Time-Evolution as an Operator in Hilbert Spaces with Applications in Quantum Mechanics

Dr. A. K. Choudhary; S. Musa; S. M. Nengem; Dr. S. K. Choudhary

doi:https://doi.org/10.14445/22315373/IJMTT-V10P511

International Journal of Mathematics Trends and Technology

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Volume 10 | Number 2 | Year 2014 | Article Id. IJMTT-V10P511 | DOI : https://doi.org/10.14445/22315373/IJMTT-V10P511

Time-Evolution as an Operator in Hilbert Spaces with Applications in Quantum Mechanics

Dr. A. K. Choudhary , S. Musa , S. M. Nengem , Dr. S. K. Choudhary

Citation :

Dr. A. K. Choudhary , S. Musa , S. M. Nengem , Dr. S. K. Choudhary, "Time-Evolution as an Operator in Hilbert Spaces with Applications in Quantum Mechanics," International Journal of Mathematics Trends and Technology (IJMTT), vol. 10, no. 2, pp. 60-69, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V10P511

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, i (dψ(t))/dt=Hψ(t) in the Hilbert space is solved by the use of time-evolution. 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, , expressed by the Schrodinger equation, above.

Keywords

Time-evolution, self-adjoint operator, unitary operator, Hamiltonian operator, observable, commutator relation, Quantum mechanics.

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