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arXiv:1701.07076

Operator of Time and Properties of Solutions of Schroedinger Equation for Time Dependent Hamiltonian

Slobodan Prvanović, D. Arsenović

Published Jan 24, 2017• 3 citations
arXivHTMLPDF

Abstract

Within the framework of self-adjoint operator of time in non-relativistic quantum mechanics some properties of solutions of Schroedinger equation, related to Hilbert space formalism, are investigated for two types of time dependent Hamiltonian.
DOI
10.48550/arxiv.1701.07076

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Within the framework of self-adjoint operator of time in non-relativistic quantum mechanics the equation describing change of the state of quantum system with respect to energy is introduced. The operator of time appears to be the generator of the change of energy while the operator of energy, that is conjugate to the operator of time, generates time evolution in this proposal. Two examples, one with discrete time and the other with continuous one, are given and the generalization of Schroedinger equation is proposed.
We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation.
Let HHH be a self-adjoint operator (a Hamiltonian) on a complex Hilbert space $ \mathcal{H}$
ABSTRACT: The main aim of this article is to present the time-evolution as the unitary and self-adjoint operators on Hilbert spaces and to describe its application in the development of quantum mechanics. The initial value problems associated with the quantum mechanical Schrodinger equation, ( ) = ( ) in the Hilbert space is solved by the use of timeevolution. The importance of time-evolution is also seen as the operation of turning machine in quantum mechanics, which is regarded as the time-evolution of the machine control state. Time-evolution of a quantum mechanical system is observed to be unitary and self-adjoint operators of Hilbert space, since it corresponds to an observable, specifically energy position and momentum. It was observed that time-evolution of a quantum mechanical system is generated by a self-adjoint operator, called Hamiltonian, V K H ˆ ˆ ˆ   , expressed by the Schrodinger equation, above.
We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product.