Exponential operators, operational rules and evolution problems

Abdelwahab Amr

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).

Ravinder Rupchand Puri

C M Cheng, P. C. W. Fung

The authors present a method for finding the evolution operator for the Schrodinger equation for the Hamiltonian expressible as H(t)=a 1 (t)J 4 +a 2 (t)J 0 +a 3 (t) J - where J + , J 0 and J - are the SU(2) group generators. Such a method is applied to the disentangling technique for exponential operators which are not necessarily unitary. As a demonstration of our general approach, they solved the problem of a harmonic oscillator with a varying mass.

P. C. García Quijas, L. M. Arévalo, Aguilar Centro +4 more

There is a widespread belief in the quantum physical community, and in textbooks used to teach Quantum Mechanics, that it is a difficult task to apply the time evolution operator on an initial wave function. That is to say, because the hamiltonian operator generally is the sum of two operators, then it is a difficult task to apply the time evolution operator on an initial wave function, because it implies to apply terms like (A+B)^n. A possible solution of this problem is to factorize the time evolution operator and then apply successively the individual exponential operator on the initial wave function. However, the exponential operator does not directly factorize. In this work we present useful ways to factorizing the time evolution operator when the argument of the exponential is a sum of two operators which obey specific commutation relations. Then, we apply the exponential operator as an evolution operator for the case of elementary unidimensional potentials, like the harmonic oscillator. Also, we argue about an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed in previous papers published in this Journal. Finally, we discuss the possible insight that can be learned using this approach in teaching Quantum Mechanics.

Pedro Pereyra Padilla