Review of Mathematical Modelling of the Time Dependent Schrodinger Wave Equation using Different methods


Osang J. E., Adedokun I. O., Isreal-Cookey C., Egor A. O., Uquetan U. I., Ekpo C. M., Umuji J. I.

This paper focuses on the principle of time dependent Schrödinger equation (TDSE) which simplified so many reviewed ideas, class room teaching, research ideologies and personal study meant for advanced knowledge. In this research work,  the enlightenment of the basic concept of Schrödinger wave equation to improve knowledge about simple ways for mathematical understanding in deriving TDSE using different technique in more comprehensive approach. The research shows clearly that TDSE can be derived using time independent equation, wave mechanics, classical & Hamilton-Jacobi’s equations. Different methods and ways by different researchers/scholars have been used in the past. In this paper, we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational. Surely, a particle is what a particle detector detects. This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation. However, the result provides a comprehensive well derived derivation, derived using various approaches, which would make this research a unique one from different areas of specializations.


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