Time Dependant Momentum Operator and Time Evolution of Free Quantum Particles

In spite of outstanding achievement of the quantum theory, the theory is under never-ending judgment due to probabilistic character and certain unresolved paradoxes arising from the very fundamentals. Additionally, it is debated that a more sophisticated theory beyond the quantum theory emerges as a need for a better understanding of the atomic and sub atomic worlds. This paper, for that reason, focuses on the very fundamental structure of the quantum theory such as the physical and philosophical meanings of the wave functions, the probability and the Schrödinger Wave Equation (SWE). Specifically deals with the time evolution of the free quantum particles. The work just arises from the difference of the classical and quantum waves, specifically the comparison of the time and space dependence of classical waves and Schrödinger waves, for the free space, indicates that time dependence shows a clear dissimilarity. In order to tackle the problem, a time dependent momentum operator, for the first time, is defined and consequently an alternative TDSWE is straightforwardly derived and its philosophical implications have been discussed. Additionally, time and position dependence of the probability functions, relating free particles, have been physically and philosophically discussed and some alternative approaches and comments are forwarded. Finally, the approaches, comments and mathematical equations are in good agreement with the standard quantum theory and interestingly go further possibly tracing some signs of new pathways to resolve the mysterious quantum world. Keywords: Schrödinger wave equation, fundamentals of quantum mechanics, momentum operator, quantum probabilities, time evolution of quantum systems