In this paper, the relativistic linear quantum simple harmonic oscillator problem is solved by the method of Fourier Hermite series to derive the exact analytical solutions of the relativistic linear quantum simple harmonic oscillator. The first profound physical result of this work is the discovery of indefinitely fine corrections to the well known sequence of Schrodinger's quantum mechanical eigenenergies which become significant as the oscillator moves faster and faster compared to the speed of light in vacuo, especially the subatomic and elementary particles. the result implies corresponding hitherto unknown results in the areas of theoretical and experimental physics of oscillations and vibrations, such as Solid State Physics, Statistical and Thermal Physics and Particle Physics.
