A generalisation of Schroedinger's wave mechanics for relativistic regions of validity

A scalar wave equation is derived which can be applied to the relativistic motion of charged particles in slowly varying electromagnetic fields and which is valid in a sufficiently small energy interval, if spin effects can be neglected. Since this wave equation is a linear partial differential equation of first order in the time derivative but of second order in the space derivatives, it is not relativistically invariant. But it is possible to define a positively definite particle density from which a conservation law can be derived. The main success of this theory is a very simple generalisation of the Schrödinger theory of classical wave mechanics in such a way that it can be also applied to relativistic particles.