The Schrödinger equation, named after Austrian physicist Erwin Schrödinger, is a fundamental equation in quantum mechanics that describes the behavior of quantum systems. It is a non-relativistic equation because it does not take into account the effects of special relativity, which is a theory that describes the behavior of objects moving at speeds close to the speed of light.
The Schrödinger equation was developed in the 1920s, prior to the development of quantum field theory, which incorporates both quantum mechanics and special relativity. At the time, physicists were primarily concerned with understanding the behavior of non-relativistic particles, such as electrons in atoms or particles in a potential well.
The Schrödinger equation is based on the wave-particle duality of quantum mechanics, which treats particles as waves described by a mathematical quantity called the wave function. The equation determines how the wave function of a quantum system evolves over time and how it relates to the system's energy. It is derived by applying the principles of wave mechanics to the Hamiltonian of the system, which represents the total energy of the system.
While the Schrödinger equation successfully describes many phenomena in non-relativistic quantum mechanics, it fails to account for certain relativistic effects, such as the behavior of particles moving at high speeds or the creation and annihilation of particle-antiparticle pairs. To describe these phenomena, one needs to employ relativistic quantum mechanics or quantum field theory, which introduces additional mathematical formalism and is capable of handling the relativistic nature of particles.
In summary, the Schrödinger equation is non-relativistic because it was developed before the incorporation of special relativity into quantum mechanics. It remains a powerful tool for understanding and predicting the behavior of non-relativistic quantum systems, while relativistic effects require the use of more advanced formalisms.