Abstract
In the early development of quantum mechanics in two different forms, as ‘matrix mechanics’ by Heisenberg and as ‘wave mechanics’ by Schrödinger, it was soon realized, in particular by mathematicians, that a unified formulation of the theory could be given in terms of Hilbert space concepts.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1996) ((Translated from the original 1932 German Springer edition) A great classic, written by the creator of the mathematical formalism of quantum mechanics. The part on the theory of measurement may have been overtaken by more recent developments, but the general discussion of quantum theory, in particular of selfadjoint operators and the spectral theorem remains instructive and fresh. https://app.box.com/s/fsp81filfebei98umd6u)
Reed, M., Simon, B.: Methods of Mathematical Physics. I: Functional Analysis. Academic Press, New York (1972) (This book is because of its readability our main reference for this chapter. A textbook on functional analysis, with the theory of operators in Hilbert space as its central topic, especially written for applications in mathematical physics. It is the first of a series of four books by the same authors. The other volumes are more specialized, but contain nevertheless useful material, in particular II and IV.)
Schmüdgen, K.: Unbounded Operators in Hilbert Space. Springer, Hardcover (2012) (The best recent book on the theory of unbounded operators.)
Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space Translated from the Russian Dover (1993). A classical standard reference on Hilbert space theory. The first five chapters of this book can be found at http://www.stat.wisc.edu/~wahba/stat860/pdf2/ag1.35.pdf
Prugovecki, E.: Quantum Mechanics in Hilbert Space, 2nd edn. Dover Publications, New York (1981) (Even though its general approach may be somewhat dated, it contains a wealth of precise and clearly formulated additional material on the mathematical aspects of quantum theory. The modest price of the Dover edition makes it especially worthwhile.)
Dunford, N., Schwartz, J.T.: Linear Operators, vol. I–III. Wiley-Interscience, New York (1957, 1963, 1971) (Reissued as paperbacks in 1988, at outrageous prices. A formidable set, a monument of scholarship. Maybe not suitable for quick browsing, but it remains, after 40 years, an indispensable tool for serious research in linear operator theory.)
Helmberg, G.: Introduction to Spectral Theory in Hilbert Space North-Holland 1969. Reissued by Dover (2008) A leisurely and clear introduction
Gustafson, S.J., Michael, I.: Sigal Mathematical Concepts of Quantum Mechanics Springer, Berlin (2006) (The level of mathematical rigour in this book is variable; however its merit is that it discusses a number of interesting advanced topics)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bongaarts, P. (2015). Functional Analysis: Hilbert Space. In: Quantum Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-09561-5_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-09561-5_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09560-8
Online ISBN: 978-3-319-09561-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)