Factorizing the time evolution operator

Abstract

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 e itĤ/ 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 ψ(x, 0), for it implies to apply terms like (â +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, i. e. eâ +b = eâeb. In this work we present a useful procedure for 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 particle subject to a constant force and the harmonic oscillator. Also, we argue about an apparent paradox concerning the time evolution operator and non-spreading wave packets addressed previously in the literature.

A full quantum analysis of the Stern–Gerlach experiment using the evolution operator method: analyzing current issues in teaching quantum mechanics

“…In order to be able to use QM postulates, it is necessary to factorize the evolution operator ˆˆ = -U e tH i . The factorization can be made in many ways, see [31,32]. For example, a factorization must be used to apply the solution for the Schrödinger equation:…”

“…that is to say, a linear combination of the external and internal eigenfunctions. We do not pursue this path in this paper, instead we use the evolution operator method [31,32,39]. However, equation (22) serves the purpose of analyzing, in the following pharagraphs, the solutions given by others authors to equation (8).…”

“…here s 0 is the width of the wave packet, r is the position, and k is the wave vector. Applying the operator in equation (6) we have that the evolved state is of the form instead of that appearing in equation (30) of [1], with M given in equation (31) of the published article [1]. Notice that the only changes are in the sign inside the argument of the exponentials multiplying …”

Quantum Nonlocality and Quantum Correlations in the Stern–Gerlach Experiment

“…However, in the scientific literature, we can find proposals that treat the external degrees of freedom (EDF) as a quantum variable giving a quantum description of the evolution of the EDF; see [ 5 , 6 , 7 , 8 , 9 , 10 , 11 , 13 ]. Nevertheless, in a recent paper, we gave a complete quantum treatment of the SGE [ 5 , 6 ], as it was originally thought of by Scully et al [ 9 ], by using the evolution operator method [ 21 , 22 , 23 , 24 , 25 ] and obtaining the solution to the Schrödinger equation; this allows us to see the quantum features of the EDF. It is worth mentioning that in Figure 1 of [ 5 ], we are replacing the usual continuous path by a dotted one to stress the absence of classical paths.…”

“…Recently, we argued that the SGE could easily be used to introduce the concept of entanglement between the external and internal degrees of freedom in the teaching of quantum mechanics [ 5 , 6 ], to exemplify the entanglement generation between discrete and continuous variables and between pure and mixed states, as well. To show these properties of the SGE, we have used the evolution operator method [ 21 , 22 , 23 , 24 , 25 ]; to see an independent test of this method, see [ 26 , 27 ].…”