A new equation for the relativistic scalar field is derived from the correspondence with the classical commutator and covariance. It has the same form as the Klein Gordon equation with different coefficients.
A covariant formalism of Relativistic Quantum Mechanics is demonstrated, through it's de- velopment and application. The Relativistic Case is shown to follow a similar structure to the established Non-Relativistic formalism. Reasons for preferring the new covariant formalism over the established method are presented. Solutions to the case of a scalar particle in a one-dimensional field are presented. The Relativistic Energy Eigenfunction is derived. Results are generated from initial Gaussian states via a Green's Function method. A Green's Function for the system is derived and applied. The solution to the Quantum System is shown to follow a scaled version of the classical path. The equivalent system in the Non-Relativistic case is also analysed. The Energy Eigenfunc- tion for the time-independent Schrodinger Equation is derived. A Green's Function is derived for the time-dependent SchrÄdinger Equation and solutions to the system are found using intial Gaussian States. A short demonstration of the established method, that of the Klein-Gordon equation, is provided. Some deficiencies in the Klein-Gordon equation are shown. These deficiencies are shown to not exist in the covariant formalism.