Definition of time operator in quantum mechanics

Jakub Róg

Marcos Amaku, Francisco Antônio Bezerra Coutinho, F. Masafumi Toyama

The usual definition of the time evolution operator e−iHt/ℏ=∑n=0∞1n!(−iℏHt)n, where H is the Hamiltonian of the system, as given in almost every book on quantum mechanics, causes problems in some situations. The operators that appear in quantum mechanics are either bounded or unbounded. Unbounded operators are not defined for all the vectors (wave functions) of the Hilbert space of the system; when applied to some states, they give a non-normalizable state. Therefore, if H is an unbounded operator, the definition in terms of the power series expansion does not make sense because it may diverge or result in a non-normalizable wave function. In this article, we explain why this is so and suggest, as an alternative, another definition used by mathematicians.

V. S. Olkhovsky, Erasmo Recami, A. J. Gerasimchuk

Ossama Kullie

Attosecond science is of a fundamental interest in physics. The measurement of the tunneling time in attosecond experiments, offers a fruitful opportunity to understand the role of time in quantum mechanics (QM). We discuss in this paper our tunneling time model in relation to two time operator definitions introduced by Bauer and Aharonov–Bohm. We found that both definitions can be generalized to the same type of time operator. Moreover, we found that the introduction of a phenomenological parameter by Bauer to fit the experimental data is unnecessary. The issue is resolved with our tunneling model by considering the correct barrier width, which avoids a misleading interpretation of the experimental data. Our analysis shows that the use of the so-called classical barrier width, to be precise, is incorrect.

H. D. Zeh

The concept of time as used in various applications and interpretations of quantum theory is briefly reviewed.

Jakub Róg

Bridgman’s operationalism imposes a set of methodological requirements on its proponents, which become untenable when applied to the definition of time within the formalism of non-relativistic quantum mechanics. This paper explores the operational definition of time and investigates how far it can be aligned with the core principles of operationalism. To this end, Busch’s (2008) framework, which distinguishes three roles of time in the quantum mechanical formalism, is employed. One of these roles introduces demands that operationalism cannot meet without compromising its essential principles. As a result, it seems reasonable to conclude that operationalism and quantum mechanics are incompatible (at least in the context of the problem of time in quantum mechanics).