On the Boundary Conditions in Deformed Quantum Mechanics with Minimal Length Uncertainty

Abstract

We find the coordinate space wave functions, maximal localization states, and quasiposition wave functions in a GUP framework that implies a minimal length uncertainty using a formally self-adjoint representation. We show how the boundary conditions in quasiposition space can be exactly determined from the boundary conditions in coordinate space.

1. Introduction

The existence of a minimal length uncertainty proportional to the Planck length ℓ_𝑃~10⁻³⁵ m is one of the common predictions of various candidates of quantum gravity. This idea in the context of the Generalized (Gravitational) Uncertainty Principle (GUP) has attracted much attention in recent years and many papers have appeared in the literature to address the effects of this minimal length on various quantum mechanical systems [1–14].

It is pointed out by Mead that gravity amplifies the Heisenberg’s measurement uncertainty which makes it impossible to measure distances more accurate than Planck’s length [15]. In fact, since the increase of the energies to probe small distances considerably disturbs the spacetime structure because of the gravitational effects, the spatial uncertainty eventually increases at energy scales as large as the Planck scale. This minimal length can be considered as a fundamental property of quantum spacetime, a natural UV regulator, and a solution for the trans-Planckian problem. Since the string theory with large or warped extra dimensions can lower the Planck scale into the TeV range, this fundamental length scale also moved into the reach of the Large Hadron Collider.

The thought experiments that support the minimal length proposal include the Heisenberg microscope with Newtonian gravity and its relativistic counterpart [15], limit to distance measurements [16], limit to clock synchronization, and limit to the measurement of the blackhole horizon [4]. Moreover, different approaches to quantum gravity such as string theory, loop quantum gravity, and loop quantum cosmology, quantized conformal fluctuations [17, 18], asymptotically safe gravity [19], and noncommutative geometry all indicate a fundamental limit to the resolution of structure.

Based on the Heisenberg’s microscope and taking into account both the normal and the gravitational uncertainties one finds [20] 1Δ𝑋≳2Δ𝑃+𝐺Δ𝑃, (1) where 𝐺=ℓ²_𝑃 is the gravitational constant. As Adler and Santiago observed this GUP is invariant under ℓ_𝑃Δ𝑃↔1/ℓ_𝑃Δ𝑃 and therefore has a momentum inversion symmetry. Because of the universality of the gravity, this correction modifies all Hamiltonians for the quantum systems near the Planck scale.

Recently, an experimental scheme is suggested by Pikovski et al. to test the presence of the minimal length scale in the context of quantum optics [21]. They used quantum optical control and optical interferometric techniques for direct measurement of the canonical commutator deformations of a massive object. This experiment does not need the Planck-scale accuracy of position measurement and can be reached by the current technology. Some attempts have been also made to test possible quantum gravitational phenomena using astronomical observations [22, 23].

In this paper, we consider a GUP that implies a minimal length uncertainty proportional to the Planck length. We find the exact coordinate space wave functions and quasiposition space wave functions using a formally self-adjoint representation. We first obtain the eigenfunctions of the position operator and the maximal localization states. Then we discuss how the boundary conditions can be imposed consistently in both coordinate space and quasiposition space.

2. The Generalized Uncertainty Principle

Consider the following one-dimensional deformed commutation relation [13]: [𝑋,𝑃]=𝑖ℏ1+𝛽𝑃², (2) where for 𝛽=0 we recover the well-known commutation relation in ordinary quantum mechanics and Δ𝑋Δ𝑃≥(ℏ/2)(1+𝛽(Δ𝑃)²). Since Δ𝑋 cannot be made arbitrarily small, the absolutely smallest uncertainty in positions for this GUP is (Δ𝑋)_min=ℏ𝛽. To proceed further, consider the following representation [11]: 𝑋=𝑥,𝑃=tan𝛽𝑝𝛽, (3) which exactly satisfies (2). This representation is formally self-adjoint subject to the inner product: 𝜓∣𝜙=^+𝜋/2𝛽_{−𝜋/2𝛽}d𝑝𝜓^∗𝑝𝜙𝑝 (4) and preserves the ordinary nature of the position operator.

The operator 𝐴 with dense domain 𝒟(𝐴) is self-adjoint if 𝒟(𝐴)=𝒟(𝐴^†) and 𝐴=𝐴^†. However, for the position operator 𝑋 in the momentum space, we have^+𝜋/2𝛽_{−𝜋/2𝛽}d𝑝𝜓^∗𝑝𝜕𝑖ℏ𝜙𝑝=𝜕𝑝_{+𝜋/2𝛽−𝜋/2𝛽}𝑝d𝑝𝑖ℏ𝜕𝜓𝜕𝑝^∗𝜙𝑝+𝑖ℏ𝜓^∗𝑝𝜙𝑝_{𝑝=+𝜋/2𝛽}−𝑖ℏ𝜓^∗𝑝𝜙𝑝_{𝑝=−𝜋/2𝛽}, (5)where 𝜙(𝑝) vanishes at 𝑝=±𝜋/2𝛽 and 𝜓^∗(𝑝) takes arbitrary values at the boundaries. Indeed, the adjoint of the position operator 𝑋^†=𝑖ℏ𝜕/𝜕𝑝 has the same form but it acts on a different space of functions𝒟(𝑋)=𝜙,𝜙^′∈ℒ²−𝜋2𝛽,+𝜋2𝛽;𝜙+𝜋2𝛽=𝜙−𝜋2𝛽,𝒟𝑋=0_†=𝜓,𝜓^′∈ℒ²−𝜋2𝛽,+𝜋2𝛽;nootherrestrictionon𝜓. (6) Therefore, 𝑋 is merely symmetric (𝑋=𝑋^†), but it is not a true self-adjoint operator. On the other hand, the momentum operator is a self-adjoint operator which can be shown using the von Neumann’s theorem [11], that is, 𝑃=𝑃^† and 𝒟(𝑃)=𝒟𝑃^†=𝜙∈𝒟_max(ℝ), (7) where 𝒟_max denotes the maximal domain on which 𝑃 has a well-defined action 𝒟_max(𝑃)=𝜙∈ℒ²(ℝ)∶𝑃𝜙∈ℒ²(ℝ). (8)

In this representation, the completeness relation and scalar product can be written as 𝑝^′∣𝑝=𝛿𝑝−𝑝^′,_{+𝜋/2𝛽−𝜋/2𝛽}d𝑝𝑝𝑝=1. (9)(10) Also the eigenfunctions of the position operator in momentum space are given by the solutions of the eigenvalue equation 𝑋𝑢_𝑥𝑝=𝑥𝑢_𝑥𝑝, (11) where 𝑢_𝑥(𝑝)=⟨𝑝∣𝑥⟩. The normalized solution is 𝑢_𝑥𝑝=𝛽𝜋𝑝exp−𝑖ℏ𝑥. (12) Note that the physical meaning of the present eigenstates is different from the ones provided in [11, 13]. Now using (10) we find the wave function in coordinate space as 𝜓(𝑥)=𝛽𝜋^+𝜋/2𝛽_{−𝜋/2𝛽}𝑒^{𝑖𝑝𝑥/ℏ}𝜙𝑝d𝑝. (13) However, since the uncertainties for the eigenfunctions of the position operator are zero, that is, Δ𝑋_|𝑥⟩=0, |𝑥⟩ cannot be the physical solution. So, following Kempf et al. we define the maximal localization states |𝜙^ML_𝜉⟩ with the following properties [13]: 𝜙^ML_𝜉|𝑋|𝜙^ML_𝜉=𝜉,Δ𝑋_|𝜙^ML_𝜉⟩=(Δ𝑋)_min=ℏ𝛽. (14)(15) These states also satisfy 𝑋−⟨𝑋⟩+⟨[𝑋,𝑃]⟩2(Δ𝑃)²(𝑃−⟨𝑃⟩)𝜙=0, (16) where ⟨[𝑋,𝑃]⟩=𝑖ℏ(1+𝛽(Δ𝑃)²+𝛽⟨𝑃⟩²). Thus, in the momentum space the above equation takes the following form:𝜕𝑖ℏ()𝜕𝑝−⟨𝑋⟩+𝑖ℏ1+𝛽Δ𝑃²+𝛽⟨𝑃⟩²2(Δ𝑃)²tan𝛽𝑝𝛽−⟨𝑃⟩𝜙𝑝=0, (17)which has the solution 𝜙𝑝−𝑖=𝒩expℏ()⟨𝑋⟩+1+𝛽Δ𝑃²+𝛽⟨𝑃⟩²2(Δ𝑃)²𝑝+()⟨𝑃⟩1+𝛽Δ𝑃²+𝛽⟨𝑃⟩²2(Δ𝑃)²lncos𝛽𝑝𝛽. (18) To find the absolutely maximal localization states we need to choose the critical momentum uncertainty Δ𝑃=1/𝛽 that gives the minimal length uncertainty and takes ⟨𝑃⟩=0; that is, 𝜙^ML_𝜉𝑝=𝒩cos𝛽𝑝𝑒^{−𝑖𝑝𝜉/ℏ}, (19) where the normalization factor is given by 𝒩=2𝛽𝜋. (20) It is straightforward to check that 𝜙^ML_𝜉(𝑝) exactly satisfies (14) and (15). Because of the fuzziness of space, these maximal localization states are not mutually orthogonal: 𝜙^ML_𝜉′∣𝜙^ML_𝜉=𝒩^2+𝜋/2𝛽_{−𝜋/2𝛽}d𝑝cos²𝛽𝑝𝑒^{−𝑖𝑝(𝜉−𝜉′)/ℏ}=8𝛽^3/2ℏ³𝜋sin𝜋𝜉−𝜉^′/2ℏ𝛽𝜉−𝜉^′3−4𝛽ℏ²𝜉−𝜉^′. (21) To find the quasiposition wave function 𝜒(𝜉), we define 𝜒(𝜉)≡𝜙^ML_𝜉∣𝜙, (22) where in the limit 𝛽→0 it goes to the ordinary position wave function 𝜒(𝜉)=⟨𝜉∣𝜙⟩. Now the transformation of the wave function in the momentum representation into its counterpart quasiposition wave function is 𝜒(𝜉)=𝒩^+𝜋/2𝛽_{−𝜋/2𝛽}d𝑝cos𝛽𝑝𝑒^{𝑖𝑝𝜉/ℏ}𝜙𝑝,=1√2𝜓𝜉+ℏ𝛽+𝜓𝜉−ℏ𝛽. (23)(24) So the quasiposition wave function at ⟨𝑋⟩=𝜉 is the superposition of the coordinate space wave functions at 𝜉+ℏ𝛽 and 𝜉−ℏ𝛽. In other words, the quasiposition wave function is the result of the interference of two coordinate space wave functions.

3. Boundary Conditions

In this section, we discuss how the boundary conditions in quasiposition space can be determined by fixing the boundary conditions in coordinate space.

3.1. Dirichlet Boundary Condition

Consider the following Dirichlet boundary condition in coordinate space 𝜓(𝑥)_𝑥=𝜉0=0, (25) which gives ^+𝜋/2𝛽_{−𝜋/2𝛽}𝑒^{𝑖(𝑝/ℏ)𝜉0}𝜙𝑝d𝑝=0. (26) Now, using (24) and to first of the GUP parameter, we find 𝜒(𝜉)_𝜉=𝜉0=ℏ²𝛽√2𝜓^′′𝜉₀+𝒪𝛽², (27) which fixes the quasiposition wave functions at 𝜉=𝜉₀. Note that, for the following class of the coordinate space solutions 𝜓(𝑥)=𝐴sin(𝜔𝑥)+𝐵cos(𝜔𝑥),sin^𝑛(𝜔𝑥),cos^𝑛(𝜔𝑥), (28) we exactly have 𝜒(𝜉)_𝜉=𝜉0=𝜓(𝑥)_𝑥=𝜉0=0, (29) where 𝜉₀ are the zeros of (28). So, for these cases, the quasiposition wave functions obey the same boundary conditions as coordinate space wave functions.

3.2. Neumann Boundary Condition

The Neumann boundary condition determines the values that the derivative of a wave function is to take on the boundary of the domain. Let us consider the following boundary condition in coordinate space: 𝜓^′(𝑥)_𝑥=𝜉0=0, (30) where prime denotes the derivation with respect to the argument. This equation is equivalent to ^+𝜋/2𝛽_{−𝜋/2𝛽}𝑒^{𝑖(𝑝/ℏ)𝜉0}𝑝𝜙𝑝d𝑝=0. (31) Also, using (23) we have 𝜒^′(𝜉)=1√2𝜓^′𝜉+ℏ𝛽+𝜓^′𝜉−ℏ𝛽. (32) Therefore, to first order we obtain 𝜒^′(𝜉)_𝜉=𝜉0=ℏ²𝛽√2𝜓^′′′𝜉₀+𝒪𝛽². (33) Moreover, for the coordinate space wave functions presented in (28) we exactly find 𝜒^′(𝜉)_𝜉=𝜉0=𝜓^′(𝑥)_𝑥=𝜉0=0. (34) So, both 𝜒(𝜉) and 𝜓(𝑥) satisfy the same Neumann boundary condition.

Now let us elaborate on the correspondence between the uncertainties in position and the imposition of localized boundary conditions. In the GUP framework, it is not possible to measure the position of a particle more accurate than (Δ𝑋)_min. So we cannot define the potentials with infinitely sharp boundaries. In fact, the position of these boundaries can be only specified within this uncertainty. As it is shown in [11], the potentials with infinitely sharp boundaries such as the particle in a box potential cannot be properly defined in the GUP framework with respect to ordinary quantum mechanics.

4. Conclusions

In this paper, we have investigated the issue of the boundary conditions in deformed quantum mechanics which implies a minimal length uncertainty proportional to the Planck length. We found the coordinate space wave functions, maximal localization states, and quasiposition wave functions using a formally self-adjoint representation. We indicated that the position operator 𝑋 is merely symmetric and the momentum operator 𝑃 is truly self-adjoint which agrees with [13]. The maximal localization states are the physical states and obey the minimal length uncertainty; that is, Δ𝑋_|𝜙^ML_𝜉⟩=ℏ𝛽. We showed that the boundary conditions in coordinate space specify the boundary conditions in quasiposition space and found the exact relations for both Dirichlet and Neumann boundary conditions. Also, for a particular class of solutions, the boundary conditions are found to be the same in coordinate and quasiposition spaces. In fact, because of (15) the quasiposition wave functions ⟨𝜙^ML_𝜉∣𝜙⟩ and their boundary conditions contain the standard physical interpretation.

Acknowledgment

The authors would like to thank the referees for invaluable comments and important suggestions which considerably improved the quality of the paper.