Dirac -Function Potential in Quasiposition Representation of a Minimal-Length Scenario

Eur. Phys. J. C (2018) 78:179 https://doi.org/10.1140/epjc/s10052-018-5659-6

Regular Article - Theoretical Physics

Dirac δ-function potential in quasiposition representation of a minimal-length scenario

M. F. Gusson, A. Oakes O. Gonçalves, R. O. Francisco, R. G. Furtado, J. C. Fabrisa, J. A. Nogueirab Departamento de Física, Universidade Federal do Espírito Santo, Vitória, ES 29075-910, Brazil

Received: 26 April 2017 / Accepted: 22 February 2018 / Published online: 3 March 2018 © The Author(s) 2018. This article is an open access publication

Abstract A minimal-length scenario can be considered as Although the first proposals for the existence of a mini- an effective description of quantum gravity effects. In quan- mal length were done by the beginning of 1930s [1–3], they tum mechanics the introduction of a minimal length can were not connected with quantum gravity, but instead with be accomplished through a generalization of Heisenberg’s a cut-off in nature that would remedy cumbersome diver- uncertainty principle. In this scenario, state eigenvectors of gences arising from quantization of systems with an infinite the position operator are no longer physical states and the number of degrees of freedom. The relevant role that grav- representation in momentum space or a representation in a ity plays in trying to probe a smaller and smaller region of quasiposition space must be used. In this work, we solve the the space-time was recognized by Bronstein [4] already in Schroedinger equation with a Dirac δ-function potential in 1936; however, his work did not attract a lot of attention. quasiposition space. We calculate the bound state energy and It was only in 1964 that Mead [5,6] once again proposed a the coefficients of reflection and transmission for the scatter- possible connection between gravitation and minimal length. ing states. We show that√ leading corrections are of order of Hence, we can assume that gravity may lead to an effec- the minimal length (O( β))and the coefficients of reflection tive cut-off in the ultraviolet. Furthermore, if we are con- and transmission are no longer the same for the Dirac delta vinced that gravitational effects are in order when a minimal well and barrier as in ordinary quantum mechanics. Further- length is introduced then a minimal-length scenario could more, assuming that the equivalence of the 1s state energy of be thought of as an effective description of quantum gravity the hydrogen atom and the bound state energy of the Dirac effects [7]. δ-function potential in the one-dimensional case is kept in a As far as we know, the introduction of a minimal-length minimal-length scenario, we also find that the leading cor- scenario can be carried out in three different ways [7– rection term for the ground state energy of the hydrogen atom 9]: a generalization of the Heisenberg uncertainty principle −25 1 is of the order of the minimal length and Δxmin ≤ 10 m. (GUP), a deformation of special relativity (DSR) and a modification of the dispersion relation (MDR). Various problems connected with the minimal length have been studied in the context of non-relativistic quantum 1 Introduction mechanics. Among them are the harmonic oscillator [10–14], the hydrogen atom [15–21], step and barrier potentials [22– Gravity quantization has become a huge challenge for theo- 24], finite and infinite square wells [25,26]. In the relativistic retical physicists. Despite enormous efforts made, so far, it context, the Dirac equation has been studied in [27–33]. The was not possible to obtain a theory which can be considered Casimir effect has also been studied in a minimal-length sce- suitable and not even a consensus approach. Nevertheless, nario in [34–38]. most of the candidate theories for gravity quantization seem An interesting problem in quantum mechanics is the Dirac to have one common point: the prediction of the existence of δ-function potential. In general, the Dirac δ-function poten- a minimal length, that is, a limit for the precision of a length tial is used as a pedagogical exercise. Nevertheless, it has also measurement. been used to model physical quantum systems [39]. Maybe because the attractive Dirac δ-function potential is one of a e-mail: [email protected] 1 It is named doubly special relativity because of the existence of two b e-mail: [email protected] universal constants: the speed of light and the minimal length. 123 179 Page 2 of 7 Eur. Phys. J. C (2018) 78 :179 the simplest quantum systems which displays both a bound tainty principle (GUP). Since state and a set of continuous states, it has been used to model |[x ˆ, pˆ]| atomic and molecular systems [40–45]. In addition, the short- ΔxΔp ≥ , (1) range interactions in condensed matter with a large scattering 2 length can actually be modeled as a Dirac δ-function poten- a generalization of the uncertainty principle corresponds to a tial [46–50]. In quantum field theory, in order to treat the modification in the algebra of the operators. There are differ- Casimir effect more realistically, the boundary conditions ent suggestions of the modification of the commutation rela- 1 σ( )φ2( ) tion between the position and momentum operators which are replaced by the interaction potential 2 x x , where σ(x) represents the field of the material of the borders (back- implement a minimal-length scenario. Weare concerned with ground field). Hence, for sharply localized borders the back- the most usual of them, proposed by Kempf [10,11], which ground field can be modeled by a Dirac δ-function [51,52]. in a one-dimensional space is given by The Dirac δ-function potential by its very nature is a chal- [ˆ, ˆ]:= ¯ + β ˆ2 , lenging problem in a minimal-length scenario. x p ih 1 p (2) Ferkous [53] and Samar and Tkachuck [54] have inde- where β is a parameter related to the minimal length. The pendently calculated the bound state energy in “momen- commutation relation (2) corresponds to the GUP tum space”. In the two papers cited, the authors√ have found β β h¯ a√ correction for the expression of energy in and ΔxΔp ≥ 1 + β(Δp)2 + βˆp2 , (3) ( β ∼ Δxmin), but with different coefficients, therefore hav- 2 ing disagreeing outcomes. Samar and Tkachuck claim that which implies the existence of a non-zero minimal uncer- (−∞, ∞) √ this is because Ferkous considers p to belong to , tainty in the position Δxmin = h¯ β. π π whereas they consider p to belong to − √ , √ .Inthis 2 β 2 β Unfortunately, in this scenario the eigenstates of the posi- work, we propose to solve the problem of a non-relativistic tion operator are not physical sates2 and, consequently, the particle of mass m in the presence of the Dirac δ-function representation in position space can no longer be used, that is, potential in quasiposition space. Since the quasiposition an arbitrary state vector |ψ cannot be expanded in the basis space representation is used we can consider the cases of of state eigenvectors of the position operator {|x}. Hence the bound states and scattering states as well. We find the same obvious way to go ahead is to make use of the representation expression for the energy of the bound state as obtained by in momentum space: Ferkous. In addition, assuming that the equality between ∂ψ(˜ p) the 1s state energy of the hydrogen atom and the bound p|ˆx|ψ=ih¯ + βp2 , 1 ∂ (4) state energy of the Dirac δ-function potential in the one- p ˜ dimensional case when the coefficient of the δ-potential is p|ˆp|ψ=pψ(p). (5) replaced by the fine structure constant [41] is kept in a However, the representation in momentum space is not minimal-length scenario, we find that the leading correc- suitable in some cases, such as, for example, when the wave tion for the ground state energy of hydrogen atom is of the √ function has to satisfy a boundary condition at specifics order of the minimal length (O( β)), differently from the points. So, the representation in quasiposition space [56], values commonly found in the literature using perturbative methods [15–18,55], but according to the results obtained x ML|ˆx|ψ(t)=xψqp(x, t), (6) by Fityo et al. [20] and Bouaziz and Ferkous [21]usinga ∂2 ∂ψqp( , ) ML 2 x t non-perturbative approach. x |ˆp|ψ(t)=−ih¯ 1 − βh¯ , (7) ∂x2 ∂x The rest of this paper is organized as follows. In Sect. 2 we show how to introduce a minimal-length scenario and find to first order in the β parameter, is more appropriate.3 |x ML the time-independent Schroedinger equation in a quasiposi- are state vectors of maximal localization which satisfy [10] tion space representation. In Sect. 3 we solve the modified ML|ˆ| ML= , ∈ , Schroedinger equation and find the bound state energy and x x x x with x (8) (Δ ) = Δ = β, the coefficients of reflection and transmission for the scatter- x |x ML xmin h¯ (9) ing states. We present our conclusions in Sect. 4. 2 That is because the uncertainty ΔA of an operator Aˆ in any of its state eigenvectors |ψA must be zero, which is not the case for the position 2 Minimal-length scenario operator, since Δxmin > 0. 3 ˆ =ˆ Pedram√ [56] has proposed a representation in which x xo and ( β ˆ ) In quantum theory, a minimal-length scenario can be accom- ˆ = tan √ po ˆ ˆ p β ,wherexo and po are ordinary operators of position and plished by imposing a non-zero minimal uncertainty in the momentum, which obey the canonical commutation relation [ˆxo, pô]= measurement of position which leads to a generalized uncer- ih¯ . 123 Eur. Phys. J. C (2018) 78 :179 Page 3 of 7 179 and require that the second and first derivatives are continuous at = ML ML x 0. Consequently, Eq. (15) turns into [39,58] x |x =1. (10) 3 3 β d ϕII(0) d ϕI (0) mV0 The time-independent Schroedinger equation for a non- − = ϕ(0). (16) 3 dx3 dx3 h¯ 4 relativistic particle of mass m in the quasiposition space representation takes the form As is well known, taking into account the sign of the energy, two cases can then arise: (i) bound states when E < 0 h¯ 2 d2ϕqp(x) h¯ 4 d4ϕqp(x) − + β and (ii) scattering states when E > 0. 2m dx2 3m dx4 +V (x)ϕqp(x) = Eϕqp(x). (11) 3.1 Bound states The above modified Schroedinger equation shows that GUP effects are performed by the fourth-order derivative In this case, the general solution of Eq. (14) is given by 4 term. This term modifies the probability current as follows kx −kx ϕ , (x) = A , e + B , e [23]: I II I II I II kβ x −kβ x + C , e + D , e , (17) ih¯ ∂ψ ∂ψ∗ I II I II J =− ψ∗ − ψ 2m ∂x dx where, to first order in β, 3 3 3 ∗ iβh¯ ∗ ∂ ψ ∂ ψ 1 + ψ − ψ k := k 1 + βh¯ 2k2 , (18) m ∂x3 ∂x3 0 3 0 ∂2ψ∗ ∂ψ ∂2ψ ∂ψ∗ + − , (12) 3 1 2 2 ∂ 2 ∂ ∂ 2 ∂ kβ := 1 − βh¯ k , (19) x x x x 2h¯ 2β 3 0 5 but it does not modify the probability density, and

ρ =|ψ|2. (13) 2m|E| k0 := . (20) h¯ 2 3 Dirac δ-function potential The coefficients can be found, except by one normalization constant, requiring that the solutions remain finite when In this section, we consider a non-relativistic particle of mass x →±∞and the continuity of the solution and of its first m in the presence of the Dirac delta-function potential in a and second derivatives at x = 0. We arrive at the result ⎧ minimal-length scenario. According to Eq. (11)wehave ⎨ ϕ ( ) = kx − k kβ x , < I x Ae kβ Ae x 0 2 2ϕ( ) 4 4ϕ( ) (21) h¯ d x h¯ d x ⎩ −kx k −kβ x − + β − V δ(x)ϕ(x) ϕII(x) = Ae − Ae , x > 0, 2m dx2 3m dx4 0 kβ = Eϕ(x), (14) where A is the normalization constant. From Eq. (16) we can find the bound state energy up to where V0 > 0 is a constant. − order β as Integrating Eq. (14) between and (with arbitrarily small and positive), and then taking the limit → 0, we mV2 β m2V 3 m3V 4 =− 0 + 2 0 − β 0 , obtain E 2 3 2 4 (22) 2h¯ 3 h¯ h¯ dϕ (0) dϕ (0) 2 d3ϕ (0) d3ϕ (0) II − I − βh¯ 2 II − I which is in agreement with Ferkous’s result [53]. It is inter- dx dx 3 dx3 dx3 esting to note that the first correction brought about√ by the 2mV0 introduction of a minimal-length scenario is O( β). + ϕ(0) = 0, (15) h¯ 2 For an electron, the relative difference between the bound state energy arising from the introduction of a minimal length where ϕI (x) and ϕII(x) are the solutions of Eq. (14)for x < 0 and x > 0, respectively. and the absolute value of the ordinary energy of the bound Since the third derivative of ϕ(x) at x = 0 has a finite state is showed as a function of the minimal length for the discontinuity (that is to say, a jump by a finite amount), we energy about 1 eV in Fig. 1 and as a function of E0 (1 eV −20 ≤ E0 ≤ 1 keV) for Lmin = 10 minFig.2.InFig.1 −17 4 From now on, we are going to omit the qp superscript of the wave we choose the 10 m upper value for the minimal length function for the sake of simplicity. because it is in accordance with that commonly found in 5 That is because the authors assume that there are no changes in the the literature [15,55,57] and it is consistent with the one time-dependent part of the Schroedinger equation. at the electroweak scale [9,22,23]. For the Planck length, 123 179 Page 4 of 7 Eur. Phys. J. C (2018) 78 :179

we have ⎧ ik ⎨ ikx β A −ikx A kβ x ϕI (x) = Ae + e − e , x < 0, k b b (25) ⎩ − ϕ ( ) = aA −ikx − A kβ x , > , II x b e b e x 0 where 2βh¯ 4k β 2 2 a := 1 + kβ + k , (26) 3mV0 kβ b := a − i , (27) k and A is a normalization constant. Consequently, the reflection and transmission coefficients ΔE Fig. 1 Bound state: E as a function of Lmin in units of meter, for are given by ≈ 0 E0 1eV 2 kβ 1 = R 2 2 (28) k 2βh¯ 4k k 1 + β k2 + k2 + β 3mV0 β k and β ¯ 4 2 2 h kβ 2 2 2 1 + kβ + k kβ 3mV0 = . T 2 2 (29) k 2βh¯ 4k k 1 + β k2 + k2 + β 3mV0 β k Note that R + T = 1, as it must. It is instructive to write the reflection and transmission coefficients up to first corrections. We have ⎡ ⎤ ΔE −1 −1 Fig. 2 Bound state: as a function of E0 in units of eV, for Lmin = 2 2 E0 2h¯ |E| 2β 2mV0 mV −20 R = 1+ ⎣1− 1+ 0 ⎦ 10 m 2 ¯ ¯ 2| | mV0 3 h 2h E

Δ (30) E ≈ 8.4 × 10−26, unfortunately a virtually unmeasurable E0 and effect in quantum gravity using current technology. ⎡ ⎤ − − 2 1 β 2 3 2 1 mV0 ⎣ 2 m V0 mV0 ⎦ 3.2 Scattering states T = 1+ 1+ 1+ . 2h¯ 2|E| 3 h¯ 3|E| 2h¯ 2|E| In this case, the general solution of Eq. (14) is given by (31)

ikx −ikx The above results show that the reflection and the trans- ϕI,II(x) = AI,IIe + BI,IIe mission coefficients are no longer the same in the cases of kβ x −kβ x + CI,IIe + DI,IIe , (23) a delta-function well (V0 > 0) and a delta-function bar- < where rier (V0 0). Therefore the presence of a minimal length decreases the chances of tunneling. It is also interesting to note that the first correction brought 3 1 2 2 kβ := 1 + βh¯ k . (24) 2h¯ 2β 3 0 about√ by the introduction of a minimal-length scenario is O( β) in the same way as in the bound state energy. Now we demand there is no reflected wave function for Figures 3 and 4 show the relative difference between the x > 0, consequently BII = 0. From the requirement that transmission coefficient arising from the introduction of a solutions remain finite when x →±∞we have DI = 0 and minimal length and T0 (ordinary transmission coefficient) for CII = 0. In this case, the continuity of the solution and of the cases of a Dirac-delta well (dashed line) and of a Dirac- its first and second derivatives at x = 0 do not suffice to find delta barrier (continuous line). Figure 3 is for electrons scat- the coefficients. It is also necessary to use the discontinuity tering of energy about 1 eV and V0 = 2 eVÅ. For the Planck Δ of the third derivative at x = 0, Eq. (16). After some algebra, length, T ≈ 8.9 × 10−27, again a virtually unmeasurable T0 123 Eur. Phys. J. C (2018) 78 :179 Page 5 of 7 179

− | | 4. e−kβ |x| and e kβ x solutions are only significant for very small values of x, that is, at high energy. Thus we could assume that they lie far outside the validity range at which the Schroedinger equation may consistently work and throw them away. However, that is a naive assumption, because they lead to the emergence of traces of quantum gravity in low energy physics, as the previous results show. Note that they provide the continuity of first and second derivatives at x = 0. Δ Fig. 3 Scattering state: T as a function of L in units of meter, for 5. It is well known, at least since Frost’s work of 1954 [41], T0 min E0 ≈ 1eV that the ground state energy of the hydrogen atom (1s state) is identical to the bound state energy of a Dirac δ- function potential in the one-dimensional case when V0 is replaced by the fine structure constant, α. Thus, assuming that this identity is kept in a minimal-length scenario,6 the result (22) predicts a leading correction√ for ground state energy of the hydrogen atom of O( β), whereas the result commonly found in the literature using perturbative methods is of O(β) [15–18,55]. It is important to add that using a non-pertubative approach Fityo et al. [20] and Bouaziz and√ Ferkous [21] have also found a first correction of O( β). ΔT Fig. 4 Scattering state: T as a function of Lmin in units of meter, for Now, we can make a rough estimate of an upper bound ≈ 0 E0 1MeV for the minimal-length value comparing our result with experimental data [55]. Using the data obtained in Ref. [59], in which the accuracy of about 4, 2 × 10−14 eV has effect. Figure 4 is for proton scattering of energy about 1 been obtained, we find that Δx ≤ 10−25 m. Hence, = × −2 min MeV and V0 3 10 MeVÅ. For the Planck length, in the case of the proton scattering, from the previous ΔT − − ≈ 4.7 × 10 17. Note that L ∼ 10 17 m results in ΔT −17 −25 T0 min subsection, we find ∼ 10 for L ∼ 10 m, T0 min significant effects, which may be an indication that Lmin is which is a more representative result. far from the electroweak scale.

3.3 Remarks 4 Conclusion

1. It is easy to see that in the limit β → 0 we recover In this work, we solve, in quasiposition space, the Schroe- the results known for the Dirac δ-function potential in dinger equation for a Dirac δ-function potential. Our result ordinary quantum mechanics. for the bound state energy is in agreement with that calculated 2. A more detailed analysis shows that kβ and kβ do not by Ferkous in momentum space. Moreover, we find that for − | | a leading correction for the reflection and transmission coef- vanish even if m = 0. Therefore, e−kβ |x| and e kβ x ficients of the scattering states, the bound state energy and − | | − 3 |x| −kβ |x| kβ x → 2h¯ β solutions still persist since e , e e ground state√ of the hydrogen atom are of order of the minimal when m = 0. Consequently, this leads us to presume that length, O( β). We also show that in the presence of a mini- such solutions are “background solutions” caused by the mal length the coefficients of reflection and transmission for introduction of an effective description of the effects of the Dirac delta-function well and the Dirac delta-function quantum gravity. However, since their coefficients in Eqs. barrier are no longer the same. There is a decrease in the (21) and (25) vanish when m = 0, they are not present chances of tunneling. in the bound state and the scattering states solutions. Although different physical systems can be modeled by a 3. It is important to point out that now the first derivative at Dirac δ-function potential, we have to ask ourselves about the x = 0 is no longer discontinuous. However, in the limit validity of the results, since the Dirac δ-function potential is β → 0 the discontinuity at x = 0 is recovered. Moreover, already an approximation to an actual physical system. That if the term of O(β2) is considered in the Schroedinger equation the third derivative will turn into a continuous 6 Since the symmetry of the 1s state must remain the same in the two one at x = 0, and so on. cases. 123 179 Page 6 of 7 Eur. Phys. J. C (2018) 78 :179 is, are the minimal-length effects smaller than the ones due to 17. S. Benczik, L.N. Chang, D. Minic, T. Takeuchi, Hydrogen-atom themodelingbytheDiracδ-function potential? Probably the spectrum under a minimal-length hypothesis. Phys. Rev. A 72, answer is yes, though it is difficult to be sure. What we can 012104 (2005) 18. M.M. Stetsko, V.M. Tkachuk, Perturbation hydrogen-atom spec- claim is that the estimates of a upper bound for the minimal- trum in deformed space with minimal length. Phys. Rev. A 74, length value are acceptable, in the sense that even though the 012101 (2006) corrections for a more realistic potential can be greater than 19. K. 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