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Combinatorial Analysis of the Exponential Map for Time-Evolution Operator

Ali Mert CEYLAN, Vecdi Aytaç

Published Mar 31, 2023• 0 citations
Publisher

Abstract

Abstract In this study, we present a novel formalization for exponential approximation. This formalization is for discrete analysis of continuous time-dependent perturbation operator of quantum mechanics, compared to multiplying exponentials. This leads us to Galton arrays for better representation and algebraic operations over exponential multiplications. Rotations on Galton arrays allow us to perform various algebraic operations and intuitively explain commutative and noncommutative operators equality in exponential approximations. Proposed algebraic operations over Galton arrays will provide an easier understanding and a novel way to formalize and visualize multiplying exponentials. But most importantly, it focused on the timeevolution operator to be written in the proposed formalization of timeordered multiplication of exponentials with noncommuting operators.

DOI
10.21203/rs.3.rs-2271337/v2

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Abstract In this study, we present a novel formalization for exponential approximation. This formalization is for discrete analysis of continuous time-dependent perturbation operator of quantum mechanics, compared to multiplying exponentials. This leads us to Galton arrays for better representation and algebraic operations over exponential multiplications. Rotations on Galton arrays allow us to perform various algebraic operations and intuitively explain commutative and noncommutative operators equality in exponential approximations. Proposed algebraic operations over Galton arrays will provide an easier understanding and a novel way to formalize and visualize multiplying exponentials. But most importantly, it focused on the timeevolution operator to be written in the proposed formalization of timeordered multiplication of exponentials with noncommuting operators.
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Abstract In this study, we present a novel formalization for exponential approximation. This formalization is for discrete analysis of continuous time-dependent perturbation operator of quantum mechanics, compared to multiplying exponentials. This leads us to Galton arrays for better representation and algebraic operations over exponential multiplications. Rotations on Galton arrays allow us to perform various algebraic operations and intuitively explain commutative and noncommutative operators equality in exponential approximations. Proposedalgebraic operations over Galton arrays will provide an easier understanding and a novel way to formalize and visualize multiplying exponentials. But most importantly, it focused on the time evolution operator to be written in the proposed formalization of time ordered multiplication of exponentials with noncommuting operators. MSC Classification(2020): 81Q05 , 81Q15 , 81R60 , 81S05 , 81T75 , 82C99 , 52B70
We present a novel combinatorial proof of the \textbf{Time-Ordered Exponential} formula for the \textbf{Time-Evolution Operator} based on the \textbf{Heaviside Formulation Of The Time-Ordering Operator}. Unlike conventional derivations that rely on induction our approach decomposes the \textbf{Symmetric Group} into permutations that produce decreasing time sequences and those that do not using a simple discriminant function. This decomposition together with an \textbf{Interchange Rule } between time-ordering and multiple integrals yields the \textbf{Dyson–Chen Identity} directly. The proof is self-contained and avoids recursion entirely. It appears in the literature as a problem in Advanced Quantum Mechanics (Springer, 2012, Problem 13.1).
An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.
The approximation of a short-time evolution operator for a molecular system with the Hamiltonian is constructed in terms of the evolution operator of an unperturbed system, a perturbative potential V and commutators of and V . A description of a powerful method for such a construction based on the Lie-algebra technique is given. It is demonstrated that efficient representation for the evolution operator can be obtained using approximations like where . This relation is accurate up to the third order in time. The simplest approximations of this type are listed.
The map from the commutative algebra to the noncommutative algebra of operators is the essential operation of Feyn- man's operational calculus for noncommuting operators. Thus formulas which simplify this operation are meaningful to the subject. In a recent paper the procedure for for iterative disentangling has been es- tablished in the setting of Feynman's operationalcalculus for time indepen- dent operatorsA1, · · · ,An and associatedprobabilitymeasures µ1, · · · ,µn. The main purpose for this paper is to extend the procedure for methods for iterative to time dependent operators.