An advance in the arithmetic of the Lie groups as an alternative to the forms of the Campbell-Baker-Hausdorff-Dynkin theorem

Abstract The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators X and Y , according to the Campbell-Baker-Hausdorff-Dynkin theorem, e X + Y is not equivalent to e X e Y , but is instead given by the we...

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Bibliographic Details
Published inPhysica scripta Vol. 99; no. 8; pp. 85219 - 85225
Main Authors Kim, Sunghyun, Liu, Zhichen, Klemm, Richard A
Format Journal Article
LanguageEnglish
Published IOP Publishing 01.08.2024
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Summary:Abstract The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators X and Y , according to the Campbell-Baker-Hausdorff-Dynkin theorem, e X + Y is not equivalent to e X e Y , but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators { X , Y , Z }, such that [ X , Y ] = κ Z for scalar κ and cyclic permutations, here it is proven that e a X + b Y is equivalent to e p Z e q X e − p Z for scalar p and q . Extensions for e a X + b Y + c Z are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields.
Bibliography:PHYSSCR-128777.R2
ISSN:0031-8949
1402-4896
DOI:10.1088/1402-4896/ad5e11