Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Spin matrix exponentials and transmission matrices

Author: L. Young
Journal: Quart. Appl. Math. 19 (1961), 25-30
MSC: Primary 78.00
DOI: https://doi.org/10.1090/qam/119823
MathSciNet review: 119823
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The three Pauli spin matrices $ {\sigma _i}\left( {i = 1,2,3} \right)$ occur in the mechanical, especially quantum mechanical, theory of rotation in three-dimensional space. The three spin matrix exponentials are here defined as exp $ \left( {{\sigma _i}x} \right)$, where $ x$ is the independent variable. Transmission matrices can be expressed in terms of spin matrix exponentials, thereby permitting a more systematic treatment of transmission line circuits.

References [Enhancements On Off] (What's this?)

  • [1] Leo Young, Concerning Riblet's theorem, Trans. IRE MTT-7, 477-478 (1959)
  • [2] R. E. Collin, Theory and design of wide-band multisection quarter-wave transformers, Proc. IRE 43, 179-185 (1955)
  • [3] H. J. Riblet, General synthesis of quarter-wave impedance transformers, Trans. IRE MTT-5, 36-43 (1957)
  • [4] Leo Young, Tables for cascaded homogeneous quarter-wave transformers, Trans. IRE MTT-7, 233-237 (1959); Trans. IRE MTT-8, 243-244 (1960)
  • [5] P. I. Richards, Resistor-transmission-line circuits, Proc. IRE 36, 217-220 (1948)
  • [6] Paul I. Richards, A special class of functions with positive real part in a half-plane, Duke Math. J. 14 (1947), 777–786. MR 0022261
  • [7] Leo Young, The quarter-wave transformer prototype circuit, Trans. IRE MTT-8, 483-489 (1960)
  • [8] Leo Young, Optimum quarter-wave transformers Trans. IRE MTT-8, 478-482 (1960)
  • [9] Leo Young, Design of microwave stepped transformers with applications to filters, Doctor of Engineering Dissertation, The Johns Hopkins University, Baltimore, Md., April 1959
  • [10] See almost any book on Quantum Mechanics (e.g. P. A. M. Dirac, The principles of quantum mechanics, Oxford University Press, 3rd ed. p. 149)
  • [11] Herbert Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, Mass., 1951. MR 0043608
  • [12] W. T. Payne, Elementary spinor theory, Amer. J. Phys. 20 (1952), 253–262. MR 0047392, https://doi.org/10.1119/1.1933190
  • [13] H. A. Wheeler, Wheeler monographs, vol. I, Wheeler Laboratories, Great Neck, New York 1953, Monograph No. 4, Geometric relations in circle diagrams of transmission-line impedance
  • [14] G. A. Deschamps, New chart for the solution of transmission-line and polarization problems, Trans. IRE MTT-1, 5-13 (1953), or Electrical Communication 30, 247-254 (1953)
  • [15] E. Folke Bolinder, Note on impedance transformations by the isometric circle method, Trans. IRE MTT-6, 111-112 (1958), where references to some of Bolinder's earlier papers are given
  • [16] W. T. Payne, Spinor theory of four-terminal networks, J. Math. Physics 32 (1953), 19–33. MR 0055196
  • [17] M. C. Pease, The analysis of broad-band microwave ladder networks, Proc. IRE 38, 180-183 (1950), Appendix
  • [18] G. L. Ragan, Microwave transmission circuits, M. I. T. Rad. Lab. Ser., vol. 9, McGraw-Hill Book Co., New York 1948
  • [19] E. F. Bolinder, Note on the matrix representation of linear two-port networks, Trans. IRE CT-4, 337-9 (1957)
  • [20] Leo Young, Transformation matrices, Trans. IRE CT-5, 147-148 (1958)
  • [21] Leo Young, Inhomogeneous quarter-wave transformers of two sections, Trans. IRE MTT, scheduled for Nov. 1960
  • [22] The counterpart to Eq. (16) is $ {\Gamma _1} = {\Gamma _2} = \Gamma $, which arises with symmetrical networks. In this case, the transmission coefficient and reflection coefficient vectors are orthogonal. See Leo Young, A theorem on lossless symmetrical networks, Trans. IRE CT-7, 75 (1960)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 78.00

Retrieve articles in all journals with MSC: 78.00

Additional Information

DOI: https://doi.org/10.1090/qam/119823
Article copyright: © Copyright 1961 American Mathematical Society

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website