Content uploaded by Homa Shababi
Author content
All content in this area was uploaded by Homa Shababi on May 08, 2018
Content may be subject to copyright.
April 19, 2018 15:5 MPLA S0217732318500682 page 1
Modern Physics Letters A
Vol. 33, No. 12 (2018) 1850068 (14 pages)
c
World Scientific Publishing Company
DOI: 10.1142/S0217732318500682
Some applications of the most general form of the higher-order GUP
with minimal length uncertainty and maximal momentum
Homa Shababi∗
Young Researchers and Elites Club, Science and Research Branch,
Islamic Azad University, Tehran, Iran
h.shababi@srbiau.ac.ir
Won Sang Chung
Department of Physics and Research Institute of Natural Science,
College of Natural Science, Gyeongsang National University,
Jinju 660-701, Republic of Korea
mimip4444@hanmail.net
Received 6 December 2017
Revised 28 March 2018
Accepted 29 March 2018
Published 20 April 2018
In this paper, using the new type of D-dimensional nonperturbative Generalized Un-
certainty Principle (GUP) which has predicted both a minimal length uncertainty and
a maximal observable momentum,1first, we obtain the maximally localized states and
express their connections to [P. Pedram, Phys. Lett. B 714, 317 (2012)]. Then, in the
context of our proposed GUP and using the generalized Schr¨odinger equation, we solve
some important problems including particle in a box and one-dimensional hydrogen
atom. Next, implying modified Bohr–Sommerfeld quantization, we obtain energy spec-
tra of quantum harmonic oscillator and quantum bouncer. Finally, as an example, we
investigate some statistical properties of a free particle, including partition function and
internal energy, in the presence of the mentioned GUP.
Keywords: Quantum gravity; generalized uncertainty principle; minimal length un-
certainty; maximal momentum; particle in a box; hydrogen atom; Bohr–Sommerfeld
quantization; quantum harmonic oscillator; quantum bouncer.
PACS No.: 04.60.-m
1. Introduction
Various approaches to quantum gravity such as string theory, loop quantum gravity
and noncommutative geometry predict the existence of a minimal measurable length
of the order of the Planck length.3–12 According to these theories near the Planck
∗Corresponding author
1850068-1
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 2
H. Shababi & W. S. Chung
scale, the Heisenberg Uncertainty Principle should be replaced by the so-called
Generalized Uncertainty Principle (GUP). Let us speak more precisely. The effects
of gravity play an important role in the limit of small distances around the Planck
length, lp=G/c3≈10−35 m, where Gis Newton’s constant, is the Planck
constant, and cis the speed of light. Equivalently, these effects are dominant in the
limit of high energies near the Planck energy Ep=c5/G ≈1.956 ×109J. As it
is known, in nonrelativistic quantum mechanics, a particle with an arbitrary energy
can be accurately localized and its exact location can be obtained. According to the
quantum field theory and based on Heisenberg Uncertainty Principle, only particles
with infinite energy can be localized. However, in quantum gravity theories that
try to combine the effects of gravity with quantum mechanics, the localization of
particles completely disappear. In general relativity, it is possible to access the path
of particles that is impossible to obtain in quantum mechanics. In fact, in quantum
mechanics, determination of exact position of particles requires infinite momentum
uncertainty. Now, in order to achieve the ultimate theory of quantum gravity, one
of the results of combination of gravity and quantum mechanics is introducing the
minimal measurable length of the order of the Planck length. In this case, particles
lose their point-like description. Thus, to find the position of particles, a minimal
length uncertainty proportional to the Planck length should be taken into account
and the sharp localization condition for particles is relaxed.11 Thus, the ordinary
Heisenberg uncertainty principle should be replaced. This modification is called
the GUP.
In other words, the ordinary Heisenberg uncertainty principle ΔXΔP≥/2is
incompatible with the quantization of gravity because ΔXgoes to zero in the high
momentum limit, which demands infinite energy for measurement of position to
high accuracy. So, the ordinary Heisenberg uncertainty principle should be replaced
by GUP. According to the notation of Kempf, Mangano and Mann (KMM),12 the
GUP relation can be written as
ΔX≥
2ΔP+const×ΔP, (1)
which the first term in the RHS, comes from the ordinary uncertainty principle,
while the second term appears when we consider energies near the Planck energy.
The second term is shown to be related to the existence of a minimal observable
length of the order of the Planck length. Equation (1) was first proposed in the
context of string theory.5,8,18,19 Thus, to incorporate the concept of minimal meas-
urable length into quantum mechanics, one should deform the ordinary Heisenberg
algebra, [X, P ]=i, as follows:
[X, P ]=i(1 + βP2),(2)
which suggests the existence of the fundamental minimal length as
(ΔX)0=β=√α0lp,(3)
where β=α0lp
2,andα0is of the order of unity.
1850068-2
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 3
GUP with minimal length uncertainty and maximal momentum
Also, some generalizations of Eq. (2) were studied by some authors in the fol-
lowing form:
[X, P ]=iF(P),(4)
where F(P) is the deforming map giving the minimal length. Also, some dif-
ferent choices of F(P) lead to the following modified communication relations
as1,2,13,38,40,42 follows:
Ref. 38 ⇒[X, P ]=ieaP 2,
Ref. 40 and 42 ⇒[X, P ]=i(1 + βP2j),j=1,2,3,... ,
Ref. 40 ⇒[X, P ]=i(1 + βP 2+γP 4),
Ref. 13 ⇒[X, P ]=i1−√β+2βP 2,
Ref. 2 ⇒[X, P ]= i
1−βP2,
Ref. 1 ⇒[X, P ]= i
(1 −βP2)2,
where some of them also imply the existence of the maximal momentum in addition
to the minimal measurable length.1,2,13
Up to now, much progresses have been made in the study of GUP formalism
and its applications (see Refs. 11, 12, 14–37 and references therein).
In this paper, using the most general form of F(P), namely
F(P)= 1
(1 −βP2)N+1 ,N=0,1,2,... , (5)
which leads to the higher-order GUP with minimal length uncertainty and maximal
momentum1
[X, P ]= i
(1 −βP2)N+1 ,(6)
first, we investigate maximally localized states. Then, by this GUP and in the pres-
ence of the modified Schr¨odinger equation, we find eigenfunctions and eigenvalues
for a particle in a box and one-dimensional hydrogen atom. Next, using modified
Bohr–Sommerfeld quantization, the energy spectrum of quantum harmonic oscilla-
tor and quantum bouncer is obtained. Moreover, as an example, in this framework,
some statistical properties of a free particle are obtained. We show that our results
are a general form of their counterparts in Refs. 2 and 43 which were obtained by
the higher-order GUP with minimal length uncertainty and maximal momentum.
2. Coordinate Realization
In this section, we define the coordinate representation for algebra (6) as
X=x, P =h(p),(7)
1850068-3
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 4
H. Shababi & W. S. Chung
where p=−id
dx . Now, the communication relation is obtained as
[x, h(p)] = ihF (h(p)) ,(8)
which gives the following p:
p=
N+1
l=0 N+1
l(−1)lβlh2l+1
2l+1,(9)
and up to the second-order of β, it is concluded that
h≈p+N+1
3βp3+(N+ 1)(7N+ 10)
30 β2p5.(10)
It should be noted that if we set N= 0, it leads to Pedram’s model.2
3. Maximally Localized States
As it is known, in the seminal paper of KMM, the solutions of the following equation
indicate the maximally localized states11:
X−X+[X, P ]
2(ΔX)2(P−P)|ψ=0,(11)
where [X, P ]=iF (P)andF(P)∼(1+βP2). Also, according to another framework
which was proposed by Detournay and collaborators, the solutions of the follow-
ing Euler–Lagrange equation in momentum space express the maximally localized
states39
[−(F(p)∂p)2−ξ2+2a(iF (p)∂p−ξ)+2b(ν(p)−γ)−μ2]ψ(p)=0,(12)
in which ν(p) is an arbitrary function with finite expectation value, aand b
are Lagrange multipliers, γ=ψ|ν(p)|ψ
ψ|ψ,ξ=ψ|X|ψ
ψ|ψand μ2≡(ΔX)2
min =
min ψ|X2−ξ2|ψ
ψ|ψ.39 Then, a function, namely, z(p) was defined as
z(p)=p
0
F−1(q)dq , (13)
in which z(+Pmax)=α+>0andz(−Pmax)=α−<0.
It is worth mentioning that the issue of maximally localized states in the pres-
ence of minimal length and maximal momentum was first studied in Ref. 30. Next,
according to Ref. 2, this issue was investigated with F(P)=/(1 −βP2) obtaining
z(p)=−1p−β
3p3,(14)
which then concluded that α±=±2
3√β.
Now, we investigate maximally localized states in the presence of our proposed
GUP (6) which is expressed as
z(p)=−1
N+1
l=0 N+1
l(−1)lβlp2l+1
2l+1,(15)
1850068-4
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 5
GUP with minimal length uncertainty and maximal momentum
and (α±)=±1
√β√π(N+1)!
2(3/2+N)! .Note that if we set N= 0 in (15), we obtain its
counterpart in Ref. 2.
4. Some Applications of the Schr¨odinger Equation in the Presence
of the Proposed GUP
In the recent paper,1as one of the applications of our proposed GUP (6), we investi-
gated the cosmological constant and compared its massless type with the previous
models.2,11 In the following subsections, to find more applications of this GUP,
we study some more problems, including particle in a box and one-dimensional
hydrogen atom.
4.1. Particle in a box
Consider a particle with mass mwhich is confined in an infinite one-dimensional
box with length Land the following potential:
V(x)=0,0<x<L,
∞,elsewhere .(16)
As we know, the problem of one-dimensional particle in a box, in the presence of
a minimal measurable length, was first studied in Ref. 31. Also, so far this issue has
been examined by various authors in the presence of different GUPs (see Refs. 40,
42, 43 and references therein).
Now, in the presence of our general proposed GUP (6), for a particle in a box,
the generalized Schr¨odinger equation is obtained as
−2
2m
∂2ψn
∂x2+4β
3m(N+1)∂4ψn
∂x4−6β2
90m(N+ 1)(26N+ 35) ∂6ψn
∂x6+O(β3)=Enψn,
(17)
where for satisfying Eq. (17), the eigenfunction ψnis obtained as
ψn=2
Lsin nπx
L.(18)
So, the energy spectrum reads
En=2
2mnπ
L2
+4β
3m(N+1)
nπ
L4
+6β2
90m(N+ 1)(26N+ 35)nπ
L6
.(19)
These results which are fully compatible with their counterparts in Refs. 2
and 40 indicate that in the framework of our mentioned GUP, there is no change
in the eigenfunctions but there is a positive shift in the energy spectrum which
depends on the values of GUP parameter β. Also, if we set N= 0, the energy
spectrum (19) is obtained equivalent to Ref. 2.
1850068-5
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 6
H. Shababi & W. S. Chung
4.2. One-dimensional hydrogen atom
As another application, in this section we investigate one-dimensional hydrogen
atom eigenvalue problem which is given by
P2φ−α
Xφ=Eφ. (20)
In momentum space, the action of the inverse operator 1/X is defined as
1
Xφ(p)=−i
p
−p0
(1 −βq2)N+1φ(q)dq +c, (21)
where p0=1/√βand cis a constant. So, the Schr¨odinger equation in momentum
space is expressed as
p2φ(p)+ i
αp
−p0
(1 −βq2)N+1φ(q)dq −αc =−φ(p),(22)
where we set =−E. Now, differentiating Eq. (22) with respect to p,weobtain
φ(p)+ 1
p2+2p+i
α(1 −βp2)N+1φ(p)=0,(23)
and the solution of (23) reads
φ(p)= A
p2+e−i
αΘN(p),(24)
where
ΘN(p)=β+1
pF11
2,−N,1,3
2;βp2,−p2
−βp2F11
2,−N;3
2;βp2,(25)
F1(a, b, b,c;x, y ) are Appell’s hypergeometric functions of two variables
F1(a, b, b,c;x, y )= ∞
m=0
∞
n=0
(a)m+n(b)m(b)n
(c)m+n
xmyn,(26)
and
2F1(a, b;c;x)= ∞
n=0
(a)n(b)n
n!(c)n
xn,(27)
is the generalized hypergeometric function in which (a)nis the Pochhammer symbol
in the following form:
(a)n=a(a+1)(a+2)···(a+n−1) for n≥1,(a)0=1.(28)
Thus, the probability density in momentum space is obtained as
|φ(p)|2=A2
p2+2,(29)
1850068-6
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 7
GUP with minimal length uncertainty and maximal momentum
where Ais the normalization constant in the form of
A−2=√πΓ(2 + N)
2√β2(1 + β)2β
Γ(N+3/2) +1−(2N+1)β
Γ(N+5/2)
×2F11
2,1; 5
2+N;−1/β.(30)
Now, following the procedure of Hermitian condition in Ref. 41, the quantization
condition for energy reads
√πΓ(N+1)
2√βnΓ(3/2+N)−βn+(1+βn)2F11
2,1; 3
2+N;−1/βn=nπ
α,
n=1,2,3,... . (31)
5. Some Applications of the Bohr Sommerfeld Quantization in the
Presence of the Proposed GUP
In this section, we study the effects of GUP (6) on Bohr–Sommerfeld quantization.
As it is known, Bohr–Sommerfeld quantization rule reads46
−X
F(P)dP =2π(n+δ),n=0,1,2,... , (32)
where [X, P ]=iF(P)andδ=1/2 when the boundaries have no vertical lines,
while δ=3/4 denotes the boundaries have one vertical line.
Now, we want to study the behavior of Bohr–Sommerfeld quantization in the
presence of minimal length uncertainty and maximal momentum, i.e. with F(P)=
1
(1−βP2)N+1 .
So, this obtains
−(1 −βP2)N+1XdP =2π(n+δ),n=0,1,2,... . (33)
In the following subsections, using this modified quantization (33), we investi-
gate two important problems, namely quantum harmonic oscillator and quantum
bouncer.
5.1. Harmonic oscillator
Consider a particle with mass mwhich is confined in a harmonic potential V(X)=
1
2mω2. The Hamiltonian of this system is expressed as
H=P2+X2.(34)
Henceforth, for the sake of simplicity, we put m=1/2andω=2.SinceH(P, X )=
E, it obtains
X=E−P2.(35)
1850068-7
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 8
H. Shababi & W. S. Chung
So, using Bohr–Sommerfeld quantization condition (33), we find
2√E
−√E
(1 −βP2)N+1E−P2dP =2πn+1
2,n=0,1,2,... . (36)
Thus, we have
En2F11
2;−1−N;2;βEn=−2n+1
2,(37)
in which for small values of β, it is concluded that
En≈(2n+1)+ N+1
42β(2n+1)
2.(38)
It should be noted that for N= 0 and large amounts of n, our result in Eq. (38)
is asymptotically compatible with its counterpart in Ref. 11. Also, in Eq. (38), if we
set N= 0, it leads to the energy spectrum of the quantum one-dimension harmonic
oscillator which was obtained by Fityo.46
5.2. Quantum bouncer
Let us consider a bouncing particle on an ideal reflecting floor in the Earth’s grav-
itational field, which its potential is
V(X)=λX , X > 0,
∞,X<0,(39)
λ=mg =g/2, mis the mass of the particle and gis the acceleration caused by
the gravitational attraction of the Earth. From H(P, X)=E,weobtain
X=1
λ(E−P2).(40)
Now, according to Bohr–Sommerfeld quantization condition (33), we have
2√E
0
(1 −βP2)N+1 1
λ(E−P2)dP =2πn+3
4,n=0,1,2,... , (41)
which leads to the following solution:
2
λ
N+1
k=0 N+1
k(−β)kEn2k+1En
2k+1−1
2k+3En2=2πn+3
4,
(42)
and for small values of β, we obtained
En≈3λπ
2n+3
42/3
+2β(N+1)
15 3λπ
2n+3
44/3
.(43)
1850068-8
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 9
GUP with minimal length uncertainty and maximal momentum
As it is obvious in Eq. (43), the effects of minimal length uncertainty and max-
imal momentum lead to the positive shift in the energy spectrum of a quantum
bouncer. Also, in comparison to the case in which the assumption of a maximum
momentum is ignored, this positive shift is smaller and this is due to the fact that
when an upper bound on the momentum is imposed, we actually eliminate the con-
tribution of highly excited states. It should be noted that, the perturbative effects of
the minimal length and maximal momentum on the quantum bouncer spectrum are
studied in Refs. 25, 26 and 49. Moreover, this problem in the context of the KMM
GUP is exactly solved in Ref. 51. Now, it is worth mentioning that if we set N=0
in (43), the energy spectrum of quantum bouncer for the case50 F(p)= 1
1−βP2is
obtained.
6. Some Thermodynamical Properties
In this section, as an example, we study some thermodynamical properties of a free
particle in the framework of GUP (6). In this regard, we need the invariant phase
space volume which is expressed as1
dDxdDp
[F(p)]D−1[F(p)+p2g(p)] .(44)
Let us explain why Eq. (44) gives an invariant volume element. As it is known,
the deformed phase space volume should be invariant under the time evolution
of the system.1,44,45,47,48 To this end, consider F(p)andg(p) with the following
definitions:
F(p)=1+βF1(p),g(p)=βg1(p),(45)
where βis a small amount. So, up to the first-order of β, the terms including g2,
gF,(F)2,..., can be ignored. In this step, we consider evolution of the coordinates
and momenta during the small time δt, which are obtained as
x
i=x1+δxi,p
i=pi+δpi,(46)
where
δxi={xi,H}F,g δt ={xi,p
j}F,g
∂H
∂pj
+{xi,x
j}F,g
∂H
∂xjδt, (47)
δpi=−{xj,p
i}F,g
∂H
∂xjδt. (48)
Now, after this infinitesimal evolution, the infinitesimal phase space volume reads
dDxdDp=JdDxdDp,(49)
1850068-9
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 10
H. Shababi & W. S. Chung
where the Jacobian Jto the first-order of δt is expressed as
J=1+∂δxi
∂xi
+∂δpi
∂pi,
=1+∂
∂xiδxi
δt +∂δpi
∂piδpi
δt δt ,
=1+(1 −D)h−F
p−(1 + D)g−pgpj
∂H
∂xj
δt . (50)
Thus, using Eq. (49), we obtain
dDxdDp=1−DF
p+2g+pgpj
∂H
∂xj
δtdDxdDp,(51)
where we ignore the terms proportional to Fg. According to Eq. (51), dDxdDp
is not invariant under the time evolution. So, we want the invariant phase space
volume to take the following form:
W(p)dDxdDp,(52)
which should obey
W(p)dDxdDp=W(p)dDxdDp.(53)
Then, let us assume that W(p)isgivenby
W(p)=[F(p)]μ[σ(p)]ν.(54)
Since, p=(pi+δpi)2≈p+1
ppiδpi,weobtain
σ(p)=σp+1
ppiδpi,
=σ(p)+1
ppiδpiσ(p),
=σ(p)1−σ
pσ (F+p2g)pj
∂H
∂xj
δt,(55)
and
F(p)=F(p)1−F
pF (F+p2g)pj
∂H
∂xj
δt.(56)
Thus, we have
W(p)dDxdDp=1−1
pνσ
σ+μF
F(F+p2g)+DF+(p2g)pj
∂H
∂xj
δt
×W(p)dDxdDp.(57)
1850068-10
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 11
GUP with minimal length uncertainty and maximal momentum
Now, if we set ν=−1andμ=1−D, (57) will obey Eq. (53). So, it is concluded
that
σ=F+p2g. (58)
Therefore, the invariant phase space volume (44) is obtained. Moreover, since the
deformed measure determines the number of quantum states in the phase space, so,
in order to satisfy the Liouville theorem, the deformed volume should be invariant
under the time evolution of the system and consequently the number of microstates
remains unchanged. In the semiclassical regime, the volume of the phase space
determines the number of microstates which according to the Liouville theorem,
should be invariant under the time evolution.47,48
Now, after these explanations, we continue our problem. In D-dimensional
spherical coordinate systems, the state density in momentum space reads
D(p)dp =VA
D−1pD−1dp
hD[F(p)]D−1[F(p)+p2g(p)] ,(59)
where
AD−1=2πD/2
Γ(D/2) .(60)
As it is known, for a system with N0non-interacting particles, the partition function
is given by
ZN0=ZN0
1,(61)
where
Z1=d3xd3p
[F(p)]2[F(p)+p2g(p)]e−bH(x,p).(62)
H(x,p) is the Hamiltonian of the system which is defined for a free particle as
H=1
2mp2=1
2m(p2
x+p2
y+p2
z),(63)
where b=1/kBT,kBis Boltzmann constant and Tis temperature.
Now, for the free particle, we investigate some thermodynamical quantities in-
cluding partition function and internal energy in the presence of Eq. (6). To this
end, we put Eq. (63) into Eq. (62), which leads to the following partition function
(2m=1):
Z1=4πV 1/√β
0
dp p2(1 −βp2)N+3e−bp2.(64)
Here, using the new variable √βp =u, the above equation can be written as
Z1=4πV β−3/2IN+3 ,(65)
where
IM=1
0
du u2(1 −u2)Me−bu2/β .(66)
1850068-11
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 12
H. Shababi & W. S. Chung
Now, let us define
IM=CMe−b/β +DMerf b
β,(67)
and differentiating from both sides of the above equation, the following recurrence
relations are obtained
CM+1 =−β2
bC
M+1
√πb−1/2β1/2DM,(68)
DM+1 =DM−β2
bD
M,(69)
where “prime” denotes the derivation with respect to β.
Thus, the modified internal energy is calculated as
U=−∂
∂b ln ZN0=N0⎡
⎢
⎣
A3e−b
β+A4erf b
β
A1e−b
β+A2erf b
β⎤
⎥
⎦,(70)
where
A1=CN+3 ,(71)
A2=DN+3 ,(72)
A3=∂bCN+3 +b
β2CN+3 −1
√πb1/2β−3/2DN+3 ,(73)
A4=∂bDN+3 .(74)
7. Conclusion
In this paper, we investigated some problems in the presence of the most general
type of D-dimensional nonperturbative Generalized Uncertainty Principle, which
was admitted both a minimal length uncertainty and a maximal observable momen-
tum. In this regard, first we obtained maximally localized states which concluded
that if we set N= 0, our result would lead to its counterpart which was obtained in
Pedram’s paper.2Next, in the presence of this GUP, we calculated eigenfunctions
and eigenvalues of the particle in a box which concluded that there is no change
in the eigenfunctions but there is a positive shift in the energy spectrum with the
dependence on β. Then we obtained energy eigenvalues and eigenfunctions for one-
dimensional hydrogen atom in terms of Appell’s and generalized hypergeometric
functions. Moreover, we investigated the behavior of Bohr–Sommerfeld quantiza-
tion in the presence of minimal length uncertainty and maximal momentum. Using
this modified quantization, we studied two important problems, including quantum
harmonic oscillator and quantum bouncer. Our results concluded that the energy
spectra are bounded from above which was compatible with their counterparts in
1850068-12
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 13
GUP with minimal length uncertainty and maximal momentum
previous works. Finally, as an further investigation and as an example, we studied
some thermodynamical properties of the free particle including partition function
and internal energy in the framework of our proposed GUP which implied that all
modified terms were dependant on β.
Acknowledgments
The authors are grateful to the referee for their constructive comments which con-
siderably improved the quality of the paper.
References
1. H. Shababi and W. S. Chung, Phys. Lett. B 70, 445 (2017).
2. P. Pedram, Phys. Lett. B 714, 317 (2012).
3. G. Veneziano, EPL 2, 199 (1986).
4. E. Witten, Phys. Today 49, 24 (1996).
5. D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 216, 41 (1989).
6. D. Amati, M. Ciafaloni and G. Veneziano, Nucl. Phys. B 347, 550 (1990).
7. D. Amati, M. Ciafaloni and G. Veneziano, Nucl. Phys. B 403, 707 (1993).
8. K. Konishi, G. Paffuti and P. Provero, Phys. Lett. B 234, 276 (1990).
9. L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995).
10. M. Maggiore, Phys. Lett. B 319, 83 (1993).
11. A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D 52, 1108 (1995).
12. A. Kempf and G. Mangano, Phys. Rev. D 55, 7909 (1997).
13. A. F. Ali, S. Das and E. C. Vagenas, Phys. Lett. B 678, 497 (2009).
14. H. S. Snyder, Phys. Rev. 71, 38 (1947).
15. C. N. Yang, Phys. Rev. 72, 874 (1947).
16. C. A. Mead, Phys. Rev. 135, 849 (1964).
17. F. Karolyhazy, Nuovo Cimento A 42, 390 (1966).
18. D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 197, 81 (1987).
19. D. J. Gross and P. F. Mende, Phys. Lett. B 197, 129 (1987).
20. S. Capozziello, G. Lambiase and G. Scarpetta, Int. J. Theor. Phys. 39, 15 (2000).
21. M. Bo jowald and A. Kempf, Phys. Rev. D 86, 085017 (2012).
22. F. Scardigli, Phys. Lett. B 452, 39 (1999).
23. R. J. Adler and D. I. Santiago, Mod. Phys. Lett. A 14, 1371 (1999).
24. F. Scardigli and R. Casadio, Class. Quantum Grav. 20, 3915 (2003).
25. P. Pedram, K. Nozari and S. H. Taheri, JHEP 1103, 093 (2011).
26. K. Nozari and P. Pedram, EPL 92, 50013 (2010).
27. K. Nozari, M. Moafi and F. Rezaee Balef, Adv. High Energy Phys. 2013, 252178
(2013).
28. M. Asghari, P. Pedram and K. Nozari, Phys. Lett. B 725, 451 (2013).
29. J. Vahedi, K. Nozari and P. Pedram, Gravit. Cosmol. 18, 211 (2012).
30. K. Nozari and A. Etemadi, Phys. Rev. D 85, 104029 (2012).
31. K. Nozari and T. Azizi, arXiv:quant-ph/0507018.
32. K. Nozari, P. Pedram and M. Molkara, Int. J. Theor. Phys. 51, 1268 (2012).
33. K. Nozari, M. Khodadi and M. A. Gorji, EPL 112, 60003 (2016).
34. H. Shababi, J. Theor. Phys. 1, 236 (2012).
35. H. Shababi, J. Theor. Phys. 1, 163 (2012).
36. H. Shababi and P. Pedram, Int. J. Theor. Phys. 55, 2813 (2016).
1850068-13
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
April 19, 2018 15:5 MPLA S0217732318500682 page 14
H. Shababi & W. S. Chung
37. P. Pedram, M. Amirfakhrian and H. Shababi, Int. J. Mod. Phys. D 24, 1550016
(2015).
38. K. Nouicer, Phys. Lett. B 646, 63 (2007).
39. S. Detournay, Cl. Gabriel and Ph. Spindel, Phys. Rev. D 66, 125004 (2002).
40. H. Shababi, P. Pedram and W. S. Chung, Int. J. Mod. Phys. A 31, 1650101 (2016).
41. P. Pedram, J. Phys. A 45, 505304 (2012).
42. P. Pedram, Int. J. Mod. Phys. D 19, 2003 (2010).
43. P. Pedram, Phys. Lett. B 718, 638 (2012).
44. L. Chang, D. Minic, N. Okamura and T. Takeuchi, Phys. Rev. D 65, 125028 (2002).
45. P. Wang, H. Yang and X. Zhang, JHEP 1008, 043 (2010).
46. T. V. Fityo, I. O. Vakarchuk and V. M. Tkachuk, J. Phys. A:Math. Gen. 39, 379
(2005).
47. M. A. Gorji, K. Nozari and B. Vakili, Phys. Rev. D 89, 084072 (2014).
48. M. Khodadi, K. Nozari and E. N. Saridakis, Class. Quantum Grav. 35, 015010 (2017).
49. F. Brau and F. Buisseret, Phys. Rev. D 74, 036002 (2006).
50. P. Pedram, Eur. Phys. J. C 73, 2609 (2013).
51. P. Pedram, Int. J. Theor. Phys. 51, 1901 (2012).
1850068-14
Mod. Phys. Lett. A 2018.33. Downloaded from www.worldscientific.com
by UNIVERSITY OF NEW ENGLAND on 04/23/18. For personal use only.
