Position in Models of Quantum Mechanics with a Minimal Length †

Proceeding Paper Position in Models of Quantum Mechanics with a Minimal Length †

Pasquale Bosso

Department of Physics and Astronomy, University of Lethbridge, 4401 University Drive, Lethbridge, AB T1K 3M4, Canada; [email protected] † Presented at the 1st Electronic Conference on Universe, 22–28 February 2021; Available online: https://ecu2021.sciforum.net/.

Abstract: Candidate theories of quantum gravity predict the presence of a minimal measurable length at high energies. Such feature is in contrast with the Heisenberg Uncertainty Principle. Therefore, phenomenological approaches to quantum gravity introduced models spelled as modifications of quantum mechanics including a minimal length. The effects of such modification are expected to be relevant at large energies/small lengths. One first consequence is that position eigenstates are not included in such models due to the presence of a minimal uncertainty in position. Further- more, depending on the particular modification of the position–momentum commutator, when such models are considered from momentum space, the position operator is changed, and a measure factor appears to let the position operator be self-adjoint. As a consequence, the (quasi-)position representation acquires numerous issues. For example, the position operator is no longer a multiplicative operator, and the momentum of a free particle does not correspond directly to its wave number. Here, we will review such issues, clarifying aspects of minimal length models, with particular reference to the representation of the position operator. Furthermore, we will show how such a (quasi-)position description of quantum mechanical models with a minimal length affects results concerning simple systems. Keywords: quantum gravity phenomenology; generalized uncertainty principle; quantum mechan- Citation: Bosso, P. Position in ics; position; de Broglie relations Models of Quantum Mechanics with a Minimal Length. Phys. Sci. Forum 2021, 2, 35. https://doi.org/ 10.3390/ECU2021-09275 1. Introduction Academic Editor: Paolo Giacomelli Several approaches to quantum gravity, as well as gedanken experiments in black holes and high energy physics, suggest the presence of a minimal measurable length [1–8]. Published: 22 February 2021 Such minimal length may be due to structural properties of space time (e.g., causal dynam- ical triangulation, loop quantum gravity, etc.) or to a fundamental minimal uncertainty Publisher’s Note: MDPI stays neutral in position (e.g., string theory, re-elaborations of the Heisenberg microscope including with regard to jurisdictional claims in gravity, etc.). Regardless of its nature, such minimal length is in contrast with one of the published maps and institutional affil- cornerstones of quantum mechanics, viz. The Heisenberg uncertainty principle. Therefore, iations. a modification of quantum mechanics accounting for such minimal measurable length is often considered in phenomenological approaches to quantum gravity. This modification is referred to as the generalized uncertainty principle (GUP). Several different approaches to GUP have been proposed in the past. In fact, GUP can be cast as a modification of the Copyright: © 2021 by the author. uncertainty relation between position and momentum, with no necessary modification of Licensee MDPI, Basel, Switzerland. the representations of the corresponding operators [8–10]; alternatively, modifications of This article is an open access article Poisson brackets have been considered in the classical sector, inducing modified classical distributed under the terms and dynamics [11–14]; finally, a modified commutation relation between position and momen- conditions of the Creative Commons tum is studied, related with a modified uncertainty relation via the Schrödinger–Robertson Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Phys. Sci. Forum 2021, 2, 35. https://doi.org/10.3390/35 https://www.mdpi.com/journal/psf Phys. Sci. Forum 2021, 2, 35 2 of 7

relation [15–18]. In this work, we will consider this last approach. Speciﬁcally, we will consider a generic commutation relation of the form

[qˆ, pˆ] = i} f (pˆ). (1)

To study such relation and its implications, we will use [16] as a guideline. In particular, based on the work [19], we will study the action of the position operator in models with a minimal length described by Equation (1). It is worth pointing out at this stage that, as Equation (1) introduces a minimal uncertainty in position, position eigenstates are not included in such a model. We will thus see how a new representation is possible, following the ideas developed in [16] and the implications of such alternative perspective.

2. Quasi-Position Representation Let us start from the momentum–space representation, in which the momentum operator acts on a wavefunction ψ(p) by multiplying by the momentum variable p. In this space, a possible representation for the position operator is

d qˆψ(p) = i} f (p) ψ(p). (2) dp

As mentioned in the introduction, position eigenstates are not present in such model. Thus, we do not have a meaningful representation in which the operator qˆ acts multiplicatively. However, we can introduce maximally localized states as those states with the smallest position uncertainty possible, and write any state as a superposition of such maximally localized states. For practical reasons, we will specialize our analysis to the case in which the function f (p) in Equation (1) is given as f (p) = 1 − 2δp + δ2 + e p2, (3)

with δ e δ = 0 , e = 0 . (4) 2 2 MPlc MPlc

Here, MPl is the Planck mass, c is the speed of light in vacuum, and δ0 and e0 are di- mensionless parameters. Similar results in a more general setting can be found in [19]. A function of minimal uncertainty in position can be found, as usually implemented in quantum mechanics, as the function ψhqi(p) of position expectation value hqi fulﬁlling the following equation −i}h f (p)i (qˆ − hqi)ψhqi = (pˆ − hpi)ψhqi. (5) (∆p)2 Using Equations (2) and (3), we ﬁnd a differential equation of solution 1 hqip0(p) ψhqi(p) = p exp −i , (6) 1 − 2δp + (δ2 + e)p2 }

where p0(p) is deﬁned as " # Z p dp0 1 −δ + δ2 + e p 1 δ p0(p) = 0 = √ arctan √ + √ arctan √ . (7) 0 f (p ) e e e e

It is important to notice that, in general, the quantity p0 is bounded on subset of R. In fact, in the particular case of Equation (3), we ﬁnd

1 δ π 1 δ π p0 ∈ √ arctan √ − √ , √ arctan √ + √ . (8) e e 2 e e e 2 e Phys. Sci. Forum 2021, 2, 35 3 of 7

Such property is in agreement with the existence of a minimal length as, as we will see below, the two extremes correspond to the shorter wavelength allowed in the model. Furthermore, it is worth noticing that such an interval is not symmetric with respect to 0. This is a consequence of the linear term in p in Equation (3), which introduces a difference between the left and right direction. Considering the function in Equation (6) as a function of both p and ξ = hqi and using it as a kernel for an integral transform, we obtain the following two transforms, which generalize the Fourier transform in the case of GUP.

T−1[φ](ξ) = √ 1 R ∞ dp ψ(p, ξ)φ(p), 2π} −∞ f (p) − (9) T[φ](p) = √ 1 R ∞ dξ[ψ(p, ξ)] 1φ(ξ). 2π} −∞ Such transform allows for representing a state as a superposition of maximally localized states of different position expectation values ξ. We can then use them to ﬁnd the momentum eigenstates in such new “quasi-position” representation −1 1 1 ξ p0(pe) φp(ξ) = T [δ(p − p)](ξ) = √ exp i . (10) e e 2 2 3/2 2π} [1 − 2δpe + (δ + e)pe ] } We thus see that a momentum eigenstate, i.e., a free-particle state, in quasi-position space corresponds to a plane wave of wave number k = p0(pe)/}. Therefore, GUP changes the de Broglie relation between wave number and momentum. Furthermore, in models in which f (p) is not an even function, e.g., δ0 6= 0, we have two different wave numbers for left- and right-moving waves or, equivalently, a different dispersion relation for left- and right-moving particles. Furthermore, the same transformations allow for writing the position and momentum operators in quasi-position space. In fact, we ﬁnd qˆ = ξ + i} δ2 + e pˆ − δ , √ h√ δ i etan epˆ0−arctan √ +δ (11) pˆ = e pˆ = −i d δ2+e , 0 } dξ

We then see that the position operator in such representation is not a multiplicative operator, as in standard quantum mechanics. This result is due to the rich structure of models with a minimal length.

3. Examples We are now going to consider two special systems in which a minimal length, via the results found in the previous section, has interesting consequences.

3.1. Particle in a Box Let us consider a one-dimensional box of side L. Such a system, similarly to what is implemented in ordinary quantum mechanics, can be described in terms of a discontinuous potential, which is 0 between ξ = 0 and ξ = L, and inﬁnite anywhere else. Proceeding as usual, one considers a superposition of left- and right- moving waves with same momentum eigenvalue p up to a sign, and imposes the relevant boundary conditions. However, as in GUP models the wavelengths of left- and right- moving waves are, in general, different, we cannot directly conclude that a solution of the Schrödinger equation is a standing wave with L a semi-integer multiple of the wavelength. In fact, after the necessary calculation, one may ﬁnd that, in this case, the condition is that the sum of wavelengths of the left- and right-moving waves be a divider of the box width or, in terms of the quantity p0,

2nπ} [p (p ) − p (−p )] = , (12) 0 n 0 n L Phys. Sci. Forum 2021, 2, 35 4 of 7

Phys. Sci. Forum 2021, 2, 35 4 of 7

where is the momentum eigenvalue associated with the ’s energy state. The relation above can then be inverted to find the possible values of the momentum eigenvalue. where pn is the momentum eigenvalue associated with the n’s energy state. The relation However, as is bounded, as shown in the previous section, Equation (12) admits only above can then be inverted to find the possible values of the momentum eigenvalue. a finite number of integers . The maximum number of possible energy eigenstates is in However, as p is bounded, as shown in the previous section, Equation (12) admits only 0 ⁄ afact finite related number with of the integers ratio n. The, with maximum the Planck number length. of possible Specifically, energy one eigenstates finds for is the in maximum number fact related with the ratio L/lPl, with lPl the Planck length. Specifically, one finds for the maximum number =⌊ ⌋. (13) L nmax = b √ c. (13) 2lPl e0 In Figure 1, two different choices for the parameters and are compared to the standardIn Figure case.1 ,The two vertical different dashed choices lines for corre the parametersspond to theδ higher0 and e energy0 are compared state allowed to the in standardeach model. case. Such The verticaleffect becomes dashed linesmore correspond relevant the to thenarrower higher the energy box stateis, for allowed example, in each model. Such effect becomes more relevant the narrower the box is, for example, when when =20 and =1, one find that only 10 energy states are allowed. Conversely, L = 20l and e = 1, one find that only 10 energy states are allowed. Conversely, one can one canPl find that0 boxes narrower than a particular width do not allow any state, that is, find that boxes narrower than a particular width do not allow any state, that is, when when √ L < 2l e0. (14) <2Pl. (14)

FigureFigure 1. 1.Energy Energy times times mass mass of a particleof a particle in a box in of a widthbox of 1 m.width The green1 m. The line representsgreen line the represents ordinary case,the ordinary and the blue case, and and orange the blue lines and represent orange two lines different represent GUP two parametrizations. different GUP The dashed vertical lines represent the maximum integer n allowed for that particular model. parametrizations. The dashed vertical lines represent the maximum integer n allowed 3.2.for that Potential particular Barrier model.

As a second example, let us consider a potential barrier of width L and height U0 between3.2. Potentialξ = Barrier0 and ξ = L. As in ordinary quantum mechanics, considering a particle approachingAs a second the barrier example, from thelet left,us cons a genericider a energy potential eigenstate barrier is of given width by a superpositionand height ofbetween a left- and=0 aright-moving and =. As wave in ordinary on the left quantum of the barrier, mechanics, corresponding considering to a a reflected particle andapproaching an incoming the wave, barrier respectively; from the aleft, left- a and generic a right-moving energy eigenstate wave in the is regiongiven ofby the a barrier;superposition and a right-moving of a left- and wave a right-moving on the right wave of the on barrier, the left corresponding of the barrier, to corresponding a transmitted wave.to a reflected Imposing and the an corresponding incoming wave, boundary respectively; conditions, a left- and one cana right-moving then find a relationwave in forthe theregion transmission of the barrier; and and reflection a right-moving coefficients. wave Carrying on the outright the of calculations,the barrier, corresponding one can find thatto a resonancestransmitted in wave. the transmission Imposing the amplitudes corresponding appear boundary at different conditions, energies one than can in then the standardfind a relation case. Furthermore, for the transmission contrary to ordinary and reflection quantum mechanics,coefficients. only Carrying a finite numberout the ofcalculations, resonances areone allowed. can find As that inthe resonances case of a particlein the intransmission a box, resonances amplitudes are related appear with at thedifferent sum of energies the wavelengths than in the of standard left- and case right-moving. Furthermore, waves contrary in the regionto ordinary of the quantum barrier. Specifically,mechanics, aonly resonance a finite is number obtained of whenresonances the width are ofallowed. the barrier As in is the a half-integer case of a particle multiple in ofa box, such resonances a sum. In terms are related of the quantitywith the psum0, we of have the wavelengths of left- and right-moving waves in the region of the barrier. Specifically, a resonance is obtained when the width 2nπ of the barrier is a half-integerp multiplep0 − pof −suchp0 a= sum. In, terms of the quantity (15), we 0 n 0 n L have 0 with p n the momentum eigenvalue of the stationary states corresponding to a resonance, i.e., ′ −−′ = , (15) q pn = 2m(En − U0). (16)

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with ′ the momentum eigenvalue of the stationary states corresponding to a resonance, i.e.,

In this case as well, as p0 acquires = 2 values − on. a limited interval, we can conclude that(16) only ﬁnitely many resonances are allowed, as shown in Figure2. The maximum number of resonancesIn this depend case as onwell, the as ratio acquiresL/lPl and values is given on a by limited interval, we can conclude that only finitely many resonances are allowed, as shown in Figure 2. The maximum number L of resonances depend on the ratio n⁄ =andb is given√ c .by (17) max 2l e Pl 0 =⌊ ⌋. (17)

FigureFigure 2. 2.Resonance Resonance energies energies with with a a minimal minimal length length (blue (blue and and orange orange dots) dots) and and in in the the standard standard case (greencase (green dots). dots). The vertical The vertical lines represent lines represent the maximum the maximum number number of resonances of resonances allowed allowed in the models. in the models. Here, a barrier of width = 20 and height 2 = 1202 ℏ ⁄ has been used. Here, a barrier of width L = 20lPl and height U0 = 120} /mL has been used. 4.4. ConclusionsConclusions TheThe generalized generalized uncertainty uncertainty principle principle is is a quantuma quantum model model describing describing the the existence existence of aof minimal a minimal uncertainty uncertainty in position. in position. In this paper, In this we consideredpaper, we suchconsidered a model modifyingsuch a model the commutationmodifying the relation commutation between positionrelation andbetw momentum.een position Due and to themomentum. presence of Due a minimal to the uncertaintypresence of in a position, minimal it uncertainty is not possible in toposition define, a it position is not representation,possible to define and wea position had to resortrepresentation, to an alternative and we description, had to resort often to referredan alternative to as “quasi-position”. description, often However, referred suchto as methodology“quasi-position”. introduces However, various such issues methodology that we haveintroduces tried tovarious clarify issues here. Inthat particular, we have thetried position to clarify operator here. doesIn particular, not act multiplicatively the position operator in such does new not representation. act multiplicatively Rather, itin containssuch new a term representation. proportional toRather, the momentum it contains operator a term and proportional another which to isthe an momentum imaginary constant.operator Suchand ananother imaginary which constant is an imaginary is related toconstant. the position Such expectation an imaginary value constant on states is ofrelated minimal to the uncertainty position inexpectation position. Furthermore, value on states by studyingof minimal momentum uncertainty eigenstates in position. in theFurthermore, quasi-position by studying representation, momentum we saw eigensta that theytes can in bethe described quasi-position as plane representation, waves with awe wave saw number that they which can isbe notdescribed proportional as plane to thewaves momentum with a wave eigenvalue, number suggesting which is not a modificationproportional of to de the Broglie momentum relations. eigenvalue, In fact, left- andsuggesting right-moving a modification waves, characterized of de Broglie by therelations. same momentum In fact, left- eigenvalue, and right-moving are described waves, by characterized different wavelengths. by the same Furthermore, momentum ineigenvalue, general, the are wave described number by of different a free wave wavele acquiresngths. its Furthermore, value on a limited in general, interval, the thuswave introducingnumber of aa maximumfree wave waveacquires number its value or a minimalon a limited wavelength, interval, compatible thus introducing with the a introduction of a minimal length. All this has interesting implications when quantum maximum wave number or a minimal wavelength, compatible with the introduction of systems are analyzed, as in the two examples presented here. In fact, we have seen that, a minimal length. All this has interesting implications when quantum systems are in contrast to ordinary quantum mechanics, only a finite number of stationary states are analyzed, as in the two examples presented here. In fact, we have seen that, in contrast allowed in a box. Furthermore, depending on the width of the box and on the parameters to ordinary quantum mechanics, only a finite number of stationary states are allowed in of the adopted model, a box may even present no state at all. As a second example, a box. Furthermore, depending on the width of the box and on the parameters of the we considered the case of a potential barrier. Here, a minimal length also has relevant adopted model, a box may even present no state at all. As a second example, we implications. In fact, we found a finite number of resonance energies for a wave crossing considered the case of a potential barrier. Here, a minimal length also has relevant the barrier. Such number depends on the width of the barrier and on the parameters of the implications. In fact, we found a finite number of resonance energies for a wave crossing model. Furthermore, when no resonance is present, the reflection coefficient is strongly the barrier. Such number depends on the width of the barrier and on the parameters of suppressed, even for energies below the potential energy of the barrier, as seen in Figure3. Again,the model. these lastFurthermore, features are when compatible no resonance with the presenceis present, of athe minimal reflection length. coefficient is strongly suppressed, even for energies below the potential energy of the barrier, as seen

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Phys. Sci. Forum 2021, 2, 35 6 of 7 in Figure 3. Again, these last features are compatible with the presence of a minimal length.

FigureFigure 3. 3.Transmission Transmission coefficient coefficient as aas function a function of the of ratiothe ratio between between the energy the energy of the incomingof the incoming wave wave and the barrier height. Here, a barrier of width =1 and height2 =2 120 ℏ ⁄ has and the barrier height. Here, a barrier of width L = 1lPl and height U0 = 120} /mL has been used. been used. It is worth noticing that the transmission coefficient is equal to 1, or close to 1, even It is worth noticing that the transmission coefficient is equal to 1, or close to 1, even when E < U0. when <. The present analysis, although limited to a one-dimensional case and to a specific model,The serves present to shedanalysis, some although light on limited the issues to a concerningone-dimensional a description case and of to quantum a specific mechanicsmodel, serves with ato minimal shed some length. light In fact,on the this issues is often concerning the playground a description of phenomenological of quantum studiesmechanics in quantum with gravitya minimal [20– 22length.]. A detailed In fact, and carefulthis is analysis often isthe therefore playground necessary of inphenomenological order to potentially studies compare in quantum such features gravity with [20–22]. experimental A detailed observations. and careful analysis is therefore necessary in order to potentially compare such features with experimental Institutionalobservations. Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Institutional Review Board Statement: Not applicable. 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