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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Published in | Physica scripta Vol. 99; no. 8; pp. 85219 - 85225 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
IOP Publishing
01.08.2024
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Subjects | |
Online Access | Get full text |
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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. |
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Bibliography: | PHYSSCR-128777.R2 |
ISSN: | 0031-8949 1402-4896 |
DOI: | 10.1088/1402-4896/ad5e11 |