Spectra of Harmonic Oscillators with GUP and Extra Dimensions

In this paper, we address the spectra of simple harmonic oscillators based on the generalized uncertainty principle (GUP) with a Kaluza-Klein compactified extra dimension. We show that in this scenario, to make the results compatible with experiments, the minimal length scale equals to the radius of compact extra dimension.


Introduction
All quantum gravity theories, such as string theory and loop gravity theory, predict that there exits a minimal observable length, which proportional to the Planck length [1]- [7].This length scale can be easily justified by combining black hole physics with quantum uncertainty principle.Quantum mechanics is free of the theory of gravity.However, when the energy scale approaches Planck scale, the effect of quantum gravity cannot be neglected.It means that quantum mechanics should be modified to accommodate the influences from gravity.Or, it implies that we need a gravitational quantum mechanics.The simplest way is to modify the Heisenberg Uncertainty Principle.The uncertainty principle 1 2 x p ∆ ∆ ≥ shows that the large p ∆ could make the position distance x ∆ arbitrarily small.When the gravity is included in the theory, the x ∆ acquires a minimal length, caused by the emerged horizon of mini black hole.To realize this minimal length scale, we introduce a simple model, the so-called the Generalized Uncertainty Principle (GUP).Incorporating GUP into black holes has been discussed in a lot of papers [8] [9] [10] [11] [12].It is a toy model of quantum gravity theory, which is expected to reveal some features of the ultimate theory.Nearly all problems in quantum mechanics could be discussed with this modified commutator to model the system bathed in strong gravitational background.The modified fundamental commutator ( 1 c = = ) is [13]- [25] [ ] ( ) This new commutator relation includes the different conditions, which could describe the Heisenberg Uncertainty Principle in quantum mechanics by ( ) f p = and also includes the nonzero position distance in high energy scale.
In our requirement, ( ) Note that this formalism is special case of generalized modification that [ ] ( ) . Since the momentum has not minimal scale, it is restricted by choosing 0 α = .Therefore, this special expression gives the genera- lized uncertainty principle (GUP) ( ) Generalizing the commutator to high dimensional space and expressing it as tensor formalism, one obtains that 1 and the correspondent higher dimensional GUP model is where 2 i i p p p = ∑ .This uncertainty principle easily deduces the minimal ob- servable length that ( ) In [26], the energy levels of a free particle including an extra dimension was discussed.The results showed that the minimum observable length equals to the compactification radius of the extra dimension.However, if the particle experiences an non-vanishing potential, what is the result likely to be?In this paper, we extend our discussion to the harmonic oscillator in the extra dimension and consider the GUP's modified energy levels.The result is agree with previous work.The present works of extra dimensions can be refereed in [27]- [36].
This paper is organized as follows.In Section 2, the perturbation method based on GUP is briefly reviewed.In Section 3, two dimensional harmonic oscillator including an compact extra dimension is discussed, and the energy spec- The generic expression is ( ) Journal of High Energy Physics, Gravitation and Cosmology trum is obtained for the large and the small extra dimension.In Section 4, based the perturbation method and GUP, the energy spectrum including quantum gravitational effect is obtained.In Section 5, we summarize the conclusion obtained in the Section 4.

The Perturbation Method Based on GUP
In our scenario, we require the wave function satisfies the periodicity ( ) ( ) , where x represents the usual Euclidean space and y represents the compactified space.Obviously, the function can be separated into two parts , where Θ represents the wave function in compactified space, and we have ( ) ( ) . Therefore, the energy spectrum of Euclidean and compactified space can be obtained respectively.After combining these two parts, we can find the total energy of the system.Following this step, we use the minimal length uncertainty relations, GUP specially, to rewrite the wave function.The modified operators are given by 0 , where are usual canonical operators.The unperturbed Schrödinger equation reads Modifying the commutator relation, we can obtain the new Hamiltonian of this quantum system ( ) the first-order corrections to the energy eigenvalues are given as .

Review of Harmonic Oscillator in Compactified Extra Dimension
Considering the two dimensional harmonic oscillator which includes one of dimensional Euclidean space and another is compactified extra space, we need to solve the Schrödinger equation ( ) The Schrödinger equation can be solved by the method of separation of variables, ( ) ( ) ( ) . In this section, we do Journal of High Energy Physics, Gravitation and Cosmology not discuss the effects of GUP in Equation (10) . Only the harmonic oscillator with periodic condition is reviewed in this section.In the next section, we will discuss the GUP modification of Equation (10), and the perturbation method will be applied since it is convenient enough for seeing some features of quantum gravity.Now, we rewrite Equation (10) as Since the harmonic oscillator lives in the compactified space, the physical meaning of the equation will be changed.The harmonic oscillator in Euclidean space x only satisfies the boundary condition by ( ) 0 φ ±∞ = .However, it should satisfy ( ) ( ) and add an additional restriction for the potential by ( ) ( ) in the extra dimension y, which identifies any two points that the length differ by 2πR .However, the potential of simple harmonic oscillator does not satisfy this condition, we need to construct a new potential of harmonic oscillator with periodicity.Bezerra and Rego-Monteiro had discussed the harmonic oscillator on a circle and constructed a new wave function with periodic oscillator potential [37].The momentum operator G in a compactified space is given by [37] Therefore, the Schrödinger equation in the extra space is constructed with the unitary operator where  is the eigenvalue of extra space.In Equation ( 13), the unitary operator W describes the periodic potential in the extra space, the parameter K is used for reducing the Equation ( 13) to the simple harmonic oscillator when ρ → ∞ .In order to determine the value of K, we set 0 α = and simplify Equation (13)   as ( ) , Equation ( 14) becomes ( ) with ( ) ( ) and set 15) can be rewritten as The solutions of Mathieu's equation cannot be expressed in terms of hypergeometric functions.We address two extremal situations in the following discussions. 1) case.This situation corresponds to the large extra dimension.The characteristic values j a of Mathieu function, which represent energy spectrum of harmonic oscillator in the compactified space, are given by [39] [40] ( ) ( ) ( ) where 2 1 the spectrum in ( 15) is given by ( ) ( ) ( ) It includes the usual energy spectrum of simple harmonic oscillator and the energy spectrum which comes from the quantization of extra dimension.
2) 1 q case.This situation corresponds to the small extra dimension.

When 1 q
, the solution to Equation (18) has two branches, one is even, the other is odd [41].In our model, η has the periodicity π , so we only adopt the even solutions in [41].We rearrange the notation of the even solutions as follows where 0,1, m = , and [40] The normalization of wave function

( )
, q η Θ requires the functions From Equation (22) and Equation ( 23), the Mathieu Equation ( 18) can fall into into two categories ( ) Corresponding to the wave function ( 22) and ( 23), we write the energy spectrum as More specifically, for 3 m ≤ , we have the energy spectrum that [42]   ( ) ( ) Notice that when l → ∞ , ( ) Combining the spectrums in the part of extra dimension and the part of Euclidean space, one can easily find the total energy spectrum of simple harmonic oscillator:

The Energy Spectrum with GUP
Now, we begin to modify the commutator relations.The Schrödinger equation is given by Equation ( 7), and p is determined by Equation ( 6).The first-order corrections of the energy eigenvalues are given by Equation ( 9).We only consider the small extra dimension, that is, 1 q . Then we rewrite Equation ( From Equation (10) and Equation ( 18), the potential is Then using Equations ( 22)~( 25), we have ( ) where Therefore, we obtain the total energy spectrum with Now, for analyzing the result, we should calculate the value of l Λ and l Ξ .
Then Equation (58) should be rewritten as This expression has same expression with our previous work [26].According to the analysis in [26], for the small extra space, the second term is larger than the first term since it has the term 2 1 ρ .In the high energy case, the leading order of spectrum, which includes the first excited state of extra dimension effect, can be written as

Conclusion
In summary, we have discussed the GUP effect of harmonic oscillator in the compactified extra dimension.We use the modified momentum operator and modified harmonic oscillator potential to build the periodic Hamiltonian.The correspondent wave is so-called Matheu function.Using the operators of GUP to modify this periodic Hamiltonian, we find the modified Hamiltonian.
For obtaining the modified energy levels, we use the perturbation method for Matheu function.Keeping the leading order of the energy levels, we obtain the same energy eigenvalue formation with our previous work [26].This result shows the minimum observable length equals the compactification radius of the extra dimension.

B 2 y equals 1 .
. R. Mu et al.Journal of High Energy Physics, Gravitation and CosmologyWhen ρ → ∞ , Equation (17) must reduce to the simple harmonic oscillator.This means the coefficient of p is a positive function and which can be expanded in ity phenomenons in intermediate energy scale.In GUP model, we choose the first two terms such that