h n′ | ni n ing bandlimited themselves, satisfy the generalized sam-
n=X−∞ pling theorem expressed in Eq. (16) authomatically and
thereforereflectinherentlytheU(1)symmetryunderthe
in a reliable manner, where ψ(xθ) = ψθ are sam-
n n continuous shifts of the spatial grid determined by the
pled values of the coordinate wavefunction on the
discrete eigenvaluesofthe variousself-adjointextensions
arbitrarily chosen grid xθ . Here figures the ma-
trix element xθ′ = xx{θ n=} aΠ(x,xθ) of the uni- of the coordinate operator.
h n′ | ni n
tary transformation from the discrete coordinate
representation characterized by θ to the one char-
IV. BANDLIMITED POTENTIALS
acterizedbyθ′,whichisdirectlyrelatedtothepro-
jector Πˆ onto the bandlimited Hilbert space. Thus
Letushereillustratethemanneronecouldobservethe
Eq. (15) recasted into the form
formally local potential V(x) in the framework of Quan-
ψ(x)=aΠ(x,x )ψθ (16) tumMechanicswithfinitebandwidth. Thebestonecan
n n dotoconstructmaximallylocalizedstatesϕ (x)centered
x¯
at arbitrary positions x¯ [16–19] and detect the potential
can be interpreted as the particular case of the
exerted on it. One should however be aware of the fact
samplingtheoreminthe bandlimitedHilbertspace
thatthereexistonlyacountablesetofphysicallydistigu-
[28], a generalization of Shannon’s sampling theo-
rem [59]. With a similar logic identifying x = xθ′ ishable positions xθn, those of the eigenvalues of an arbi-
n′ trarily chosenself-adjoint extension xˆθ of the coordinate
and y = xθ′′ as the n′-th and n′′-th eigenval-
n′′ operator, which form a grid of spacing a = π/K. Thus
ues of the appropriate self-adjoint extensions xˆθ′ we can sample the potential only at the grid points. Let
and xˆθ′′, respectively, one is enabled to reexpress ϕθ be the sequence of the maximally localized states
any matrix element (ΠOΠ)(x,y) = xΠˆOˆΠˆ y of {| ni}
h | | i centered on the grid points. Thus, potential values
an arbitrary operator Oˆ in terms of the matrix
Oθ = xθ ΠˆOˆΠˆ xθ = xθ Oˆ xθ as V¯θ = ϕθ ΠˆVˆΠˆ ϕθ (18)
nm h n| | mi h n| | mi n h n| | ni
∞ can only be observed at a discrete set of points of a grid
O(x,y)=a2 Π(x,xθ)Oθ Π(xθ ,y) (17) with spacing a. The maximally localized states ϕθ
n nm m | ni
n,m=−∞ belong to the subspace of bandlimited wavefunctions,
X
Πˆ ϕθ = ϕθ , so that Eq. (18) reduces to
implyingO(xθ,xθ )=Oθ . Therefore,toanyvec- | ni | ni
n m nm
tor of the bandlimited Hilbert space represented V¯θ = ϕθ Vˆ ϕθ . (19)
n h n| | ni
by the vector ψθ ℓ2 in the discrete coordinate
{ n} ∈ According to Shannon’s basic sampling theorem on ban-
representation there corresponds a continuous
Rθ dlimited real functions [59], a continuous potential V¯(x¯)
coordinatewavefunctionψ(x), andtoanyoperator
ΠˆOˆΠˆ mapping the bandlimited Hilbert space into wreictohnfistnriutectbeadnfdrwomidtihtsKva(lkuxes∈V¯[−θK=,KV¯(])xθca)ntapkeernfecotnlytbhee
itself, i.e. to any matrix of the discrete coordinate n n
set of equidistant points xθ spaced 2π =a apart:
representation θ correspondsthekernelO(x,y)of { n} 2K
R
the continuous coordinate representation. ∞
V¯(x¯)=a V¯θΠ(x¯ xθ). (20)
2. Being aware of the construction of the coordinate n − n
wavefunction ψ(x) = ∞ x = xθ′ 1ˆ1ψ via n=X−∞
n=−∞h n| | i
Eq. (15), one can insert any of the decomposi- Now one has to show that the reconstructed potential
tions of the unit operaPtor 1ˆ1 = ∞ xθ xθ V¯(x¯) is bandlimited and does not depend on the partic-
n=−∞| nih ni
over associated with any of the representations ular choice of the grid, i.e. that of the self-adjoint ex-
H P
. This means, on the one hand, that the result- tension of the coordinate operator. One can choose the
θ
Ring wavefunction ψ(x) associated in the naive co- sampled values V¯θ on the grid xθ =na+θ shifted by
n { n }
ordinate representation to the bandlimited vector any constant θ [0,a). Let (x¯) be the function which
∈ V
6
takesthevalues (xθ)=V¯θ foranyn Zandθ [0,a). The reconstructed bandlimited continuous potential is
V n n ∈ ∈
It is generally not bandlimited, implying then given by Eq. (G6) in App. G and is shown in
∞ dl Fig. 1.
(x)= eilx˜(l). (21)
V 2π V
Z−∞
Let us now reconstruct a potential V¯(x¯) from the sam-
pled values V¯θ for given θ and ask how far the resulted
n 0.8
function depends on the particular choice of θ, i.e. that
oftheparticularchoiceoftheself-adjointextensionofthe
0.6
coordinate operator xˆ. According to the reconstruction
formula in Eq (20) we get 0
V 0.4
∞ -x)/
V¯(x¯)=a V(xθn)Π(x¯−xθn) (22) -V( 0.2
n=−∞
X
which implies
0
V¯(x¯+θ)
∞ -0.2
=a (na+θ)Π(x¯ na) -10 -5 0 5 10
V − -
x/a
n=−∞
X
∞ dl K dq ∞
=a ˜(l) eilθ+iqx¯ ei(l−q)na.(23) FIG. 1. Sampled values (boxes) and the reconstructed ban-
Z−∞ 2πV Z−K 2π n=−∞ dlimitedcontinuouspotential(solidline)forV(x)=V0aδ(x).
X ThenumericalresultobtainedbyterminatingthesuminEq.
Making use of the sum (G2), one obtains (20) at n = ±50 and the analytic one given by Eq. (G4)
cannot be distinguished on thefigure.
K dl
V¯(x¯+θ)= ˜(l)eil(x¯+θ) = (x¯+θ) (24)
K
2πV V
Z−K
i.e. thereconstructedfunctionV¯(x¯)= (x¯)whichisin-
K
dependentofthechoiceoftheparticulaVrself-adjointrep- It is peaked taking values of the order V0 in the inter-
rOenseentcaatnionnoto,foxˆf aconudrsise,braencdolnimstirtuecdt, VthKe(x¯fu)n=cti(oΠˆnV)((x¯x¯)). mvaalnx¯ne∈ro[−ut21siad,e21oaf]tahnadtifnatlelsrvoaffl.rTahpeidclyhairnacatneroisstcicillwataovrey-
which contains modes outside of the band [ K,K].VBe- length of the oscillations is 2a.
low we shall take the sampled potential va−lues on the Another example is that the finite jump at x = 0 of
grid xn = na (for θ = 0) and, for the sake of simplicity, the potential step V(x) = V0Θ(−x) becomes smeared
suppress the upper indices θ. out. The sampled values taken at xn =na are
InApp. Fwehavedeterminedthemaximallylocalized
0
stateϕ˜˜x¯(kx)inthe wavevectorrepresentationbymaking V¯n =V0 dxϕna(x)2. (27)
| |
use of the method of Detournay, Gabriel, and Spindel Z−∞
[18], see Eq. (F7). Rewriting it into the coordinate rep-
ThevaluesV¯ aremonotonicallydecreasingwithincreas-
resentation, one finds n
ing n. Moreover, the relation ϕ (x)2 = ϕ (x na)2
na 0
ϕ (x)= K dkxeikxxϕ˜˜ (k ) holds for the integrand, implyi|ng | | − |
x¯ 2π x¯ x
Z−K −na
a 1 1 V¯n =V0 dxϕ0(x)2. (28)
= 2[Π(x−x¯+ 2a)+Π(x−x¯− 2a)], (25) Z−∞ | |
r
implyingapositioninaccuracyof ϕ xˆ2 x¯2 ϕ =a2/4. Becauseϕ0(x)isevenandnormalizedto1,onegetsV¯0 =
h x¯| − | x¯i 1V , andV¯ =V (1 r ) with r = nadxϕ (x)2 for
The wavefunction(25) is real, has a maximum at x=x¯, 2 0 ±n 0 2± n n 0 | 0 |
varies slightly in the smallinterval x [x¯ 1a,x¯+ n>0,where triviallyrn increasesstrictlymonotonically
∈Ix¯ ≡ − 2 fromr =0to1withngoingtoinfinitRy sothatV¯ takes
1a] centered at the point x¯ and falls off rapidly in an 0 n
2 the value V for large negative index n and decreases
oscillatory manner outside of the interval . Obviously 0
Ix¯ strictly monotonically through the value V¯ = 1V at
suchastateenablesonetodetectabandlimitedpotential 0 2 0
V¯(x¯)smearedoutascomparedtoV(x). Forexample,the n=0 to zero when n increases to infinitely large integer
values. Solongthelocalizedstateϕ (x)almostentirely
Dirac-deltalike potential, V(x)=V aδ(x) is observedas na
0 extends over a region where the potential is V or zero,
a broadened one with finite height V and finite width, 0
0 it detects its original value, but when it ‘feels’ the sud-
V¯ =V aϕ (0)2 = V0a2[Π(x 1a)+Π(x + 1a)]2. den jump of the potential step it detects monotonically
n 0 | xn | 2 n− 2 n 2 decreasing values when the increasing sequence x runs
n
(26) through x = 0. The sampled values already smear out
7
thesuddenjumpoveraregionofwidtha. Reconstruction
of the bandlimited potential V¯(x¯) from the sample shall
1
make the fall of the potential around x = 0 oscillatory.
Instead of evaluating analytically the reconstructed po-
0.8
tentialitselfweshallillustratetheoscillationsintroduced
bythereconstructiononamoresimpleexample. Forthat 0 0.6
V
purposeletuschoosethesample n = (xn =an)taken )/
V V -x
of the simple step function V(x) = V0Θ(−x) for which -V( 0.4
the well-known integral representation
V ∞ e−ilx 0.2
0
(x)= dl (29)
V 2πi l iǫ
Z−∞ − 0
holds. Thebandlimitedpotentialreconstructedfromthis
-10 -5 0 5 10
sample Vn is then given as x-/a
V K e−ilx
¯(x)= 0 dl . (30) FIG. 2. Sampled values (boxes), the reconstructed bandlim-
V 2πiZ−K l−iǫ itedcontinuouspotential(solidline)anditsanalyticapproxi-
Let us evaluate this integral by closing the straightline mation (dashedline) for thepotential stepV(x)=V0Θ(−x).
Thenumericalresult (solid line) hasbeenobtained bytermi-
section [−K,K] on the real axis via a half circle C− of nating thesum in Eq. (20) at n=±50.
radius K on the lower half of the complex l plane, along
whichone has l=Ke−iα withα running fromzeroto π,
V e−ilx pairwisecommuting,andtheir commonsetofeigenfunc-
¯(x)= 0 dl tions are the plane waves eikxx. The projector acts on
V 2πi l
ZC− plane waves as
V π
= 0 dαe−iKxcosα−Kxsinα. (31) ∞
2π Z0 dyΠ(x,y)eikxy =Θ(K−|kx|)eikxx (35)
Changingthe integrationvariablefromαto β =α 1π, Z−∞
−2 so that the subset of plane waves with wavevector k
one can recast the integral in the form x
[ K,K],canonicalmomentump (k )=F−1(α~k )/α∈
x x x
V π/2 (− , ) and energy E = [F−1(α~k )]2/(2mα2) ∈
¯(x)= 0 dβcos(Kxsinβ)e−Kxcosβ −∞ ∞ kx x ∈
V π [0, ) eigenvalues, respectively, form the set of com-
Z10 Si (Kx) mo∞neigenfunctionsoftheoperatorsΠˆHˆfreeΠˆ,ΠˆkˆxΠˆ and
=V0 2 − π . (32) ΠˆpˆxΠˆ in bandlimited Quantum Mechanics. For canon-
(cid:18) (cid:19) ical momenta much less than the Planck momentum,
The reconstructed potential tends to V for x , i.e. for p (1/α), one can expand the deforma-
0 x
zero for x + taking the value 1V at x = 0→an−d∞it tion func|tion| ≪in powers of the magnitude of u = αp
→ ∞ 2 0 x
falls in an oscillatory manner from the value V to zero. as f(u) = 1 + f u + f u2 + ... which implies the
0 1 2
The reconstructed potential and its analytic approxima- expansions F(u) =| u| 1f uu + 1(f2 f )u3 + ...,
− 2 1 | | 3 1 − 2
tion in Eq. (32) are demonstrated in Fig. 2. F−1(v)=v+1f v v +1(f2+2f )v3+...withv =α~k
2 1 | | 6 1 2 x
andthewell-knownmodificationoftheenergyeigenvalue
due to the GUP
~2k2 1
V. MOTION OF A WAVEPACKET IN FREE Ekx ≈ 2mx 1+f1α~|kx|+12(7f12+8f2)α2~2kx2+...
SPACE (cid:18) (cid:19)
(36)
for k K. As to the plane waves the only effect due
x
The Schr¨odingerequationforthe free motionofa par- | | ≪
to the finite band width is the lack of plane waves with
ticle in one-dimensional space, say the x axis is
wavevectorsfrom the UV region.
∞ Now let us seek the solution of the time-dependent
i~∂tψ(x,t)= dy(ΠHfreeΠ)(x,y)ψ(y,t) (33) Schr¨odinger equation (33) in the form of a wavepacket,
Z−∞ i.e. that of a superposition from plane waves belonging
with the Hamiltonian to the finite band of wavevectors,
1
Hfree(x,y)= 2mα2[F−1(−iα~∂x)]2δ(x−y). (34) ψ(x,t)= K dkxa(kx,t)eikxx, (37)
2π
Z−K
The HamiltonianHˆfree, the operatorsofthe wavevector [p (k )]2
kˆx and the canonicalmomentum pˆx =F−1(α~kˆx)/α are i~∂ta(kx,t)= x2mx a(kx,t). (38)
8
Inorderto chooseaninitialconditiona (k )=a(k ,t= with s = k k¯, p¯ = p (k¯), f¯ = f(α2p¯2) > 1,
0 x x x x
0) consistent with the generalized uncertainty relation f¯′ = f′(α2p¯2),−f′(u) = df(u)/du. The time-dependent
impliedbyEq. (1),onehasto be rathercautious. Inthe wavefunction (41) can then be rewritten as
wavevector representation the coordinate operator xˆ is
i p¯2
firenperdesiennttehdebdyenthsee dfoormmaailndifferentLia2l[opKe,rKato]rofi∂skqxuadree- ψ(x,t)=N exp −~2mt+ik¯x
Dx ⊂ − (cid:18) (cid:19)
integrable wavefunctions ψ˜˜(k ), consisting of absolutely K+k¯ ds
x e−A(t)s2+iB(x,t)s. (44)
cLo2n[tinKu,oKus].fTunhcetpiohnyssicwahlodsoemdaeinrivativescoanlssoistbinelgonofgthtoe ×Z−K+k¯ 2π
phys
wav−efunctions satisfying the geneDralized uncertainty re- Letusnowmakeuseofσk K andconsiderthelimiting
lationshouldbeasubsetofthe domain , . cases(i) k¯=0 and(ii) k¯ =≪K. As a goodapproximation
x phys x
WeshouldchoosetheinitialconditionfrDomtDhatph⊆ysDical wecanreplacethedefiniteintegralsbyimproperonesfor
K ∞ 0
domain, a (k ) . For xˆ being a symmetric oper- case(i) dk dk andforcase(ii) dk
0 x ∈Dphys −K x ⇒ −∞ x −2K x ⇒
ator, only two types of boundary conditions are allowed 0
dk . Because of the inequality Re A(t) > 0, the
[61]. In the set of functions satisfying ψ˜˜(K)= Cψ˜˜( K) in−t∞egralxRremains conRvergentand one ends upRwith
− R
with C = 1 the coordinate operator is self-adjoint, but
itnheoucor|ocr|adsienathteisospheorualtdorbweeerxeclduidagedonbaelcianutsheeincotohradtincaastee ψ(x,t)∝A−12(t)exp −~i 2p¯m2 t+ik¯x− B42A(x(t,)t) (45)
(cid:18) (cid:19)
representation and the GUP in Eq. (1) could not be
implying the Gaussian spatial distribution
satisfied. Thus the wavefunctions ψ˜˜(k ) should
x phys
satisfy the other type of boundary cond∈itDions, namely (x v¯t)2
ψ(x,t)2 A−1(t)exp − , (46)
Dirichlet’s boundary conditions ψ˜˜( K) = ψ˜˜(K) = 0 for | | ∝ (cid:18)− 2σx2(t) (cid:19)
−
which xˆ is not self-adjoint. Therefore we choose an ini-
centeredatthepositionx¯=v¯tattimetandmovingwith
tial condition a0(kx) satisfying the boundary conditions the speed v¯=f¯p¯/m>p¯/m. Therefore,the center ofthe
a ( K)=0,
0 ± wavepacketmoveswithalargerspeedinthebandlimited
casethaninordinaryQuantumMechanics. Thevariance
[p (k ) p¯]2
a (k )= exp x x − , (39) of the position distribution is given by
0 x N − 4σ2
(cid:18) p (cid:19) A(t)2 ~2f¯2 t2
corresponding to a Gaussian wavepacket with the mean σx2(t)= R| e A|(t) = 4σ2 1+ τ2 , (47)
wavevectork¯, the meancanonicalmomentum p¯=p (k¯), p (cid:20) (cid:21)
x
the variance σp, and the normalization factor . With- where the characteristic time for the spreading of the
out loss of generality one can assume k¯ > 0 (i.Ne. p¯> 0) wavepacketis given as
and σ K~. The solution of Eq. (38) with the initial
p ≪ f¯
condition (39) is
τ =τ <τ (48)
0f¯+2α2p¯2f¯′ 0
i [p (k )]2
x x
a(kx,t)=a0(kx)exp(cid:18)−~ 2m t(cid:19) (40) imn~/te(2rmσ2s) ionfotrhdeinacrhyarQacutaenrtisutmicMspecrehaadniincgs. Wtimeeseeτ0tha=t
p
the finite band width can cause a much faster spread of
andtheevolutionofthewavepacketisgivenbythetime-
thewavepacketwhenitsmeanwavevectorapproachesthe
dependent wavefunction
limiting value K.
K dk i [p (k )]2
x x x
ψ(x,t)= a (k )exp t+ik x .
2π 0 x −~ 2m x
Z−K (cid:18) (cid:19) VI. STATIONARY STATES OF A PARTICLE IN
(41)
A POTENTIAL
Forthesakeofsimplicityletusassumethatthedefor-
mationfunctionf(u)isanalyticatu=0. Forsufficiently
sharp distribution, σ K~ one can expand the expo- Let us discuss now the problem of stationary states of
nent at the mean k¯ aps ≪ aparticleinapotentialV(x). EvenifGUP doesnotim-
ply a finite wavevector cutoff, it results in the modified
i p¯2 kinetic energy operator Hˆfree, introduced in the previ-
Exp. = t+ik¯x A(t)s2+iB(x,t)s (42)
−~2m − ous section, so that the Hamiltonian can be written as
Hˆ = Hˆfree +Vˆ. In order to determine the low-energy
where statesthenon-degeneratestationaryperturbationexpan-
~2f¯2 iβ t β t sionhasbeenwidelyappliedwithanexpansioninpowers
2 1
A(t)= + , B(x,t)=x , ofthesmallparameterα(seee.g. [22–24,37–51]). When
4σ2 2m~ − 2m~
p additionallyevenafinitebandwidthisenforcedbyGUP,
β =2~p¯f¯, β =~2f¯(f¯+2α2p¯2f¯′), (43) the projectedHamiltonianΠˆ(Hˆfree+Vˆ)Πˆ figures in the
1 2
9
stationary Schr¨odinger equation (13). Now we shall use well potential are given in App. B. The expressions for
the perturbation expansion for the low-energy states in asymptotically large depth V of the potential, i.e. for
0
another manner, without expanding in the small param- states with ǫ /V 1 are also given. The matrix ele-
n 0
≪
eterα. Namely,we shallsimply saythatthe wholeGUP ments h , v and t contributing additively to the
nn nn nn
effect, including the effect of projection to states with fi- energy shift can be expressed in terms of the various
nite band width, is a perturbation and account it for in pieces ϕ (x) (i = I, II, III) of the wavefunction de-
i
the first order. Therefore we split the projected Hamil- fined in the intervals I , respectively (c.f. App B). Since
i
tonian as the operator hˆ is local, while the operators vˆ and tˆare
ΠˆHˆΠˆ =Hˆ +Hˆ′, (49) nonlocal due to the projection operator, we can write
0 their matrix elements in the form:
whereHˆ =Tˆ +Vˆ istheHamiltonianinordinaryQuan-
0 0 III
tum Mechanics with the usual kinetic energy operator h = h , h = dxϕ∗(x)h ϕ (x),
Tˆ0 =~2kˆx2/(2m) and nn Xi=I i i ZIi i x i
Hˆ′ =hˆ+vˆ+tˆ (50) III
v = v , v = dx dyϕ∗(x)v(x,y)ϕ (y),
nn i,j i,j i j
represents the perturbation caused by GUP and the re- i,j=I ZIi ZIj
X
striction to finite band width. The latter consists of the III
pwuidreelyGiUnPtheeffleitcetrahˆtu=reHaˆn−dtHˆh0ep=iecHˆesfrvˆee=−ΠˆTVˆˆ0ΠˆdiscVˆusasnedd tnn =i,j=Iti,j, ti,j =ZIidxZIj dyϕ∗i(x)t(x,y)ϕj(y),
tˆ= ΠˆHˆfreeΠˆ Tˆ responsible for the additiona−l modi- X
0 (54)
−
fication of the potential and the kinetic energy operator
due to the restriction of the states to those with finite where
band width. Obviously, the projection alters the local
1 1
potential and kinetic energy into nonlocal quantities. h = [F−1( iα~∂ )]2 ( ~2∂2) ,
x 2m α2 − x − − x
Let ϕn be the complete set of eigenstates of the un- (cid:18) (cid:19)
perturb{ed}Hamiltonian Hˆ0, v(x,y)= 1[V(x)+V(y)][Π(x,y) δ(x y)],
2 − −
Hˆ0|ϕni=ǫn|ϕni, (51) t(x,y)= 1 [F−1( iα~∂ )]2[Π(x,y) δ(x y)]
then the energy levels E = ǫ +∆ǫ of the perturbed 2mα2 − x − −
n n n
system are shifted by (55)
∆ǫ = ϕ Hˆ′ ϕ =h +v +t , (52) are the appropriate kernels. Hermitian symmetry of the
n n n nn nn nn
h | | i operatorshˆ,vˆ,andtˆ,reflectionsymmetryofthepotential
wϕhertˆeϕhnnre=prehsϕennt|hˆt|hϕenein,evrngnys=hifthsϕcna|vˆu|sϕendib,yapnudretnGnU=P to x = L/2 and being the operator [F−1(α~kˆx)]2 even
h n| | ni lead to the symmetry relations
effect and by the projection of the potential and that of
the kinetic energy, respectively. h =h ,
I III
v =(v )∗, v =v , v = v ,
j,i i,j III,III I,I III,II I,II
±
t =(t )∗, t =t , t = t . (56)
VII. TOY MODEL: PARTICLE IN A BOX j,i i,j III,III I,I III,II I,II
±
Here the signs correspondto eigenstatescharacterized
A. Energy shifts of stationary states by the wa±vevectorsk . Furthermore, v =0 trivially
± II,II
because V(x)=0 for x I .
II
∈
We shall apply the method described in the previous
section to a toy model, a particle bounded in a square-
well potential B. Shift due to pure GUP effect
V(x)=V [Θ( x)+Θ(x L) (53)
0 − − We call pure GUP effect the energy shift hnn of sta-
of width L and let go finally the depth of the potential tionary states because of the modification of the canon-
well V to infinity. Although a sudden jump of the po- ical momentum and that of the kinetic energy from ~kˆ
0 x
tential is unrealistic in bandlimited Quantum Mechanics and (~kˆ )2/(2m) in ordinary Quantum Mechanics to
x
asemphasizedin[58],butinourtreatmentthatproblem α−1F−1(α~kˆ ) and (2mα2)−1[F−1(α~kˆ )]2 when GUP
x x
shall be cured by projection that makes potential edges
isinwork. Makinguseofthe resultsofApp. Conefinds
effectively smeared out over a range of the order a, as
thatthe functions ϕ (x) (i=I, ,II, III)areeigenfunc-
i
argued in Sect. IV previously.
tions of the kinetic energy operator. Then one gets
When GUP effects (including finite band width) are
neglected, the solutions ϕn(x) corresponding to the un- h = 1 1 [F−1(α~k )]2 ~2k2 dxϕ (x)2,
perturbedboundstateswithenergyǫn <V0inthesquare II 2m(cid:18)α2 ± − ±(cid:19)ZIII | II |
10
h = 1 1 [F−1( iα~κ)]2+(~κ )2 dxϕ (x)2 C. Shift due to finite band width
I 2m α2 − ± | I |
(cid:18) (cid:19)ZII
=h (57) Finite band width, i.e. the existence of the finite UV
III
wavevector cutoff K results in the absence of quantum
and the energy shift due to pure GUP effect
fluctuations with UV wavevectors k > K, that is ex-
x
| |
pressed in our approach by the projection of the opera-
sink L
h = (1+ ± ) [F−1(α~k )]2 (α~k )2 tors of potential and kinetic energies. The energy shift
nn ± ±
" k±L (cid:18) − (cid:19) vnn of the n-th energy level caused by the replacement
of the potential by its projected counterpart can be ex-
ρ 2
+| ±| [F−1( iα~κ±)]2+(α~κ±)2 pressed in terms of the independent integrals vI,I, vI,II
2κ±L(cid:18) − (cid:19)# and vI,III when the symmetry relations discussed above
sink L ρ 2 −1 are accounted for. Here vI,I and vI,III are real because
2mα2 1+ ± + | ±| . (58) ϕI(x) and ϕIII(x) are real functions. As discussed in
×" (cid:18) k±L 2κ±L(cid:19)# App. D the leading order contribution comes from the
integralv inthelimitofinfinitepotentialdepth,while
I,II
Let us consider now the limit κ , the case of a
± the other independent integrals v are suppressed like
particle in a box. In that limit ρ 2→ ∞k2/κ2 and, con- i,j
| ±| ∼ ± ± powers of 1/κ± as compared to it. One finds (c.f. Eq.
sequently, one obtains finite energy shift if and only if
(D12))thatv vanishesfortheenergylevelsnevenand
the limit lim [F−1( iα~κ)]2κ−3 = C remains fi- nn
κ→∞ ∞ fortheenergylevelsnodditisgivenas(c.f. Eqs. (D10),
−
nite. We shall assume that only such deformation func-
(D11), and (D12))
tions are physically reasonable for which C =0, which
∞
means that the tails of the wavefunction in the outer ~2k 2k
regionsII andIIII ofthe square-wellpotential give van- vnn ≈ 2mL Kπ[4sin2(νπ)+(nπ)2cos(2νπ)
ishingcontributionstothekineticenergywhenthedepth (cid:18)
of the potential becomes infinite. The deformation func- + n4 ]+ (k/K)2 . (62)
tion f = 1 + α2p2 with F−1(u) = tanu satisfies that O O
x (cid:19)
condition because the limit lim tanh u = 1 is fi- (cid:0) (cid:1) (cid:0) (cid:1)
u→±∞
± The vanishing of v for even n is a consequence of the
nite. Finally, we end up with the pure GUP energy shift nn
particular form of the wavefunction in a square-well po-
h =R ǫ with
nn h n
tential, namely the alternating sign of the tail of the
F−1(α~k ) 2 wavefunction in the region III with the alteration of
±
Rh = α~k −1 (59) even and odd n values in the numeration of the station-
(cid:18) ± (cid:19) arystateswithincreasingenergy. Theratiooftheenergy
for any deformation function being reasonable in the shift v for odd n to the unperturbed energy ǫ of the
nn n
above discussed sense. For highly excited states char- stationary state n,
acterized by wavevectorsk K close to the UV cutoff
±
≈ v 2
the ratio Rh explodes which signals simply that the per- R = nn [4sin2(νπ)+(nπ)2cos(2νπ)
v
turbation expansion seases to work. For low-lying states ǫn ≈ KLπ
for which our approach is applicable, the expansion in + n4 ]+ (ℓ /L)2 (63)
P
the small parameter α~k =nα~π/L yields O O
±
turns out to ta(cid:0)ke v(cid:1)alues o(cid:0)f the ord(cid:1)er (ℓ /L).
α~π 1 α~π P
R f n+ 7f2+8f n2+... .(60) Thus, the potential energy shift due to the UV cutoff
h ≈ L 1 12 1 2 L seems to be many orders of magnitude larger – at least
(cid:20) (cid:18) (cid:19) (cid:21)
for the lowest energy levels – as compared to the energy
Forhα=ℓ andf =1+α2p2 onegetsf =0andf =1
P x 1 2 shift caused by pure GUP, because it holds R /R
and the ratio h v ≈
(ℓ /L). It is remarkable that R oscillates strongly
P v
2 ℓ 2 Owith the fine-tuning of the length of the box L confin-
R P n2+ (nℓ /L)2 (61)
h P ing the particle. The variation of ν in the interval [0,1]
≈ 3 2L O
(cid:18) (cid:19) corresponds to the tiny change of the box size L in a
(cid:0) (cid:1)
raising quadratically with increasing n and being inde- range of (ℓ )= (a), the size of the grid constant, as
P
O O
pendent of the mass of the particle in the box. Thus we well as that of the maximal accuracy ∆x of position
min
recoveredtheresultobtainedinRefs. [60]and[51](given determination. Therefore, an averaging over ν might be
after Eq. (14) for j = 1). An order-of-magnitude esti- more reliable when one wants to incorporate the indef-
mate gives(ℓP/L)2 10−40, 10−50, and10−58 forboxes initeness of the size of the box, a direct consequence of
ofthesizeL=10−15≈m(sizeofanucleon),10−10 m(size the impossibility to determine positions more precisely
ofaH-atom),and10−6m(thewavelengthofinfraredra- than the distance ∆x . This yields then
min
diation),respectively. Soevenforthefirstfewthousands
of energy levels the pure GUP correction remains a tiny 1 4
dνR [1+ n4 ]+ (KL)−2 . (64)
correction. v ≈ KLπ O O
Z0
(cid:0) (cid:1) (cid:0) (cid:1)