New Approach to Nonperturbative Quantum Mechanics with Minimal Length Uncertainty
aa r X i v : . [ h e p - t h ] J a n New Approach to Nonperturbative Quantum Mechanics with Minimal LengthUncertainty
Pouria Pedram ∗ Department of Physics, Science and Research Branch,Islamic Azad University, Tehran, Iran (Dated: October 15, 2018)The existence of a minimal measurable length is a common feature of various approaches toquantum gravity such as string theory, loop quantum gravity and black-hole physics. In this scenario,all commutation relations are modified and the Heisenberg uncertainty principle is changed to the so-called Generalized (Gravitational) Uncertainty Principle (GUP). Here, we present a one-dimensionalnonperturbative approach to quantum mechanics with minimal length uncertainty relation whichimplies X = x to all orders and P = p + βp to first order of GUP parameter β , where X and P arethe generalized position and momentum operators and [ x, p ] = i ~ . We show that this formalism is anequivalent representation of the seminal proposal by Kempf, Mangano, and Mann and predicts thesame physics. However, this proposal reveals many significant aspects of the generalized uncertaintyprinciple in a simple and comprehensive form and the existence of a maximal canonical momentum ismanifest through this representation. The problems of the free particle and the harmonic oscillatorare exactly solved in this GUP framework and the effects of GUP on the thermodynamics of thesesystems are also presented. Although X , P , and the Hamiltonian of the harmonic oscillator allare formally self-adjoint, the careful study of the domains of these operators shows that only themomentum operator remains self-adjoint in the presence of the minimal length uncertainty. Wefinally discuss the difficulties with the definition of potentials with infinitely sharp boundaries. PACS numbers: 04.60.Bc
I. INTRODUCTION
The unification of general relativity with the laws ofquantum mechanics is one of the oldest wishes of the-oretical physicists from the birth of quantum mechan-ics. We can mention the canonical quantization [1] andthe path integral quantization of gravity [2] as two well-known but old proposals which tried to present a quanti-zation scheme for gravity. However, from the field theo-retical viewpoint, the theory of relativity is not renormal-izable and leads to ultraviolet divergencies. Moreover,around the Planck energy scale, the effects of gravity areso important that they would result in discreteness ofthe spacetime manifold. This argument is based on thefact that, when we try to probe small distances with highenergies, it will significantly disturb the spacetime struc-ture by the gravitational effects. However, the theory canbe renormalizable by introducing a minimal observablelength as an effective cutoff in the ultraviolet domain.The existence of a minimum measurable length is oneof the common aspects of various candidates of quan-tum gravity such as string theory, loop quantum gravity,and quantum geometry. Within a string-theoretical ar-gument, we can say that a string cannot probe distancessmaller than its length. Moreover, some Gedanken exper-iments in black-hole physics and noncommutativity of thespacetime manifold all agree on the existence of a mini-mal observable distance of the order of the Planck length ℓ P l = p G ~ /c ≈ − m , where G is Newton’s constant ∗ [email protected] [3–7]. In fact, the finite resolution of spacetime points isa consequence of finite time measurement. In principle,one can probe very short distances in D X can be madearbitrarily small by letting ∆ P to grow correspondingly.However, for energies close to the Planck energy, the par-ticle’s Schwarzschild radius and its Compton wavelengthbecome approximately in the order of the Planck length.So, in order to merge the idea of the minimal lengthinto quantum mechanics, we need to modify the ordi-nary uncertainty principle to the so-called GeneralizedUncertainty Principle (GUP). Indeed, the notion of min-imal length should quantum mechanically be describedas a minimal uncertainty in position measurements. Theintroduction of this idea has drawn much attention in re-cent years and many papers have been appeared in theliterature to address the effects of GUP on various quan-tum mechanical systems and phenomena [8–24].In this paper, we present a nonperturbative approachto one-dimensional gravitational quantum mechanicswhich implies a minimal length uncertainty so that thegeneralized position operator does not change to all or-ders, that is, X = x and the generalized momentum oper-ator is given by P = p + βp to first order of the GUP pa-rameter. In this formalism the generalized position andmomentum operators satisfy [ X, P ] = i ~ (1 + βP ) where x and p are the ordinary position and momentum opera-tors [ x, p ] = i ~ . We show that this proposal is equivalentwith Kempf, Mangano, and Mann (KMM) representa-tion, but it only modifies the kinetic part of the Hamil-tonian and has no effect on the potential part. Moreover,this representation agrees with perturbative approachesand predicts the presence of a maximal canonical mo-mentum p max . Here, we consider the problems of thefree particle and the harmonic oscillator in the context ofthe generalized uncertainty principle and obtain the ex-act eigenvalues and corresponding eigenfunctions. Then,we discuss the consequences of the minimal uncertaintyin position measurement on the partition function, meanenergy, and heat capacity of these systems. The difficul-ties with potentials with infinitely sharp boundaries arealso presented. II. THE GENERALIZED UNCERTAINTYPRINCIPLE
According to the Heisenberg uncertainty relation, inprinciple, we can separately measure the position andmomentum of particles with arbitrary precision. Thus, ifthere is a genuine lower bound on the results of the mea-surements, the Heisenberg uncertainty relation should bemodified. Here we consider a generalized uncertaintyprinciple which results in a minimum observable length∆ X ∆ P ≥ ~ (cid:0) β (∆ P ) + ζ (cid:1) , (1)where β is the GUP parameter and ζ is a positive con-stant that depends on the expectation value of the mo-mentum operator. We also have β = β / ( M P l c ) where M P l is the Planck mass and β is of the order of one.Note that the deviation from the Heisenberg picture takesplace in the high energy limit where the quantum gravityeffects are dominant. So, for the energies much smallerthan the Planck energy M P l c ∼ GeV, we shouldrecover the famous Heisenberg uncertainty relation. Itis straightforward to check that the above inequality re-lation (1) implies the existence of an absolute minimallength uncertainty as (∆ X ) min = ~ √ β . In the contextof string theory, we can interpret this length as the stringlength. Accordingly, the string’s length is proportional tothe square root of the GUP parameter. In one-dimension,the above uncertainty relation can be obtained from a de-formed commutation relation, namely[ X, P ] = i ~ (1 + βP ) , (2)where, for β = 0 we recover the well-known commuta-tion relation in ordinary quantum mechanics. Now us-ing Eqs. (1) and (2) we can find the relation between ζ and the expectation value of the momentum operatori.e. ζ = β h P i . As Kempf, Mangano, and Mann havesuggested in their seminal paper, in momentum spacerepresentation, we can write X and P as [25] P φ ( p ) = pφ ( p ) , (3) Xφ ( p ) = i ~ (cid:0) βp (cid:1) ∂ p φ ( p ) , (4) where X and P are symmetric operators on the densedomain S ∞ with respect to the following scalar product h ψ | φ i = Z + ∞−∞ d p βp ψ ∗ ( p ) φ ( p ) , (5)where R + ∞−∞ d p βp | p ih p | = 1 and h p | p ′ i = (cid:0) βp (cid:1) δ ( p − p ′ ). In this representation the position operator is merelysymmetric, but P is self-adjoint [25]. With this defini-tion, the commutation relation (2) is exactly satisfied.Also, in quasiposition representation this formulation re-sults in [25] P ψ ( x ) = tan (cid:0) − i ~ √ β∂ x (cid:1) √ β ψ ( x ) , (6) Xψ ( x ) = x + β tan (cid:0) − i ~ √ β∂ x (cid:1) √ β ! ψ ( x ) . (7)Note that, for the general potential, expressing the posi-tion operator as a combination of ordinary position andmomentum operators results in a complicated high-ordergeneralized Schr¨odinger equation. So, finding the solu-tions even for the simple potentials would not be an easytask.To overcome this problem, we propose the followinggeneralized position and momentum operators X = x, (8) P = tan (cid:0) √ βp (cid:1) √ β , (9)where x and p obey the canonical commutation relation[ x, p ] = i ~ . X and P are symmetric operators on thedense domain S ∞ of functions decaying faster than anypower( h ψ | X ) | φ i = h ψ | ( X | φ i ) and ( h ψ | P ) | φ i = h ψ | ( P | φ i ) , (10)but now with respect to the scalar product: h ψ | φ i = Z + π √ β − π √ β dp ψ ∗ ( p ) φ ( p ) , (11)The symmetry of P (9) is obvious. The symmetry of X (8) can be seen by performing partial integrations Z + π √ β − π √ β dp ψ ∗ ( p ) (cid:18) i ~ ∂∂p (cid:19) φ ( p )= Z + π √ β − π √ β d p (cid:18) i ~ ∂ψ ( p ) ∂p (cid:19) ∗ φ ( p ) , (12)which is valid for the functions vanishing at ± π √ β . In-deed, the symmetry property of the position and mo-mentum operators ensures that all expectation valuesare real. This definition exactly satisfies the condition[ X, P ] = i ~ (1 + βP ) and agrees with the well-knownrelations [26], namely X = x, (13) P = p (cid:18) β p (cid:19) , (14)to the first order of the GUP parameter. Note that to O ( β ), the definitions (6) and (7) result in X = x + βp and P = p (cid:0) β p (cid:1) which differ with (13) and (14).Now, we show that our proposal and KMM represen-tation are equivalent in essence. Indeed, they are relatedby the following canonical transformation: X → h (cid:16)p βP (cid:17)i X, (15) P → arctan (cid:16)p βP (cid:17) / p β, (16)which transforms (8) and (9) into (3) and (4) subjectedto condition (2). We can interpret P and p as follows: p isthe momentum operator at low energies ( p = − i ~ ∂/∂x )while P is the momentum operator at high energies.Obviously, this procedure affects all Hamiltonians inadopted quantum mechanics.Note that for an operator A which is “formally” self-adjoint ( A = A † ) such as (8) and (9), this does notprove that A is truly self-adjoint because in general thedomains D ( A ) and D ( A † ) may be different. The op-erator A with dense domain D ( A ) is said to be self-adjoint if D ( A ) = D ( A † ) and A = A † . For instance,similar to KMM representation, X is merely symmetricbut not self-adjoint. To see this note that in this repre-sentation and in the momentum space the wave func-tion φ ( p ) have to vanish at the end of the p interval( − π/ √ β < p < π/ √ β ), because the tangent functiondiverges there. So, X is a derivative operator i ~ ∂/∂p onan interval with Dirichlet boundary conditions. But thismeans that X cannot be self-adjoint because all candi-dates for the eigenfunctions of X , (the plane waves, whichare even normalizable) are not in the domain of X be-cause they do not obey Dirichlet boundary conditions.Calculating the domain of the adjoint of X shows that itis larger than that of X , so X is indeed not self-adjointi.e. Z + π √ β − π √ β d p ψ ∗ ( p ) (cid:18) i ~ ∂∂p (cid:19) φ ( p )= Z + π √ β − π √ β d p (cid:18) i ~ ∂ψ ( p ) ∂p (cid:19) ∗ φ ( p )+ i ~ ψ ∗ ( p ) φ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p =+ π √ β − i ~ ψ ∗ ( p ) φ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = − π √ β . (17)Now since φ ( p ) vanishes at p = ± π √ β , ψ ∗ ( p ) can take anyarbitrary value at the boundaries. The above equationimplies that X is symmetric, but it is not a self-adjointoperator. Although its adjoint X † = i ~ ∂/∂p has thesame formal expression, it acts on a different space of functions, namely D ( X ) = (cid:26) φ, φ ′ ∈ L (cid:18) − π √ β , + π √ β (cid:19) ; φ (cid:18) + π √ β (cid:19) = φ (cid:18) − π √ β (cid:19) = 0 (cid:27) , (18) D ( X † ) = (cid:26) ψ, ψ ′ ∈ L (cid:18) − π √ β , + π √ β (cid:19) ;no other restriction on ψ (cid:27) . (19)As it is also shown in Ref. [27], any operator X whichobeys the uncertainty relation (1) is merely symmetric.On the other hand, since there are no Dirichlet boundaryconditions on the wave functions in the position space( −∞ < x < ∞ ), P is still self-adjoint. In the next sectionand after finding the momentum eigenfunctions, we provethe self-adjointness property of P using von Neumann’stheorem.To proceed further, let us consider the following Hamil-tonian H = P m + V ( X ) , (20)which using Eqs. (8) and (9) can be written exactly andalso perturbatively as H = tan (cid:0) √ βp (cid:1) βm + V ( x ) , (21)= H + ∞ X n =3 ( − n − n (2 n − n − B n m (2 n )! β n − p n − , (22)where H = p / m + V ( x ) and B n is the n th Bernoullinumber. So the corrected terms in the modified Hamil-tonian are only momentum dependent and proportionalto p n − for n ≥
3. As we shall explicitly show, thepresence of these terms leads to a positive shift in theparticle’s energy spectrum. Note that, in general, evenfor the self-adjoint position and momentum operators,it is by no means obvious that the resulting Hamilto-nian will be self-adjoint until and unless the potentialterm is specified and the appropriate domain is chosen.It is worth to mention that all our calculations are inone-dimensional space. Indeed, in higher dimensions itis necessary to have noncommutativity of coordinates inorder to satisfy the Jacobi identity as done by KMM[25]. In one-dimensional space, the Jacobi identity is au-tomatically satisfied. Also, one may relax the point sizeproperty of the particle as in the string theory. So wecan interpret Eq. (21) as the Schr¨odinger equation forthe particle with size ∼ ~ √ β , where the effect of thenonzero size effectively appears in the kinetic part of theHamiltonian.In the quantum domain, this Hamiltonian results in thefollowing generalized Schr¨odinger equation in the quasi-position representation: − ~ m ∂ ψ ( x ) ∂x + ∞ X n =3 α n ~ n − β n − ∂ n − ψ ( x ) ∂x n − + V ( x ) ψ ( x ) = E ψ ( x ) , (23)where α n = 2 n (2 n − n − B n / m (2 n )! and thesecond term is due to the GUP corrected terms in (22).Among infinite possible canonical transformations(CTs), our proposal (8) and (9) has some useful andnovel properties. First, it does not change the natureof the position operator and, consequently, the poten-tial term and only modifies the momentum or the ki-netic operator. So, among several CTs, only this onepreserves the ordinary nature of the position operator.Second, this formalism lets us to write the Hamiltonian as H = H + βH + β H + ... , where H = p / m + V ( x ) isthe ordinary Hamiltonian and H , H , ... contain only themomentum operator. So, using the perturbation theory,the unperturbed eigenfunctions satisfy H | ψ i = E | ψ i and we can find h H i , h H i , ... in an straightforward man-ner as done for various cases such as Ref. [26]. In otherCTs like the KMM proposal, we cannot decompose theHamiltonian in such configurations. So, in this sense,this proposal is compatible with perturbative represen-tations. Third, this proposal predicts the existence of amaximal canonical momentum. In fact, the particularform of the kinetic part of the Hamiltonian (21) impliesthe existence of a maximal momentum: p max = π √ β = πM P l c √ β , (24)which mimics the recent GUP proposal predicting thepresence of both a minimal length uncertainty and amaximal momentum uncertainty through a doubly spe-cial relativity consideration [30–33]. There, the gener-alized momentum has an upper bound proportional to M P l c/α where α similar to β is of the order of unity.However, for our case, the generalized momentum P hasno upper bound and it is not physically equivalent withaforementioned GUP. Therefore, the idea of a maximum“canonical” momentum naturally arises from our repre-sentation. III. GUP AND THE FREE PARTICLE
In ordinary quantum mechanics, the free particle wavefunction u p ( x ) is defined as the eigenfunction of the mo-mentum operator P op P op u p ( x ) = p u p ( x ) , (25)where p is the eigenvalue. The momentum operator hasthe following representation in the quasiposition space P op = ~ i ∂∂x . (26) So, from Eq. (25) we have ~ i ∂u p ( x ) ∂x = pu p ( x ) , (27)which has the following solution u p ( x ) = 1 √ π ~ exp (cid:18) ipx ~ (cid:19) , (28)where the constant of integration is chosen to satisfy Z ∞−∞ u ∗ p ( x ) u p ( x ′ )d p = δ ( x − x ′ ) . (29)In GUP scenario, to find the momentum eigenfunctionin the position space, we write the momentum operator(9) as P op = tan (cid:0) − i ~ √ β∂ x (cid:1) / √ β which results in thefollowing eigenvalue equationtan (cid:0) − i ~ √ β∂ x (cid:1) √ β u p ( x ) = pu p ( x ) . (30)Now, let us consider a class of solutions which satisfiesEqs. (27) and (30) at the same time, but with differenteigenvalues [ p → p ′ in Eq. (27)], i.e., u p ( x ) = A ( p ) exp (cid:18) ip ′ x ~ (cid:19) , (31)where p ′ = f ( p ). Inserting this solution in Eq. (30) re-sults in tan (cid:0) √ βp ′ (cid:1) / √ β = p or p ′ = 1 √ β arctan (cid:16)p βp (cid:17) , (32)so we have u p ( x ) = A ( p ) exp (cid:20) i ~ √ β arctan (cid:16)p βp (cid:17) x (cid:21) . (33)To obtain A ( p ), we demand that the momentum eigen-function satisfies Eq. (23) of Ref. [25] as the modifiedversion of (29) which results in A ( p ) = (cid:20) π ~ (cid:0) βp (cid:1) (cid:21) − / . (34)Thus, we finally obtain the momentum eigenfunctions as u p ( x ) = 1 p π ~ (1 + βp ) exp (cid:20) i ~ √ β arctan (cid:16)p βp (cid:17) x (cid:21) , (35)which, to the first order agrees with the solution pre-sented in Ref. [22] i.e. u p ( x ) = (cid:18) − βp π ~ (cid:19) / exp (cid:20) i ~ (cid:18) p − β p (cid:19) x (cid:21) . (36)Note that this solution for β → / (cid:0) βp (cid:1) in the definition of the scalar product(5) indeed appeared in the momentum-dependent nor-malization coefficient of the momentum eigenfunctions,namely, | A ( p ) | ∼
11 + βp . (37)At this point, we can use the following theorem tocheck the self-adjointness property of the position andmomentum operators [28, 29] Theorem 1. (von Neumann’s theorem)
For an operator A with deficiency indices ( n + , n − ) there are three possi-bilities:1. If n + = n − = 0 , then A is self-adjoint (this is anecessary and sufficient condition).2. If n + = n − = n ≥ , then A has infinitely manyself-adjoint extensions, parameterized by a unitary n × n matrix.3. If n + = n − , then A has no self-adjoint extension. To use von Neumann’s theorem, we have to find thewave functions φ ± given by P † φ ± ( x ) = tan (cid:0) − i ~ √ β∂ x (cid:1) √ β φ ± ( x ) = ± iλφ ± ( x ) . (38)So using Eq. (33) we have φ ± ( x ) = C ± exp (cid:20) ∓ ~ √ β tanh − (cid:16)p βλ (cid:17) x (cid:21) . (39)Since the operator P is defined on the whole real axiswhere φ ± diverge at x → ∓∞ and consequently are notnormalizable, none of the functions φ ± belong to theHilbert space L ( R ) and therefore the deficiency indicesare (0 , D ( P ) = D ( P † ) = { φ ∈ D max ( R ) } , (40)where D max denotes the maximal domain on which theoperator P has a well defined action, i.e., D max ( P ) = (cid:8) φ ∈ L ( R ) : P φ ∈ L ( R ) (cid:9) . Using the same procedurefor the position operator X on the finite interval, it isstraightforward to check that both φ ± ( p ) = C ± e ∓ λp be-long to L ( − πβ − / , πβ − / ) and the deficiency in-dices are (1 , IV. GUP AND THE HARMONIC OSCILLATOR
In this section, we study the classical and quantum me-chanical solutions of the harmonic oscillator in the GUPframework and present its semiclassical results. More-over, we study the effects of the minimal length uncer-tainty on the thermodynamic aspects of the harmonicoscillator in both classical and quantum domains.
A. Classical Description
Let us consider the Hamiltonian of a particle of mass m confined in a quadratic potential H ( HO ) = tan (cid:0) √ βp (cid:1) βm + 12 mω x , (41)which using the Hamiltonian equations results in˙ x = tan( √ βp ) sec ( √ βp ) √ βm , (42)˙ p = − mω x. (43)So, in the GUP formalism, the velocity ˙ x is not equalto p/m , but it tends to p/m as β goes to zero. UsingEqs. (42) and (43) we obtain¨ p + ω tan( √ βp ) sec ( √ βp ) √ β = 0 . (44)If we set the initial conditions as x (0) = a and p (0) = 0,it is straightforward to check that the above equationadmits the following solutions p ( t ) = ± √ β arctan η r η ) cot (cid:16)p η ωt (cid:17) , (45) x ( t ) = ∓ a p η cot (cid:16)p η ωt (cid:17)r η ) cot (cid:16)p η ωt (cid:17) , (46)where η = √ βmωa . So the actual frequency of the har-monic oscillator in GUP scenario increases with respectto the absence of GUP as ¯ ω = p βm ω a ω ≥ ω . Infact, this frequency depends on GUP parameter, parti-cle’s mass, and the initial position. Moreover, as β in-creases, the particle is often located at the end points ± a and the accessible phase space decreases with respect tothe absence of GUP (see Fig. 1). B. Semiclassical description
Before studying the corresponding generalizedSchr¨odinger equation, it is worthwhile to find the quan-tized energy spectrum using the semiclassical scheme.The Wentzel-Kramers-Brillouin (WKB) quantizationrule, represented succinctly by the formula I p d q = (cid:18) n + 12 (cid:19) h, n = 0 , , . . . , (47)allows us to find the approximate energy spectrum andin ordinary quantum mechanics gives the exact results.Using Eq. (41) we find I p d x = 2 √ β Z a − a arctan (cid:16)p βmω p a − x (cid:17) d x = 2 π p βm ω a − βmω , (48) t -4-224 x H t L t -4-224 p H t L -4 -2 2 4 x -4-224 p FIG. 1. The temporal behavior of x and p , and the phase space of the harmonic oscillator for β = 0 (blue line) and β = 1 (redline). We set m = ω = 1 and a = 5 (¯ ω = √ which results in the following semiclassical energy spec-trum E ( SC ) n = 12 mω a n , = − γ ~ ω + ~ ω (cid:18) n + 12 (cid:19) (1 + γ/
2) + 12 ~ ωγn , (49)where γ = βm ~ ω . As we have expected, E ( SC ) n tends to ~ ω ( n + 1 /
2) as β goes to zero. However, contrary to theordinary formulation where E ( SC ) n is equal to the exactenergy spectrum, it does not give the exact spectrum inthe GUP formalism. This is due to the fact that fromthe Hamiltonian (41) we expect that the energy spectrumdepends on numerous powers of β , but E ( SC ) n only rep-resents a linear dependence of the GUP parameter. Inthe next section, we show this fact by a rigorous mathe-matical proof. However, it can be considered as a goodapproximation which is related to the correct quadraticdependence on the quantum number. C. Quantum Description
For the case of the harmonic oscillator, because of thequadratic form of the potential V ( x ) = 1 / mω x , weobtain a second-order differential equation in the momen-tum space, namely − ∂ φ ( p ) ∂p + tan (cid:0) √ γp (cid:1) γ φ ( p ) = ¯ ǫ φ ( p ) , (50)where p → √ m ~ ω p , γ = m ~ ωβ , and ¯ ǫ = E ~ ω . In termsof the new variable z = √ γp , we obtain (cid:20) − ∂ ∂z + ν ( ν −
1) tan ( z ) − ǫ ( ν ) (cid:21) φ ( z ; ν ) = 0 , (51)where by definition ν = 12 (cid:18) r γ (cid:19) , ǫ ( ν ) = ¯ ǫγ , (52) and the boundary condition is φ ( z ; ν ) (cid:12)(cid:12)(cid:12)(cid:12) z = ± π/ = 0 . (53)The above differential equation is exactly solvable andthe eigenfunctions can be obtained in terms of Gausshypergeometric functions where we briefly present thesolutions [34].To find the even parity states, let us use the substitu-tion ξ = sin ( z ) which leads to ξ (1 − ξ ) ∂ φ ( ξ ; ν ) ∂ξ + (cid:18) − ξ (cid:19) ∂φ ( ξ ; ν ) ∂ξ + (cid:20) ∆( ν, ǫ ) − ν ( ν − − ξ (cid:21) φ ( ξ ; ν ) = 0 , (54)where ∆( ν, ǫ ) = [ ν ( ν −
1) + ǫ ( ν )]. Now, to get rid ofthe regular singularity of the last term we search for thesolution of the form φ ( ξ ; ν ) = (1 − ξ ) a Y ( ξ ; ν ) , (55)where a satisfy the algebraic equation a − a − ν ( ν −
1) = 0 . (56)So we obtain the Gauss hypergeometric equation for thevariable Y ( ξ ; ν ) ξ (1 − ξ ) Y ′′ + (cid:18) − ( α + β + 1) ξ (cid:19) Y ′ − αβY = 0 , (57)subjected to α + β = 2 a and αβ = a − ∆( ν, ǫ ). Thisequation admits two independent solutions. However,the physically acceptable solution which vanishes at theboundary lim ξ → Y ( ξ ; ν ) = 0 is Y ( ξ ; ν ) = A ( ν )(1 − ξ ) ν/ F (cid:18) α, β ; ν + 12 ; 1 − ξ (cid:19) , (58)where A ( ν ) is the normalization constant. The analytic-ity and the convergence of the hypergeometric functionfor all ξ ∈ [0 ,
1] results in α or β = − k, k = 0 , , , . . . . (59)So we obtain the even parity eigenfunctions Y k ( ξ ; ν ) = A k ( ν )(1 − ξ ) ν/ F (cid:18) − k, ν + k ; ν + 12 ; 1 − ξ (cid:19) , (60)and the eigenvalues ǫ k ( ν ) = 4 k ( ν + k ) + ν, k = 0 , , , . . . . (61)Finally, in terms of the original variable p we have φ k ( p ; γ ) = A k ( ν ) [cos( √ γp )] (cid:16) q γ (cid:17) / × F (cid:18) − k, ν + k ; ν + 12 ; cos ( √ γp ) (cid:19) . (62)To find the antisymmetric solutions let us define φ ( z ; ν ) = sin( z ) ϕ ( z ; ν ) , (63)where φ is an even function of z . By substitution of thissolutions in the original equation we have (cid:20) − ∂ ∂z − x ) ∂∂z + ν ( ν −
1) tan ( z ) + 1 − ǫ ( ν ) (cid:21) × ϕ ( z ; ν ) = 0 , (64)where by choosing ξ = sin ( z ) can be written as (cid:20) ξ (1 − ξ ) ∂ ∂ξ + (cid:18) − ξ (cid:19) ∂∂ξ + ∆( ν, ǫ ) − − ν ( ν − − ξ (cid:21) ϕ ( ξ ; ν ) = 0 . (65)Similar to the procedure for the even states let us define φ ( ξ ; ν ) = (1 − ξ ) ν/ U ( ξ ; ν ) , (66)which converts Eq. (65) to the Gauss hypergeometricequation ξ (1 − ξ ) U ′′ + (cid:18) − (¯ α + ¯ β + 1) ξ (cid:19) U ′ − ¯ α ¯ βU = 0 , (67)where ¯ α = ( ν + 1) − p ∆( ν, ǫ ) and ¯ β = ( ν + 1) + p ∆( ν, ǫ ). As before we set ¯ α = − k and find the eigenen-ergies ǫ k +1 ( ν ) = (2 k + 1)(2 ν + 2 k + 1) + ν, k = 0 , , . . . , (68)for the antisymmetric eigenfunctions U k +1 ( ξ ; ν ) = B k ( ν ) p ξ (1 − ξ ) ν/ × F (cid:18) − k, ν + k + 1; ν + 12 ; 1 − ξ (cid:19) . (69)In terms of the original variables we have φ k +1 ( p ; γ ) = B k ( ν ) sin( √ γp ) [cos( √ γp )] (cid:16) q γ (cid:17) / × F (cid:18) − k, ν + k + 1; ν + 12 ; cos ( √ γp ) (cid:19) . (70) n E n (cid:144) Ñ Ω FIG. 2. Comparing E n / ~ ω (red line) and E ( SC ) n / ~ ω (blueline) for γ = 2 with the ordinary harmonic oscillator spectrum(green line). Note that we can combine Eqs. (61) and (68) in a singleformula to express the full spectrum, namely ǫ n ( ν ) = n (2 ν + n ) + ν for n = 0 , , , . . . or E n ( γ ) = ~ ω (cid:18) n + 12 (cid:19) (cid:16)p γ / γ/ (cid:17) + 12 ~ ωγn , (71)in terms of γ . So, as we have expected, this result exactlycoincides with the spectrum of the harmonic oscillator inthe formalism proposed by Kempf, Mangano, and Mann.In Fig. 2, we have depicted the energy spectrum of theharmonic oscillator in GUP framework (71), its semiclas-sical approximation (49), and its spectrum in ordinaryquantum mechanics. The efficiency of the semiclassicalsolution is manifest in the figure. In fact, to first orderof the GUP parameter, E n is equal to E ( SC ) n up to apositive constant, namely E n ≃ E ( SC ) n + 18 γ ~ ω. (72)To check the self-adjointness property of H ( HO ) , it isnatural to present the sesquilinear form for ψ and φ as2 iB ( ψ, φ ) = h Hψ | φ i − h ψ | H | φ i , = Z + π √ β − π √ β d p ( Hψ ( p )) ∗ φ ( p ) − Z + π √ β − π √ β d p ψ ∗ ( p ) Hφ ( p ) , = − mω ~ "Z + π √ β − π √ β d p ψ ′′ ( p ) ∗ φ ( p ) − Z + π √ β − π √ β d p ψ ∗ ( p ) φ ′′ ( p ) , = 12 mω ~ " ψ ∗ ( p ) φ ′ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = π √ β − ψ ′∗ ( p ) φ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = π √ β − ψ ∗ ( p ) φ ′ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = − π √ β + ψ ′∗ ( p ) φ ( p ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = − π √ β . (73)On the other hand, using the explicit form of the solu-tions (62) and (70), it is straightforward to check that thefirst derivative of the solutions as well as φ ( p ; ν ) vanishesat the boundaries i.e. φ ′ ( p ; ν ) (cid:12)(cid:12)(cid:12)(cid:12) p = ± π √ β = 0 . (74)Therefore, ψ ∗ ( p ) and ψ ′∗ ( p ) can take arbitrary valuesat the boundaries. This means that the domain of theadjoint of H is larger than that of H , so the Hamiltonianis symmetric but not self-adjoint. The domains are D ( H ) = (cid:26) φ ∈ D max (cid:18) − π √ β , + π √ β (cid:19) ; φ (cid:18) + π √ β (cid:19) = φ (cid:18) − π √ β (cid:19) = φ ′ (cid:18) + π √ β (cid:19) = φ ′ (cid:18) − π √ β (cid:19) = 0 (cid:27) , (75) D ( H † ) = (cid:26) ψ ∈ D max (cid:18) − π √ β , + π √ β (cid:19) ;no other restriction on ψ (cid:27) . (76)Note that this result is not surprising because even theHamiltonian of the one-dimensional particle in a box isnot a truly self-adjoint operator as well [29]. D. Classical partition function
In statistical mechanics, the canonical partition func-tion of N identical, one-dimensional oscillators which en-codes the statistical properties of a thermodynamic sys-tem can be written in the classical domain as Z ( b ) = 1 N ! h N Z exp [ − bH ( p · · · p N , x · · · x N )] × d p · · · d p N d x · · · d x N , (77)where b ≡ /k B T , k B denotes Boltzmann’s constant and T is the temperature. For N noninteracting oscillators,the total partition function can be obtained from thesingle-particle partition function as Z ( b ) = 1 N ! h N (cid:20)Z + ∞−∞ exp (cid:20) − bmω x (cid:21) d x (cid:21) N × "Z + π √ β − π √ β exp (cid:20) − b βm tan ( p βp ) (cid:21) d p N , (78)= 1 N ! h N (cid:18) πk B Tmω (cid:19) N/ × π exp (cid:16) βmk B T (cid:17) erfc (cid:16) √ βmk B T (cid:17) √ β N , (79)where erfc( x ) is the complementary error function. Us-ing the asymptotic expansion of the complementary errorfunction for large x , namelyerfc( x ) = e − x x √ π " ∞ X n =1 ( − n · · · · · (2 n − x ) n , (80) k B T E (cid:143) (cid:144) N FIG. 3. The classical mean energy of N harmonic oscillatorsin thermal equilibrium versus temperature for β = 1 (red line)and β = 0 (blue line). We set m = 1. we can write the partition function in terms of powers of k B T as Z ( b ) = 1 N ! (cid:18) k B T ~ ω (cid:19) N ∞ X n =1 (2 n − − βmk B T ) n ! N , (81)where for β → E C = − ∂∂b ln Z = N k B T s k B T πβm exp ( − / βmk B T )erfc (cid:16)p / βmk B T (cid:17) − βm , (82)= N (cid:18) k B T + P ∞ n =1 n (2 n − − βm ) n ( k B T ) n +1 P ∞ n =1 (2 n − − βmk B T ) n (cid:19) , (83)which goes to N k B T for β →
0. Therefore, as indi-cated in Fig. 3, in the presence of GUP, the mean energydecreases with respect to β = 0. The reason for thereduction of mean energy with respect to β = 0 is a con-sequence of the reduction of phase space volume (surface)due to possible definition of a rescaled ~ . In fact the vol-ume of the fundamental cell increases in the presence ofthe minimal length uncertainty relation and the numberof degrees of freedom reduces consequently. Moreover,it modifies the Helmholtz free energy A = − k B T ln Z and the entropy S = k B ln Z + E/T as well. The aboveequation shows that the equipartition theorem fails in theGUP scenario. Although the averaged potential satisfiesthe equipartition theorem i.e. h / mω x i = k B T /
2, thekinetic part yields the smaller value h K i < k B T / ∂E∂T , decreases with respectto the absence of GUP, namely C β =0 V < C β =0 V . (84)Note that, for the case of the ideal gas we can write thepartition function as Z ( b ) = V N N ! h N π exp (1 / βmk B T ) erfc (cid:16)p / βmk B T (cid:17) √ β N , (85)so, using the definition of pressure P = b ∂ ln Z∂V , we re-cover the ordinary ideal gas equation of state
P V = N k B T . However, the corresponding heat capacity willbe modified as mentioned above. E. Quantum partition function
In the quantum statistical mechanics, the partitionfunction for a single oscillator is given by Z ( b ) = ∞ X n =0 exp ( − bE n ) , (86)where the energy eigenvalues are defined in Eq. (71).Now, Z ( b ; γ ) = e − (1 / b ~ ω (cid:16) √ γ / γ/ (cid:17) × ∞ X n =0 exp (cid:20) − b ~ ω (cid:18)(cid:16)p γ / γ/ (cid:17) n + 12 γn (cid:19)(cid:21) , (87)= e − (1 / b ~ ω (cid:16) √ γ / γ/ (cid:17) P ( b ; γ ) , (88)where we defined P ( b ; γ ) ≡ P ∞ n =0 exp h − b ~ ω (cid:16)(cid:16)p γ / γ/ (cid:17) n + γn (cid:17)i .So we have P ( b ; 0) = − exp( − b ~ ω ) and Z ( b ; 0) = exp( − (1 / b ~ ω )1 − exp( − b ~ ω ) . Also, the mean energy of the oscillator isgiven by E = − ∂∂b ln Z = 12 ~ ω (cid:16)p γ / γ/ (cid:17) − P ′ ( b ; γ ) P ( b ; γ ) , (89)= ~ ω (cid:16)p γ / γ/ (cid:17) ×
12 + P ∞ n =0 (cid:18) n + n √ /γ (cid:19) e − ~ ωkBT (cid:16)(cid:16) √ γ / γ/ (cid:17) n + γn (cid:17) P ∞ n =0 e − ~ ωkBT (cid:16)(cid:16) √ γ / γ/ (cid:17) n + γn (cid:17) , (90)where prime denotes the derivative with respect to b . Themean energy of the harmonic oscillator in the quantumdomain and in the GUP formalism is depicted in Fig. 4which shows a modified minimum value in the low tem-perature limit E ≃ ~ ω (cid:16)p γ / γ/ (cid:17) . (91)To compare the classical and quantum results in the high-temperature limit, we can write Eq. (82) as E C N ~ ω = 12 k B T ~ ω + s k B T / ~ ω πγ exp (cid:16) − / γk B T/ ~ ω (cid:17) erfc (cid:16)q / γk B T/ ~ ω (cid:17) − γ . (92) k B T (cid:144) Ñ Ω E (cid:143) (cid:144) Ñ Ω FIG. 4. The quantum mechanical mean energy of the har-monic oscillator E n / ~ ω versus k B T / ~ ω for γ = 1 (red line)and γ = 0 (blue line). k B T (cid:144) Ñ Ω E (cid:143) (cid:144) Ñ Ω FIG. 5. The classical (dashed line) and quantum mechanical(solid line) mean energy of the harmonic oscillator for γ = 0 . γ = 0 (blue line). In Fig. 5, the classical (92) and the quantum mechanical(90) mean energy of the harmonic oscillator for γ = 0 . V. GUP AND THE POTENTIALS WITHSHARP BOUNDARIES
In the GUP scenario, we cannot measure the positionof a particle with an uncertainty less than (∆ X ) min . So,in principle, it is not possible to properly define the po-tentials with infinitely sharp boundaries (It is well knownthat such sharp boundaries cannot be also defined intheories with space-time uncertainty [35, 36]). Indeed,the position of the boundaries can be only determinedwithin this uncertainty. However, one may argue that ina first-step analysis, the assumption of sharp boundarieswould be an acceptable approximation. But the valid-ity of this approximation requires that the uncertaintyin the energy spectrum due to the boundaries’ positionuncertainty to be much smaller than the GUP energycorrection.To investigate this point, we study the problem of a0particle in a box as an example of a potential with in-finitely sharp boundaries and compare both energy cor-rections. So, let us consider a particle with mass m con-fined in an infinite one-dimensional box with length LV ( x ) = (cid:26) < x < L, ∞ elsewhere . (93)The corresponding eigenfunctions should satisfy the fol-lowing generalized Schr¨odinger equation − ~ m ∂ ψ n ( x ) ∂x + ∞ X j =3 α j ~ j − β j − ∂ j − ψ ( x ) ∂x j − = E n ψ n ( x ) , (94)for 0 < x < L and they also meet the boundary condi-tions ψ n (0) = ψ n ( L ) = 0. Because of the boundary con-ditions, the eigenfunctions do not change with respect tothe absence of the GUP ( β = 0) [22]. This fact leadsus to consider the following additional condition for theeigenfunctions − ~ m ∂ ψ n ( x ) ∂x = ε n ψ n ( x ) , < x < L, (95)where ε n = n π ~ mL . If this condition is also satisfied, wecan write the second term in Eq. (94) in terms of ψ n ( x )i.e. ∂ j − ψ n ( x ) ∂x j − = − mε n ~ ∂ j − ψ n ( x ) ∂x j − = · · · = (cid:18) − mε n ~ (cid:19) j − ψ n ( x ) . (96)So, we have − ~ m ∂ ψ n ( x ) ∂x + ∞ X j =3 α j ~ j − β j − ∂ j − ψ n ( x ) ∂x j − = ε n + ∞ X j =3 | α j | β j − (2 m ) j − ε j − n ψ n ( x ) . (97)Now, comparing Eqs. (94) and (97) shows that E n = tan (cid:0) √ mβε n (cid:1) mβ , (98)= ε n + 43 βmε n + 6845 β m ε n + 496315 β m ε n + · · · , = ~ mL (cid:20) n π n π (cid:18) (∆ X ) min L (cid:19) + 17 n π (cid:18) (∆ X ) min L (cid:19) + · · · (cid:21) . (99)This GUP corrected energy spectrum can be also ob-tained using the Wilson-Sommerfeld quantization rulegiven by I p d q = nh, n = 1 , , . . . , (100) with two conjugate variables p and q and the integer n .Since the potential is constant (zero) inside the box, wehave I p d x = 2 L √ β arctan (cid:16)p βmE n (cid:17) . (101)So the semiclassical spectrum is E ( SC ) n = tan (cid:0) √ βnπ ~ /L (cid:1) mβ , (102)which exactly coincides with the quantum mechanicalspectrum given by Eq. (98). These results show thatthe GUP energy correction is of order of (cid:16) (∆ X ) min L (cid:17) .Now let us find the energy correction due to the un-certainty in the position of the well’s walls∆ E n ≃ (cid:12)(cid:12)(cid:12)(cid:12) dε n dL (cid:12)(cid:12)(cid:12)(cid:12) (∆ X ) min = n π ~ mL (cid:18) (∆ X ) min L (cid:19) , (103)which is first order in (∆ X ) min /L . Therefore, the GUPenergy correction is much smaller than ∆ E n and cannotbe detected in the presence of the minimal length. Thisresult confirms that the particle in a box potential cannotbe defined in the GUP framework as in ordinary quantummechanics. This conclusion can be also generalized toother potentials with infinitely sharp boundaries. VI. CONCLUSIONS
In this paper, we proposed a nonperturbative gravita-tional quantum mechanics in agreement with the exis-tence of a minimal length uncertainty relation. In thisformalism the generalized Hamiltonian takes the form H = tan (cid:0) √ βp (cid:1) / (2 βm ) + V ( x ), where x and p are theordinary position and momentum operators. We showedthat this approach is equivalent with KMM representa-tion and we found the corresponding canonical transfor-mation. This representation has some advantages: First,it modifies only the kinetic part (momentum operator)and the potential term (position operator) remains un-changed. Second, this formalism is compatible with per-turbative schemes. Third, this representation predictsthe existence of a maximal canonical momentum propor-tional to M P l c/ √ β . Because of the universality of theGUP effects, this formalism can potentially be tested invarious quantum mechanical systems, of which we havestudied just a few cases.We thoroughly studied the case of the harmonic oscil-lator in classical and quantum domains. In the classicaldomain, we found the trajectory of the oscillating par-ticle and showed that the GUP modified frequency ofthe oscillator depends on mass, initial position and theGUP parameter. Also, for large β the particle is oftenlocated around the end points. In the quantum domain,we obtained the exact energy eigenvalues and the eigen-functions and showed that they are in agreement with1those obtained in Ref. [25]. Moreover, the quadratic de-pendence of the energy spectrum on the state numberis confirmed using the semiclassical approximation. Toaddress the effects of the generalized uncertainty prin-ciple on the thermodynamic properties of the harmonicoscillator, we found the partition functions and the meanenergies in both classical and quantum limits. We showedthat, in the presence of the GUP and at the fixed tem-perature, the mean energy and the heat capacity of theoscillator reduce in comparison with those of the ordinaryclassical and quantum mechanics.Also, we have indicated that X and H ( HO ) are merelysymmetric, but P is a truly self-adjoint operator. Notethat these results for X and P agree with those ofKMM representation [25]. However, the difference isthat in this representation all these operators are for-mally self-adjoint, i.e., A = A † ( A ∈ { X, P, H ( HO ) } ), but D ( A ) = D ( A † ) for A ∈ { X, H ( HO ) } and D ( P ) = D ( P † ).On the other hand, in KMM representation only P isformally and truly self-adjoint. The problems with thepotentials with sharp boundaries are finally discussed.We showed that for this type of potentials, the GUP en-ergy correction is much smaller than the uncertainty inthe energy spectrum due to the boundaries’ position un-certainty. ACKNOWLEDGMENTS
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