Skip to main content
SpringerLink
Log in

Basics of Quantum Mechanics

  • Chapter
Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems

Part of the book series: Theoretical and Mathematical Physics ((TMP))

  • 2216 Accesses

Abstract

Quantum mechanics was started after works by Heisenberg (Z. Phys. 43:172, 1927), Schrödinger (Naturwissenschaften 48:52, 1935) and Dirac around 1925. Together with relativity theory it is the most fundamental theory in physics today. There are two important points in quantum mechanics: (1) Quantum mechanics is a statistical theory. (2) Every quantum system assigns a Hilbert space.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Accardi, L.: Topics in quantum probability. Phys. Rep. 71, 169–192 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  2. Accardi, L., Frigerio, A., Lewis, J.: Quantum stochastic processes. Publ. Res. Inst. Math. Sci. 18, 97–133 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  3. Accardi, L.: The probabilistic roots of the quantum mechanical paradoxes. In: Diner, S., Lochak, G., Selleri, F. (eds.) The Wave–Particle Dualism, pp. 297–330. Reidel, Dordrecht (1984)

    Google Scholar 

  4. Accardi, L.: Quantum Probability and Applications II: Proceedings Workshop Held Heidelberg. Springer, Berlin (1985)

    MATH  Google Scholar 

  5. Accardi, L., Lu, Y.G., Volovich, I.V.: Quantum Theory and Its Stochastic Limit. Springer, Berlin (2002)

    MATH  Google Scholar 

  6. Araki, H.: Mathematical Theory of Quantum Fields. Oxford University Press, London (1999)

    MATH  Google Scholar 

  7. Belavkin, V.P.: Conditional entropy and capacity of quantum channels. In: Proc. of VIII-th Conference on Coding Theory and Information Transmission, Moscow-Kuibishev, pp. 15–19 (1982)

    Google Scholar 

  8. Belavkin, V.P.: Quantum stochastic calculus and quantum nonlinear filtering. J. Multivar. Anal. 42, 171–201 (1992)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Belavkin, V.P., Ohya, M.: Quantum entropy and information in discrete entangled states. Infin. Dimens. Anal., Quantum Probab. Relat. Top. 4(2), 137–160 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bogoliubov, N.N., Logunov, A.A., Oksak, A.I., Todorov, I.T.: General Principles of Quantum Field Theory. Nauka, Moscow (1987)

    Google Scholar 

  11. Bush, P., Lahti, P., Mittelstaedt, P.: The Quantum Theory of Measurement. Springer, Berlin (1996)

    Google Scholar 

  12. Davies, E.B., Lewis, J.T.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Davies, E.B.: Quantum Theory of Open System. Academic Press, San Diego (1976)

    Google Scholar 

  14. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, London (1958)

    MATH  Google Scholar 

  15. Emch, G.G.: Algebraic Methods in Statistical Mechanics and Quantum Field Theory. Wiley, New York (1972)

    MATH  Google Scholar 

  16. Faddeev, L.D., Yakubovsky, O.A.: Lectures on Quantum Mechanics for Mathematical Students, 2nd edn. NITS, Moscow (2001)

    Google Scholar 

  17. Feynman, R.P.: The Feynman Lectures on Physics. Quantum Mechanics, vol. 3. Addison-Wesley, Reading (1965)

    MATH  Google Scholar 

  18. Gill, R.: On quantum statistical inference. J. R. Stat. Soc. B 65, 1–31 (2003)

    Article  MathSciNet  Google Scholar 

  19. Haag, R.: Local Quantum Physics. Fields, Particles, Algebras. Springer, Berlin (1996)

    MATH  Google Scholar 

  20. Heisenberg, W.: Principle of indeterminacy. Z. Phys. 43, 172 (1927)

    ADS  Google Scholar 

  21. Hiai, F., Petz, D.: The proper formula for relative entropy and its asymptotics in quantum probability. Commun. Math. Phys. 143, 99–114 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Series in Statistics and Probability, vol. 1. North-Holland, Amsterdam (1982)

    MATH  Google Scholar 

  23. Holevo, A.S.: Statistical Structure of Quantum Theory. Lect. Notes Phys., vol. 67. Springer, Berlin (2001)

    MATH  Google Scholar 

  24. Ingarden, R.S., Kossakowski, A., Ohya, M.: Information Dynamics and Open Systems. Kluwer Academic, Dordrecht (1997)

    MATH  Google Scholar 

  25. Khrennikov, A.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer Academic, Dordrecht (1997)

    MATH  Google Scholar 

  26. Khrennikov, A., Volovich, I.V.: Local realism, contextualism and loopholes in Bell’s experiments. In: Foundations of Probability and Physics-2. Ser. Math. Model., vol. 5, pp. 325–344. Vaxjo University Press, Vaxjo (2002)

    Google Scholar 

  27. Khrennikov, A., Volovich, I.V.: Einstein, Podolsky and Rosen versus Bohm and Bell. quant-ph/0211078 (2002)

  28. Kraus, K.: General state changes in quantum theory. Ann. Phys. 64(2), 139–177 (1985)

    Google Scholar 

  29. Landau, L.D., Lifshitz, E.M.: Quantum Mechanics. Pergamon, Elmsford (1965)

    MATH  Google Scholar 

  30. Mackey, G.W.: Mathematical Foundations of Quantum Mechanics. Benjamin, Elmsford (1963)

    MATH  Google Scholar 

  31. Messian, A.: Mechanique Quantique. Dunod, Paris (1959)

    Google Scholar 

  32. Ohya, M.: Note on quantum proability. L. Nuovo Cimento 38(11), 203–206 (1983)

    MathSciNet  Google Scholar 

  33. Omnes, R.: The Interpretation of Quantum Mechanics. Princeton Series in Physics. Princeton University Press, Princeton (1994)

    MATH  Google Scholar 

  34. Peres, A.: Quantum Theory: Concepts and Methods. Kluwer Academic, Dordrecht (1993)

    MATH  Google Scholar 

  35. Petz, D.: The Algebra of Canonical Communication Relation. Leuven University Press, Leuven (1990)

    Google Scholar 

  36. Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley, Reading (1985)

    Google Scholar 

  37. Schrödinger, E.: Foundations of quantum mechanics. Naturwissenschaften 48, 52 (1935)

    Google Scholar 

  38. Segal, I.B.: Mathematical Foundations of Quantum Field Theory. Benjamin, Elmsford (1964)

    Google Scholar 

  39. Streater, R.F., Wightman, A.S.: PCT, Spin, Statistics and All That. Benjamin, New York (1964)

    Google Scholar 

  40. Sudarshan, E.C.G., Mathews, P.M., Rau, J.: Stochastic dynamics of quantum-mechanical systems. Phys. Rev. 121, 920–924 (1961)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  41. Tomonaga, S.: Quantum Mechanics, vols. I, II. North-Holland, Amsterdam (1962)

    Google Scholar 

  42. Varadarajan, V.S.: Geometry of Quantum Theory. Springer, New York (1985)

    MATH  Google Scholar 

  43. Volovich, I.V.: Quantum cryptography in space and Bell’s theorem. In: Khrennikov, A. (ed.) Foundations of Probability and Physics: Proceedings of the Conference, Vaxjo, Smoland, Sweden, 25 November–1 December 2000. QP-PQ: Quantum Probability and White Noise Analysis, vol. 13, pp. 364–373. World Scientific, Singapore (2002)

    Google Scholar 

  44. Volovich, I.V.: Seven principles of quantum mechanics. quant-ph/0212126 (2002)

  45. Volovich, I.V., Trushechkin, A.S.: On quantum compressed states on interval and uncertainty relation for nanoscopic systems. Proc. Steklov Math. Inst. 265, 1–31 (2009)

    Article  MathSciNet  Google Scholar 

  46. von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955)

    MATH  Google Scholar 

  47. Wheeler, J.A., Zurek, W.H.: Quantum Theory and Measurement. Princeton University Press, Princeton (1983)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Information Sciences, Tokyo University of Science, Yamazaki 2641, 278-8510, Noda, Japan

    Masanori Ohya

  2. Mathematical Physics, Steklov Mathematical Institute, Gubkin St 8, 119991, Moscow, Russia

    Igor Volovich

Authors
  1. Masanori Ohya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Volovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masanori Ohya .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Ohya, M., Volovich, I. (2011). Basics of Quantum Mechanics. In: Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0171-7_5

Download citation

Publish with us

Policies and ethics

Search

Navigation