research article minimal length effects on tunnelling from

Research ArticleMinimal Length Effects on Tunnelling from SphericallySymmetric Black Holes

Benrong Mu12 Peng Wang2 and Haitang Yang23

1School of Physical Electronics University of Electronic Science and Technology of China Chengdu 610054 China2Center for Theoretical Physics College of Physical Science and Technology Sichuan University Chengdu 610064 China3Kavli Institute for Theoretical Physics China (KITPC) Chinese Academy of Sciences Beijing 100080 China

Correspondence should be addressed to Benrong Mu mubenronguestceducn

Received 16 October 2014 Revised 1 January 2015 Accepted 2 January 2015

Academic Editor Elias C Vagenas

Copyright copy 2015 Benrong Mu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We investigate effects of the minimal length on quantum tunnelling from spherically symmetric black holes using the Hamilton-Jacobi method incorporating theminimal lengthWe first derive the deformedHamilton-Jacobi equations for scalars and fermionsboth of which have the same expressions The minimal length correction to the Hawking temperature is found to depend on theblack holersquos mass and the mass and angular momentum of emitted particles Finally we calculate a Schwarzschild black holersquosluminosity and find the black hole evaporates to zero mass in infinite time

1 Introduction

The classical theory of black holes predicts that nothingincluding light could escape from the black holes HoweverStephen Hawking first showed that quantum effects couldallow black holes to emit particles The formula of Hawkingtemperature was first given in the frame of quantum fieldtheory [1] After that various methods for deriving Hawkingradiation have been proposed Among them is a semiclassicalmethod of modeling Hawking radiation as a tunneling effectproposed by Kraus and Wilczek [2 3] which is known asthe null geodesic method Later the tunneling behaviorsof particles were investigated using the Hamilton-Jacobimethod [4ndash6] Using the null geodesic method and theHamilton-Jacobi method much fruit has been achieved [7ndash18] The key point of the Hamilton-Jacobi method is usingWKB approximation to calculate the imaginary part of theaction for the tunneling process

On the other hand various theories of quantum gravitysuch as string theory loop quantum gravity and quantumgeometry predict the existence of a minimal length [19ndash21]The generalized uncertainty principle (GUP) [22] is a simpleway to realize this minimal length

An effective model of the GUP in one-dimensionalquantum mechanics is given by [23 24]

119871119891119896 (119901) = tanh(119901

119872119891

) (1)

119871119891120596 (119864) = tanh( 119864

119872119891

) (2)

where the generators of the translations in space and time arethe wave vector 119896 and the frequency 120596 119871119891 is the minimallength and 119871119891119872119891 = ℏ The quantization in positionrepresentation 119909 = 119909 leads to

119896 = minus119894120597119909 120596 = 119894120597119905 (3)

Therefore the low energy limit 119901 ≪ 119872119891 including order of11990131198723119891 gives

119901 = minus119894ℏ120597119909(1 minusℏ2

1198722119891

1205972119909) (4)

119864 = 119894ℏ120597119905(1 minusℏ2

1198722119891

1205972119905) (5)

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 898916 8 pageshttpdxdoiorg1011552015898916

2 Advances in High Energy Physics

where we neglect the factor 13 From (1) it is noted thatalthough one can increase 119901 arbitrarily 119896 has an upper boundwhich is 1119871119891 The upper bound on 119896 implies that particlescould not possess arbitrarily small Compton wavelengths120582 = 2120587119896 and that there exists a minimal length sim 119871119891Furthermore the deformed Klein-GordonDirac equationsincorporating (4) and (5) have already been obtained in [23]which will be briefly reviewed in Section 2 So [23] provides away to incorporate the minimal length with special relativitya good starting point for studying Hawking radiation astunnelling effect

The black hole is a suitable venue to discuss the effectsof quantum gravity Incorporating GUP into black holes hasbeen discussed in a lot of papers [25ndash30]The thermodynam-ics of black holes has also been investigated in the frameworkof GUP [29 30] In [31] a new form of GUP is introduced

1199010= 1198960

119901119894= 119896119894(1 minus 120572k + 21205722k2)

(6)

where119901119886 is themodified fourmomentum 119896119886 is the usual fourmomentum and 120572 is a small parameterThemodified velocityof photons two-dimensional Klein-Gordon equation theemission spectrum due to the Unruh effect is obtained thereRecently the GUP deformed Hamilton-Jacobi equation forfermions in curved spacetime have been introduced and thecorrected Hawking temperatures have been derived [32ndash36]The authors consider the GUP of form

119909119894 = 1199090119894

119901119894 = 1199010119894 (1 + 1205731199012)

(7)

where 1199090119894 and 1199010119894 satisfy the canonical commutation rela-tions Fermionsrsquo tunnelling black hole thermodynamics andthe remnants are discussed there

In this paper we investigate scalars and fermions tunnel-ing across the horizons of black holes using the deformedHamilton-Jacobi method which incorporates the minimallength via (4) and (5) Our calculation shows that thequantum gravity correction is related not only to the blackholersquos mass but also to the mass and angular momentum ofemitted particles

The organization of this paper is as follows In Section 2from the modified fundamental commutation relation wegeneralize the Hamilton-Jacobi method in curved spacetimeIn Section 3 incorporatingGUPwe investigate the tunnellingof particles in the black holes In Section 4 we investigatehow a Schwarzschild black hole evaporates in our modelSection 5 is devoted to our conclusion In this paper we takeGeometrized units 119888 = 119866 = 1 where the Planck constantℏ is square of the Planck Mass 119898119901 We also assume that theemitted particles are neutral

2 Deformed Hamilton-Jacobi Equations

To be generic we will consider a spherically symmetricbackground metric of the form

1198891199042= 119891 (119903) 119889119905

2minus1198891199032

119891 (119903)minus 1199032(1198891205792+ sin21205791198891206012) (8)

where 119891(119903) has a simple zero at 119903 = 119903ℎ with 1198911015840(119903ℎ) being finite

and nonzero The vanishing of 119891(119903) at point 119903 = 119903ℎ indicatesthe presence of an event horizon In this section we will firstderive the deformed Klein-GordonDirac equations in flatspacetime and then generalize them to the curved spacetimewith the metric (8)

In the (3 + 1) dimensional flat spacetime the relationsbetween (119901119894 119864) and (119896119894 120596) can simply be generalized to

119871119891119896119894 (119901) = tanh(119901119894

119872119891

)

119871119891120596 (119864) = tanh( 119864

119872119891

)

(9)

where in the spherical coordinates one has for

= minus119894 (119903120597

120597119903+120579

119903

120597

120597120579+

120601

119903 sin 120579120597

120597120601) (10)

Expanding (9) for small arguments to the third order gives theenergy and momentum operator in position representation

119864 = 119894ℏ120597119905(1 minusℏ2

1198722119891

1205972119905)

=ℏ

119894[119903(120597119903 minus

ℏ21205973119903

1198722119891

) + 120579(120597120579

119903minusℏ21205973120579

11990331198722119891

)

+120601(

120597120601

119903 sin 120579minus

ℏ21205973120601

1199033sin31205791198722119891

)]

(11)

where we also omit the factor 13 Substituting the aboveenergy and momentum operators into the energy-momen-tum relation the deformed Klein-Gordon equation satisfiedby the scalar field with the mass119898 is

1198642120601 = 119901

2120601 + 119898

2120601 (12)

where 1199012 = sdot Making the ansatz for 120601

120601 = exp (119894119868ℏ) (13)

and substituting it into (12) one expands (12) in powers of ℏand then finds that the lowest order gives the deformed scalarHamilton-Jacobi equation in the flat spacetime

(120597119905119868)2(1 +

2 (120597119905119868)2

1198722119891

) minus (120597119903119868)2(1 +

2 (120597119903119868)2

1198722119891

)

minus(120597120579119868)2

1199032(1 +

2 (120597120579119868)2

11990321198722119891

)

minus

(120597120601119868)2

1199032sin2120579(1 +

2 (120597120601119868)2

1199032sin21205791198722119891

) = 1198982

(14)

which is truncated at O(11198722119891)

Advances in High Energy Physics 3

Similarly the deformed Dirac equation for a spin-12fermion with the mass119898 takes the form of

(1205740119864 + 120574 sdot minus 119898)120595 = 0 (15)

where 1205740 1205740 = 2 120574119886 120574119887 = minus2120575119886119887 and 1205740 120574119886 = 0 with theLatin index 119886 running over 119903 120579 and120601Multiplying (1205740119864+ 120574sdot+

119898) by (15) and using the gamma matrices anticommutationrelations the deformed Dirac equation can be written as

1198642120595 = (119901

2+ 1198982) 120595 minus

[120574119886 120574119887]

2119901119886119901119887120595

(16)

To obtain the Hamilton-Jacobi equation for the fermion theansatz for 120595 takes the form of

120595 = exp(119894119868ℏ) V (17)

where V is a vector function of the spacetime Substituting(17) into (16) and noting that the second term on RHS of(16) does not contribute to the lowest order of ℏ we find thatthe deformed Hamilton-Jacobi equation for a fermion is thesame as the deformed one for a scalar with the same massnamely (14) Note that one can use the deformed Maxwellrsquosequations obtained in [23] to get the deformed Hamilton-Jacobi equation for a vector boson However for simplicitywe just stop here

In order to generalize the deformed Hamilton-Jacobiequation (14) to the curved spacetime with the metric(8) we first consider the Hamilton-Jacobi equation withoutGUP modifications In [37] we show that the unmodifiedHamilton-Jacobi equation in curved spacetime with 1198891199042 =119892120583]119889119909

120583119889119909

] is

119892120583]120597120583119868120597]119868 minus 119898

2= 0 (18)

Therefore the unmodified Hamilton-Jacobi equation in themetric (8) becomes

(120597119905119868)2

119891 (119903)minus 119891 (119903) (120597119903119868)

2minus(120597120579119868)2

1199032minus

(120597120601119868)2

1199032sin2120579= 1198982 (19)

On the other hand the unmodified Hamilton-Jacobi equa-tion in flat spacetime can be obtained from (14) by taking119872119891 rarr infin

(120597119905119868)2minus (120597119903119868)

2minus(120597120579119868)2

1199032minus

(120597120601119868)2

1199032sin2120579= 1198982

(20)

Comparing (19) with (20) one finds that theHamilton-Jacobiequation in the metric (8) can be obtained from the one inflat spacetime by making replacements 120597119903119868 rarr radic119891(119903)120597119903119868 and120597119905119868 rarr 120597119905119868radic119891(119903) in the no GUP modifications scenarioTherefore by making replacements 120597119903119868 rarr radic119891(119903)120597119903119868 and120597119905119868 rarr 120597119905119868radic119891(119903) the deformed Hamilton-Jacobi equation

in flat spacetime (14) leads to the deformedHamilton-Jacobiequation in the metric (8) which is to O(11198722119891)

(120597119905119868)2

119891 (119903)(1 +

2 (120597119905119868)2

119891 (119903)1198722119891

) minus 119891 (119903) (120597119903119868)2(1 +

2119891 (119903) (120597119903119868)2

1198722119891

)

minus(120597120579119868)2

1199032(1 +

2 (120597120579119868)2

11990321198722119891

)

minus

(120597120601119868)2

1199032sin2120579(1 +

2 (120597120601119868)2

1199032sin21205791198722119891

) = 1198982

(21)

3 Quantum Tunnelling

In this section we investigate the particlesrsquo tunneling at theevent horizon 119903 = 119903ℎ of themetric (8)whereGUP is taken intoaccount Since metric (8) does not depend on 119905 and 120601 120597119905 and120597120601 are killing vectors Taking into account the Killing vectorsof the background spacetime we can employ the followingansatz for the action

119868 = minus120596119905 +119882 (119903 120579) + 119901120601120601 (22)

where 120596 and 119901120601 are constants and they are the energyand the 119911-component of angular momentum of emittedparticles respectively Inserting (22) into (21) we find that thedeformed Hamilton-Jacobi equation becomes

1199012119903 (1 +

2119891 (119903) 1199012119903

1198722119891

)

=1

1198912 (119903)[1205962(1 +

21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982+ 120582)]

(23)

where 119901119903 = 120597119903119882 119901120579 = 120597120579119882 and

120582 =1199012120579

1199032(1 +

21199012120579

11990321198722119891

) +

1199012120601

1199032sin2120579(1 +

21199012120601

1199032sin21205791198722119891

) (24)

Since themagnitude of the angularmomentumof the particle119871 can be expressed in terms of 119901120579 and 119901120601

1198712= 1199012120579 +

1199012120601

sin2120579 (25)

one can rewrite 120582 as

120582 =1198712

1199032+ O(

1

1198722119891

) (26)


Solving (23) for 119901119903 to O(11198722119891) gives

120597119903119882∓

= plusmn1

119891 (119903)radic1205962(1 +

21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982 + 120582)

times radic1 minus2

1198722119891119891 (119903)

[1205962(1 +21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982 + 120582)]

(27)

where +minus represent the outgoingingoing solutions In orderto get the imaginary part of119882plusmn we need to find residue of theRHSof (27) at 119903 = 119903ℎ by expanding theRHS in a Laurent serieswith respect to 119903 at 119903 = 119903ℎ We then rewrite (27) as

120597119903119882∓

= plusmn1

119891 (119903)52radic1205962(119891 (119903) +

21205962

1198722119891

) minus 1198912 (119903) (1198982 + 120582)

times radic1198912 (119903) minus2

1198722119891

[1205962(119891 (119903) +21205962

1198722119891

) minus 1198912 (119903) (1198982 + 120582)]

(28)

Using119891(119903) = 1198911015840(119903ℎ)(119903minus119903ℎ)+(11989110158401015840(119903ℎ)2)(119903minus119903ℎ)

2+O((119903minus119903ℎ)

3)

one can single out the 1(119903 minus 119903ℎ) term of the Laurent series

120597119903119882∓ simplusmn119886minus1

119903 minus 119903ℎ

(29)

where we have

119886minus1 =120596

1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)] + O(1

1198724119891

) (30)

Using the residue theory for semicircles we obtain for theimaginary part of119882plusmn to O(1119872

2119891)

Im119882plusmn = plusmn120587120596

1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)] (31)

However when one tries to calculate the tunneling rateΓ from Im119882plusmn there is so called ldquofactor-two problemrdquo [38]One way to solve the ldquofactor-two problemrdquo is introducinga ldquotemporal contributionrdquo [14ndash16 39 40] To consider aninvariance under canonical transformations we also followthe recent work [14ndash16 39 40] and adopt the expression∮119901119903119889119903 = int119901

+119903 119889119903 minus int119901

minus119903 119889119903 for the spatial contribution to

Γ The spatial and temporal contributions to Γ are given asfollows

Spatial Contribution The spatial part contribution comesfrom the imaginary part of 119882(119903) Thus the spatial partcontribution is proportional to

exp [minus1ℏIm∮119901119903119889119903]

= exp [minus1ℏIm(int119901

+119903 119889119903 minus int119901

minus119903 119889119903)]

= expminus 2120587120596

1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)]

(32)

Temporal Contribution As pointed in [15 38ndash40] the tem-poral part contribution comes from the ldquorotationrdquo whichconnects the interior region and the exterior region ofthe black hole Thus the imaginary contribution from thetemporal part when crossing the horizon is Im(120596Δ119905outin) =120596(1205872120581) where 120581 = 1198911015840(119903ℎ)2 is the surface gravity at the eventhorizonThen the total temporal contribution for a round tripis

Im (120596Δ119905) =2120587120596

1198911015840 (119903ℎ) (33)

Therefore the tunnelling rate of the particle crossing thehorizon is

Γ prop exp [minus1ℏ(Im (120596Δ119905) + Im∮119901119903119889119903)]

= expminus 4120587120596

1198911015840 (119903ℎ)[1 +

1

1198722119891

(1198982+1198712

1199032ℎ

)]

(34)

This is the expression of Boltzmann factor with an effectivetemperature

119879 =1198911015840(119903ℎ)

4120587

ℏ

1 + (11198722119891) (1198982 + (1198712119903

2ℎ))

(35)

where 1198790 = ℏ1198911015840(119903ℎ)4120587 is the original Hawking temperature

For the standard Hawing radiation all particles veryclose to the horizon are effectively massless on accountof infinite blueshift Thus the conformal invariance of thehorizon make Hawing temperatures of all particles thesame The mass angular momentum and identity of theparticles are only relevant when they escape the potentialbarrier However if quantum gravity effects are consideredbehaviors of particles near the horizon could be differentFor example if we send a wave packet which is governed bya subluminal dispersion relation backwards in time towardthe horizon it reaches a minimum distance of approachand then reverses direction and propagate back away fromthe horizon instead of getting unlimited blueshift towardthe horizon [41 42] Thus quantum gravity effects mightmake fermions and scalars experience different (effective)


Hawking temperatures However our result shows that in ourmodel the tunnelling rates of fermions and scalars dependon their masses and angular momentums but independentof the identities of the particles to O(11198722119891) In other wordseffective Hawking temperatures of fermions and scalars arethe same to O(11198722119891) in our model as long as their massesenergies and angular momentums are the same

4 Thermodynamics of Black Holes

For simplicity we consider the Schwarzschild metric with119891(119903) = 1 minus 2119872119903 with the black holersquos mass 119872 The eventhorizon of the Schwarzschild black hole is 119903ℎ = 2119872 In thissection we work with massless particles Near the horizon ofthe the black hole angular momentum of the particle 119871 sim

119901119903ℎ sim 120596119903ℎ Thus one can rewrite 119879

119879 sim1198790

1 + 212059621198722119891

(36)

where 1198790 = ℏ8120587119872 for the Schwarzschild black hole Asreported in [43] the authors obtained the relation 120596 ≳

ℏ120575119909 between the energy of a particle and its positionuncertainty in the framework of GUP Near the horizon ofthe the Schwarzschild black hole the position uncertainty ofa particle will be of the order of the Schwarzschild radius ofthe black hole [44] 120575119909 sim 119903ℎ Thus one finds for 119879

119879 sim1198790

1 + 1198984119901211987221198722119891

(37)

where we use ℏ = 1198982119901 Using the first law of the black

hole thermodynamics we find that the corrected black holeentropy is

119878 = int119889119872

119879

sim119860

41198982119901

+

41205871198982119901

1198722119891

ln( 119860

16120587)

(38)

where119860 = 41205871199032ℎ = 16120587119872

2 is the area of the horizonThe log-arithmic term in (38) is thewell known correction fromquan-tum gravity to the classical Bekenstein-Hawking entropywhich has appeared in different studies of GUP modifiedthermodynamics of black holes [27ndash29 45ndash51] In general theentropy for the Schwarzschild black hole of mass119872 in fourspacetime dimensions can be written in form of

119878 =119860

4+ 120590 ln( 119860

16120587) + O(

1198722119891

119860) (39)

where 120590 = 211987221198911198722 in our paper Neglecting the terms

O(1198722119891119860) in (39) there could be three scenarios dependingon the sign of 120590

(1) 120590 = 0 This case is just the standard Hawking radi-ation The black holes evaporate completely in finitetime

(2) 120590 lt 0 The entropy 119878 as function of mass developsa minimum at some value of119872min This predicts theexistence of black hole remnants Furthermore theblack holes stop evaporating in finite time This iswhat happened in [27ndash29 45ndash51] which is consistentwith the existence of a minimal length

(3) 120590 gt 0 This is a subtle case In the remainder ofthe section we will investigate how the black holesevaporate in our model

For particles emitted in a wave mode labelled by energy120596 and 119871 we find from (34) that [52]

(Probability for a black hole to emit

a particle in this mode)

= exp(minus120596119879) times (Probability for a black hole to absorb

a particle in the same mode) (40)

where 119879 is given by (35) Neglecting back-reaction detailedbalance condition requires that the ratio of the probabilityof having 119873 particles in a particular mode with 120596 and 119871

to the probability of having 119873 minus 1 particles in the samemode is exp(minus120596119879) One then follows the standard textbookprocedure to get the average number 119899120596119871 in the mode

119899120596119871 = 119899 (120596

119879) (41)

where we define

119899 (119909) =1

exp 119909 minus (minus1)120598 (42)

and 120598 = 0 for bosons and 120598 = 1 for fermions In [53] countingthe number of modes per frequency interval with periodicboundary conditions in a large container around the blackhole Page related the expected number emitted permode 119899120596119871to the average emission rate per frequency interval 119889119899120596119871119889119905by

119889119899120596119871

119889119905= 119899120596119871

119889120596

2120587ℏ (43)

Following Pagersquos argument we find that in our model

119889119899120596119871

119889119905= 119899120596119871

120597120596

120597119901119903

119889119901119903

2120587ℏ= 119899120596119871

119889120596

2120587ℏ (44)

where 120597120596120597119901119903 is the radial velocity of the particle and thenumber of modes between the wavevector interval (119901119903 119901119903 +119889119901119903) is1198891199011199032120587ℏ where119901119903 = 120597119903119868 is the radial wavevector Sinceeach particle carries off the energy 120596 the total luminosity isobtained from 119889119899120596119871119889119905 by multiplying by the energy 120596 andsumming up over all energy 120596 and 119871

119871 = sum

119897=0

(2119897 + 1) int120596119899120596119897

119889120596

2120587ℏ (45)


where 1198712 = (119897 + 1)119897ℏ2 and the degeneracy for 119897 is (2119897 +

1) However some of the radiation emitted by the horizonmight not be able to reach the asymptotic region We need toconsider the greybody factor |119879119894(120596)|

2 where 119879119894(120596) representsthe transmission coefficient of the black hole barrier which ingeneral can depend on the energy 120596 and angular momentum119897 of the particle Taking the greybody factor into account wefind for the total luminosity

119871 = sum

119897=0

(2119897 + 1) int1003816100381610038161003816119879119894 (120596)

1003816100381610038161003816

2120596119899120596119897

119889120596

2120587ℏ (46)

Usually one needs to solve the exact wave equations for|119879119894(120596)|

2 which is very complicated On the other handone can use the geometric optics approximation to estimate|119879119894(120596)|

2 In the geometric optics approximation we assume120596 ≫ 119872 and high energy waves will be absorbed unlessthey are aimed away from the black hole Hence we have|119879119894(120596)|

2= 1 for all the classically allowed energy 120596 and

angular momentum 119897 of the particle In this approximationthe Schwarzschild black hole is just like a black sphere ofradius 119877 = 3

32119872 [54] which puts an upper bound on

119897(119897 + 1)ℏ2

119897 (119897 + 1) ℏ2⩽ 27119872

21205962 (47)

Note that we neglect possible modifications from GUP to thebound since we are interested in the GUP effects near thehorizon Thus the luminosity is

119871 = int

infin

0

120596119889120596

2120587ℏ3int

2711987221205962

0119899[

120596

1198790

(1 +1

1198722119891

119897 (119897 + 1) ℏ2

21198722)]

sdot 119889 [119897 (119897 + 1) ℏ2]

=119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 [119906 (1 + 119886

2119906119909)] 119889119909

(48)

where we define 119906 = 1205961198790 119910 = 119897(119897 + 1)ℏ211987221205962 and 119886 =

1198790radic2119872119891 = 119898

21199018

radic2120587119872119891119872 For 119872 ≫ 11989821199018

radic2120587119872119891 wehave 119886 ≪ 1 and hence the luminosity is

119871 asymp27

321205872ℏ311987940119860int

infin

01199063119899 (119906) 119889119906 (49)

which is just Stefanrsquos law for black holes Therefore for largeblack holes they evaporate in almost the same way as in Case1 until 119872 sim 119898

21199018

radic2120587119872119891 when the term 1198862119906119909 starts to

dominate in (48) Then the luminosity is approximated by

119871 sim119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 (11988621199062119909) 119889119909

=

211987221198724119891

1205871198986119901

int

infin

0V3119889Vint

27

0119899 (V2119909) 119889119909

(50)

where V = 119886119906 Not worrying about exact numerical factorsone has for the evaporation rate

119889119872

119889119905= minus119871 sim minus119860

1198722

1198722119891

(51)

where 119860 gt 0 is a constant Solving (51) for 119872 gives 119872 sim

1198722119891119860119905 The evaporation rate considerably slows down when

black holesrsquo mass 119872 sim 11989821199018

radic2120587119872119891 The black hole thenevaporates to zero mass in infinite time However the GUPpredicts the existence of a minimal length It would makemuchmore sense if there are black hole remnants in the GUPmodels How can we reconcile the contradiction When wewrite down the deformed Hamilton-Jacobi equation (21) weneglect terms higher than O(11198722119891) However when 119872 sim

11989821199018

radic2120587119872119891 our effective approach starts breaking downsince contributions fromhigher order terms become the sameimportant as these from terms O(11198722119891) Thus one then hasto include these higher order contributions For example ifthere are higher order corrections to (51) as in

119889119872

119889119905sim minus119860

1198722

1198722119891

+ 1198611198723

1198723119891

(52)

where 119861 gt 0 one can easily see that there exists a minimummass119872min sim 119872119891radic119860119861 for black holes In another word theO(11198722119891) terms used in our paper are not sufficient enough toproduce black hole remnants predicted by the GUP To do soone needs to resort to higher order terms if the full theory isnot available It is also noted that when at late stage of a blackhole with 119886 ≳ 1 (36) becomes

119879 sim1198790

1198984119901211987221198722119891

sim 119872 (53)

which means that the tempreture of the black hole goes tozero as the mass goes to zero

The GUP is closely related to noncommutative geometryIn fact when the GUP is investigated in more than onedimension a noncommutative geometric generalization ofposition space always appears naturally [22] On the otherhand quantum black hole physics has been studied in thenoncommutative geometry [55] In [56] a noncommutativeblack holersquos entropy received a logarithmic correction with120590 lt 0 However [57 58] showed that the correctionsto a noncommutative schwarzschild black holersquos entropymight not involve any logarithmic terms In either case thetempreture of the noncommutative schwarzschild black holereaches zero in finite time with remnants left

5 Conclusion

In this paper incorporating effects of the minimal lengthwe derived the deformedHamilton-Jacobi equations for bothscalars and fermions in curved spacetime based on the mod-ified fundamental commutation relations We investigatedthe particlesrsquo tunneling in the background of a sphericallysymmetric black hole In this spacetime configurations weshowed that the corrected Hawking temperature is not onlydetermined by the properties of the black holes but alsodependent on the angular momentum and mass of theemitted particles Finally we studied how a Schwarzschildblack hole evaporates in our model We found that the blackhole evaporates to zero mass in infinite time


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge useful discussionswith Y He Z Sun and H W Wu This work is supportedin part by NSFC (Grant nos 11005016 11175039 and 11375121)and SYSTF (Grant no 2012JQ0039)

References

[1] S W Hawking ldquoParticle creation by black holesrdquo Communica-tions in Mathematical Physics vol 43 no 3 pp 199ndash220 1975Erratum inCommunications inMathematical Physics vol 46 p206 1976

[2] P Kraus and F Wilczek ldquoSelf-interaction correction to blackhole radiancerdquo Nuclear Physics B vol 433 no 2 pp 403ndash4201995

[3] P Kraus and F Wilczek ldquoEffect of self-interaction on chargedblack hole radiancerdquo Nuclear Physics B vol 437 no 1 pp 231ndash242 1995

[4] K Srinivasan and T Padmanabhan ldquoParticle production andcomplex path analysisrdquo Physical Review D vol 60 no 2 ArticleID 024007 20 pages 1999

[5] M Angheben M Nadalini L Vanzo and S Zerbini ldquoHawkingradiation as tunneling for extremal and rotating black holesrdquoJournal of High Energy Physics vol 2005 no 5 article 014 2005

[6] R Kerner and R BMann ldquoTunnelling temperature and Taub-NUT black holesrdquo Physical Review D vol 73 no 10 Article ID104010 2006

[7] S Hemming and E Keski-Vakkuri ldquoThe Spectrum of strings onBTZ black holes and spectral ow in the SL(2R) WZW modelrdquoNuclear Physics B vol 626 no 1-2 pp 363ndash376 2002

[8] A J M Medved ldquoRadiation via tunneling from a de Sittercosmological horizonrdquo Physical ReviewD vol 66 no 12 ArticleID 124009 7 pages 2002

[9] E C Vagenas ldquoSemiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via self-gravitationrdquoPhysics Letters Section B Nuclear Elementary Particle and High-Energy Physics vol 533 no 3-4 pp 302ndash306 2002

[10] M Arzano A J Medved and E C Vagenas ldquoHawkingradiation as tunneling through the quantum horizonrdquo Journalof High Energy Physics vol 2005 article 037 2005

[11] S-Q Wu and Q-Q Jiang ldquoRemarks on Hawking radiation astunneling from the BTZ black holesrdquo Journal of High EnergyPhysics vol 3 article 79 2006

[12] M Nadalini L Vanzo and S Zerbini ldquoHawking radiation astunnelling the119863-dimensional rotating caserdquo Journal of PhysicsA Mathematical and Theoretical vol 39 no 21 pp 6601ndash66082006

[13] B Chatterjee A Ghosh and P Mitra ldquoTunnelling from blackholes and tunnelling into white holesrdquo Physics Letters B vol 661no 4 pp 307ndash311 2008

[14] V Akhmedova T Pilling A de Gill and D Singleton ldquoTempo-ral contribution to gravitationalWKB-like calculationsrdquoPhysicsLetters B vol 666 no 3 pp 269ndash271 2008

[15] E T Akhmedov T Pilling and D Singleton ldquoSubtleties in thequasi-classical calculation of Hawking radiationrdquo International

Journal of Modern Physics D Gravitation Astrophysics Cosmol-ogy vol 17 no 13-14 pp 2453ndash2458 2008

[16] V Akhmedova T Pilling A de Gill and D SingletonldquoComments on anomaly versus WKBtunneling methods forcalculating Unruh radiationrdquo Physics Letters B vol 673 no 3pp 227ndash231 2009

[17] R Banerjee and B R Majhi ldquoQuantum tunneling and backreactionrdquo Physics Letters B vol 662 no 1 pp 62ndash65 2008

[18] D Singleton E C Vagenas T Zhu and J R Ren ldquoInsightsand possible resolution to the information loss paradox via thetunneling picturerdquo Journal of High Energy Physics vol 2010 no8 article 089 2010 Erratum to Journal of High Energy Physicsvol 2011 no 1 article 021 2011

[19] P K Townsend ldquoSmall-scale structure of spacetime as theorigin of the gravitational constantrdquo Physical Review D vol 15article 2795 1977

[20] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[21] K Konishi G Paffuti and P Provero ldquoMinimum physicallength and the generalized uncertainty principle in stringtheoryrdquo Physics Letters B vol 234 no 3 pp 276ndash284 1990

[22] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 pp 1108ndash1118 1995

[23] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stocker ldquoCollider signatures in the Planck regimerdquoPhysics Letters B vol 575 no 1-2 pp 85ndash99 2003

[24] S F Hassan and M S Sloth ldquoTrans-Planckian effects ininflationary cosmology and themodified uncertainty principlerdquoNuclear Physics B vol 674 no 1-2 pp 434ndash458 2003

[25] A F Ali ldquoNo existence of black holes at LHC due to minimallength in quantum gravityrdquo Journal of High Energy Physics vol2012 article 67 2012

[26] B Majumder ldquoQuantum black hole and the modified uncer-tainty principlerdquo Physics Letters B vol 701 no 4 pp 384ndash3872011

[27] A Bina S Jalalzadeh and A Moslehi ldquoQuantum black holein the generalized uncertainty principle frameworkrdquo PhysicalReview D vol 81 no 2 Article ID 023528 2010

[28] P Chen and R J Adler ldquoBlack hole remnants and dark matterrdquoNuclear Physics BmdashProceedings Supplements vol 124 pp 103ndash106 2003

[29] L Xiang and X QWen ldquoBlack hole thermodynamics with gen-eralized uncertainty principlerdquo Journal of High Energy Physicsvol 2009 no 10 article 46 2009

[30] W Kim E J Son and M Yoon ldquoThermodynamics of a blackhole based on a generalized uncertainty principlerdquo Journal ofHigh Energy Physics vol 2008 no 1 article 35 2008

[31] B R Majhi and E C Vagenas ldquoModified dispersion relationphotonrsquos velocity and unruh effectrdquo Physics Letters B vol 725no 4-5 pp 477ndash480 2013

[32] D ChenHWu andH Yang ldquoFermionrsquos tunnellingwith effectsof quantum gravityrdquo Advances in High Energy Physics vol 2013Article ID 432412 6 pages 2013

[33] D Chen H Wu and H Yang ldquoObserving remnants byfermionsrsquo tunnelingrdquo Journal of Cosmology and AstroparticlePhysics vol 2014 no 3 article 36 2014

[34] D Y Chen Q Q Jiang P Wang and H Yang ldquoRemnantsfermionsrsquo tunnelling and effects of quantum gravityrdquo Journal ofHigh Energy Physics vol 2013 no 11 article 176 2013


[35] D Chen and Z Li ldquoRemarks on remnants by fermionsrsquotunnelling from black stringsrdquoAdvances in High Energy Physicsvol 2014 Article ID 620157 9 pages 2014

[36] Z Y liu and J R Ren ldquoFermions tunnelling with quantumgravity correctionrdquo httparxivorgabs14037307

[37] M Benrong P Wang and H Yang ldquoDeformed Hamilton-Jacobi method in covariant quantum gravity effective modelsrdquohttparxivorgabs14085055

[38] E T Akhmedov V Akhmedova and D Singleton ldquoHawkingtemperature in the tunneling picturerdquo Physics Letters B vol 642no 1-2 pp 124ndash128 2006

[39] B D Chowdhury ldquoProblems with tunneling of thin shells fromblack holesrdquo Pramana vol 70 no 1 pp 3ndash26 2008

[40] E T Akhmedov V Akhmedova D Singleton and T PillingldquoThermal radiation of various gravitational backgroundsrdquo Inter-national Journal of Modern Physics A vol 22 no 8-9 pp 1705ndash1715 2007

[41] W G Unruh ldquoSonic analogue of black holes and the effects ofhigh frequencies on black hole evaporationrdquo Physical Review Dvol 51 article 2827 1995

[42] S Corley and T Jacobson ldquoHawking spectrum and highfrequency dispersionrdquo Physical Review D vol 54 no 2 pp1568ndash1586 1996

[43] G Amelino-Camelia M Arzano Y Ling and G MandanicildquoBlack-hole thermodynamics with modified dispersion rela-tions and generalized uncertainty principlesrdquo Classical andQuantum Gravity vol 23 no 7 pp 2585ndash2606 2006

[44] J D Bekenstein ldquoBlack holes and entropyrdquo Physical Review Dvol 7 pp 2333ndash2346 1973

[45] BMajumder ldquoBlack hole entropy and themodified uncertaintyprinciple a heuristic analysisrdquo Physics Letters B vol 703 no 4pp 402ndash405 2011

[46] K Nozari and S Saghafi ldquoNatural cutoffs and quantum tunnel-ing fromblack hole horizonrdquo Journal ofHigh Energy Physics vol2012 no 11 article 005 2012

[47] R Banerjee and S Ghosh ldquoGeneralised uncertainty principleremnant mass and singularity problem in black hole thermody-namicsrdquo Physics Letters B vol 688 no 2-3 pp 224ndash229 2010

[48] M Cavaglia and S Das ldquoHow classical are TeV-scale blackholesrdquo Classical and Quantum Gravity vol 21 no 19 pp 4511ndash4522 2004

[49] Y S Myung Y-W Kim and Y-J Park ldquoBlack hole thermody-namics with generalized uncertainty principlerdquo Physics LettersB vol 645 no 5-6 pp 393ndash397 2007

[50] K Nouicer ldquoQuantum-corrected black hole thermodynamicsto all orders in the Planck lengthrdquo Physics Letters B vol 646no 2-3 pp 63ndash71 2007

[51] B Majumder ldquoBlack hole entropy with minimal length intunneling formalismrdquo General Relativity and Gravitation vol45 no 11 pp 2403ndash2414 2013

[52] J B Hartle and S W Hawking ldquoPath-integral derivation ofblack-hole radiancerdquo Physical Review D vol 13 no 8 pp 2188ndash2203 1976

[53] D N Page ldquoParticle emission rates from a black hole masslessparticles from an uncharged nonrotating holerdquo Physical ReviewD vol 13 no 2 pp 198ndash206 1976

[54] R M Wald General Relativity University of Chicago ChicagoIll USA 1984

[55] P Nicolini ldquoNoncommutative black holes the final appeal toquantum gravity a reviewrdquo International Journal of ModernPhysics A vol 24 no 7 pp 1229ndash1308 2009

[56] R Banerjee B R Majhi and S Samanta ldquoNoncommutativeblack hole thermodynamicsrdquo Physical Review D vol 77 no 12Article ID 124035 2008

[57] R Banerjee B R Majhi and S K Modak ldquoNoncommutativeSchwarzschild black hole and area lawrdquo Classical and QuantumGravity vol 26 no 8 Article ID 085010 2009

[58] E Spallucci A Smailagic and P Nicolini ldquoNon-commutativegeometry inspired higher-dimensional charged black holesrdquoPhysics Letters B vol 670 no 4-5 pp 449ndash454 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


where we neglect the factor 13 From (1) it is noted thatalthough one can increase 119901 arbitrarily 119896 has an upper boundwhich is 1119871119891 The upper bound on 119896 implies that particlescould not possess arbitrarily small Compton wavelengths120582 = 2120587119896 and that there exists a minimal length sim 119871119891Furthermore the deformed Klein-GordonDirac equationsincorporating (4) and (5) have already been obtained in [23]which will be briefly reviewed in Section 2 So [23] provides away to incorporate the minimal length with special relativitya good starting point for studying Hawking radiation astunnelling effect

The black hole is a suitable venue to discuss the effectsof quantum gravity Incorporating GUP into black holes hasbeen discussed in a lot of papers [25ndash30]The thermodynam-ics of black holes has also been investigated in the frameworkof GUP [29 30] In [31] a new form of GUP is introduced

1199010= 1198960

119901119894= 119896119894(1 minus 120572k + 21205722k2)

(6)

where119901119886 is themodified fourmomentum 119896119886 is the usual fourmomentum and 120572 is a small parameterThemodified velocityof photons two-dimensional Klein-Gordon equation theemission spectrum due to the Unruh effect is obtained thereRecently the GUP deformed Hamilton-Jacobi equation forfermions in curved spacetime have been introduced and thecorrected Hawking temperatures have been derived [32ndash36]The authors consider the GUP of form

119909119894 = 1199090119894

119901119894 = 1199010119894 (1 + 1205731199012)

(7)

where 1199090119894 and 1199010119894 satisfy the canonical commutation rela-tions Fermionsrsquo tunnelling black hole thermodynamics andthe remnants are discussed there

In this paper we investigate scalars and fermions tunnel-ing across the horizons of black holes using the deformedHamilton-Jacobi method which incorporates the minimallength via (4) and (5) Our calculation shows that thequantum gravity correction is related not only to the blackholersquos mass but also to the mass and angular momentum ofemitted particles

The organization of this paper is as follows In Section 2from the modified fundamental commutation relation wegeneralize the Hamilton-Jacobi method in curved spacetimeIn Section 3 incorporatingGUPwe investigate the tunnellingof particles in the black holes In Section 4 we investigatehow a Schwarzschild black hole evaporates in our modelSection 5 is devoted to our conclusion In this paper we takeGeometrized units 119888 = 119866 = 1 where the Planck constantℏ is square of the Planck Mass 119898119901 We also assume that theemitted particles are neutral

2 Deformed Hamilton-Jacobi Equations

To be generic we will consider a spherically symmetricbackground metric of the form

1198891199042= 119891 (119903) 119889119905

2minus1198891199032

119891 (119903)minus 1199032(1198891205792+ sin21205791198891206012) (8)

where 119891(119903) has a simple zero at 119903 = 119903ℎ with 1198911015840(119903ℎ) being finite

and nonzero The vanishing of 119891(119903) at point 119903 = 119903ℎ indicatesthe presence of an event horizon In this section we will firstderive the deformed Klein-GordonDirac equations in flatspacetime and then generalize them to the curved spacetimewith the metric (8)

In the (3 + 1) dimensional flat spacetime the relationsbetween (119901119894 119864) and (119896119894 120596) can simply be generalized to

119871119891119896119894 (119901) = tanh(119901119894

119872119891

)

119871119891120596 (119864) = tanh( 119864

119872119891

)

(9)

where in the spherical coordinates one has for

= minus119894 (119903120597

120597119903+120579

119903

120597

120597120579+

120601

119903 sin 120579120597

120597120601) (10)

Expanding (9) for small arguments to the third order gives theenergy and momentum operator in position representation

119864 = 119894ℏ120597119905(1 minusℏ2

1198722119891

1205972119905)

=ℏ

119894[119903(120597119903 minus

ℏ21205973119903

1198722119891

) + 120579(120597120579

119903minusℏ21205973120579

11990331198722119891

)

+120601(

120597120601

119903 sin 120579minus

ℏ21205973120601

1199033sin31205791198722119891

)]

(11)

where we also omit the factor 13 Substituting the aboveenergy and momentum operators into the energy-momen-tum relation the deformed Klein-Gordon equation satisfiedby the scalar field with the mass119898 is

1198642120601 = 119901

2120601 + 119898

2120601 (12)

where 1199012 = sdot Making the ansatz for 120601

120601 = exp (119894119868ℏ) (13)

and substituting it into (12) one expands (12) in powers of ℏand then finds that the lowest order gives the deformed scalarHamilton-Jacobi equation in the flat spacetime

(120597119905119868)2(1 +

2 (120597119905119868)2

1198722119891

) minus (120597119903119868)2(1 +

2 (120597119903119868)2

1198722119891

)

minus(120597120579119868)2

1199032(1 +

2 (120597120579119868)2

11990321198722119891

)

minus

(120597120601119868)2

1199032sin2120579(1 +

2 (120597120601119868)2

1199032sin21205791198722119891

) = 1198982

(14)

which is truncated at O(11198722119891)



(1205740119864 + 120574 sdot minus 119898)120595 = 0 (15)



1198642120595 = (119901

2+ 1198982) 120595 minus

[120574119886 120574119887]

2119901119886119901119887120595

(16)


120595 = exp(119894119868ℏ) V (17)



120583119889119909

] is

119892120583]120597120583119868120597]119868 minus 119898

2= 0 (18)


(120597119905119868)2

119891 (119903)minus 119891 (119903) (120597119903119868)

2minus(120597120579119868)2

1199032minus

(120597120601119868)2

1199032sin2120579= 1198982 (19)


(120597119905119868)2minus (120597119903119868)

2minus(120597120579119868)2

1199032minus

(120597120601119868)2

1199032sin2120579= 1198982

(20)



(120597119905119868)2

119891 (119903)(1 +

2 (120597119905119868)2

119891 (119903)1198722119891

) minus 119891 (119903) (120597119903119868)2(1 +

2119891 (119903) (120597119903119868)2

1198722119891

)

minus(120597120579119868)2

1199032(1 +

2 (120597120579119868)2

11990321198722119891

)

minus

(120597120601119868)2

1199032sin2120579(1 +

2 (120597120601119868)2

1199032sin21205791198722119891

) = 1198982

(21)



119868 = minus120596119905 +119882 (119903 120579) + 119901120601120601 (22)


1199012119903 (1 +

2119891 (119903) 1199012119903

1198722119891

)

=1

1198912 (119903)[1205962(1 +

21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982+ 120582)]

(23)

where 119901119903 = 120597119903119882 119901120579 = 120597120579119882 and

120582 =1199012120579

1199032(1 +

21199012120579

11990321198722119891

) +

1199012120601

1199032sin2120579(1 +

21199012120601

1199032sin21205791198722119891

) (24)


1198712= 1199012120579 +

1199012120601

sin2120579 (25)


120582 =1198712

1199032+ O(

1

1198722119891

) (26)



120597119903119882∓

= plusmn1

119891 (119903)radic1205962(1 +

21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982 + 120582)

times radic1 minus2

1198722119891119891 (119903)

[1205962(1 +21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982 + 120582)]

(27)


120597119903119882∓

= plusmn1

119891 (119903)52radic1205962(119891 (119903) +

21205962

1198722119891

) minus 1198912 (119903) (1198982 + 120582)


1198722119891

[1205962(119891 (119903) +21205962

1198722119891

) minus 1198912 (119903) (1198982 + 120582)]

(28)


2+O((119903minus119903ℎ)

3)


120597119903119882∓ simplusmn119886minus1

119903 minus 119903ℎ

(29)

where we have

119886minus1 =120596

1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)] + O(1

1198724119891

) (30)


2119891)


1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)] (31)


+119903 119889119903 minus int119901




exp [minus1ℏIm∮119901119903119889119903]


+119903 119889119903 minus int119901

minus119903 119889119903)]

= expminus 2120587120596

1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)]

(32)


Im (120596Δ119905) =2120587120596

1198911015840 (119903ℎ) (33)



= expminus 4120587120596

1198911015840 (119903ℎ)[1 +

1

1198722119891

(1198982+1198712

1199032ℎ

)]

(34)


119879 =1198911015840(119903ℎ)

4120587

ℏ

1 + (11198722119891) (1198982 + (1198712119903

2ℎ))

(35)








119879 sim1198790

1 + 212059621198722119891

(36)



119879 sim1198790

1 + 1198984119901211987221198722119891

(37)



119878 = int119889119872

119879

sim119860

41198982119901

+

41205871198982119901

1198722119891

ln( 119860

16120587)

(38)

where119860 = 41205871199032ℎ = 16120587119872


119878 =119860

4+ 120590 ln( 119860

16120587) + O(

1198722119891

119860) (39)













119899120596119871 = 119899 (120596

119879) (41)

where we define

119899 (119909) =1

exp 119909 minus (minus1)120598 (42)


119889119899120596119871

119889119905= 119899120596119871

119889120596

2120587ℏ (43)


119889119899120596119871

119889119905= 119899120596119871

120597120596

120597119901119903

119889119901119903

2120587ℏ= 119899120596119871

119889120596

2120587ℏ (44)


119871 = sum

119897=0

(2119897 + 1) int120596119899120596119897

119889120596

2120587ℏ (45)





119871 = sum

119897=0

(2119897 + 1) int1003816100381610038161003816119879119894 (120596)

1003816100381610038161003816

2120596119899120596119897

119889120596

2120587ℏ (46)







119897(119897 + 1)ℏ2

119897 (119897 + 1) ℏ2⩽ 27119872

21205962 (47)


119871 = int

infin

0

120596119889120596

2120587ℏ3int

2711987221205962

0119899[

120596

1198790

(1 +1

1198722119891

119897 (119897 + 1) ℏ2

21198722)]

sdot 119889 [119897 (119897 + 1) ℏ2]

=119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 [119906 (1 + 119886

2119906119909)] 119889119909

(48)


1198790radic2119872119891 = 119898

21199018

radic2120587119872119891119872 For 119872 ≫ 11989821199018


119871 asymp27

321205872ℏ311987940119860int

infin

01199063119899 (119906) 119889119906 (49)


21199018



119871 sim119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 (11988621199062119909) 119889119909

=

211987221198724119891

1205871198986119901

int

infin

0V3119889Vint

27

0119899 (V2119909) 119889119909

(50)


119889119872

119889119905= minus119871 sim minus119860

1198722

1198722119891

(51)





11989821199018


119889119872

119889119905sim minus119860

1198722

1198722119891

+ 1198611198723

1198723119891

(52)


119879 sim1198790

1198984119901211987221198722119891

sim 119872 (53)



5 Conclusion





Acknowledgments


References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





(1205740119864 + 120574 sdot minus 119898)120595 = 0 (15)



1198642120595 = (119901

2+ 1198982) 120595 minus

[120574119886 120574119887]

2119901119886119901119887120595

(16)


120595 = exp(119894119868ℏ) V (17)



120583119889119909

] is

119892120583]120597120583119868120597]119868 minus 119898

2= 0 (18)


(120597119905119868)2

119891 (119903)minus 119891 (119903) (120597119903119868)

2minus(120597120579119868)2

1199032minus

(120597120601119868)2

1199032sin2120579= 1198982 (19)


(120597119905119868)2minus (120597119903119868)

2minus(120597120579119868)2

1199032minus

(120597120601119868)2

1199032sin2120579= 1198982

(20)



(120597119905119868)2

119891 (119903)(1 +

2 (120597119905119868)2

119891 (119903)1198722119891

) minus 119891 (119903) (120597119903119868)2(1 +

2119891 (119903) (120597119903119868)2

1198722119891

)

minus(120597120579119868)2

1199032(1 +

2 (120597120579119868)2

11990321198722119891

)

minus

(120597120601119868)2

1199032sin2120579(1 +

2 (120597120601119868)2

1199032sin21205791198722119891

) = 1198982

(21)



119868 = minus120596119905 +119882 (119903 120579) + 119901120601120601 (22)


1199012119903 (1 +

2119891 (119903) 1199012119903

1198722119891

)

=1

1198912 (119903)[1205962(1 +

21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982+ 120582)]

(23)

where 119901119903 = 120597119903119882 119901120579 = 120597120579119882 and

120582 =1199012120579

1199032(1 +

21199012120579

11990321198722119891

) +

1199012120601

1199032sin2120579(1 +

21199012120601

1199032sin21205791198722119891

) (24)


1198712= 1199012120579 +

1199012120601

sin2120579 (25)


120582 =1198712

1199032+ O(

1

1198722119891

) (26)



120597119903119882∓

= plusmn1

119891 (119903)radic1205962(1 +

21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982 + 120582)

times radic1 minus2

1198722119891119891 (119903)

[1205962(1 +21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982 + 120582)]

(27)


120597119903119882∓

= plusmn1

119891 (119903)52radic1205962(119891 (119903) +

21205962

1198722119891

) minus 1198912 (119903) (1198982 + 120582)


1198722119891

[1205962(119891 (119903) +21205962

1198722119891

) minus 1198912 (119903) (1198982 + 120582)]

(28)


2+O((119903minus119903ℎ)

3)


120597119903119882∓ simplusmn119886minus1

119903 minus 119903ℎ

(29)

where we have

119886minus1 =120596

1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)] + O(1

1198724119891

) (30)


2119891)


1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)] (31)


+119903 119889119903 minus int119901




exp [minus1ℏIm∮119901119903119889119903]


+119903 119889119903 minus int119901

minus119903 119889119903)]

= expminus 2120587120596

1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)]

(32)


Im (120596Δ119905) =2120587120596

1198911015840 (119903ℎ) (33)



= expminus 4120587120596

1198911015840 (119903ℎ)[1 +

1

1198722119891

(1198982+1198712

1199032ℎ

)]

(34)


119879 =1198911015840(119903ℎ)

4120587

ℏ

1 + (11198722119891) (1198982 + (1198712119903

2ℎ))

(35)








119879 sim1198790

1 + 212059621198722119891

(36)



119879 sim1198790

1 + 1198984119901211987221198722119891

(37)



119878 = int119889119872

119879

sim119860

41198982119901

+

41205871198982119901

1198722119891

ln( 119860

16120587)

(38)

where119860 = 41205871199032ℎ = 16120587119872


119878 =119860

4+ 120590 ln( 119860

16120587) + O(

1198722119891

119860) (39)













119899120596119871 = 119899 (120596

119879) (41)

where we define

119899 (119909) =1

exp 119909 minus (minus1)120598 (42)


119889119899120596119871

119889119905= 119899120596119871

119889120596

2120587ℏ (43)


119889119899120596119871

119889119905= 119899120596119871

120597120596

120597119901119903

119889119901119903

2120587ℏ= 119899120596119871

119889120596

2120587ℏ (44)


119871 = sum

119897=0

(2119897 + 1) int120596119899120596119897

119889120596

2120587ℏ (45)





119871 = sum

119897=0

(2119897 + 1) int1003816100381610038161003816119879119894 (120596)

1003816100381610038161003816

2120596119899120596119897

119889120596

2120587ℏ (46)







119897(119897 + 1)ℏ2

119897 (119897 + 1) ℏ2⩽ 27119872

21205962 (47)


119871 = int

infin

0

120596119889120596

2120587ℏ3int

2711987221205962

0119899[

120596

1198790

(1 +1

1198722119891

119897 (119897 + 1) ℏ2

21198722)]

sdot 119889 [119897 (119897 + 1) ℏ2]

=119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 [119906 (1 + 119886

2119906119909)] 119889119909

(48)


1198790radic2119872119891 = 119898

21199018

radic2120587119872119891119872 For 119872 ≫ 11989821199018


119871 asymp27

321205872ℏ311987940119860int

infin

01199063119899 (119906) 119889119906 (49)


21199018



119871 sim119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 (11988621199062119909) 119889119909

=

211987221198724119891

1205871198986119901

int

infin

0V3119889Vint

27

0119899 (V2119909) 119889119909

(50)


119889119872

119889119905= minus119871 sim minus119860

1198722

1198722119891

(51)





11989821199018


119889119872

119889119905sim minus119860

1198722

1198722119891

+ 1198611198723

1198723119891

(52)


119879 sim1198790

1198984119901211987221198722119891

sim 119872 (53)



5 Conclusion





Acknowledgments


References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120597119903119882∓

= plusmn1

119891 (119903)radic1205962(1 +

21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982 + 120582)

times radic1 minus2

1198722119891119891 (119903)

[1205962(1 +21205962

119891 (119903)1198722119891

) minus 119891 (119903) (1198982 + 120582)]

(27)


120597119903119882∓

= plusmn1

119891 (119903)52radic1205962(119891 (119903) +

21205962

1198722119891

) minus 1198912 (119903) (1198982 + 120582)


1198722119891

[1205962(119891 (119903) +21205962

1198722119891

) minus 1198912 (119903) (1198982 + 120582)]

(28)


2+O((119903minus119903ℎ)

3)


120597119903119882∓ simplusmn119886minus1

119903 minus 119903ℎ

(29)

where we have

119886minus1 =120596

1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)] + O(1

1198724119891

) (30)


2119891)


1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)] (31)


+119903 119889119903 minus int119901




exp [minus1ℏIm∮119901119903119889119903]


+119903 119889119903 minus int119901

minus119903 119889119903)]

= expminus 2120587120596

1198911015840 (119903ℎ)[1 +

2

1198722119891

(1198982+1198712

1199032ℎ

)]

(32)


Im (120596Δ119905) =2120587120596

1198911015840 (119903ℎ) (33)



= expminus 4120587120596

1198911015840 (119903ℎ)[1 +

1

1198722119891

(1198982+1198712

1199032ℎ

)]

(34)


119879 =1198911015840(119903ℎ)

4120587

ℏ

1 + (11198722119891) (1198982 + (1198712119903

2ℎ))

(35)








119879 sim1198790

1 + 212059621198722119891

(36)



119879 sim1198790

1 + 1198984119901211987221198722119891

(37)



119878 = int119889119872

119879

sim119860

41198982119901

+

41205871198982119901

1198722119891

ln( 119860

16120587)

(38)

where119860 = 41205871199032ℎ = 16120587119872


119878 =119860

4+ 120590 ln( 119860

16120587) + O(

1198722119891

119860) (39)













119899120596119871 = 119899 (120596

119879) (41)

where we define

119899 (119909) =1

exp 119909 minus (minus1)120598 (42)


119889119899120596119871

119889119905= 119899120596119871

119889120596

2120587ℏ (43)


119889119899120596119871

119889119905= 119899120596119871

120597120596

120597119901119903

119889119901119903

2120587ℏ= 119899120596119871

119889120596

2120587ℏ (44)


119871 = sum

119897=0

(2119897 + 1) int120596119899120596119897

119889120596

2120587ℏ (45)





119871 = sum

119897=0

(2119897 + 1) int1003816100381610038161003816119879119894 (120596)

1003816100381610038161003816

2120596119899120596119897

119889120596

2120587ℏ (46)







119897(119897 + 1)ℏ2

119897 (119897 + 1) ℏ2⩽ 27119872

21205962 (47)


119871 = int

infin

0

120596119889120596

2120587ℏ3int

2711987221205962

0119899[

120596

1198790

(1 +1

1198722119891

119897 (119897 + 1) ℏ2

21198722)]

sdot 119889 [119897 (119897 + 1) ℏ2]

=119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 [119906 (1 + 119886

2119906119909)] 119889119909

(48)


1198790radic2119872119891 = 119898

21199018

radic2120587119872119891119872 For 119872 ≫ 11989821199018


119871 asymp27

321205872ℏ311987940119860int

infin

01199063119899 (119906) 119889119906 (49)


21199018



119871 sim119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 (11988621199062119909) 119889119909

=

211987221198724119891

1205871198986119901

int

infin

0V3119889Vint

27

0119899 (V2119909) 119889119909

(50)


119889119872

119889119905= minus119871 sim minus119860

1198722

1198722119891

(51)





11989821199018


119889119872

119889119905sim minus119860

1198722

1198722119891

+ 1198611198723

1198723119891

(52)


119879 sim1198790

1198984119901211987221198722119891

sim 119872 (53)



5 Conclusion





Acknowledgments


References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








119879 sim1198790

1 + 212059621198722119891

(36)



119879 sim1198790

1 + 1198984119901211987221198722119891

(37)



119878 = int119889119872

119879

sim119860

41198982119901

+

41205871198982119901

1198722119891

ln( 119860

16120587)

(38)

where119860 = 41205871199032ℎ = 16120587119872


119878 =119860

4+ 120590 ln( 119860

16120587) + O(

1198722119891

119860) (39)













119899120596119871 = 119899 (120596

119879) (41)

where we define

119899 (119909) =1

exp 119909 minus (minus1)120598 (42)


119889119899120596119871

119889119905= 119899120596119871

119889120596

2120587ℏ (43)


119889119899120596119871

119889119905= 119899120596119871

120597120596

120597119901119903

119889119901119903

2120587ℏ= 119899120596119871

119889120596

2120587ℏ (44)


119871 = sum

119897=0

(2119897 + 1) int120596119899120596119897

119889120596

2120587ℏ (45)





119871 = sum

119897=0

(2119897 + 1) int1003816100381610038161003816119879119894 (120596)

1003816100381610038161003816

2120596119899120596119897

119889120596

2120587ℏ (46)







119897(119897 + 1)ℏ2

119897 (119897 + 1) ℏ2⩽ 27119872

21205962 (47)


119871 = int

infin

0

120596119889120596

2120587ℏ3int

2711987221205962

0119899[

120596

1198790

(1 +1

1198722119891

119897 (119897 + 1) ℏ2

21198722)]

sdot 119889 [119897 (119897 + 1) ℏ2]

=119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 [119906 (1 + 119886

2119906119909)] 119889119909

(48)


1198790radic2119872119891 = 119898

21199018

radic2120587119872119891119872 For 119872 ≫ 11989821199018


119871 asymp27

321205872ℏ311987940119860int

infin

01199063119899 (119906) 119889119906 (49)


21199018



119871 sim119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 (11988621199062119909) 119889119909

=

211987221198724119891

1205871198986119901

int

infin

0V3119889Vint

27

0119899 (V2119909) 119889119909

(50)


119889119872

119889119905= minus119871 sim minus119860

1198722

1198722119891

(51)





11989821199018


119889119872

119889119905sim minus119860

1198722

1198722119891

+ 1198611198723

1198723119891

(52)


119879 sim1198790

1198984119901211987221198722119891

sim 119872 (53)



5 Conclusion





Acknowledgments


References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119871 = sum

119897=0

(2119897 + 1) int1003816100381610038161003816119879119894 (120596)

1003816100381610038161003816

2120596119899120596119897

119889120596

2120587ℏ (46)







119897(119897 + 1)ℏ2

119897 (119897 + 1) ℏ2⩽ 27119872

21205962 (47)


119871 = int

infin

0

120596119889120596

2120587ℏ3int

2711987221205962

0119899[

120596

1198790

(1 +1

1198722119891

119897 (119897 + 1) ℏ2

21198722)]

sdot 119889 [119897 (119897 + 1) ℏ2]

=119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 [119906 (1 + 119886

2119906119909)] 119889119909

(48)


1198790radic2119872119891 = 119898

21199018

radic2120587119872119891119872 For 119872 ≫ 11989821199018


119871 asymp27

321205872ℏ311987940119860int

infin

01199063119899 (119906) 119889119906 (49)


21199018



119871 sim119879401198722

2120587ℏ3int

infin

01199063119889119906int

27

0119899 (11988621199062119909) 119889119909

=

211987221198724119891

1205871198986119901

int

infin

0V3119889Vint

27

0119899 (V2119909) 119889119909

(50)


119889119872

119889119905= minus119871 sim minus119860

1198722

1198722119891

(51)





11989821199018


119889119872

119889119905sim minus119860

1198722

1198722119891

+ 1198611198723

1198723119891

(52)


119879 sim1198790

1198984119901211987221198722119891

sim 119872 (53)



5 Conclusion





Acknowledgments


References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Acknowledgments


References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics

































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article minimal length effects on tunnelling from

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