Schrödinger's equation, which is a fundamental equation in non-relativistic quantum mechanics, is not valid for relativistic particles because it does not incorporate the principles of special relativity. Schrödinger's equation was developed before the discovery of relativistic effects and is based on non-relativistic concepts such as a fixed and independent time variable.
To make the equation applicable to relativistic particles, we need to incorporate the principles of special relativity, which describe the behavior of objects moving at speeds close to the speed of light. This led to the development of a different equation known as the relativistic wave equation or the Dirac equation.
The Dirac equation is a relativistic wave equation that describes the behavior of particles with spin, such as electrons, and it combines quantum mechanics and special relativity. It was formulated by Paul Dirac in 1928 as an extension of Schrödinger's equation.
The Dirac equation includes terms that account for relativistic effects, such as time dilation and the variation of mass with velocity. It also introduces the concept of antiparticles, allowing for the description of particles and their corresponding antiparticles, such as the electron and the positron.
Therefore, to make Schrödinger's equation applicable to relativistic particles, we need to replace it with the Dirac equation, which incorporates the principles of both quantum mechanics and special relativity.