# The Minimal Length and the Shannon Entropic Uncertainty Relation

Authors: Pouria Pedram
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relationX,P=iħ1+βP2, where β is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycielski (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, that is, X=x and P=tan⁡βp/β, where [x,p]=iħ, the BBM inequality is still valid in the form Sx+Sp≥1+ln⁡π as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.


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## Body

## 1. Introduction

The existence of a minimal observable length proportional to the Planck lengthlP=Għ/c3≈10-35m is motivated by various proposals of quantum gravity such as string theory, loop quantum gravity, noncommutative geometry, and black-hole [1–3]. Indeed, several schemes have been established to investigate the effects of the minimal length range from astronomical observations [4, 5] to table-top experiments [6]. In particular, a measurement method is proposed recently to detect this fundamental length scale which is based on the possible deviations from ordinary quantum commutation relation at the Planck scale within the current technology [6].Deformed commutation relations have attracted much attention in recent years and several problems range from classical to quantum mechanical systems have been studied exactly or approximately in the context of the generalized (gravitational) uncertainty principle (GUP). Among these investigations in quantum domain, we can mention the harmonic oscillator [7–9], Coulomb potential [10–12], singular inverse square potential [13], coherent states [14], Dirac oscillator [15], Lamb’s shift, Landau levels, tunneling current in scanning tunneling microscope [16], ultracold neutrons in gravitational field [17, 18], Casimir effect [19], relativistic quantum mechanics [20–22], and cosmological problems [23–26]. On the other hand, in the classical domain, deformed classical systems in phase space [27, 28], Keplerian orbits [29], composite systems [30], and the thermostatistics [31, 32] have been investigated in the presence of the minimal length.In the last decade, many applications of information theoretic measures such as entropic uncertainty relations, as alternatives to Heisenberg uncertainty relation, appeared in various quantum mechanical systems [33–48]. The concept of statistical complexity was introduced by Shannon in 1948 [49], where the term uncertainty can be considered as a measure of the missing information. In particular, it is shown that the information entropies such as Shannon entropy may be used to replace the well-known quantum mechanical uncertainty relation. The first entropic uncertainty relation for position and momentum observables was proposed by Hirschmann [50] and is later improved by Beckner, Bialynicki-Birula, and Mycielski (BBM) [51–54].In the context of the generalized uncertainty principle, there exists two questions. (i) Are the wave functions in position space and momentum space related by the Fourier transform in arbitrary GUP algebra in the form[X,P]=iħf(X,P)? (ii) If in a particular algebra these wave functions are not related by the Fourier transform, is the BBM inequality still valid? Note that the validity of the BBM entropic uncertainty relation depends on both the algebra and the used representation. For instance, consider the modified commutation relation in the form [X,P]=iħ(1+βP2+αX2) which implies a minimal length and a minimal momentum proportional to ħβ and ħα, respectively. This form of GUP has no formally self-adjoint representation in the form X=x and P=f(p). So the momentum space and coordinate space wave functions are not related by the Fourier transform in any representation and the BBM uncertainty relation does not hold in this framework.The well-known Robertson uncertainty principle for two noncommuting observables is given by(1)ΔAΔB≥12ψA,Bψ,where [A,B] denotes the commutator of A and B and ΔA and ΔB are their dispersions. However, this uncertainty relation suffers from two serious shortcomings [55, 56]. (i) For two noncommuting observables of a finite N-dimensional Hilbert space, since the right-hand side of (1) depends on the wave function ψ, it is not a fixed lower bound. Indeed, if ψ is the eigenstate of the observable A or B, the right-hand side of (1) vanishes and there is no restriction on ΔA or ΔB by this uncertainty relation. (ii) The dispersions cannot be considered as suitable measures for the uncertainty of two complementary observables with continuous probability densities. This problem is more notable when their corresponding probability densities contain several sharp peaks. Among various proposals for the uncertainty relations that are not suffered from these shortcomings, we can mention the information-theoretical entropy instead of the dispersions which is a proper measure of the uncertainty.In this paper, we study the effects of the minimal length on the entropic uncertainty relation. In this scenario, the position and momentum operators obey the modified commutation relation[X,P]=iħ1+βP2, where β is the deformation parameter. Using formally self-adjoint representation of the algebra, we show that the coordinate space and momentum space wave functions are related by the Fourier transform and consequently the BBM inequality is preserved. However, as we will show, in the quasiposition representation the momentum space and quasiposition space wave functions are not related by the Fourier transformation and the BBM inequality does not hold. As an application, we obtain the generalized Schrödinger equation for the harmonic oscillator and exactly solve the corresponding differential equation in momentum space. Then, we find information entropies for the two lowest energy eigenstates and explicitly show the validity of the BMM inequality in the presence of the minimal length.


## 2. The Generalized Uncertainty Principle

In one-dimension, the deformed commutation relation reads [7](2)X,P=iħ1+βP2,which results in ΔXΔP≥(ħ/2)1+β(ΔP)2 (generalized uncertainty principle) and for β→0 the well-known commutation relation in ordinary quantum mechanics is recovered. Notice that, ΔX cannot take arbitrarily small values and the absolutely smallest uncertainty in position is given by (ΔX)min=ħβ.Now, consider theformally self-adjoint representation [9](3)X=x,P=tan⁡βpβ,where [x,p]=iħ, -π/2β<p<π/2β, and it exactly satisfies (2). In this representation, the ordinary nature of the position operator is preserved and the inner product of states takes the following form:(4)ψ∣ϕ=∫-π/2β+π/2βψ∗pϕpdp.Note that although the position operator obeysX†=X=iħ∂/∂p in momentum space, we have D(X)⊂D(X†). So X is merely symmetric and it is not a true self-adjoint operator. However, based on the von Neumann’s theorem, for the momentum operator we obtain P=P† and D(P)=D(P†)=ϕ∈DmaxR [9]. Thus, P is indeed a self-adjoint operator. Moreover, the scalar product and the completeness relation read(5)p′∣p=δp-p′,(6)∫-π/2β+π/2βppdp=1.In momentum space, the eigenfunctions of the position operator are given by the solutions of the eigenvalue equation(7)Xuxp=xuxp.Here, ux(p)=p∣x which can be expressed as(8)uxp=12πexp⁡-ipħx.Now, using (6) and (8), coordinate space wave function can be written as(9)ψx=12π∫-π/2β+π/2βeipx/ħϕpdp.Moreover, ϕ(p) is given by the inverse Fourier transform of the coordinate space wave function; namely,(10)ϕp=12π∫-∞+∞e-ipx/ħψxdx.To this end, by taking ϕ(p)=0 for p>π/2β we can formally extend the domain of the momentum integral (9) to -∞<p<∞ without changing the coordinate space wave function ψ(x). Therefore, ψ(x) is the Fourier transform of ϕ(p). Now, using the Babenko-Beckner inequality [51, 52] and following Bialynicki-Birula and Mycielski [53] we obtain (see [54] for details)(11)Sx+Sp≥1+ln⁡π,where(12)Sx=-∫-∞+∞ψx2ln⁡ψx2dx,Sp=-∫-∞+∞ϕp2ln⁡ϕp2dp,subject to ϕ(p)=0 for p>π/2β. Note that, in this representation, the expression for the entropic uncertainty relation is similar to the ordinary quantum mechanics. However, as we will see in the next section, the presence of the minimal length modifies the Hamiltonian, its solutions, and the values of Sx and Sp. But since ψ(x) and ϕ(p) are still related by the Fourier transform, the lower bound for the entropic uncertainty relation will not be modified.


## 3. Quasiposition Representation

Another possible representation that exactly satisfies (2) is [7](13)Xϕp=iħ1+βp2∂pϕp,Pϕp=pϕp.The corresponding scalar product and completeness relations read(14)p′∣p=1+βp2δp-p′,(15)∫-∞+∞11+βp2ppdp=1.Now, since the measure in the integral (15) is not flat, the momentum space entropy(16)Sp=-∫-∞+∞11+βp2ϕp2ln⁡ϕp2dp,has no proper form of the continuous Shannon entropy relation in this representation.The quasiposition wave function is defined as [7](17)ϕξ≡ψξml∣ϕ,where(18)ψξmlp=2βπ1+βp2-1/2e-iξtan-1βp/ħβdenotes the maximal localization states. These states satisfy 〈ψξmlXψξml〉=ξ and (ΔX)|ψξml〉=ħβ and are not mutually orthogonal; that is, ψξ′ml∣ψξml≠δ(ξ-ξ′). In this representation, the relation between the momentum space wave functions and quasiposition wave functions is given by(19)ψξ=2βπ∫-∞+∞11+βp23/2eiξtan-1βp/ħβψpdp.Thus, the quasiposition wave functions are not Fourier transform of the momentum space wave functions and the quasiposition entropy(20)Sξ=-8πħ2β-1∭-∞+∞eiξ-ξ′tan-1βp/ħβψ∗ξψξ′ln⁡ψξ′2dpdξdξ′does not represent the continuous Shannon entropy and contains plenty of overcounting (because of the nonorthogonality of ψξml) that should be avoided. These results show that the information entropies Sp (16) and Sξ (20) are not proper measures of uncertainty. However, the information entropies (12) based on formally self-adjoint representation do not suffer from these shortcomings and they can be considered as proper measures of uncertainty in the presence of the minimal length.


## 4. Quantum Oscillator

The Hamiltonian of the harmonic oscillator is given byH=P2/2m+(1/2)mω2X2. So the generalized Schrödinger equation in momentum space using the representation (3) reads(21)-12mħ2ω2d2ϕpdp2+tan2βp2mβϕp=Eϕp.Using the new variable ξ=βp, the above equation can be written as(22)d2ϕξdξ2+ϵ-Vcos2ξϕξ=0,where V=mβħω-2 and ϵ=V(1+2mβE). Now, taking(23)ϕnξ=Pnscosλ⁡ξresults in(24)1-s2d2Pnsds2-s1+2λdPnsds+ϵ-λ2Pns=0,where s=sin⁡ξ and(25)V=λλ-1,λ=121+1+4m2β2ħ2ω2.It is known that the solutions of the above equation for ϵ=(n+λ)2 are given by the Gegenbauer polynomials Cnλ(s). Therefore, the exact solutions read(26)ϕnp=NnCnλsinβpcosλβp,(27)En=ħωn+121+η24+η2+12ħωηn2,n=0,1,2,…,where η=mβħω, Nn is the normalization coefficient, and the Gegenbauer polynomials are defined as [57](28)Cnλs=∑k=0n/2-1kΓn-k+λΓλk!n-2k!2sn-2k.Note that for β→0 we obtain the ordinary energy spectrum of the harmonic oscillator; that is, En=ħωn+1/2. The Gegenbauer polynomials also satisfy the following useful formula [57]:(29)∫-111-x2ν-1/2Cnνx2dx=π21-2νΓn+2νn!n+νΓν2,Re⁡ν>-12.Since s=sin⁡ξ and Nn is given by the normalization condition ∫-π/2β+π/2β‍|ϕ(ξ)|2dp=1, we find(30)Nn=βn!n+λΓλ2π21-2λΓn+2λ.The solutions can be also written in terms of relativistic Hermite polynomials using the relation [58](31)Hnλλu=n!λn/21+u2n/2Cnλu1+u2,where Hnλ(z) denotes relativistic Hermite polynomials. Thus, we obtain Cnλsin(βp)=(λn/2/n!)cosn(βp)Hnλλtan⁡(βp) which results in(32)ϕnp=Nnλn/2n!cosλ+n⁡βpHnλλtan⁡βp.For the small values of the deformation parameter we have (m=ħ=1)(33)limβ→0⁡λ=1ωβ,β⟶0⟺λ⟶∞,limλ→∞⁡cosλ+n⁡βp=exp⁡-p22ω,limλ→∞⁡λtan⁡βp=pω,limλ→∞⁡Hnλz=Hnz,Ref.58,where Hn denotes Hermite polynomials. So in this limit the solutions read(34)limβ→0⁡ϕnp=e-p2/2ωHnp/ω2nn!πω,which are normalized eigenstates of the ordinary harmonic oscillator as we have expected.


## 5. Information Entropy

The information entropies for the position and momentum spaces can be now calculated for the harmonic oscillator in the GUP framework using (12). In ordinary quantum mechanics and in the position space, the information entropy can be obtained analytically for some quantum mechanical systems. However, since the momentum wave functions are derived from the Fourier transform, the corresponding momentum information entropies are rather difficult to obtain. For our case, as we will show, we find Sp analytically for the two lowest energy states and obtain Sx numerically. Thus, because of the difficulty in calculating Sx and Sp, we only consider the two lowest energy eigenstates.First consider the Fourier transform of the momentum space ground state (26) which gives the following state in the position space (ħ=1):(35)ψ0x=λΓλ2π2βΓ2λsin⁡π/2x/β-λx/β-λF2112xβ-λ,-λ,12xβ-λ-2,1.Also, for the first excited state (n=1) we have(36)ψ1x=iλλ+1Γλ2π2βΓ2λ+1sin⁡π/2x/β-λ+1x/β-λ+1F2112xβ-λ+1,-λ,12xβ-λ-1,1-sin⁡π/2x/β-λ-1x/β-λ-1F2112xβ-λ-1,-λ,12xβ-λ-3,1.Figure 1 shows the resulting ground state and first excited state in position space for β={0,0.5,1}. Notice that, for β→0, the solutions tend to the simple harmonic oscillator wave functions; that is, limβ→0ψn(x)=ω/2nn!πe-ωx2/2Hnωx.Figure 1
Plots of the position space wave functions form=ħ=ω=1 and n=0,1.For the ground state, the analytical expression forSp reads(37)Sp0=λHλ-λHλ-1/2+ln⁡π-ln⁡βλΓλΓλ+1/2,where Hλ denotes the harmonic number Hn=∑k=1n‍1/k. It is easy to check that, for the small values of β, the momentum information entropy tends to the ordinary harmonic oscillator information entropy; namely,(38)limβ→0⁡Sp0=limβ→0⁡Sx0=121+ln⁡π.For the first excited state, Sp is given by(39)Sp1=1+λHλ+1-λHλ-1/2+ln⁡π-2-ln⁡βΓλ+22Γλ+1/2,and for β→0 it reads(40)limβ→0⁡Sp1=limβ→0⁡Sx1=12ln⁡π+ln⁡2+γ-12,where γ≈0.5772 is the Euler constant.The information entropy densities are defined asρs(x)=|ψx|2ln⁡|ψ(x)|2 and ρs(p)=|ϕp|2ln⁡|ϕ(p)|2. The behavior of ρs(x) and ρs(p) is illustrated in Figures 2 and 3 for n=0,1 and several values of the deformation parameter. Now, using the numerical values for Sx, we obtain the left-hand side of (11). In Table 1, we have reported the position and momentum space information entropies for β={0.1,0.5,1} and showed that they obey the BBM inequality. These results indicate that the position space information entropy increases with the GUP parameter β and vice versa for the momentum space information entropy but their sum stays above the value 1+ln⁡π.Table 1
The numerical results that establish the BBM entropic uncertainty relation form=ħ=ω=1 and various values of β.
n
β
S
x
S
p
S
x
+
S
p
1
+
ln
⁡
π
0
0.1
1.12153
1.02361
2.14515
2.14473
0.5
1.30251
0.85220
2.15471
2.14473
1.0
1.49095
0.68153
2.17248
2.14473
1
0.1
1.40656
1.24992
2.65648
2.14473
0.5
1.62672
0.97566
2.60238
2.14473
1.0
1.84423
0.74892
2.59315
2.14473Figure 2
Plots of the momentum space entropy densities form=ħ=ω=1 and n=0,1.Figure 3
Plots of the position space entropy densities form=ħ=ω=1 and n=0,1.


## 6. Conclusions

In this paper, we studied the Shannon entropic uncertainty relation in the presence of a minimal measurable length proportional to the Planck length. We showed that using the formally self-adjoint representation, since the coordinate space and momentum space wave functions are related by the Fourier transformation, the measure in the information entropic integral is flat and the lower bound that is predicted by the BBM inequality is guaranteed in the GUP framework. It is worth mentioning that the BBM inequality does not hold for all wave functions. In fact, its validity depends on both the deformed algebra and its representation. As we have indicated, this inequality does not hold in quasiposition representation of our algebra[X,P]=iħ1+βP2. As another example, we mentioned the algebra [X,P]=iħ1+αX2+βP2 that implies both a minimal length and a minimal momentum. Since this algebra has no formally self-adjoint representation in the form X=x and P=f(p), the momentum space and coordinate space wave functions are not related by the Fourier transform and the BBM uncertainty relation is not valid for this form of GUP. For the case of the harmonic oscillator, we exactly solved the generalized Schrödinger equation in momentum space and found the solutions in terms of the Gegenbauer polynomials. Also, for the two lowest energy eigenstates, we obtained the solutions in position space in terms of the hypergeometric functions. Then, the analytical expressions for the information entropies are found in the momentum space with proper limiting values for β→0. Using the numerical values for the position information entropy, we explicitly showed that the BBM inequality holds for various values of the deformation parameter. To check the validity of the BBM inequality for other potentials, we need to solve the generalized Schrödinger equation which contains higher order differential terms. However, for the small anharmonic potential terms, the perturbation theory can be used to find the approximate solutions. Also, For other types of GUPs, if a formally self-adjoint representation is viable, the BBM inequality is still valid.



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*Source: 5101389-2016-04-17.xml*