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relativistic massless harmonic oscillator we also provide a graphical presentation of numerical integration of equations of motion. Section III is devoted to the quantization of the massless relativistic harmonic oscillator. 

## II. THE ANALYSIS OF THE CLASSICAL RELATIVISTIC MASSLESS HAR- MONIC OSCILLATOR 

The Hamiltonian of the relativistic massless particle subject to the potential 1 $_{2}$κ $^{2}$x2 is given by 

<img src='content_image/343081.jpg'>

where x and p are the position and the momentum of a particle, | p | = √ p 2 is the norm of the vector p (so c | p | is the kinetic energy of the particle), κ is a constant and c is the speed of light. Therefore, the Hamilton’s equations are 

<img src='content_image/343080.jpg'>

We point out that an immediate consequence of Eq. (2.2) is ˙ x 2 = c $^{2}$, that is the length of velocity is c as should be for a massless particle. The familiar integrals of the motion in a central field [18] are the energy E and the angular momentum J : 

<img src='content_image/343078.jpg'>

<img src='content_image/343079.jpg'>

As a result of the conservation of the angular momentum J the motion is planar and we can restrict, without loss of generality, to the case of a particle moving in the ( x $^{1}$, x$^{2}$) plane. On passing to the polar coordinates x = ( x $^{1}$, x$^{2}$) = ( r cos ϕ, r sin ϕ ) and p = ( p $^{1}$, p$^{2}$) = ( p cos θ, p sin θ ), where r = | x | , and p = | p | , we obtain from Eq. (2.2) the following system: 

<img src='content_image/343077.jpg'>

