Two-dimensional Klein-Gordon Oscillator in the presence of a minimal length

TL;DR

This paper explores the effects of a minimal length on a two-dimensional Klein-Gordon oscillator, deriving energy eigenvalues and wave functions in momentum space, highlighting modifications to quantum behavior due to minimal length considerations.

Contribution

It introduces a novel analysis of the Klein-Gordon oscillator incorporating minimal length, providing explicit solutions for energy levels and wave functions in momentum space.

Findings

Energy eigenvalues are explicitly calculated.

Wave functions are expressed in hyper-geometric functions.

Minimal length impacts the quantum properties of the oscillator.

Abstract

Minimal length of a two-dimensional Klein-Gordon oscillator is investigated and illustrates the wave functions in the momentum space. The energy eigenvalues are found and the corresponding wave functions are calculated in terms of hyper-geometric functions.

Equations

$$\left\{\underbrace{\left(p_x+i{m}_0\omega x\right)\left(p_x-i{m}_0\omega x\right)}+\underset{\cap }{\underbrace{\left(p_y+i{m}_0\omega y\right)\left(p_y-i{m}_0\omega y\right)}}-\frac{E^2-m_0^2{c}^4}{c^2}\right\}{\psi }_{KG}=0,$$

$$\cup$$

$$=p_x^2+m_0^2{\omega }^2{x}^2-m_0\omega \hslash ,$$

$$\cap$$

$$=p_x^2+m_0^2{\omega }^2{x}^2-m_0\omega \hslash .$$

$$x=i\hslash \frac{\partial }{\partial p_x},y=i\hslash \frac{\partial }{\partial p_x},$$

$${\hat{p}}_x=p_x,{\hat{p}}_y=p_y,$$

$$p_x=p\cos \theta ,p_y=p\sin \theta ,\text{with}{p}^2=p_x^2+p_y^2,$$

$$\hat{x}=i\hslash \frac{\partial }{\partial p_x}=i\hslash \left(\cos \theta \frac{d}{dp}-\frac{\sin \theta }{p}\frac{d}{d\theta }\right),$$

$$\hat{y}=i\hslash \frac{\partial }{\partial p_y}=i\hslash \left(\sin \theta \frac{d}{dp}+\frac{\cos \theta }{p}\frac{d}{d\theta }\right),$$

$$\cup =p^2{\cos }^2\theta -m_0^2{\omega }^2{\hslash }^2\frac{{\partial }^2}{\partial p_x^2}-\hslash m_0\omega ,$$

$$\cap =p^2{\sin }^2\theta -m_0^2{\omega }^2{\hslash }^2\frac{{\partial }^2}{\partial p_y^2}-\hslash m_0\omega .$$

$$\left(p^2-m_0^2{\omega }^2{\hslash }^2\left(\frac{{\partial }^2}{\partial p_x^2}+\frac{{\partial }^2}{\partial p_y^2}\right)-2\hslash m_0\omega -\varsigma \right){\psi }_{KG}=0.$$

$$\frac{{\partial }^2}{\partial p_x^2}+\frac{{\partial }^2}{\partial p_y^2}=\frac{{\partial }^2}{\partial p^2}+\frac{1}{p^2}\frac{{\partial }^2}{\partial {\vartheta }^2}+\frac{1}{p}\frac{\partial }{\partial p},$$

$$\left\{p^2-{\lambda }^2\left(\frac{{\partial }^2}{\partial p^2}+\frac{1}{p^2}\frac{{\partial }^2}{\partial {\theta }^2}+\frac{1}{p}\frac{\partial }{\partial p}\right)-2\lambda -\varsigma \right\}{\psi }_{KG}=0,$$

$$\varsigma =\frac{E^2-m_0^2{c}^4}{c^2}.$$

$$\frac{\partial }{\partial p_x}$$

$$\frac{\partial }{\partial p_y}$$

$${\psi }_{KG}\left(p,\theta \right)=f\left(p\right)e^{i|l|\theta },$$

$$\left(\frac{d^{2}f\left(p\right)}{d{p}^2}+\frac{1}{p}\frac{df\left(p\right)}{dp}-\frac{l^2}{p^2}f\left(p\right)\right)+\left({\kappa }^2-k^2{p}^2\right)f\left(p\right)=0,$$

$${\kappa }^2=\frac{2\lambda +\zeta }{{\lambda }^2},k^2=\frac{1}{{\lambda }^2}.$$

$$f\left(p\right)=p^{|l|}{e}^{-\frac{k}{2}{p}^2}F\left(p\right),$$

$$F^{{}^{\prime \prime }}+\left(\frac{2\left|l\right|+1}{p}-2kp\right)F^{{}^{\prime }}-\left[2k\left(\left|l\right|+1\right)-{\kappa }^2\right]F=0,$$

$$t\frac{d^{2}F}{d{t}^2}+\left\{\left|l\right|+1-t\right\}\frac{dF}{dt}-\frac{1}{2}\left\{\left|l\right|+1-\frac{{\kappa }^2}{4k}\right\}F=0,$$

$$a=\frac{1}{2}\left(\left|l\right|+1\right)-\frac{{\kappa }^2}{4k}.$$

$${\psi }_{KG}\left(p,\theta \right)=C_{n,|l|}{p}^{|l|}{e}^{-\frac{k}{2}{p}^2}{}_1{F}_1\left(-n;\left|l\right|+1;k{p}^2\right)e^{i|l|\theta },$$

$$E_n=\pm m{c}_0^2\sqrt{1+2rN}.$$

$$\left[{\hat{x}}_i,{\hat{p}}_i\right]=i\hslash {\delta }_{ij}\left(1+\beta p^2\right),$$

$$\hat{x}=i\hslash \left(1+\beta p^2\right)\frac{d}{d{p}_x},{\hat{p}}_x=p_x,$$

$$\hat{y}=i\hslash \left(1+\beta p^2\right)\frac{d}{d{p}_y},{\hat{p}}_y=p_y.$$

$$\left\{p_x^2+p_y^2+m^2{\omega }^2\left(x^2+y^2\right)+im\omega \left[x,p_x\right]+im\omega \left[y,p_y\right]-\varsigma \right\}{\psi }_{KG}=0.$$

Full text

I introduction

Recently, there have been growing interest in obtaining exact solutions of relativistic wave equations. In particular exact solutions of the Klein–Gordon equation with various vector and scalar potentials.

The Dirac relativistic oscillator is an important potential both for theory and application. It was for the first time studied by Ito et al(1, 1). They considered a Dirac equation in which the momentum $\vec{p}$ is replaced by $\vec{p}-im\beta\omega\vec{r}$, with $\vec{r}$ being the position vector, $m$ the mass of particle, and $\omega$ the frequency of the oscillator. The interest in the problem was revived by Moshinsky and Szczepaniak (2, 2), who gave it the name of Dirac oscillator (DO) because, in the non-relativistic limit, it becomes a harmonic oscillator with a very strong spin-orbit coupling term. Physically, it can be shown that the (DO) interaction is a physical system, which can be interpreted as the interaction of the anomalous magnetic moment with a linear electric field (3, 3, 4). The electromagnetic potential associated with the DO has been found by Benitez et al (5, 5). The Dirac oscillator has attracted a lot of interest both because it provides one of the examples of the Dirac’s equation exact solvability and because of its numerous physical applications (see (6, 6) and reference therein). Recently, Franco-Villafane et al (7, 7) exposed the proposal of the first experimental microwave realization of the one-dimensional DO. The experiment relies on a relation of the DO to a corresponding tight-binding system. The experimental results obtained, concerning the spectrum of the one-dimensional DO with and without the mass term, are in good agreement with those obtained in the theory. In addition, Quimbay et al (8, 8, 9) show that the Dirac oscillator can describe a naturally occurring physical system. Specifically, the case of a two-dimensional Dirac oscillator can be used to describe the dynamics of the charge carriers in graphene, and hence its electronic properties. Also, the exact mapping of the DO in the presence of a magnetic field with a quantum optics leads to consider the DO as a theory of an open quantum systems coupled to a thermal bath ( see Ref (6, 6) and references therein).

The unification between the general theory of relativity and the quantum mechanics is one of the most important problems in theoretical physics. This unification predicts the existence of a minimal measurable length on the order of the Planck length. All approaches of quantum gravity show the idea that near the Planck scale, the standard Heisenberg uncertainty principle should be reformulated. The minimal length uncertainty relation has appeared in the context of the string theory, where it is a consequence of the fact that the string cannot probe distances smaller than the string scale $\hbar\sqrt{\beta}$, where $\beta$ is a small positive parameter called the deformation parameter. This minimal length can be introduced as an additional uncertainty in position measurement, so that the usual canonical commutation relation between position and momentum operators becomes$\left[\hat{x},\hat{p}\right]=i\hbar\left(1+\beta p^{2}\right)$. This commutation relation leads to the standard Heisenberg uncertainty relation $\triangle\hat{x}\triangle\hat{p}\geq i\hbar\left(1+\beta\left(\triangle p\right)^{2}\right)$, which clearly implies the existence of a non-zero minimal length $\triangle x_{\mbox{min}}=\hbar\sqrt{\beta}$. This modification of the uncertainty relation is usually termed the generalized uncertainty principle (GUP) or the minimal length uncertainty principle(11, 11, 12, 13, 14).

Nowadays, the reconsideration of the relativistic quantum mechanics in the presence of a minimal measurable length have been studied extensively. In this context, many papers were published where a different quantum system in space with Heisenberg algebra was studied. They are: the Abelian Higgs model (15, 15), the thermostatics with minimal length (16, 16), the one-dimensional Hydrogen atom (17, 17), the casimir effect in minimal length theories (18, 18), the effect of minimal lengths on electron magnetism (19, 19), the Dirac oscillator in one and three dimensions (20, 20, 21), the solutions of a two-dimensional Dirac equation in presence of an external magnetic field (22, 22), the non-commutative phase space Schrodinger equation(23, 23), Schrodinger equation with Harmonic potential in the presence of a Magnetic Field (24, 24).

The purpose of this work is to investigate the formulation of a two-dimensional Klein-Gordon oscillator by solving fundamental equations in the framework of relativistic quantum mechanics with minimal length. The problem describes a relativistic particle moving in the relativistic harmonic oscillator called the Dirac oscillator.

The paper is organized as follows. In section II, we exposed the solutions of our problem within habitual quantum mechanics, using the new method developed by Menculini et al (10, 10) and Jana et al(25, 25). Then, section III will be devoted to the our case, i.e., the solution of a two dimensional Klein-Gordon oscillator in the framework of relativistic quantum mechanics with minimal length. Finally, in section V, we present the conclusion.

II The solutions within habitual quantum mechanics

A two-dimensional Klein-Gordon oscillator is

[TABLE]

with

[TABLE]

[TABLE]

Now, for the sake of simplicity, we bring the problem into the momentum space. Recalling that

[TABLE]

[TABLE]

and passing onto polar coordinates with the following definition (10, 10)

[TABLE]

[TABLE]

[TABLE]

Eqs. (2) and (3) transform into

[TABLE]

[TABLE]

In this case, Eq. (1) becomes

[TABLE]

By using Eqs. (6), (7) and (8), we have

[TABLE]

and consequently, we obtain

[TABLE]

with $\lambda=m\omega\hbar$ and

[TABLE]

Here we have used that

[TABLE]

[TABLE]

Now, when we choose

[TABLE]

Eq. (13) is transformed into

[TABLE]

with

[TABLE]

Now, Putting(25, 25)

[TABLE]

the differential equation

[TABLE]

is obtained for $F\left(p\right)$ which by using, instead of $p$, the variable $xt=kp^{2}$, is transformed into the Kummer equation

[TABLE]

whose solution is the confluent series ${}_{1}F_{1}\left(a;\left|l\right|+1;t\right)$, with

[TABLE]

The confluent series becomes a polynomial if and only if $a=-n,\,\left(n=0,1,2,\right)$. We then have the solutions

[TABLE]

[TABLE]

with $N=2n+\left|l\right|$ is the principal quantum number, and $r=\frac{\hbar\omega}{m_{0}c^{2}}$. This form of energy is in a good agreement with that obtained in the literature (11, 11).

III The solutions in the presence of a minimal length

In the minimal length formalism, the Heisenberg algebra is given by (12, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24)

[TABLE]

where $\beta>0$ is the minimal length parameter. A representation of $\hat{x}_{i}$ and $\hat{p}_{i}$ which satisfies Eq. (25), may be taken as

[TABLE]

[TABLE]

In this case, the KG oscillator equation is

[TABLE]

By using the equations (26) and (27), Eq. (28) becomes

[TABLE]

Now, when we put that

[TABLE]

with $j=0,\pm 1,\pm 2,\ldots$, then Eq. (29) transformed into

[TABLE]

with

[TABLE]

Now, according to the following substitution (25, 25)

[TABLE]

[TABLE]

we have

[TABLE]

with

[TABLE]

We note here that the function $\rho(p)=p^{-\frac{1}{2}}$.

Now, if we put

[TABLE]

the term $V\left(p\right)$ becomes

[TABLE]

and consequently, the final form of our differential equation is

[TABLE]

where $\bar{\varsigma}=\varsigma+\frac{1}{\beta}$. Eq. (39) brings into

[TABLE]

where

[TABLE]

with

[TABLE]

[TABLE]

Thus, we have

[TABLE]

where $U_{0}=\alpha^{2}$ with $\alpha=\lambda\sqrt{\beta}$. Eq. (44) is the well-known Schrodinger equation in a Pöschl-Teller potential with (25, 25)

[TABLE]

and with the following conditions $\zeta_{1}>1$ and $\zeta_{2}>1$.

By comparison Eq. (39) with Eq. (44), we have

[TABLE]

[TABLE]

Now, in order to solve Eq. (39), we introduce the new variable

[TABLE]

In this case, Eq. (39) can be written by

[TABLE]

With the new wave function $\varphi$, defined by

[TABLE]

we arrive at

[TABLE]

The general solution of this equation is

[TABLE]

with

[TABLE]

With the condition $a=-n$, we obtain

[TABLE]

In order to obtain the energy spectrum, it should be to note that in the limit $\beta\rightarrow 0$, the energy spectrum should be covert to no-GUP result.

Thus, the exact form of $\zeta_{1}$ and $\zeta_{2}$ are

[TABLE]

[TABLE]

where $j\neq 0$.

With the aid of Eqs. (46), (47) and (54), we obtain the final form of the spectrum of energy, which is given by

[TABLE]

with

[TABLE]

We can see that where $\beta\rightarrow 0,$ $\Sigma\rightarrow 1$, and the energy spectrum is covert to no-GUP result (Eq. (24)).

The corresponding wave function is

[TABLE]

with $N$ is the constant of normalization.

IV Conclusion

In this paper, we have exactly solved the Klein-Gordon oscillator in two dimensions in the framework of relativistic quantum mechanics with minimal length. The eigensolutions are obtained using a method developed in refs. (10, 10, 25) to solve a two-dimensional Dirac equation and Klein-Gordon equations. We firstly consider the case of the Klein-Gordon oscillator within the ordinary quantum mechanics: our results are in good agreement with those obtained in the literature. Then we have extended it in the case of the presence of a minimal length. The energy levels, for both cases, show a dependence on $N^{2}$ in the presence of the minimal length, which describes a hard confinement. In the limit where $\beta\rightarrow 0$, we recover the energy spectrum of no-GUP.