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

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Abstract: The three Pauli spin matrices occur in the mechanical, especially quantum mechanical, theory of rotation in three-dimensional space. The three spin matrix exponentials are here defined as exp , where 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.

**[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 , 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)

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DOI:
https://doi.org/10.1090/qam/119823

Article copyright:
© Copyright 1961
American Mathematical Society