Two-State Quantum Systems Revisited: A Clifford Algebra Approach

We revisit the topic of two-state quantum systems using the Clifford Algebra in three dimensions C l 3 . In this description, both the quantum states and Hermitian operators are written as elements of C l 3 . By writing the quantum states as elements of the minimal left ideals of this algebra, we co...

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Bibliographic Details
Published in:Advances in applied Clifford algebras Vol. 31; no. 2
Main Authors: Amao, Pedro, Castillo, Hernan
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.04.2021
Springer Nature B.V
Subjects:
Algebra
Applications of Mathematics
Eigenvalues
Eigenvectors
Larmor precession
Mathematical and Computational Physics
Mathematical Methods in Physics
Operators (mathematics)
Particle spin
Physics
Physics and Astronomy
Theoretical
Geometric algebra
Clifford algebras
Primary 81V45
Two-state quantum systems
Secondary 15A66
ISSN:0188-7009, 1661-4909
Online Access:Get full text
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Summary:We revisit the topic of two-state quantum systems using the Clifford Algebra in three dimensions C l 3 . In this description, both the quantum states and Hermitian operators are written as elements of C l 3 . By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system. The geometric interpretation of the Hermitian operators enables us to introduce an algebraic method to diagonalize these operators in C l 3 . We then use this approach to revisit the problem of a spin-1/2 particle interacting with an external arbitrary constant magnetic field, obtaining the same results as in the conventional theory. However, Clifford algebra reveals the underlying geometry of these systems, which reduces to the Larmor precession in an arbitrary plane of C l 3 .
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ISSN:0188-7009
1661-4909
DOI:10.1007/s00006-020-01116-1