Relativistic massless harmonic oscillator

A detailed study of the relativistic classical and quantum mechanics of the massless harmonic oscillator is presented.

Figure 1

The periodic solution of the system (2.2) obtained by numerical integration. The initial data are ${\mathit{x}}_0=(0.479000,0.000000)$ m, ${\mathit{p}}_0=(0.000000,1.290805)$ Js${\mathrm{m}}^{-1}$, the parameter ${\kappa }^2=1$ $\mathrm{J}{\mathrm{m}}^{-2}$, and $c=1$ $\mathrm{m}{\mathrm{s}}^{-1}$.

Figure 2

The plot of the radius $r=\left|\mathit{x}\right|$ (solid line) and the length of momentum $p=\left|\mathit{p}\right|$ (dotted line) versus time obtained by the numerical integration of (2.11). The initial condition is the same as in Fig. 1.

Figure 3

The plot of the coordinate (solid line) and the momentum (dotted line) of a massless oscillating particle versus time given by Eqs. (2.23) and (2.24), respectively, where $c=1$ $\mathrm{m}{\mathrm{s}}^{-1}$, ${\kappa }^2=1$ $\mathrm{J}{\mathrm{m}}^{-2}$, $r_{\max }=1$ m, and $T=4$ s.

Figure 4

The plot of the quantum probability density ${\rho }_n(r)=|{\psi }_n(\mathit{x})|{}^2$ (solid line), where ${\psi }_n(\mathit{x})$ is the wave function (3.12) and the classical probability density ${\rho }_{\mathrm{cl}}(r)$ (dashed line) given by (3.13), where $r_{\max }=r_{\max }(E_n)=\frac{\sqrt{2E_n}}{\kappa }$. We set $c=1$ $\mathrm{m}{\mathrm{s}}^{-1}$, ${\kappa }^2=1$ $\mathrm{J}{\mathrm{m}}^{-2}$, and $\hslash =1$ Js.