Skip to main content
SpringerLink
Log in
Book cover

Quantum Entanglement in Electron Optics pp 47–90Cite as

Theory

  • Chapter
  • First Online:

  • 1455 Accesses

Part of the book series: Springer Series on Atomic, Optical, and Plasma Physics ((SSAOPP,volume 67))

Abstract

It is obvious from the discussion given in Chap. 2 that for a proper study of the entanglement properties of a system of more than one particle, one needs to know the density matrix of the quantum state one is interested in. For an ab-initio calculation of such a density matrix, the corresponding density operator is always required. This chapter presents useful methodologies for calculating the density operators and density matrices for the processes (1.1)–(1.8).

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
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

Learn about institutional subscriptions

Notes

  1. 1.

    For meanings of the symbols \(\mathcal{K}_{\mathrm{p}}\) and F p—present in (3.2) and elsewhere in this monograph—see (3.6) and the related discussion.

  2. 2.

    Equation (3.3c) is appropriate for describing a γ r in a definite state of polarization (i.e., LP, RCP, LCP, or UP). Density matrix for an arbitrarily polarized photon is given in the Appendix B on pages 265–268.

  3. 3.

    \(\hat{\boldsymbol{\xi }}_{0}\:\:\, = \hat{\boldsymbol{e}}_{z} :\) linear polarization along the OZ-axis;\(\hat{\boldsymbol{\xi }}_{+1} = - \frac{1} {\sqrt{2}}(\hat{\boldsymbol{e}}_{x} + i\hat{\boldsymbol{e}}_{y})\): right circular polarization;\(\hat{\boldsymbol{\xi }}_{-1} = \frac{1} {\sqrt{2}}(\hat{\boldsymbol{e}}_{x} - i\hat{\boldsymbol{e}}_{y})\): left circular polarization.

  4. 4.

    For a definition of trace of a matrix, see, for example, (3.8) on page 254.

  5. 5.

    A somewhat detailed, but quantitative, discussion of this topic is given in the Appendix C on pages 269–271.

  6. 6.

    It is for this important reason that these diagonal elements must simultaneously meet the requirements (3.17) and (3.18).

  7. 7.

    The diagonal elements of a density matrix are called [180–182] population as well. See, for example, Appendix C for further discussion.

  8. 8.

    For an n e-electron atom with nuclear charge Ze, the SOI is given by [10, 60, 184, 186]

    $$\displaystyle\sum _{i}^{n_{e}}\,\xi (r_{ i})\,\boldsymbol{\ell}_{i} \cdot \boldsymbol{s}_{i}\qquad \text{with}\qquad \xi (r_{i}) = \frac{{\hslash }^{2}} {2m_{e}^{2}{c}^{2}}\,\frac{Z{e}^{2}} {4\pi \epsilon _{0}} \, \frac{1} {r_{i}^{3}}.$$

    Here, while r i is the distance of the i-th electron from the nucleus, ℓ ​i and s ​i are its orbital and spin angular momenta, respectively. Electrons only in an incomplete shell contribute to this sum as the contributions of those in a full shell add to zero [59].

  9. 9.

    For a hydrogenic atom [i.e., n e = 1 in footnote (8) in the present Chap. 3] with atomic number Z, the ratio of the SOI to the Coulomb interaction [i.e.,\(\frac{Z\,{e}^{2}} {4\pi \epsilon _{0}} \frac{1} {r}\)] is [58] ≃  (Zα)2, where α = 1 ∕ 137 is the fine structure constant.

  10. 10.

    See the discussion given on pages 50 in the last paragraph of Sect. 3.1.1.

  11. 11.

    A molecule with all of its nuclei in a straight line is called a linear molecule [10, 60, 68–70, 185].

  12. 12.

    Parity of a molecular state is shown by a +/- superscript on the symbol representing this state.

  13. 13.

    This line is called also the inter-nuclear axis in a linear molecule. Its direction is given by the unit vector \(\hat{\boldsymbol{e}}_{z_{m}}\) along the polar OZ m-axis of the MF.

  14. 14.

    The projection of R along the inter-nuclear axis is always zero. Hence, \(\hat{\boldsymbol{e}}_{z_{m}} \cdot \boldsymbol{N}_{} = \hat{\boldsymbol{e}}_{z_{m}}\) ⋅L \(\equiv \) Λ.

  15. 15.

    It, in other words, means that on the inversion through the origin “O” of the co-ordinate system in Fig. 9.1 of the position vectors of all the particles (i.e., electrons and nuclei) present in a RLM \(\mathfrak{T}\), state (3.48a) does not change its sign if s + \(\mathfrak{p}_{0} + N_{0}\) is even; otherwise (i.e., for s + \(\mathfrak{p}_{0} + N_{0}\) odd), the sign of (3.48a) is reversed.

  16. 16.

    In expression (3.49) and elsewhere, \(\mathcal{D}_{\lambda _{p}\,m_{p}}^{\ell_{p}}(\omega _{m})\) rotates photoelectron’s orbital angular momentum vector ℓ ​p from MF to SF in Fig. 9.1; whereas \({\Bigl [\mathcal{D}_{\mu _{p}\,\nu _{p}}^{\frac{1} {2} }{(\omega _{p})\Bigr ]}}^{{\ast}}\) is used for taking e ​p’s spin quantization direction from OZ-axis to \(\hat{\boldsymbol{u}}_{p}\) in SF.

  17. 17.

    An expression for \(\vert {1}^{{+}^{{\ast}} }\rangle\) in the present situation is readily obtained on the replacement 0 \(\rightarrow {1}^{{+}^{{\ast}} }\) everywhere in (3.48).

  18. 18.

    In order to obtain an expression for \(\vert {2}^{+}\rangle\) representing the dication \({\mbox{ $\mathfrak{T}$}}^{2+}\) of the RLM \(\mathfrak{T}\) in the absence of SDIs in Hund’s coupling scheme (b), one merely needs to replace 0 by 2 + everywhere in (3.48).

  19. 19.

    The replacement p → a every where in (3.49) will give us a spin–orbital for the Auger electron e a in the absence of SDIs in Hund’s case (b).

  20. 20.

    These circumstances are similar to those of atomic cases wherein spin–orbit coupling vanishes identically, or is negligibly small, whenever an atom’s orbital and/or spin angular momentum is nil (i.e., atom’s electronic state is S and/or has spin multiplicity one), or it has a small atomic number Z.

  21. 21.

    Spin-entanglement generated in either of the two processes (1.2) and (1.3) taking place in a RLM with the third possibility in which only SOI is important, while SRI is negligible, has not yet probably been considered. Such situations are properly described in Hund’s case (c) [68, 69, 185].

  22. 22.

    For [68, 75, 185] Λ 0 = 0, spin S 0 is no longer coupled to the inter-nuclear axis. It, in other words, means, Σ 0 = 0 and, hence, Ω 0 = 0. In such a situation, therefore, Hund’s case (a) reduces [68, 75, 185] to its scheme (b).

  23. 23.

    These two states are obtained on making everywhere the respective replacements 0 \(\rightarrow {1}^{{+}^{{\ast}} }\) and 0 \(\rightarrow {2}^{+}\) of the subscripts in (3.62).

  24. 24.

    Replacement of the subscript p by a at every place in (3.64) will immediately provide an expression for | μ a{u}a k ​a⟩ in Hund’s case (a).

References

  1. B.H. Bransden, C.J. Joachain, Physics of Atoms and Molecules, 2nd edn. (Benjamin Cummings, New York, 2003)

    Google Scholar 

  2. M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G.M.  D’Ariano, C. Macchiavello, Phys. Rev. Lett. 91, 227901 (2003)

    Article  ADS  Google Scholar 

  3. S. Parida, N. Chandra, R. Ghosh, Eup. Phys. J. D 65, 303 (2011)

    Article  ADS  Google Scholar 

  4. S. Parida, N. Chandra, Phys. Lett. A 373, 1852 (2009); Phys. Rev. A 79, 062501 (2009)

    Article  ADS  Google Scholar 

  5. C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, Vol. I & II (John Wiley, New York, 1978)

    MATH  Google Scholar 

  6. L.I. Schiff, Quantum Mechanics, 3rd rev. ed. (McGraw-Hill, New York, 1968)

    Google Scholar 

  7. M. Weissbluth, Atoms and Molecules (Academic, New York, 1978)

    Google Scholar 

  8. K. Blum, Density Matrix Theory and Applications, 3rd edn. (Springer, Berlin, 2012)

    Book  Google Scholar 

  9. U. Fano, Revs. Mod. Phys. 29, 74 (1957)

    Article  ADS  Google Scholar 

  10. A. de-Shalit, I. Talmi, Nuclear Shell Theory (Dover, New York, 2004)

    Google Scholar 

  11. A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, 1996)

    MATH  Google Scholar 

  12. R.N. Zare, Angular Momentum (Wiley-Interscience, 1988)

    Google Scholar 

  13. M. Mizushima, Theory of Rotating Diatomic Molecules (Wiley, 1975)

    Google Scholar 

  14. M. Tinkham, Group Theory and Quantum Mechanics (Dover, 2003)

    Google Scholar 

  15. H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin, 1957)

    Book  Google Scholar 

  16. P. Auger, Commn. Royal Acad. Sci. Paris 178, 929 (1924)

    Google Scholar 

  17. P. Auger, Commn. Royal Acad. Sci. Paris 178, 1535 (1924)

    Google Scholar 

  18. J. Xie, R.N. Zare, J. Chem. Phys. 93, 3033 (1990)

    Article  ADS  Google Scholar 

  19. M. Seevinck, Jos Uffink, Phys. Rev. A 78, 032101 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  20. N. Chandra, in Frontiers in Atomic. Molecular, and Optical Physics, eds. S.S. Bhattacharyya, S.C. Mukherjee, Special publication for the 75th year of the Indian Journal of Physics, 3, 279 (2003)

    Google Scholar 

  21. V.L. Jacobs, J. Phys. B 5, 2257 (1972)

    Article  ADS  Google Scholar 

  22. N. Chandra, Chem. Phys. 108, 301 (1986)

    Article  Google Scholar 

  23. P. Auger, Commn. Royal Acad. Sci. Paris 182, 776 (1926)

    Google Scholar 

  24. P. Auger, Commn. Royal Acad. Sci. Paris 177, 169 (1923)

    Google Scholar 

  25. T. Aberg, G. Howat, Handbuch der Physik 31, 469 (1982)

    ADS  Google Scholar 

  26. V.B. Berestetski, E.M. Lifshitz, L.P. Pitaevskii, Relatistic quantum Theory (Pergmon Press, N.Y.), Part 1, pp. 136

    Google Scholar 

  27. G. Breit, H.A. Bethe, Phys. Rev. 93, 888 (1954)

    Article  ADS  MathSciNet  Google Scholar 

  28. M. Dubé, P.C.E. Stamp, Int. J. Mod. Phys. B 12, 1191 (1998)

    Article  ADS  Google Scholar 

  29. M. Thorwart, P. Hänggi, Phys. Rev. A 65, 012309 (2002)

    Article  ADS  Google Scholar 

  30. B. Lohmann, Angle and Spin Resolved Auger Emission: Theory and Applications to Atoms and Molelcues (Springer Series on Atomic, Optical, and Plasma Physics, Springer-Verlag, Berlin, Vol. 46, 2009)

    Google Scholar 

  31. E.U. Condon, R. Shortley, The Theory of Atomic Spectra, (Cambridge at the University Press, England, 1970)

    MATH  Google Scholar 

  32. J.M. Brown, A.C. Carrington, Rotational Spectroscopy of Diatomic Mollecules (Cambridge University Press, Cambridge, UK, 2003)

    Book  Google Scholar 

  33. I.I. Sobelman, Atomic Spectra and Radiative Transitions, 2nd ed. (Springer, Germany, 1992)

    Book  Google Scholar 

  34. M. Rotenberg, R. Bivis, N. Metropolis, J.K. Wooten, Jr., The 3-j and 6-j Symbols (The Technology Press, MIT, Massachusetts, 1959)

    Google Scholar 

  35. N. Chandra, R. Ghosh, Phys. Rev. A 74, 052329 (2006)

    Article  ADS  Google Scholar 

  36. U. Fano, Phys. Rev. 178, 131 (1969); Addendum: Phys. Rev. 184, 250 (1969)

    Article  ADS  Google Scholar 

  37. N. Chandra, Phys. Rev. A 42, 4050 (1990)

    Article  ADS  Google Scholar 

  38. P.G. Burke, N. Chandra, F.A. Gianturco, J. Phys. B 5, 2212 (1972)

    Article  ADS  Google Scholar 

  39. N. Chandra, M. Chakraborty, J. Chem. Phys. 99, 7314 (1993)

    Article  ADS  Google Scholar 

  40. F. Hund, Handbuch der Physik 24, 561 (1933)

    Google Scholar 

  41. N. Chandra, R. Ghosh, Phys. Rev. A 69, 012315 (2004)

    Article  ADS  Google Scholar 

  42. N. Chandra, S. Sen, J. Chem. Phys. 98, 5242 (1993)

    Article  ADS  Google Scholar 

  43. R. Ghosh, Ph.D. Thesis (Indian Institute of Technology, Kharagpur, India, 2008), unpublished

    Google Scholar 

  44. N. Chandra, R. Ghosh, Rad. Phys. Chem. 75, 1808 (2006)

    Article  ADS  Google Scholar 

  45. R. Ghosh, N. Chandra, S. Parida, Eur. Phys. J. Special Topics 169, 117 (2009)

    Article  ADS  Google Scholar 

  46. N. Chandra, S. Sen, J. Chem. Phys. 102, 8359 (1995)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur, 721302, West Bengal, India

    N. Chandra

  2. c/o B.N. Panda, 1/A–46(H.I.G.), Lingraj Vihar, Pokhariput, 751020, Orissa, India

    R. Ghosh

Authors
  1. N. Chandra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Chandra .

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Chandra, N., Ghosh, R. (2013). Theory. In: Quantum Entanglement in Electron Optics. Springer Series on Atomic, Optical, and Plasma Physics, vol 67. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24070-6_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-24070-6_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24069-0

  • Online ISBN: 978-3-642-24070-6

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation