Quantum mechanics can only provide the probability of finding moving particles at the atomic level. The Schrodinger equation provides the information of a wavefunction, which gives only the quantitative prediction of probability of finding a moving particle at a particular location. Quantum mechanics appears as a strange theory, it can provide a powerful tool to explain many observed phenomena at the atomic level. Modeling of quantum mechanics using the Schrodinger equation is only one of the choices. Quantum mechanics can only provide information about moving particles in a statistical sense. For the case of “light waves,” defines the wavelength of a photon of light. This is also known as the quantum theory of light. In quantum mechanics, the wave function in the Schrodinger equation is associated with the likelihood of finding the particle at a particular time and at a particular place.
Paraxial light and atom optics are compared. To lowest order the slowly varying amplitude of a light field in a dielectric medium with a spatially dependent refractive index satisfies an equation which has the form of a Schr\"odinger equation: the ``optical Schr\"odinger equation.'' The unsystematic procedure of neglecting certain second-order derivatives is replaced by a systematic expansion which allows the calculation of consecutive corrections. The general theory is applied to harmonic motion in a graded index fiber and to tunneling between coupled fibers. The physical relations between wave evolution of massive particles and paraxially propagating waves are elucidated.
It is not possible to use the Schrödinger equation to describe the motion of a particle with relativistic effects since it is not invariant relative to the Lorentz transformation. However, the attempt to generalize quantum mechanics in the relativistic domain on the basis of the one particle theory is somehow difficult. So we find the appropriate equations on the basis of the relativistic invariance. To obtain the non-relativistic equations, we take the limit as the speed of light tends to infinity. This obviously recovers the Schrödinger equation. Under this heading, we extend the non-relativistic Schrödinger equation in the description of particles with speeds comparable with that of light. This extension should be consistent with the Lorentz transformation [ 12 ]. The characteristic feature of the relativistic wave equations is the spin which provides a useful gauge in the applicability of particular equations that describe particular kinds of particles. We have two types of relativistic equations, i.e., the equation with spin-zero due to Schrödinger and spin-half due to Dirac that describes the electron.