# Quantum Tunneling in Deformed Quantum Mechanics with Minimal Length

Authors: Xiaobo Guo; Bochen Lv; Jun Tao; Peng Wang
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

In the deformed quantum mechanics with a minimal length, one WKB connection formula through a turning point is derived. We then use it to calculate tunneling rates through potential barriers under the WKB approximation. Finally, the minimal length effects on two examples of quantum tunneling in nuclear and atomic physics are discussed.


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## Body

## 1. Introduction

Various theories of quantum gravity, such as string theory, loop quantum gravity, and quantum geometry, predict the existence of a minimal length [1–3]. For a review of a minimal length in quantum gravity, see [4]. Some realizations of the minimal length from various scenarios have been proposed. Specifically, one of the most popular models is the Generalized Uncertainty Principle (GUP) [5, 6], derived from the modified fundamental commutation relation:(1)X,P=iħ1+βP2,where β=β0lp2/ħ2=β0/c2mpl2,mpl is the Planck mass, lp is the Planck length, and β0 is a dimensionless parameter. With this modified commutation relation, one can easily find(2)ΔXΔP≥ħ21+βΔP2,which leads to the minimal measurable length:(3)ΔX≥Δmin=ħβ=β0lp.The GUP has been extensively studied recently; see, for example, [7–14]. For a review of the GUP, see [15].To study 1D quantum mechanics with the deformed commutators (1), one can exploit the following representation for X and P:(4)X=X0,P=P01+β3P02,where [X0,P0]=iħ. It can easily show that such representation fulfills the relation (1) to Oβ. Furthermore, we can adopt the position representation:(5)X0=x,P0=ħi∂∂x.Therefore for a quantum system described by(6)H=P22m+VX,the deformed stationary Schrodinger equation in the position representation is(7)d2ψxdx2-lβ2d4ψxdx4+2mE-Vxħ2ψx=0,where lβ2≡(2/3)ħ2β and terms of order β2 are neglected.If (7) with β=0 can be solved exactly, one could use the perturbation method to solve (7) by treating the term with lβ2 as a small correction. However for the general Vx, one might need other methods to solve (7). In fact, the WKB approximation in deformed space has been considered [16]. In [16], the authors considered the deformed commutation relation(8)X,P=iħfP,where fP is some function. For fP, one could solve the differential equation:(9)dPpdp=fPfor Pp, and pP denotes the inverse function of Pp. It is interesting to note that there might be more than one inverse function for Pp. However, one usually finds that there is only one inverse function pP which vanishes at P=0. The remaining ones are called “runaway” solutions, which are not physical and should be discarded [17, 18]. Then, they used the WKB approximation to show that the solution to the deformed Schrodinger equation:(10)P2ħiddxψx+2mVx-Eψx=0,was(11)ψx=1PxfPxC1exp⁡iħ∫xpPxdx+C2exp⁡-iħ∫xpPxdx,where Px=2mE-Vx. Moreover, it also showed that condition,(12)P2x≫ħddxPxfPx,had to be satisfied to make the WKB approximation valid. However, condition (12) fails near a turning point, where Px=0.For the case withfP=1+βP2, we derived one WKB connection formula through turning points and Bohr-Sommerfeld quantization rule in [19]. In this paper, we continue to consider other WKB connection formulas and calculate tunneling rates through potential barriers. The remainder of our paper is organized as follows. In Section 2, we derive one WKB connection formula and use it to find the formula for the tunneling rate through a potential barrier. Then two examples of quantum tunneling in nuclear and atomic physics are discussed in Section 3. Section 4 is devoted to our conclusions.


## 2. Tunneling through Potential Barriers

We now consider WKB description of tunneling through a potential barrierVx, which vanishes as x→±∞ and rises monotonically to its maximum Vmax at x=x0 as x approaches x0 from either the left or the right side of x0. In Figure 1, we plot the potential Vx. For a particle of energy E, there are two turning points x=x1 and x=x2,x1<x2, at which Vx=E. There are two classical allowed regions, Region I with x<x1 and Region III with x>x2. To describe tunneling, we need to choose appropriate boundary conditions in the classical allowed regions. We postulate an incident right-moving wave in Region I, where the WKB approximation solution to (10) includes a wave incident, the barrier and a reflected wave:(13)ψx=Aexp⁡-i/ħ∫xx1pPxdxPxfPx+Bexp⁡i/ħ∫xx1pPxdxPxfPx.In Region III, there is only a transmitted wave:(14)ψx=Fexp⁡i/ħ∫x2xpPxdxPxfPx.In the classically forbidden Region II, there are exponentially growing and decaying solutions:(15)ψx=Cexp⁡-1/ħ∫x1xpPxdxPxfPx+Dexp⁡1/ħ∫x1xpPxdxPxfPx=C′exp⁡1/ħ∫xx2pPxdxPxfPx+D′exp⁡-1/ħ∫xx2pPxdxPxfPx,where(16)C′=Ce-η,D′=Deη,η≡1ħ∫x1x2pPxdx.To calculate the tunneling rate, we need to use connection formulas to relate F,C/C′, and D/D′ to A. In [19], we derived one WKB connection formula around x=x1 in the case with fP=1+βP2. If D=0, we found that(17)CPxfPxexp⁡-1ħ∫x1xpPxdx⟶2CPxfPxsin⁡1ħ∫x1xpPxdx+π4,toOβ2,which gives(18)A=Ce-iπ/4.In what follows, we will derive a WKB connection formula around x=x2 to relate C′ and D′ to F and then calculate the tunneling rate through the potential barrier.Figure 1
Scattering from a potential barrier.To match WKB solutions, we need to solve the deformed Schrodinger equation (7) in the vicinity of the turning point x=x2. A linear approximation to the potential Vx near the turning point x=x2 is(19)Vx≈Vx2-V′x2x-x2,where Vx2=E. To simplify (7), a new dimensionless variable ρ could be introduced:(20)ρ=x2-x2mV′x2ħ21/3.Thus, (7) becomes(21)-α2d4ψρdρ4+d2ψρdρ2-ρψρ=0,where α2=2mV′x2/ħ22/3lβ2. The differential equation (7) can be solved by Laplace’s method [19]. Integral representations of the solutions are(22)Iρ=∫Cexp⁡ρt+α2t55-t33dt,where the contour C is chosen so that the integrand vanishes at endpoints of C. Specifically, define five sectors:(23)Θn≡2nπ+π/25,2nπ+3π/25,n=0,1,2,3,4.The contour C must originate at one of them and terminate at another.The asymptotic expressions ofIρ for large values of ρ can be obtained by evaluating the integral (22) using the method of steepest descent. To do this, we make the change of variables t=ρ1/2s and find(24)Iρ=ρ1/2∫Cexp⁡ρ3/2f±sds,where f±s≡±s-s3/3+as5/5 with + for ρ>0 and − for ρ<0, and a≡α2ρ≪1 in the physical region [19]. We will show below that there exists a steepest descent contour ranging from s=∞exp⁡7πi/5 to s=∞exp⁡9πi/5, which is the red contour in Figure 3. Such contour could let us match the asymptotic expression of Iρ at large negative value of ρ with the WKB solution (14) in Region III. Note that ∞exp⁡7πi/5∈Θ3 and ∞exp⁡7πi/5∈Θ4.The method of steepest descent is very powerful to calculate integrals of the form:(25)Ix=∫Cgzexfzdz,where C is a contour in the complex plane. We are usually interested in the behavior of Ix as x→∞. The key step of the method of steepest descent is applying Cauchy’s theorem to deform the contours C to the contours coinciding with steepest descent paths. Around a saddle point z0, where f′z0=0, there are two constant-phase (steepest) contours, on which Im⁡fz is constant, passing through z0 if f′′z0≠0. One of them is a steepest descent contour, along which Re⁡fz increases as we go towards z0. The other is a steepest ascent contour, along which Re⁡fz decreases as we go towards z0. If Ix is integrated along the steepest descent contour, the asymptotic behavior of Ix is dominated by the contribution from the saddle point z0.In Figures2 and 3, we plot saddle points (red points in figures) of f+s and f-s, respectively, and constant-phase contours passing through them. Specifically, saddle points of f+s are(26)s=±λ+≡±1-1-4a2a,s=±η+≡±1+1-4a2a,and these of f-s are(27)s=±λ-≡±1-1+4a2a,s=±η-≡±1+1+4a2a.The red contours in Figures 2 and 3 are the steepest descent contours connecting s=∞exp⁡7πi/5 to s=∞exp⁡9πi/5, along which the integral (24) is integrated. Note that red arrows on them denote the steepest contours’ directions. On the other hand, the black arrows on the constant-phase contours around saddle points denote the directions in which values of Re⁡f±s increase. Following the black arrows on the red contour in Figure 2, we find that Re⁡f+-λ+ and Re⁡f+η+ are smaller than Re⁡f+λ+. Thus for the case with ρ>0, the asymptotic expression of Iρ is dominated by the contribution from the saddle s=λ+. The method of steepest descent gives(28)I1≪ρ≪α-2~πexp⁡ρ3/2f+λ+ρ1/42f+′′λ+~π1+3/4aρ1/4exp⁡2ρ3/231+3a10,where a≪1 is used, and terms of Oa2 are neglected in the second line. For the case with ρ>0, Figure 3 shows that the asymptotic expression of Iρ is dominated by the contribution from the saddle s=-λ-, and hence we find(29)I-1≫ρ≫-α-2~πexp⁡π/4iexpρ3/2f--λ-ρ1/42f-′′-λ+~πexp⁡π/4i1-3/4aρ1/4exp⁡2iρ3/231-3a10,where terms of Oa2 are neglected in the second line.Figure 2
Saddle points and constant-phase (steepest) contours off+s. The red contour is a steepest descent contour, along which Iρ is integrated. The black arrows on the constant-phase contours around saddle points denote the directions in which values of Re⁡f+s increase.Figure 3
Saddle points and constant-phase (steepest) contours off-s. The red contour is a steepest descent contour, along which Iρ is integrated. The black arrows on the constant-phase contours around saddle points denote the directions in which values of Re⁡f-s increase.Around the turning pointx=x2,x-x2≪1, and Px~2mV′x2x-x2. In this region, we find that WKB solutions (14) and (15) become(30)ψx~F2mV′x2/ħ21/3ħ1-3a/4ρ1/4exp⁡2iρ3/231-3a10,ψx~12mV′x2/ħ21/3ħ1+3/4aρ1/4C′exp⁡2ρ3/231+3a10+D′exp⁡-2ρ3/231+3a10,where we use pP≈P-β/3P3 and terms of Oa2 are neglected, and we express x in terms of ρ using (20). In the overlap regions, where ρ≫1 and x-x2≪1, matching WKB solutions (30) with the Iρ’s asymptotic expressions (28) and (29) gives(31)C′=Fexp⁡-π4i,D′=0,which by (16) lead to C=Fexp⁡-π/4ieη and D=0. Since D=0, (18) gives(32)F=iAe-η,and the transmission probability is(33)T=F2A2~e-2η.


## 3. Examples

The dimensionless numberβ0=c2mpl2β=ħ2β/lp2 plays an important role when implications and applications of nonzero minimal length are discussed. Normally, if the minimal length is assumed to be order of the Planck length lp, one has β0~1. In [9], based on the precision measurement of STM current, an upper bound of β0 was given by β0<1021. In the following, we use (33) to study effects of GUP on α decay and cold electrons emission from metal via strong external electric field.


### 3.1.α Decay

The decay of a nucleus into anα-particle (charge 2e) and a daughter nucleus (charge Ze) can be described as the tunneling of an α-particle through a barrier caused by the Coulomb potential between the daughter and the α-particle (Figure 4) [20]. For an α-particle of energy E in the potential in Figure 4, there are two turning points, the nuclear radius R and the outer turning point b, which is determined by(34)E=Ze22πε0r⟹b=Ze22πε0E.The exponent η in (33) is(35)η=1ħ∫Rbp2mE1-Z1Z2e24πε0Erdx≈2mEħbarccos⁡Rb-Rb-R+2mEβ33barcsec⁡bR-Rb-R-2bb-RR,where m is the mass of the α-particle. At low energies (relative to the height of the Coulomb barrier at r=R), we have b≫R and then(36)η≈2me24ε0ħ1-8mEβ3πbRZE.The probability of emission of an α-particle is proportional to e-2η and hence the lifetime of the parent nucleus is about(37)τ~e2η.The density of nuclear matter is relatively constant, so R3 is proportional to the number of nucleons A. Empirically, we have(38)R~A1/3fm.Therefore, we find(39)ln⁡τ-1≈const-2me22ε0ħ1-β0EMeV1/2ZA-1/3×10-40ZE.On the other hand, a large collection of data shows that a good fit to the lifetime data obeys the Geiger–Nuttall law [21](40)ln⁡τ-1=C1-C2ZE,where C1 and C2 are constants. If the effects of GUP do not make (39) differ too much from the Geiger–Nuttall law, it will put an upper bound:(41)β0<1040.The GUP correction to the α-decay has also been considered in [22]. We both find that the effects of the GUP would increase the tunneling probability and hence decrease the lifetime τ.Figure 4
The potential energy of anα-particle in a radioactive nucleus.


### 3.2. Electron Emission from the Surface of Cold Metals

If a metal is placed in a very strong electric field, then there exists cold emission of electrons from the surface of the metal. This emission of the electrons can be explained via quantum tunneling. In [23], the shape of a tunneling barrier was assumed to be the exact triangular barrier, which has been known as the Fowler-Nordheim tunneling. Note that work must be done to remove an electron from the surface of a metal. In “free electron gas” model, one could hence take the potential energy of the electron inside the metal to be zero and for the outside to be Vx=V0>0. At the absolute zero temperature, if the Fermi energy of these electrons EF is less than V0, therefore after reaching the surface of the metal, they are reflected back into the metal. Now if the external electric field is applied toward the surface of the metal, the potential energy becomes(42)Vx=V0-eExforx>00forx<0,where E is the magnitude of the electric field. This potential is shown in Figure 5.Figure 5
The potential energy inside and outside a metallic surface when an external electric fieldE is added.We now use (33) to calculate the GUP modified transmission probability. For an electron of energy Ex≤EF<V0, there are two turning points:(43)x1=0,x2=V0-ExeE.The exponent η in (33) is(44)ηEx≡1ħ∫x1x2p2mEx-V0+eExdx≈22mV0-Ex3/23eEħ1-2mβV0-Ex5,which gives the transmission probability TEx≈e-2ηEx.Next we want to calculate the electric current density in this case. As a consequence of the GUP, the number of quantum states should be changed to [24](45)Vd3p2πħ31+βp23,where p2=pipi. Therefore, the electric current density is given by(46)J=e∫2dpxdpydpz2πħ31+βp23pxmTEx,where Ex=px2/2m. The range of variations of px, py, and pz is limited to the points inside the Fermi sphere(47)px2+py2+pz2≤2mEF.To calculate J, we use cylindrical coordinates,(48)py=ρcos⁡ϕ,pz=ρsin⁡ϕ,ρ2+px2≤2mEF,and have(49)J=4πe2πħ3∫02mEFpxmTExdpx∫02mEF-px2ρdρ1+βρ2+px23≈2πe2πħ3∫02mEFpxmTEx2mEF-px21-3β22mEF+px2dpx.To simplify the result, we change Ex to(50)ϵ=EF-Ex.Therefore, one has(51)J≈em2π2ħ3∫0EFϵTϵ1-3βm2EF-ϵdϵ.Since Tϵ decreases rapidly with increasing ϵ, therefore in Tϵ, we can expand V0-EF+ϵ3/2:(52)V0-EF+ϵ3/2≈V0-EF3/2+32ϵV0-EF1/2.We find(53)J≈em2π2ħ3exp⁡-2Q3V0-EF2Q21+4βmV0-EFQ15,where we extend the range of integration in (51) to 0,∞, and(54)Q=22mV0-EF3/2ħeE.In [25], the Fowler-Nordheim tunneling in device grade ultra-thin 3–6 nm n+poly-Si/SiO2/n-Si structures has been analyzed. Typically for this electron tunneling, we have(55)Q~10,V0-EF~1eV,m~0.5MeV. Therefore from (53), the correction due to GUP is given by(56)δJJ~10-50β0.The comparison of the calculated and experimental tunnel current was plotted in FIG. 8 of [25], which implies δJ/J≲10-2. Then the upper bound on β0 follows:(57)β0<1048.


## 3.1.α Decay

The decay of a nucleus into anα-particle (charge 2e) and a daughter nucleus (charge Ze) can be described as the tunneling of an α-particle through a barrier caused by the Coulomb potential between the daughter and the α-particle (Figure 4) [20]. For an α-particle of energy E in the potential in Figure 4, there are two turning points, the nuclear radius R and the outer turning point b, which is determined by(34)E=Ze22πε0r⟹b=Ze22πε0E.The exponent η in (33) is(35)η=1ħ∫Rbp2mE1-Z1Z2e24πε0Erdx≈2mEħbarccos⁡Rb-Rb-R+2mEβ33barcsec⁡bR-Rb-R-2bb-RR,where m is the mass of the α-particle. At low energies (relative to the height of the Coulomb barrier at r=R), we have b≫R and then(36)η≈2me24ε0ħ1-8mEβ3πbRZE.The probability of emission of an α-particle is proportional to e-2η and hence the lifetime of the parent nucleus is about(37)τ~e2η.The density of nuclear matter is relatively constant, so R3 is proportional to the number of nucleons A. Empirically, we have(38)R~A1/3fm.Therefore, we find(39)ln⁡τ-1≈const-2me22ε0ħ1-β0EMeV1/2ZA-1/3×10-40ZE.On the other hand, a large collection of data shows that a good fit to the lifetime data obeys the Geiger–Nuttall law [21](40)ln⁡τ-1=C1-C2ZE,where C1 and C2 are constants. If the effects of GUP do not make (39) differ too much from the Geiger–Nuttall law, it will put an upper bound:(41)β0<1040.The GUP correction to the α-decay has also been considered in [22]. We both find that the effects of the GUP would increase the tunneling probability and hence decrease the lifetime τ.Figure 4
The potential energy of anα-particle in a radioactive nucleus.


## 3.2. Electron Emission from the Surface of Cold Metals

If a metal is placed in a very strong electric field, then there exists cold emission of electrons from the surface of the metal. This emission of the electrons can be explained via quantum tunneling. In [23], the shape of a tunneling barrier was assumed to be the exact triangular barrier, which has been known as the Fowler-Nordheim tunneling. Note that work must be done to remove an electron from the surface of a metal. In “free electron gas” model, one could hence take the potential energy of the electron inside the metal to be zero and for the outside to be Vx=V0>0. At the absolute zero temperature, if the Fermi energy of these electrons EF is less than V0, therefore after reaching the surface of the metal, they are reflected back into the metal. Now if the external electric field is applied toward the surface of the metal, the potential energy becomes(42)Vx=V0-eExforx>00forx<0,where E is the magnitude of the electric field. This potential is shown in Figure 5.Figure 5
The potential energy inside and outside a metallic surface when an external electric fieldE is added.We now use (33) to calculate the GUP modified transmission probability. For an electron of energy Ex≤EF<V0, there are two turning points:(43)x1=0,x2=V0-ExeE.The exponent η in (33) is(44)ηEx≡1ħ∫x1x2p2mEx-V0+eExdx≈22mV0-Ex3/23eEħ1-2mβV0-Ex5,which gives the transmission probability TEx≈e-2ηEx.Next we want to calculate the electric current density in this case. As a consequence of the GUP, the number of quantum states should be changed to [24](45)Vd3p2πħ31+βp23,where p2=pipi. Therefore, the electric current density is given by(46)J=e∫2dpxdpydpz2πħ31+βp23pxmTEx,where Ex=px2/2m. The range of variations of px, py, and pz is limited to the points inside the Fermi sphere(47)px2+py2+pz2≤2mEF.To calculate J, we use cylindrical coordinates,(48)py=ρcos⁡ϕ,pz=ρsin⁡ϕ,ρ2+px2≤2mEF,and have(49)J=4πe2πħ3∫02mEFpxmTExdpx∫02mEF-px2ρdρ1+βρ2+px23≈2πe2πħ3∫02mEFpxmTEx2mEF-px21-3β22mEF+px2dpx.To simplify the result, we change Ex to(50)ϵ=EF-Ex.Therefore, one has(51)J≈em2π2ħ3∫0EFϵTϵ1-3βm2EF-ϵdϵ.Since Tϵ decreases rapidly with increasing ϵ, therefore in Tϵ, we can expand V0-EF+ϵ3/2:(52)V0-EF+ϵ3/2≈V0-EF3/2+32ϵV0-EF1/2.We find(53)J≈em2π2ħ3exp⁡-2Q3V0-EF2Q21+4βmV0-EFQ15,where we extend the range of integration in (51) to 0,∞, and(54)Q=22mV0-EF3/2ħeE.In [25], the Fowler-Nordheim tunneling in device grade ultra-thin 3–6 nm n+poly-Si/SiO2/n-Si structures has been analyzed. Typically for this electron tunneling, we have(55)Q~10,V0-EF~1eV,m~0.5MeV. Therefore from (53), the correction due to GUP is given by(56)δJJ~10-50β0.The comparison of the calculated and experimental tunnel current was plotted in FIG. 8 of [25], which implies δJ/J≲10-2. Then the upper bound on β0 follows:(57)β0<1048.


## 4. Conclusions

In this paper, we considered quantum tunneling in the deformed quantum mechanics with a minimal length. We first found one WKB connection formula through a turning point. Then the tunneling rates through potential barriers were derived using the WKB approximation. Finally, effects of the minimal length on quantum tunneling were discussed in two examples in nuclear and atomic physics,α decay and the Fowler-Nordheim tunneling. Upper bounds on β0 were given in these two examples, in which one had β0<1040 and β0<1048, respectively.In [9], the effects of the minimal length on electrons tunneling through a potential barrier of form(58)Vx=V0θx-θx-awere considered. For electrons of the mass m and energy E<V0, the GUP deformed Schrodinger equation was solved for the transmission coefficient, which was given by(59)TT0=1+4mβ2E-V023V0+2βa3ħ2mV0-E3/2,where T0 is the standard tunneling amplitude and β used in [9] is β/3 in this paper. Although the exact result for the transmission coefficient was given by (59), it is a good exercise to use (33) to calculate the transmission coefficient and compare the WKB approximated result with the exact one. In fact, there are two turning points x=0 and a for potential (58). Thus, we have(60)T~exp⁡-2ħ∫0ap2mE-V0dx.Since pP≈P-(β/3)P3, one finds(61)TT0=1+2βa3ħ2mV0-E3/2,where T0 is the standard WKB tunneling amplitude. It is noteworthy that the second term in the right-hand side of (59) is Oħ0, while the third term is Oħ-1 and hence is the leading order term of the WKB expansion. So as expected, the WKB result (61) agrees with the exact one (59) to Oħ-1. To put an upper bound on β0, the authors considered the following approximate values:(62)m~0.5MeV/c2,a~10-10m,E~10eV,V0~10eV.In this case, we find from (59) that(63)δTT0~10-48β0+10-51V0-E1eV3/2β0,which shows that when V0-E≳6eV, the leading WKB order term dominates, and hence the WKB approximation is a good one. Note that it was assumed that V0≈E in [9], and then the first term in the right-hand side of (63) dominates. However, if the leading WKB order term dominates, one can follow the argument in [9] and has(64)β0<V0-E1eV-3/2×1024,which might put a stronger bound on β0 in some other case.In the framework of the doubly special relativity, it has been shown that there is an upper bound on the momentum for a test particle [26]. To incorporate the existence of the minimal measurable length and the maximal measurable momentum, a new form of the GUP has been proposed [11, 27]:(65)X,P=iħ1-αP+2α2P2,where α≡α0/mplc is the GUP parameter and can be related to β through dimensional analysis with α2=β. Note that the relation α2=β implies that β0~α02 since β0=β/mpl2c2. The Hilbert space representation of quantum mechanics with the deformed fundamental commutation relation (65) was investigated in [28]. Comparing to the quantum mechanics with only a minimal measurable length [6], the authors of [28] found that there were several novel and interesting features in this deformed quantum mechanics. For example, it showed that the expectation value of energy in the formal position eigenvectors and the energy at the short wavelength of a test particle were no longer divergent, while both of them were divergent in the quantum mechanics with only a minimal measurable length. Moreover, the authors also found that the position operator X was symmetric but not self-adjoint, while the momentum operator P was self-adjoint due to absence of the minimal uncertainty in momentum.On the phenomenological side, the upper bounds onα0 have been studied [29, 30]. For example, the effects of the minimal length and the maximal momentum on the transition rate of ultra-cold neutrons bouncing above a mirror in the Earth’s gravitational field were investigated in [29], and the accuracy of Nesvizhevsky experiments put an upper bound on α0: (66)α0≤1029,which implies that β0~α02≤1058. Moreover, some tighter bounds came from studying the Planck scale effects on current experiments in Superconductivity and muon experiments [30], which gave α0≤1017β0≤1034 and α0≤108β0≤1016, respectively.



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