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

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.

[1] Atkins, P.W. and Averson, W. (1974).Quanta: A handbook of concepts.Oxford university press. ISBN 0 -19- 855493-1.

[2] Bransden, B.H. and Joachain, J.C. (1983). Physics of Atoms and molecules. Longman. ISBN 0-582-44401-2.

[3] Davies, E. B (1996). Spectral theory and differential operators . Cambridge University Press.

[4] Grants, I. S. And Philips, W. R.(2008): Electromagnetism (2nd edition). Manchester Physics series. ISBN 0-471-92712-0.

[5] Kreyszig, E. (1978).Introductory Functional Analysis with applications. New York , John Wiley and sons ,Inc.

[6] Resnick,R. And Eisberg R. (1985). Quantum Physics of Atoms, molecules, solids, Nuclei and Particles (2nd edition).John Wiley and sons, New York.ISBN 978-0-471-87373-0.

[7] Pankov, A. (2001). Introduction to spectral theory of Schrӧdinger operators. State pedagogical University Vinnitsa Ukraine.

[8] Park,C.B (1994).McGraw Hill. Encyclopedia of Physics (2n ed). McGraw Hill.pp.786, 1261.

[9] Peleg, Y., Pnini, R., Zaarur, E. and Hencht, E.(2010). Quantum Mechanics. Schuanm’s outline series (2n ed). McGraw Hill.pp 70.

[10] Teschl, G. (2009). Mathematical methods in quantum mechanics: with applications to Schrӧdinger operator.American mathematical society.

[11] Yoav, P., P. Reuven and Z. Elyahu, (1988). Schaum’s outlines of quantum mechanics. Tata MCgraw Hill publishing company limited New York.

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