The transition of energy and bound states in the continuum of fractional Schrödinger equation in gravitational field and the effect of the minimal length

Abstract

In this paper, we study the fractional Schrödinger equation in the Earth’s gravitational field. We firstly introduce a family of auxiliary functions to construct solutions to the fractional Schrödinger equation in the Planck length. These solutions include the particular solution obtained previously by using the classical “Fourier transform approach”. By analyzing the solutions, we find the transition phenomenon when the dimension of Lévy path changes from integer to non-integer: the energy changes from discrete to continuous and wave functions change from non-degenerate to degenerate. Then we study the effect of the minimal length on the fractional Schrödinger equation in the Earth’s gravitational field and the solutions. We find that the presence of the minimal length brings a perturbation to the Hamiltonian in equation but it does not change the transition phenomenon. Based on these result, we prove the existence of bound states in the continuum (BICs) for the fractional quantum system in the Earth’s gravitational field and compare our BICs with those previous ones. Moreover, we provide the energy characteristic of small mass particles.

Introduction

Over the past few decades, fractional quantum quantization methods have been put forward [1], [2], [3] as a natural generalization of standard quantum quantization methods for the dynamical process satisfying the “fractional corresponding relation” [1] when the Brownian trajectories are replaced by Lévy flights. Recently some of these theoretical results have been realized in Refs. [4], [5], [6]. These theories give birth to fractional quantum mechanics which serves as a powerful tool for describing non-Markovian evolutions with a memory effect and nonlocal quantum phenomena [7], and hence enables a study route to fractional dimensions [2], [3], [8], [9].

Fractional quantum mechanics has been studied in a large number of literature from many aspects, such as constructing mathematical models and developing fractional quantum quantization methods to solve fractional Schrödinger equations. In Refs. [1], [2], [3], [7], [10], [11], different kinds of fractional Schrödinger equations have been constructed. Then the methods of solving the fractional Schrödinger equations were developed in many cases such as Fourier transform approach in momentum representation [12], solving methods for Coulomb potential [10], comb potential [13] and infinite well potential [14], and numerical solutions of eigenvalue problem [15].

Moreover, many physical applications also attract intense interests: the path of the Lévy flights on the quantum mechanical kernel [9], the profile decomposition with angularly regular data [16], optimal control of fractional quantum dynamics [17], and polariton condensate [18]. Furthermore, fractional quantum mechanics enables us to describe some phenomena that cannot be described by classical quantum mechanics, for examples one-dimensional Lévy crystals [19], [20] and exotic BICs [8].

The main results and contributions of this paper can be divided into the following three progressive parts.

• Mathematical Models: We construct the fractional Schrödinger equation in the presence of the minimal length by the generalized uncertainty principle (GUP) and the strong fractional corresponding operator (Section 2). This fractional Schrödinger equation is different from the previous kind of fractional Schrödinger equations in Refs. [1], [2], [3], [7], [10], [11] built in the Planck length. The minimal length is expected to be comparable to the Planck length and its existence is predicted in many theories including string theory, quantum gravity and black hole physics [21], [22].

• Solving Methods: We show that the classical “Fourier transform approach” [12] can only derive a single particular solution of the fractional Schrödinger equation in the Earth’s gravitational field when the dimension of the Lévy path is non-integer. Instead of using the “Fourier transform approach”, we introduce a family of auxiliary functions to construct a family of solutions to the fractional Schrödinger equation in the Planck length (Section 3). Based on the results, we notice that the solution obtained by the Fourier transform approach in Ref. [12] just corresponds to one particular solution of the solutions we obtain.

  By analyzing the solutions, we find sudden transitions of energy and wave functions (Section 4): when the dimension of Lévy path changes from integer to non-integer, the energy changes from discrete to continuous and wave functions change from non-degenerate to degenerate.

  Furthermore, we apply the solving method mentioned above to obtain the analytic solutions to the fractional Schrödinger equations in the presence of the minimal length in the same potential (Section 5). We find that the common energy shift phenomenon [23], [24] does not occur in the presence of the minimal length. Thus the phenomenon of the sudden transitions of energy and wave functions still exists in the presence of the minimal length.

• Two applications: We prove the existence of the bound states in the continuum (BICs) phenomenon in the Earth’s gravitational field in the presence of the minimal length and provide the energy characteristic of small mass particles (Section 6).

  For the first application, we find the BICs phenomenon in the presence of the minimal length based on the above results obtained for the fractional Schrödinger equations. These BICs are different from those found previously in Refs. [25], [26], [27]. Previous BICs can only be realized in two types of ways: one type of BICs is fragile and can only be realized in specially tailored potentials [25], [28], [29], [30]; the other type of BICs is realized through controlling the interactions between particles [31], [32]. However, our BICs can be realized in a simple potential without controlling the interactions between particles. Moreover, our BICs are more stable to inevitable perturbations. To the best of our knowledge, this is the first BICs phenomenon found in the Earth’s gravitational field. We clarify that this BICs phenomenon can be considered as a characteristic phenomenon in the circumstance of fractional dimensional Lévy path. Thus it provides both a criterion to determine in advance whether an unknown quantum system can be described by fractional derivatives and a verification for successful preparation of fractional quantum systems in experiments.

  For the second application, we further analyze the dependence of the continuity of energy on the mass of particles. We summarize that the continuity of energy becomes strong when (i) the dimension of the Lévy path changes from integer to non-integer and/or (ii) the mass of the particle is small. Conversely, if the Lévy path dimension approaches to integer one and the mass of particle is very large, the energy will be asymptotically close to the discrete energy in classical quantum mechanics, i.e., the continuity of the energy becomes weak. This energy characteristic can be used to distinguish the small mass particles from other particles with large mass. Without considering couplings between particles, we provide a map of the energy characteristics with respect to different Lévy dimensions and different particles including: the Z boson, tau, muon, electron and electron neutrino. This map illustrates that the small mass particles can be markedly distinguished from other particles based on this energy characteristic.