Spatial Evolution in the Interaction Model and Schrodinger Equation in a Curved Space-Time

Abstract

This study is concerned with derivation of quantum relation in a curved space –time . The importance of this study emerges from the use of energy and momentum in many applications. This study use mathematical derivation trend.

The research problem is related to the lack of suitable consistent relations for quantum eigen equations in a curved space , beside lack of relations for the special quantum evolution in the interaction model. Therefore this research aims to use the expression of time and distance in a curved space time to find a useful expression of energy and momentum Eigen equations in a curved space .These relation are used to derive the corresponding relation in the Euclidean space . The corresponding Energy –momentum relation for both curved and Euclidian space gives a relation between energy and momentum typical to that obtained from the energy and momentum Eigen equation . The expression of mass in a curved space similar to that of the generalized relativity is also found.

Using generalized special relativity a useful expression of the perturbed momentum is found . This expression is used to describe the behavior of the quantum system in the interaction model . It is found that the spatial evolution of the Schrodinger equation in the interaction model is similar to that of time evolution , where the time differential is replaced by the space one, and the Hamiltonian by the momentum operator . The same holds for the unitary operator , where the time integral is replaced by the space one and the Hamiltonian with the momentum operator . The unitary operator in the Heisenberg picture and the spatial evolution of the quantum system was found by using simple mathematics and the ordinary laws of differentiation and integration .This expression describes successfully the spatial evolution of the quantum operator . The metrics in the curved space is found to be related to the Lorentz transformation coefficient .