Spin matrix exponentials and transmission matrices

Young, L.

doi:10.1090/qam/119823

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 ${\sigma _i}\left ( {i = 1,2,3} \right )$ 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.

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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)

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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)