Complex structure-preserving method for Schrödinger equations in quaternionic quantum mechanics (original) (raw)

Abstract

The quaternionic Schrödinger equation \(\frac{\partial }{\partial t}|f\rangle =-A|f\rangle \) is the crucial part of the study of quaternionic quantum mechanics and plays indispensable roles in related fields. One of the practical and special cases that has received more attention from mathematicians and physicists is that A is a Hermitian quaternion matrix. The problem can be equivalent to a Hermitian quaternion right eigenvalue problem \(A\alpha =\alpha \lambda \) by discretization. This paper, by means of a complex representation method, studies the Hermitian quaternion Schrödinger equation problem, and proposes a novel algebraic method (complex structure-preserving method) for right eigenvalue problems of Hermitian quaternion matrices. Moreover, the complex structure-preserving method is superior and formally simple compared to previous methods, and numerical experiments also demonstrate the effectiveness of the method.

Access this article

Log in via an institution

Subscribe and save

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Data Availibility Statement

All data generated or analyzed during this study are included in this published article.

References

  1. Pawłowski, F., Olsen, J., Jørgensen, P.: Molecular response properties from a Hermitian eigenvalue equation for a time-periodic Hamiltonian. J. Chem. Phys. 142, 114109 (2015)
    Article Google Scholar
  2. Hamilton, W.R.: On a new species of imaginary quantities, connected with the theory of quaternions. Proc. R. Ir. Acad. 2, 424–434 (1840)
    Google Scholar
  3. Leo, S.D.: Quaternions and special relativity. J. Math. Phys. 37, 2955–2968 (1996)
    Article MathSciNet Google Scholar
  4. Rastall, P.: Quaternions in relativity. Rev. Mod. Phys. 36, 820 (1964)
    Article MathSciNet Google Scholar
  5. Finkelstein, D.: Notes on quaternion quantum mechanics, Logico-Algebraic-Approach to Quantum Mechamics II (1979)
  6. Jaha, E.S., Ghouti, L.: Color face recognition using quaternion PCA (2011)
  7. Xiao, X., Zhou, Y.: Two-dimensional quaternion PCA and sparse PCA. IEEE Trans. Neural Netw. Learn. Syst. 30, 2028–2042 (2018)
    Article Google Scholar
  8. Adler, S.L.: Quaternionic quantum mechanics and quantum fields. Oxford University Press, USA (1995)
    Google Scholar
  9. Arbab, A.I.: The quaternionic quantum mechanics. Appl. Phys. Res. 3(160), (2011)
  10. Giardino, S.: Quaternionic quantum mechanics in real Hilbert space. J. Geom. Phys. 158, 103956 (2020)
    Article MathSciNet Google Scholar
  11. Cokle, J.: On systems of algebra involving more than one imaginary and on equations of the fifth degree. Philos. Mag. 35, 434–437 (1849)
    Google Scholar
  12. Segre, C.: The real representations of complex elements and extension to bicomplex systems. Math. Ann. 40, 413–467 (1892)
    Article MathSciNet Google Scholar
  13. Baez, J.: The octonions. Bull. Am. Math. Soc. 39, 145–205 (2002)
    Article MathSciNet Google Scholar
  14. Jiang, T.: Algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory. J. Math. Phys. 46, 052106 (2005)
    Article MathSciNet Google Scholar
  15. Farid, F.O., Wang, Q., Zhang, F.: On the eigenvalues of quaternion matrices. Lin. Multilin. Alg. 59, 451–473 (2011)
    Article MathSciNet Google Scholar
  16. Jiang, T., Zhang, Z., Jiang, Z.: Algebraic techniques for Schrödinger equations in split quaternionic mechanics. Comput. Math. Appl. 75, 2217–2222 (2018)
    Article MathSciNet Google Scholar
  17. Jiang, T., Zhang, Z., Jiang, Z.: Algebraic techniques for eigenvalues and eigenvectors of a split quaternion matrix in split quaternionic mechanic. Comput. Phys. Commun. 229, 1–7 (2018)
    Article MathSciNet Google Scholar
  18. Guo, Z., Jiang, T., Vasil’ev, V.I., Wang, G.: A novel algebraic approach for the Schrödinger equation in split quaternionic mechanics. Appl. Math. Lett. 108485 (2022)
  19. Guo, Z., Zhang, D., Vasil’ev, V.I., Jiang, T.: Algebraic techniques for Maxwell’s equations in commutative quaternionic electromagnetics. Eur. Phys. J. Plus. 137, 577 (2022)
    Article Google Scholar
  20. Jia, Z., Wei, M., Ling, S.: A new structure-preserving method for quaternion Hermitian eigenvalue problems. J. Comput. Appl. Math. 239, 12–24 (2013)
    Article MathSciNet Google Scholar
  21. Sangwine, S., Le Bihan, N.: Quaternion toolbox for Matlab. http://qtfm.sourceforge.net/
  22. Ma, R., Jia, Z., Bai, Z.: A structure-preserving Jacobi algorithm for quaternion Hermitian eigenvalue problems. Comput. Math. Appl. 75, 809–820 (2018)
    Article MathSciNet Google Scholar
  23. Jiang, T.: An algorithm for eigenvalues and eigenvectors of quaternion matrices in quaternionic quantum mechanics. J. Math. Phys. 45(8), 3334–3338 (2004)
    Article MathSciNet Google Scholar
  24. Golub, G.H., Van Loan, C.F.: Matrix computations, JHU press (2013)

Download references

Acknowledgements

We are grateful to the editor and the anonymous referees for providing many useful comments and suggestions, which greatly improve the performance of the original paper.

Funding

The research of T. Jiang and Z. Guo are supported by the Ministry of Science and Higher Education of the Russian Federation, supplementary agreement (No. 075-02-2023-947, February 16, 2023). The research of V. I. Vasil’ev is supported by the Russian Science Foundation grant (23-41-00037). The research of Z. Guo is supported by the Chinese Government Scholarship (CSC No. 202108370087). The research of G. Wang is supported by the Russian Science Foundation grant (23-71-30013) and the Chinese Government Scholarship (CSC No. 202008370340).

Author information

Author notes

  1. Tongsong Jiang and V. I. Vasil’ev are equally contributed to this work.

Authors and Affiliations

  1. Institute of Mathematics and Information Science, North-Eastern Federal University, Yakutsk, 677000, Russia
    Zhenwei Guo, V. I. Vasil’ev & Gang Wang
  2. School of Electronic Information, Shandong Xiandai University, Jinan Shandong, 250104, People’s Republic of China
    Tongsong Jiang
  3. School of Mathematics and Statistics, Linyi University, Linyi Shandong, 276005, People’s Republic of China
    Tongsong Jiang

Authors

  1. Zhenwei Guo
  2. Tongsong Jiang
  3. V. I. Vasil’ev
  4. Gang Wang

Contributions

Zhenwei Guo performed the data analyses and wrote the manuscript; Tongsong Jiang and V. I. Vasil’ev contributed significantly to analysis and manuscript preparation; Gang Wang contributed to the conception of the study. All authors reviewed the manuscript.

Corresponding author

Correspondence toTongsong Jiang.

Ethics declarations

Competing interests

The authors declare no competing interests.

Not Applicable.

Human and Animal Ethics

Not Applicable.

Not Applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of T. Jiang and Z. Guo are supported by the Ministry of Science and Higher Education of the Russian Federation, supplementary agreement (No. 075-02-2023-947, February 16, 2023). The research of V. I. Vasil’ev is supported by the Russian Science Foundation grant (23-41-00037). The research of Z. Guo is supported by the Chinese Government Scholarship (CSC No. 202108370087). The research of G. Wang is supported by the Russian Science Foundation grant (23-71-30013) and the Chinese Government Scholarship (CSC No. 202008370340).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Cite this article

Guo, Z., Jiang, T., Vasil’ev, V.I. et al. Complex structure-preserving method for Schrödinger equations in quaternionic quantum mechanics.Numer Algor 97, 271–287 (2024). https://doi.org/10.1007/s11075-023-01703-w

Download citation

Keywords