# Minimal Length Schrödinger Equation with Harmonic Potential in the Presence of a Magnetic Field

Authors: H. Hassanabadi; E. Maghsoodi; Akpan N. Ikot; S. Zarrinkamar
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

Minimal length Schrödinger equation is investigated for harmonic potential in the presence of magnetic field and illustrates the wave functions in the momentum space. The energy eigenvalues are reported and the corresponding wave functions are calculated in terms of hypergeometric functions.


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## Body

## 1. Introduction

The study of theories characterized by a minimal observable length is an active area of research in theoretical physics, not only because of their intrinsic interest, but also because of their existence as suggested by string theory and quantum gravity. In order to incorporate the idea of the minimal length into quantum mechanics, we need to modify the ordinary uncertainty principle to the so-called Generalized Uncertainty Principle (GUP). Some authors have worked on generalized quantum theoretical framework [1–16]. Scardigli has studied a version of GU4 obtained in string theory and in gedanken experiments carried out in quantum gravity [1]. In [4], Dadić et al. [4] have investigated deformed harmonic oscillators with minimal length uncertainty relations. Chang et al. [5] have determined the energy eigenvalues and eigenfunctions of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations in [5]. The authors have considered the effects of the deformation on classical limit of the minimal length uncertainty relation in a central force potential in [6]. Recently much attention has been devoted to the study of deformed Heisenberg algebras of different kinds with minimal length [17–21]. For example, in [17], Ali and Tawfik [17] have studied the effects of GUP, which is proposed by some approaches to quantum gravity such as string theory, black hole physics, and doubly special relativity theories (DSR), on the area law of the entropy. The authors in [18] have solved the minimal length Schrödinger equation in the presence of a nonminimal Woods-Saxon interaction. The solution of the relativistic and nonrelativistic fundamental dynamical equations is an interesting phenomenon in many fields of physics. Recently, the analytic exact solutions of the nonrelativistic equation with some potential by different method are impossible [22–28]. Many other aspects of the theory are well addressed in [29–35] and references therein. The interface of the minimal length quantum mechanics with quantum gravity is discussed in [29]. The application of the formalism to the motion of a dipole in a cosmic string background was studied in [30]. Chang et al. commented on the minimal length uncertainty relation and the essentials of string theory [31]. Harbach and his coworkers proposed the precision muon tests to investigate the minimal length scale [32]. Kothawala et al. studied the problem for space times with constant curvatures [33]. Panella considered the Casimir-Polder intermolecular forces within the framework of minimal length theories [34]. Pedram et al. analyzed the effects of minimal length on the transition rate of ultracold neutrons in the gravitational field [35]. Our work is organized as follows. In Section 2, we give a brief introduction of the Generalized Uncertainty Principle. In Section 3, the Schrödinger equation is written in the momentum space including the harmonic potential and it obtains the solution of the problem. We use the useful analytical Nikiforov-Uvarov technique in our calculations (Appendix A). Finally, the conclusion is given in Section 4.


## 2. The Generalized Uncertainty Principle

An immediate consequence of the minimal length is the Generalized Uncertainty Principle (GUP) [7]
(1)Δx≥ℏΔp+αlp2Δpℏ,
where the GUP parameterαis determined from a fundamental theory [7]. At low energies, that is, energies much smaller than the Planck mass, the second term in the right hand side of the latter vanishes and we recover the well-known Heisenberg uncertainty principle. The GUP of (1) corresponds to the generalized commutation relation [7]
(2)[xop,pop]=iℏ(1+βp2),0≤β≤1,
wherexop=x^and0≤β≤1. The limitsβ→0and β→1, respectively, correspond to the normal quantum mechanics and extreme quantum gravity. Equation (2) gives the minimal length in this case as(Δx)min=2lpα and a representation ofx^iandp^iin momentum space is taken as (Appendix B)
(3)x^i=iℏ(1+βp2)∂∂pi,p^i=pi.


## 3. The Equation in the Momentum Space

The minimal length Schrödinger equation in momentum space[Ψ(r→)→Φ(p→)]is
(4){Π22m+V(r)}Φnλ(p→)=EnλΦnλ(p→),
whereE,Π→, andm, respectively, denote the energy of the system, two-dimensional momentum operator, and the mass of the particle. By consideringΠ→=p→-(e/c)A→, the minimal length Schrödinger equation appears as
(5){(p→-(e/c)A→)22m+V(r)}Φnλ(p→)=EnλΦnλ(p→),
where the vector potentialA→can be alternatively expressed asA=(-By^/2,Bx^/2,0) and B→=Bz^ is the applied magnetic field. We consider the following harmonic potential:
(6)V(r)=V0r2.
Now, by inserting the latter into (5), we have
(7){[(px-(e/c)Ax)i^+(py-(e/c)Ay)j^]22m+V0(x^2+y^2)}×Φnλ(p→)=EnλΦnλ(p→).
A representation ofx^andy^ is taken as(8a)x^=i(1+βp2)∂∂px,(8b)y^=i(1+βp2)∂∂py,where we have definedpx=pcosθ and py=psinθ. Substitution of (8a) and (8b) into (7) gives
(9){12m[(px+ecB2i(1+βp2)∂∂py)×(px+ecB2i(1+βp2)∂∂py)+(py-ecB2i(1+βp2)∂∂px)×(py-ecB2i(1+βp2)∂∂px)(px+ecB2i(1+βp2)∂∂py)]+V0[+(i(1+βp2)∂∂py)(i(1+βp2)∂∂py)(i(1+βp2)∂∂px)(i(1+βp2)∂∂px)+(i(1+βp2)∂∂py)(i(1+βp2)∂∂py)]-Enλ12m[(px+ecB2i(1+βp2)∂∂py)}Φnλ(p→)=0.
Consider
(10)∂∂px=cosϑ∂∂p-sinϑp∂∂ϑ,∂∂py=sinϑ∂∂p+cosϑp∂∂ϑ.
Equation (9) then takes the form
(11){12m[p2+ecB2i(1+βp2)∂∂ϑ-e2B22c2β(1+βp2)p∂∂p-e2B24c2(1+βp2)2∂2∂p2-e2B24c2(1+βp2)21p∂∂p-e2B24c2(1+βp2)21p2∂2∂ϑ2]+V0[-2β(1+βp2)p∂∂pmmmmmm-(1+βp2)2∂2∂p2-(1+βp2)21p∂∂pmmmmmm-(1+βp2)21p2∂2∂ϑ2]-Enλ}Φnλ(p→)=0.
By using the solution of the form,
(12)Φnλ(p→)=ϕnλ(p)eiλϑ
reduces the above equation to the form
(13){[-e2B28mc2(1+βp2)2-V0(1+βp2)2]∂2∂p2+[-e2B24mc2β(1+βp2)p-e2B28mc2(1+βp2)21p-2βV0(1+βp2)p-V0(1+βp2)21p]∂∂p+p22m-eB4mcλ(1+βp2)+e2B28mc2(1+βp2)2λ2p2+V0(1+βp2)2λ2p2-Enλ}ϕnλ(p→)=0.
In order to obtain the solution of (13), we use the transformations=-βp2and rewrite it as follows:
(14)d2ϕnλ(s)ds2+(η-(η-γ)s)s(1-s)dϕnλ(s)ds+1s2(1-s)2(σs2+τs+ρ)ϕnλ(s)=0,
where
(15)η=12-e22mc2B28μ-V02μ,γ=e22mc2B24μ+V0μ,μ=-e22mc2B24-V0,σ=18mβ2μ-e2mcBλ8βμ+e22mc2B2λ216μ+V0λ24μ+Enλ,τ=e2mcBλ8βμ-e22mc2B2λ28μ-V0λ22μ-Enλ,ρ=e22mc2B2λ216μ+V0λ24μ.
By comparing (14) with (A.11), we get
(16)α1=η,α2=η-γ,α3=1,ξ1=-σ,ξ2=τ,ξ3=-ρ,α4=12(1-η),α5=η-γ2-1,α6=(η-γ2-1)2-σ,α7=((1-η)(η-γ)2-(1-η))-τ,α8=14(1-η)2-ρ,α9=((1-η)(η-γ)2-(1-η))-τ+14(1-η)2-ρ+(η-γ2-1)2-σ=γ24+γ2+14-τ-ρ-σ,α10=1+214(1-η)2-ρ,α11=2+2(γ24+γ2+14-τ-ρ-σmmmmm+14(1-η)2-ργ24+γ2+14-τ-ρ-σ),α12=(1-η)2+14(1-η)2-ρ,α13=η-γ2-1-(γ24+γ2+14-τ-ρ-σmmmmmmmm+14(1-η)2-ργ24+γ2+14-τ-ρ-σ).
From (A.20), we have
(17)n2+n+12+ηγ2+(2n+1)(γ24+γ2+14-τ-ρ-σmmmmmmmmmmmmmmm+14(1-η)2-ργ24)-τ-2ρml+2(14(1-η)2-ρ)(γ24+γ2+14-τ-ρ-σ)=0.
For this case, from (A.24), the wave function is
(18)ϕnλ(p)=(-βp2)(1-η)/2+(1/4)(1-η)2-ρ×(1+βp2)(γ+1)/2+(γ2/4+γ/2+1/4-τ-ρ-σ)×Pn(2(1/4)(1-η)2-ρ,2γ2/4+γ/2+1/4-τ-ρ-σ)×(1+2βp2),
wherepn(α,β)(1-2z)is the Jacobi polynomial or
(19)Pn(α,β)(1-2z)=1n!z-α(1-z)-βdndzn[zn+α(1-z)n+β].
Therefore,Φnλ(p→)can be written versus the hypergeometric function as
(20)Φnλ(p→)=(-βp2)(1-η)/2+(1/4)(1-η)2-ρ×(1+βp2)(γ+1)/2+(γ2/4+γ/2+1/4-τ-ρ-σ)×2F1(-n,n+214(1-η)2-ρmmmmml+2γ24+γ2+14-τ-ρ-σ+1;mmmmllll214(1-η)2-ρ+1;-βp2)eiλϑ.


## 4. Conclusion

Exact solution of the Schrödinger equation of harmonic oscillator in a magnetic field considered in the presence of a minimal length has been obtained in terms of hypergeometric functions. We have reported eigenfunction and eigenvalues of energy. Having found the wavefunction and the spectrum, we can simply obtain the decay rate and the bounds on the minimal length parameter. In addition, many other important quantities in various fields of physics can be systematically obtained including the charge radius, form factors, and shifts in the Landau levels. More generally, we know that the minimal length formulation is in fact a parallel approach to the noncommutative formulation of the quantum theory. Therefore, roughly speaking, the whole story about the experimental verification of the noncommutative version of quantum mechanics (including experiments with TeV photon experiments, very precise measurement of the radiation emitted during the transition 1S-2S of the hydrogen atom, quantum optics, and Lamb shift studies) can be used here to comment on the results as well.



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