Quantum arrival time for open systems

We review recent progress in understanding the arrival time problem in quantum mechanics, from the point of view of the decoherent histories approach to quantum theory. We begin by discussing the arrival time problem, focussing in particular on the role of the probability current in the expected classical solution. After a brief introduction to decoherent histories we review the use of complex potentials in the construction of appropriate class operators. We then discuss the arrival time problem for a particle coupled to an environment, and review how the arrival time probability can be expressed in terms of a POVM in this case. We turn finally to the question of decoherence of the corresponding histories, and we show that this can be achieved for simple states in the case of a free particle, and for general states for a particle coupled to an environment.

The model of weak measurements is applied to various problems, related to the\ntime problem in quantum mechanics. The review and generalization of the\ntheoretical analysis of the time problem in quantum mechanics based on the\nconcept of weak measurements are presented. A question of the time interval the\nsystem spends in the specified state, when the final state of the system is\ngiven, is raised. Using the concept of weak measurements the expression for\nsuch time is obtained. The results are applied to the tunneling problem. A\nprocedure for the calculation of the asymptotic tunneling and reflection times\nis proposed. Examples for delta-form and rectangular barrier illustrate the\nobtained results. Using the concept of weak measurements the arrival time\nprobability distribution is defined by analogy with the classical mechanics.\nThe proposed procedure is suitable to the free particles and to particles\nsubjected to an external potential, as well. It is shown that such an approach\nimposes an inherent limitation to the accuracy of the arrival time definition.\n

There have been great theoretical and practical interests in investigating time problem in quantum mechanics. Because time enters quantum mechanics as a parameter rather than a dynamical operator, people have to consider how to construct a time operator and calculate average time whenever they are faced with a time problem related to a physical process. In this paper we present a general investigation on time operator and average time.

The irreversible process of ‘wave packet collapse’ in quantum measurement contradicts the reversible characteristics of the Schrödinger equation and the entire quantum mechanics. This contradiction is actually the origin of measurement problems in quantum mechanics. How to resolve this contradiction is related to the understanding of some philosophical concepts. The Copenhagen School embraces the reversible/irreversible contradictions in quantum measurement through the principle of complementarity. Both von Neumann's measurement hypothesis and Everett's many world explanation adhere to the ‘universal position of the Schrödinger equation’, but due to their detachment from the thermodynamic mechanism analysis of quantum measurement processes, they fall into the dilemma of subjectivism or possible state materialization in ontology. Decoherence theory is a natural extension of dissipative structure theory in quantum mechanics, which helps to solve quantum measurement problems by combining the study of classical conversion mechanisms between reversibility and irreversibility in statistical mechanics. Due to the common statistical roots of the time arrow problem in classical mechanics and quantum mechanics, decoherence theory cannot be seen as a measurement theory that fully implements the ‘universal Schrödinger position’. Many contradictions in quantum mechanics may need to be cleverly resolved through various contradictions in classical physics.

We derive a system of TBA equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and they can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. Our derivation builds upon the solution of similar Riemann-Hilbert problems in the study of BPS spectra in mathcal{N} = 2 gauge theories and of minimal surfaces in AdS. We also show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics. We illustrate our general analysis with a detailed study of PT-symmetric cubic oscillators and quartic oscillators.

In June 2025, the Voprosy filosofii journal hosted a jubilee “round table” “Philo­sophical Problems of Quantum Mechanics: Historical Reality and Development Prospects,” dedicated to the centenary of the emergence of the mathematical formalism of quantum mechanics. The round table was organized by the De­partment of Philosophy of the Natural Sciences at the Institute of Philosophy of the Russian Academy of Sciences. The discussion brought together specialists in the philosophy and methodology of science, in the philosophical problems of quantum mechanics, as well as natural scientists and historians of science. Participants focused primarily on the conceptual aspects of the mathematical framework of quantum mechanics and on the question of which event should be considered the true “birthdate” of this field of knowledge. In addition, the inter­disciplinary dialogue clarified the trajectories of interaction between quantum mechanics and the social and human sciences in today’s changing world. A spe­cial topic of discussion concerned the philosophical research programs that influ­enced the development of quantum mechanics, and the extent to which quantum mechanics itself has significantly shaped these programs (including eco-philo­sophical ones). Most participants agreed that dialectics and phenomenology (par­ticularly those of G.W.F. Hegel, M. Heidegger, G.G. Shpet, and others) are acquiring special importance today for comprehending the fundamental founda­tions of quantum mechanics. Another original perspective in the debate was linked to examining the philosophical meanings of logic and hermeneutics and assessing their methodological potential in research on quantum mechanics.

Time in quantum mechanics

After more than 70 years of debate about the difficulties that one encounters in working out a coherent view of physical processes based on the standard formulation of quantum mechanics, there is now a widespread belief that such difficulties do not arise from philosophical prejudices (as has been repeatedly asserted by many of the supporters of textbook quantum mechanics) but represent precise mathematical and physical challenges which call for a physical solution. As J.S. Bell appropriately stated1 “the way ahead is unromantic in that it requires mathematical work by theoretical physicists, rather than interpretations by philosophers”. It is also encouraging, for those who share this position, to see that now there are explicit proposals indicating possible ways out in which the process of measurement (and more generally all those measurement-like processes2 “we are obliged to admit ... are going on more or less all the time, more or less everywhere”) is analysed not in terms of vague assertions or resorting to ill-defined and/or contradictory dualistic evolution processes (the linear and deterministic evolution for microsystems and the nonlinear and stochastic wave packet reduction for the macroscopic ones) but in terms of more fundamental physical concepts. In particular, approaches of this kind, which S. Goldstein3 has appropriately denoted as “observer-free formulations of quantum mechanics”, can be grouped in three categories: the pilot-wave theories: the decoherent histories and the spontaneous localization models.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTThe probability equals zero problem in quantum mechanics. Or, how does an electron get from first to second to third without touching second?Frank O. Ellison and C. A. Hollingsworth Cite this: J. Chem. Educ. 1976, 53, 12, 767Publication Date (Print):December 1, 1976Publication History Received3 August 2009Published online1 December 1976Published inissue 1 December 1976https://doi.org/10.1021/ed053p767RIGHTS & PERMISSIONSArticle Views154Altmetric-Citations5LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (3 MB) Get e-Alerts Get e-Alerts

Review of: "Fundamental Issues and Measurement Problem in Quantum Mechanics"

We examine here, by using a simple example, two implementations of the minimum error method (MEM), a least-squares minimization for scattering problems in quantum mechanics, and show that they provide an efficient, numerically stable alternative to Kohn variational principle. MEM defines an error-functional consisting of the sum of the values of (HPsi - EPsi)2 at a set of grid points. The wave function Psi, is forced to satisfy the scattering boundary conditions and is determined by minimizing the least-squares error. We study two implementations of this idea. In one, we represent the wave function as a linear combination of Chebyshev polynomials and minimize the error by varying the coefficients of the expansion and the R-matrix (present in the asymptotic form of Psi). This leads to a linear equation for the coefficients and the R-matrix, which we solve by matrix inversion. In the other implementation, we use a conjugate-gradient procedure to minimize the error with respect to the values of Psi at the grid points and the R-matrix. The use of the Chebyshev polynomials allows an efficient and accurate calculation of the derivative of the wave function, by using Fast Chebyshev Transforms. We find that, unlike KVP, MEM is numerically stable when we use the R-matrix asymptotic condition and gives accurate wave functions in the interaction region.

Consistent quantum measurements

Whilst traditional modeling and research efforts put emphasis on the effect of congestion on travel choices, recently more and more studies examine the notion of travel time uncertainty on travel decision. In particular, travel time reliability forms an important component in modeling route choice and trip scheduling. This paper considers equilibrium trip scheduling under random travel delay in a single bottleneck. Travelers form a heterogeneous population with distinctive requirements on the probability of punctual arrival and each with a different valuation toward earliness and lateness. The analysis shows that random delay plays a significant role in travel costs and introduces substantial differences in the queuing pattern, departure and arrival times, as compared with the traditional deterministic model.

The consistent histories interpretation of quantum mechanics is a reformulation of the standard Copenhagen interpretation that aims at incorporating quantum probabilities as part of the axiomatic foundations of the theory. It is not only supposed to equip quantum mechanics with clear criteria of its own experimental verification but, first and foremost, to alleviate one of the stumbling blocks of the theory ‐ the measurement problem. Since the consistent histories interpretation operates with a series of quantum events integrated into one quantum history, the measurement problem is naturally absorbed as one of the events that build up a history. The interpretation rests upon the two following assumptions, proposed already by J. von Neumann: (1) both the microscopic and macroscopic regimes are subject to the same set of quantum laws and (2) a projector operator that is assigned to each event within a history permits to transcribe the history into

This is an excellent collection of 219 solved problems in quantum mechanics. They range from the continuity equation connecting probability current density with probability density (number 1), to non-relativistic bremsstrahlung calculated as in Heitler's book (number 219).