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Theoretical and Mathematical Physics

Stationary Schrödinger equation in nonrelativistic quantum mechanics and the functional integral

Гарий Владимирович Ефимов

Published Jun 1, 2012• 4 citations
Publisher
DOI
10.1007/s11232-012-0077-7

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The momentum operator may be generalized for the correspondence of the limit of quantum commutators of observables with the classical commutators. The substitution of the momentum operator into the nonrelativistic equation for the conservation of energy yields a generalization of the stationary wave equation. Replacing also the energy operator, a generalization of the time-dependent Schrödinger equations, there exists a special class of wavefunctions that satisfies the nonrelativistic wave equation in quantum theory.
We will extend Nonrelativistic Quantum Mechanics as a theory in $L\sp2$ to a theory in the space of distributions. We will provide 3 major theories of Nonrelativist Quantum Mechanics. First, we will extend the concept of an integral kernel for the evolution operator to a distribution kernel for the $L\sp2$ transition probability amplitude. Second, we will extend the $L\sp2$ Schrodinger's equation to a distributions Schrodinger's equation. Lastly, we will rigorously prove that; Feynman's original formulation of the real time, time-sliced path integral is well defined when formulated on the $L\sp2$ transition probability amplitude.
In this paper, we analyze, by using a matrix approach, the dynamics of a nonrelativistic particle in presence of a quaternionic potential barrier. The matrix method used to solve the quaternionic Schrödinger equation allows us to obtain a closed formula for the transmission coefficient. Up to now, in quaternionic quantum mechanics, almost every discussion on the dynamics of nonrelativistic particle was motivated by or evolved from numerical studies. A closed formula for the transmission coefficient stimulates an analysis of qualitative differences between complex and quaternionic quantum mechanics and by using the stationary phase method, gives the possibility to discuss transmission times.