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. "Time-Evolution as an Operator in Hilbert Spaces with Applications in Quantum Mechanics", *International Journal of Mathematical Trends and Technology (IJMTT). *V10:60-69 June 2014. ISSN:2231-5373. www.ijmttjournal.org. Published by Seventh Sense Research Group.

**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.**References**

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**Keywords**

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