Applying Classical Geometry Intuition to Quantum Spin

Using concepts of geometric orthogonality and linear independence, we logically deduce the form of the Pauli spin matrices and the relationships between the three spatially orthogonal basis sets of the spin-1/2 system. Rather than a mathematically rigorous derivation, the relationships are found by...

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Bibliographic Details
Published inarXiv.org
Main Authors Durfee, Dallin S, Archibald, James L
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.06.2016
Subjects
Angular momentum
Coordinates
Hilbert space
Mathematical analysis
Matrix methods
Orthogonality
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Summary:Using concepts of geometric orthogonality and linear independence, we logically deduce the form of the Pauli spin matrices and the relationships between the three spatially orthogonal basis sets of the spin-1/2 system. Rather than a mathematically rigorous derivation, the relationships are found by forcing expectation values of the different basis states to have the properties we expect of a classical, geometric coordinate system. The process highlights the correspondence of quantum angular momentum with classical notions of geometric orthogonality, even for the inherently non-classical spin-1/2 system. In the process, differences in and connections between geometrical space and Hilbert space are illustrated.
ISSN:2331-8422