Minimal Length Schrödinger Equation with Harmonic Potential in the Presence of a Magnetic Field
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.
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 . In , Dadić et al.  have investigated deformed harmonic oscillators with minimal length uncertainty relations. Chang et al.  have determined the energy eigenvalues and eigenfunctions of the harmonic oscillator in arbitrary dimensions with minimal length uncertainty relations in . The authors have considered the effects of the deformation on classical limit of the minimal length uncertainty relation in a central force potential in . Recently much attention has been devoted to the study of deformed Heisenberg algebras of different kinds with minimal length [17–21]. For example, in , Ali and Tawfik  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  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 . The application of the formalism to the motion of a dipole in a cosmic string background was studied in . Chang et al. commented on the minimal length uncertainty relation and the essentials of string theory . Harbach and his coworkers proposed the precision muon tests to investigate the minimal length scale . Kothawala et al. studied the problem for space times with constant curvatures . Panella considered the Casimir-Polder intermolecular forces within the framework of minimal length theories . Pedram et al. analyzed the effects of minimal length on the transition rate of ultracold neutrons in the gravitational field . 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)  where the GUP parameteris determined from a fundamental theory . 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  whereand. The limitsand , respectively, correspond to the normal quantum mechanics and extreme quantum gravity. Equation (2) gives the minimal length in this case as and a representation ofandin momentum space is taken as (Appendix B)
3. The Equation in the Momentum Space
The minimal length Schrödinger equation in momentum spaceis where,, and, respectively, denote the energy of the system, two-dimensional momentum operator, and the mass of the particle. By considering, the minimal length Schrödinger equation appears as where the vector potentialcan be alternatively expressed as and is the applied magnetic field. We consider the following harmonic potential: Now, by inserting the latter into (5), we have A representation ofand is taken aswhere we have defined and . Substitution of (8a) and (8b) into (7) gives Consider Equation (9) then takes the form By using the solution of the form, reduces the above equation to the form In order to obtain the solution of (13), we use the transformationand rewrite it as follows: where By comparing (14) with , we get From , we have For this case, from , the wave function is whereis the Jacobi polynomial or Therefore,can be written versus the hypergeometric function as
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.
A.1. The Nikiforov-Uvarov Method
In this section, the NU method and its parametric generalization are briefly presented. The NU method, in its general form, solves the ordinary linear second-order differential equation; that is,  where the prime denotes the derivative with respect to the independent variable . Here,andmust be polynomials of at most second degree andis a polynomial with at most first degree . Using the following transformation: (A.1) is reduced to the well-known hypergeometric-type equation  where Equation (A.1) has a particular solution when the following relation is satisfied: In order to obtain the energy eigenvalues equation, the functions given below are calculated: whereis a constant. The expression under the square root in (A.6) must be the square of a polynomial of the first degree, sinceis the first-degree polynomial. Then, one can obtain thevalues by considering that the discriminant of the square root has to be zero in (A.6) . Consequently, the energy eigenvalues can be obtained by comparing (A.7) with (A.5). The functiongiven in (A.3) is a hypergeometric-type function whose solution can be written in terms of the polynomials given by the Rodrigues relation whereis the normalization constant and the weight functionsatisfies the condition On the other hand, the other factor satisfies the following logarithmic equation:
A.2. Parametric Formulation of Nikiforov-Uvarov Method
The NU method, in its parametric form, solves Here, we give only the basic ingredients of the generalized NU method. In this case, comparing (A.11) with (A.1), one can obtain Inserting the above equations into (A.6) leads to  where Considering that the discriminant of the square root has to be zero in (A.6), we obtain  with From (A.13), one can easily see that differentvalues lead to different. If we take becomes and then we find  The energy eigenvalue equation can be readily obtained by using (A.5) and (A.6) with the above results as follows : In order to obtain the wave functions, one can use the following relations : where is the Jacobi polynomial and 
A simple proof of the generalized operator form of the position momentum given in (3) is written here. Let us assume In this case, we simply have A simple comparison with (B.1) indicates the correspondence
H. Hassanabadi thanks Shahrood University for their support in the work.
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