Abstract
Potential energy inherited by a system plays a significant role in the system analysis. The potential energy operators are not generalized till date. Each system has its unique basis to explain the same. This article provides a novel approach to determine a generalized potential operator for a particle using Poisson's equation in a set basis in accordance with the corresponding characteristics manifested by the unit elements of basis. The mathematics includes the fundamental definitions of quantum physics and geometrical inference of manifolds. We shall see that the operator itself is time variant and medium dependent, which in turn depicts the evolution of the system in different mediums.
Highlights
The set of basis vectors define the coordinate system, which in turn is the tool for determining the state of any observable
The objective of the article is to establish an operator in terms of position and time such that it could provide the potential energy of a particle at fed position and time
Let me first define a coordinate system, using which we would be achieving the objective of deriving a generalized potential operator in a mathematically comfortable manner
Summary
The set of basis vectors define the coordinate system, which in turn is the tool for determining the state of any observable. Mentioning a different system, a box attached to a spring and stretched by x units from the mean position is proposed to have a potential energy of 1 kx in its corresponding coordinate 2 system Various such cases give us a non-generalized potential operators which are specific to their own corresponding systems. The objective of the article is to establish an operator in terms of position and time such that it could provide the potential energy of a particle at fed position (of any specific coordinate system) and time Note that such an operator would signify the parameters of a specific system when acted over a function. Let me first define a coordinate system, using which we would be achieving the objective of deriving a generalized potential operator in a mathematically comfortable manner Later in section (7), the operator is mentioned in terms of universal position coordinate system as well
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