Vector and Hilbert Spaces
The purpose of the chapter is to introduce Hilbert spaces, and more precisely the Hilbert spaces on the field of complex numbers, which represent the abstract environment in which Quantum Mechanics is developed. To arrive at Hilbert spaces, we proceed gradually, beginning with vector spaces, then considering inner-product vector spaces, and finally Hilbert spaces. The Dirac notation will be introduced and used. A particular emphasis is given to Hermitian and unitary operators and to the class of projectors. The eigendecomposition of such operators is seen in great detail. The final part deals with the tensor product of Hilbert spaces, which is the mathematical environment of composite quantum systems.
- 1.S. Roman, Advanced Linear Algebra (Springer, New York, 1995)Google Scholar
- 2.G. Cariolaro, Unified Signal Theory (Springer, London, 2011)Google Scholar
- 3.P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford, 1958)Google Scholar
- 4.R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1998)Google Scholar
- 5.M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)Google Scholar