Spectra of Hydrogen Atom with GUP and Extra Dimensions

doi:10.4236/jmp.2013.45B006

Paper Menu >>

Journal Menu >>

Journal of Modern Physics, 2013, 4, 29-32

doi:10.4236/jmp.2013.45B006 Published Online May 2013 (http://www.scirp.org/journal/jmp)

Spectra of Hydrogen Atom with GUP and Extra

Dimensions

Benrong Mu

School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu, Sichuan, China

Email: [email protected]

Received 2013

ABSTRACT

We argue that the Generalized Uncertainty Ptinciple (GUP) and the compact Extra dimensions will lead the hydrogen

atom has the non bounded state spectra which equal the free particle in extra dimensions. We use Generalized Uncer-

tainty Principle (GUP) to calculate the new energy spectra of hydrogen atom, and which lives in three dimensional

Euclidean spaces with one extra dimension. The result is not familiar with our known before that when n is

large enough. In our modified new spectra, we obtain that

E

222 2

,,0 (1 2)2







 , which is lager than zero.

These new spectra give us new method to obtain the existenc e of extra dimensions. Finally, we find that the spectra are

same as part of extra dimensions in planck scale.

Keywords: Hydrogen Atom; GUP; Extra Dimensions

1. Introduction

There exist many theories for solving the problem in

unifying the quantum theory and gravity, such as string

theory and loop quantum theory, which all of them pre-

dict that there exits the minimal observable length which

proportional to the Planck length cm. [1]

This minimal length was emerged by our observation

method under high energy scale, and which phenomenon

is not surprised for us. In usual quantum mechanics,

which does not consider the effect of gravity, has the

Heisenberg Uncertainty Principle which is described as

p



12xp . This uncertainty principle implies that the

large could make the position distance

p



arbi-

trary small. How ever, when we cons ider the gravity, which

becomes more important in Planck scale, cannot lead us

to neglect the effect of gravity in many special cases.

This motivation gives us the advices that when we use

high energy to observe any arbitrary small scale, it will

causes the minimal blank hole, and hence its horizon

becomes the minimal observable length naturally. In this

analysis, the Heisenberg Uncertainty Principle must be

modifed for including the effects of quantum gravity and

the one of the most important models of this case is the

Generalized Uncertainty Principle (GUP). The modified

fundamental commutator







 is [2-5]







pifp (1.1)

This new commutator relation gives us the po ssibilities

to contain the different conditions, which could describe

the usual uncertainty principle in our familiar quantum

mechanics that





1fp



and also is will includes the

nonzero position distance in high energy scale. In our

requirements,





p is a positive function and which can

be expanded by Taylor series as



1(),

ppp



 

and 22







 . Our previous work had been

introduced the running constant 0





p that would

varied with energy scales and be combined with another

quantum gravity quantity: extra dimensions [6]. These

characteristics make us to construct the low energy effec-

tive theory which will contains some quantum gravity

phenomenon in intermediate energy scale. In GUP model,

we choose the first two terms as









pi p



 (1.2)

and hence, the GUP model is



pi p





 





(1.3)

Generalizing the commutator to higher dimensions and

expressing it as tensor formalism, we obtain that





ijijiji j

pip pp

 



 

 (1.4)

and the correspondent higher dimensional GUP model is









112

iii i

xpppp p







 









(1.5)

B. R. MU

where ii. This uncertainty principle easily

deduces the minimal observable length that

ppp





min

x



 for every direction.

In this paper, we construct the toy model which the

extra dimension is only considered without any other

potentials, and use the GUP to redescribe the hydrogen

atom. This assumption is not unjustifiable in view of the

fact that the standard model charges cannot move in extra

dimensions in theory of brane world model. Now, extra

dimensions are widely accepted by theorists and there are

large works about it [7-13]. This model gives us the new

spectra, which looks same as our previous works [6],

however, have different physical meaning. In usual spec-

tra of hydrogen atom, we know the spectra is given as

nme



















(1.6)

We know that the ground state energy is 113.6eVE



.

When , we obtain the upper boun d of the energy

is zero. In our following works, when consider the term

of one extra dimension

n

2m2



, the upper bound of

hydrogen atom will be larger than zero. On the other

hand, we obtain the energy spectra of hydrogen atom is

same as the spectra which describe the particle move in

the extra dimension freely in Planck scale. To sum up, it

gives us the new detect method to find the existence of

extra dimensions in intermediate energy scale.

2. Method

In our case, we need that the wave function obeys the

periodical condition

 

,,2xy xy









, where x

represents the familiar Euclidean space and y yrepresents

the compact space. As we known, the function can be

described as the production of two parts that









 





, where represents the wave function in

compact space, and we have



 









. In this

way, the familiar wave function can be divided into two

parts. We hence can calculate the energy spectrum of

Euclidean and compact space respectively, and count up

these two parts for obtaining the total energy of the sys-

tem.



0000

iii i

pppp i





  (2.1)

where 2

00i

pp0

i and 00 j

ij i









, the usual

canonical operators. The unperturbed schrödinger equa-

tion reads



pVx E











 (2.2)

with modifying the commutator relations, we can obtain

the new Hamiltonian of this quantum system that



HHVx p



 

 (2.3)

the first order correction is then given as

 



00 2

EpmEV

mE EVV



 



















(2.4)

3. Modified Spectra of Hydrogen Atom

The schrödinger eq uation of hydrogen ato m is

2()

pVr E









(3.1)

where potential ()Vr



comes from the Coulomb's law

and be given as



Vr r







 (3.2)

As we mentioned in introduction, we should consider

the hydrogen atom with an extra dimension at first.

When we consider the extra dimension, the term of

schrödinger equation should contain the momentum in

extra dimension and which can be rewritten as

2222





  (3.3)

where the correspondent wave function is









,, .r



 



In this way, the equation be divided into two equations

which are unperturbed as follow









 (3.4)



Vr E









(3.5)

so, we have the unperturbed energy spectra of hydrogen

atom with one extra dimension, which can be given as

 

000

24 2

EEE













 











 (3.6)

Now, let us modify the commutation relations by using

GUP, we obtain the perturbed term of Hamiltonian that



. So, we find the first order correction of

energy spectra is

 







000

,,, ,

nsn n

EmE EVV



















(3.7)



,, 2

mE Err

 





 





 

 





 







B. R. MU 31

where



22 32

11 11

and 12

rna r



 (3.8)

Finally, we obtain the modified energy spectra of hy-

drogen atom that



 



,, 2

,, ,2

+4m12

ns n

EE na

EEm e

sna

Em E





























 





















 

















































(3.9)

When n is large enough, in our introduction before,

will gives us . But, we obtain new value in here

when n is large, the spectra is

E

,,02

12 2

















(3.10)

This result is too surprised. As we known, the usual

ground state of hydrogen atom is. With energy

level increase, the value of energy go up to, however,

specially, there exits the upper bound when is large

enough which equals zero. But, our result introduction

the new spectra of hydrogen atom when n is large. Above

the upper bound, we find new spectra! This result is not

end of journey, noteworthily, as our works before, we

have the relations

13.6eV





 in Planck scale. It express

that



,,0 2

12 2







 (3.11)

which is the energy spectra of extra dimension. Consid-

ering the physical meaning of the nonzero position dis-

tance in Planck scale, there exists the minimal black hole

and the horizon is that nonzero po int. This condition lead

the hydrogen atom has unbounded state in extra dimen-

sions and which is reason that we obtain the positive

spectra in final.

4. Summary

We use GUP and extra dimensions models to rewritten

the equation of hydrogen atom and obtain the modified

energy spectra which is far from the familiar solution that

has a upper bound. In this modified spectra, we find that

there also exits discrete spectra in field of >0

, which

is same as the free particle that lives in extra dimensions.

This result is not familiar in usual hydrogen atom. As this

term is contributed by the wave function in extra dimen-

sion, it introduces the new method to find extra dimen-

sions in intermediate energy scale.

REFERENCES

[1] G. Veneziano, “A Stringy Needs Just Two Constants,”

Europhysics Letters, Vol. 2, No.3, 1986, p. 199.

doi:10.1209/0295-5075/2/3/006

[2] D. J. Gross and P. F. Mende, “String Theory The Planck

Scale,” Nuclear Physics B, Vol. 303, No. 3, 1988, pp.

407-454.doi:10.1016/0550-3213(88)90390-2

[3] D. Amati, M. Ciafaloni and G. Veneziano, “Can Space-

time Be Probed Below the String Size,” Physics Letters,

B, Vol. 216, No. 1-2, 1989, pp. 41-47.

doi:10.1016/0370-2693(89)91366-X;

[4] K. Konishi, G. Pa_uti and P. Provero, “Minimum Physi-

cal Length and the Generalized Uncertainty Principle in

String Theory,” Physics Letters B, Vol. 234, No. 3, 1990,

pp. 276-284. doi:10.1016/0370-2693(90)91927-4

[5] R. Guida, K. Konishi and P. Provero, “On the Short Dis-

tance Behavior of String Theories,” Modern Physics Let-

ters A, Vol. 06, No. 16, 1991, p. 1487.

doi:10.1142/S0217732391001603

[6] M. Maggiore,“ A Generalized Uncertainty Principle in

Quantum Gravity,” Physics Letters B, Vol. 304, No. 1-2,

1993, pp. 65-69. doi:10.1016/0370-2693(93)91401-8

[7] A. Kempf, G. Mangano and R. B. Mann, “Hilbert Space

Representation of the Minimal Length Uncertainty Rela-

tion,” Physical Review D, Vol. 52, 1995, pp. 1108-1118.

doi:10.1103/PhysRevD.52.1108

[8] A. Kempf, “Non-Pointlike Particles in Harmonic Oscilla-

tors,” Journal of Physics A:Mathematical and General,

Vol. 30, No. 6, 1997, p. 2093.

doi:10.1088/0305-4470/30/6/030

[9] F. Brau, “Minimal Length Uncertainty Relation and the

Hydrogen Atom,” Journal of Physics A: Mathematical

and General, Vol. 32, No. 44, 1999.

doi:10.1088/0305-4470/32/44/308

[10] F. Brau and F. Buisseret, “Minimal Length Uncertainty

Relation and Gravitational Quantum Well,” Physical Re-

view D, Vol. 74, 2006, 036002.

doi:10.1103/PhysRevD.74.036002

[11] L. N. Chang, D. Minic, N. Okamura and T. Takeuchi,

“Exact Solution of the Harmonic Oscillator in Arbitrary

Dimensions with Minimal Length Uncertainty Relations,”

Physical Review D, Vol. 65, 2002, 125027.

doi:10.1103/PhysRevD.65.125027

[12] I. Dadic, L. Jonke and S. Meljanac, “Harmonic Oscillator

With Minimal Length Uncertainty Relations and Ladder

Operators,” Physical Review D, Vol. 67, 2003, 087701

doi:10.1103/PhysRevD.67.087701

[13] K. Nozari and T. Azizi, “Some Aspects of Gravitational

Quantum Mechanics,” General Relativity and Gravitation,

Vol. 38, No. 38, 2006, pp. 735-742.

B. R. MU

doi:10.1007/s10714-006-0262-9

[14] M. M. Stetsko, “Corrections to the ns Levels of The Hy-

drogen Atom in Deformed Space With Minimal Length,”

Physical Review A, Vol. 74, No. 6, 2006, 062105.

doi:10.1103/PhysRevA.74.062105

[15] S. Z. Benczik, “Investigations on the Minimal-Length

Uncertainty Relation,” Dissertation for Doctor of Phi-

losophy.

[16] M. V. Battisti and G. Montani, “The Big-Bang Singular-

ity in the Framework of a Generalized Uncertainty Prin-

ciple,” Physics Letters B, Vol. 656, No. 1-3, 2007, pp.

96-101.doi:10.1016/j.physletb.2007.09.012.

[17] M. V. Battisti and G. Montani, Physics Review D, Vol. 77,

2008,023518.

[18] M. V. Battisti, arXiv: 0805.1178.

[19] S. Das and E. C. Vagenas, Physics Review Letters, Vol.

101, 2008, 221301.

[20] B. Mu, H. Wu and H. Yang, arXiv:0909.3635 [hep-th].

[21] B. Mu, “Spectra of harmonic oscillators with GUP and

extra dimensions,” to appear.

[22] I. Antoniadis, “A Possible New Dimension at a Few

TeV,” Physics Letters B, Vol. 246, No. 3-4, 1990, pp.

377-384. doi:10.1016/0370-2693(90)90617-F

[23] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, “The

Hierarchy Problem and New Dimensions at a Millime-

ter,” Physics Letters B, Vol. 429, No. 34, 1998, pp.

263-272.doi:10.1016/S0370-2693(98)00466-3

[24] I. A ntoniadis, N. Arkani-Hamed, S. Di mopoulos and G. R.

Dvali, “New Dimensions at a Millimeter to a Fermi and

Superstrings at a Tev,” Physics Letters B, Vol. 436, No.

3-4,1998, pp. 257-263.

doi:10.1016/S0370-2693(98)00860-0

[25] L. Randall and R. Sundrum, “Large Mass Hierarchy from

A Small Extra Dimension,” Physical Review Letters, Vol.

83, 1999, pp. 3370-3373.

doi:10.1103/PhysRevLett.83.3370

[26] L. Randall and R. Sundrum, “An Alternative to Compac-

tification,” Physical Review Letters, Vol. 83, 1999,

pp.4690-444693. doi:10.1103/PhysRevLett.83.4690

[27] G. R. Dvali, G. Gabadadze and M. Porrati, “4D Gravity

On A Brane in 5D Minkowski Space,” Physics Letters B,

Vol. 485, No. 1-3, 2000, pp. 208-214.

doi:10.1016/S0370-2693(00)00669-9

[28] T. Appelquist, H. C. Cheng and B. A. Dobrescu, “Bounds

on Universal Extra Dimensions,” Physical Review D, Vol.

64, 2001, 035002. doi:10.1103/PhysRevD.64.035002

[29] D. Cremades, L. E. Ibanez and F. Marchesano, “Standard

Model at Intersecting D5-Branes:Lowering the String

Scale,” Nuclear Physics B, Vol. 643, No. 1-3, 2002, pp.

93-130. doi:10.1016/S0550-3213(02)00746-0

[30] C. Kokorelis, “Exact Standard Model Structures From

Intersecting D5-Branes,” Nuclear Physics B, Vol. 677,

No. 1-2, 2004, pp. 115-163.

doi:10.1016/j.nuclphysb.2003.11.007