Derivation of Euler- Lagrangian Quantum Equations for Frictional Medium to Explain Lasing and Absorption beside Concept of Work

Abstract

Euler-Lagrangian formalism was utilized in this work to obtain Schrödinger, Klein-Gordon and Dirac equation for frictional medium. Fortunately these equations are typical to that obtained by others. These equations were solved for lasing and absorption conditions. These conditions show that lasing and absorption take place due to the presence of frictional relaxation time term. This conforms to common sense and physical reality, since friction effect enables the medium to absorb energy from the incident beam. This absorbed energy can cause population inversion and lasing. The ambiguous definition of the Lagrangian and kinetic energy make the concept and notion of the principle of least action misleading and confusing. In this work the Lagrangian and kinetic energy are defined in terms of work. This gives the principle of least action its real physical meaning as reflecting the tendency of physical systems to select a trajectory which consumes less work. The generalized definition of kinetic energy in terms of the work is compatible with that of Newton, Einstein and generalized special relativity. It is very striking to note that the compatibility of this definition with the notion of force requires introducing the mass as an extra dimension extending our 4 space-time coordinates to be having 5 coordinates.