Derivation of Schrodinger and Dirac Equations

The Schrodinger equation can be derived from first principles, contrary to the assertions offered in textbooks. To be consistent with the framework of quantum mechanics the dynamics must be based on the averaged energy relation. Accordingly, each eigenfunction is a specific superposition of plane wave states that fulfills the averaged energy relation. Alternatively, starting with the averaged energy relation, the Schrodinger equation is obtained to be the condition the particle eigenfunction must satisfy, at each space-time point, in order to fulfill the averaged energy relation. The same method is applied to derive the Dirac equation involving potential terms. We emphasize the difference between the free particle equations (which have single plane wave eigenfunctions) and the complete Schrodinger and Dirac equations, and we point out the inappropriateness of the heuristic derivations.