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College Physics

Derivation of Baker-Hausdorff formula using the operator's matrix representation

HE Yon

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

Derivation of Baker- Hausdorff formula using the matrix representation is a simple method. The derivation does not refer to the differential for a parameter. Furthermore,the matrix representation for Baker- Hausdorff formula is more useful to treat the specific physical problem.

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GENERALIZED BAKER-HAUSDORFF FORMULA AND ITS APPLICATION

井思聪, 阮图南

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Anticommutator analogue of the Baker-Hausdorff lemma

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The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.

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The idea is to study the Baker-Campbell-Hausdorff formula and its recent developments to highlight its behavior. Then we use that formula to calculate the propagator in quantum mechanics if the system considered is a harmonic oscillator.

•Dec 1, 2017•0 citations

Harmonic oscillator integrals by operator methods, a simple unified approach

Wolfgang Witschel

One- and two-centre matrix elements for a wide class of operators in an harmonic oscillator basis are derived by operator methods. The advantage of the method is that all calculations are reduced to the well known Hausdorff transform. The elementary operator steps in the harmonic oscillator occupation number representation make this technique suitable for introductory quantum mechanics classes. Zusammenfassung. Fur eine Vielzahl von Operatoren wurden Ein- und Zweizentren-Matrixelemente des harmonischen Oszillators mit Operatormethoden abgeleitet. Die elementaren Operatorschritte in der Besetzungszahldarstellung wurden auf die bekannte Hausdorff-Formel zuruckgefuhrt.

European Journal of Physics•Nov 1, 1996•5 citations
The Baker--Campbell--Hausdorff (BCH) formula plays a critical role in many branches of mathematics and physics. It expresses the logarithm of the product of exponentials of non-commuting operators as an infinite series of nested commutators of the operators involved. The Zassenhaus formula is the dual of the BCH formula: the exponential of a sum of operators is written as an infinite product of exponentials involving the operators and their commutators. In practical computations, however, one typically has to truncate the expansions, and so understanding the error committed by the resulting approximations and eventually providing suitable bounds for this error is of paramount interest. In this work we present a general strategy to derive rigorous error bounds and explicit error constants for the BCH and Zassenhaus formulas when the operators involved are skew-adjoint, as is the case for quantum evolution problems.
One- and two-centre matrix elements for a wide class of operators in an harmonic oscillator basis are derived by operator methods. The advantage of the method is that all calculations are reduced to the well known Hausdorff transform. The elementary operator steps in the harmonic oscillator occupation number representation make this technique suitable for introductory quantum mechanics classes. Zusammenfassung. Fur eine Vielzahl von Operatoren wurden Ein- und Zweizentren-Matrixelemente des harmonischen Oszillators mit Operatormethoden abgeleitet. Die elementaren Operatorschritte in der Besetzungszahldarstellung wurden auf die bekannte Hausdorff-Formel zuruckgefuhrt.