Energy Uncertainty Relation Research Articles

Time–Energy Uncertainty Relation in Nonrelativistic Quantum Mechanics

The time–energy uncertainty relation in nonrelativistic quantum mechanics has been intensely debated with regard to its formal derivation, validity, and physical meaning. Here, we analyze two formal relations proposed by Mandelstam and Tamm and by Margolus and Levitin and evaluate their validity using a minimal quantum toy model composed of a single qubit inside an external magnetic field. We show that the ℓ1 norm of energy coherence C is invariant with respect to the unitary evolution of the quantum state. Thus, the ℓ1 norm of energy coherence C of an initial quantum state is useful for the classification of the ability of quantum observables to change in time or the ability of the quantum state to evolve into an orthogonal state. In the single-qubit toy model, for quantum states with the submaximal ℓ1 norm of energy coherence, C<1, the Mandelstam–Tamm and Margolus–Levitin relations generate instances of infinite “time uncertainty” that is devoid of physical meaning. Only for quantum states with the maximal ℓ1 norm of energy coherence, C=1, the Mandelstam–Tamm and Margolus–Levitin relations avoid infinite “time uncertainty”, but they both reduce to a strict equality that expresses the Einstein–Planck relation between energy and frequency. The presented results elucidate the fact that the time in the Schrödinger equation is a scalar variable that commutes with the quantum Hamiltonian and is not subject to statistical variance.

Time–Energy Uncertainty Relation for Neutrino Oscillations: Historical Development, Applications, and Future Prospects

The time–energy uncertainty relation (TEUR) plays a fundamental role in quantum mechanics, as it allows the grasping of peculiar aspects of a variety of phenomena based on very general principles and symmetries of the theory. Using the Mandelstam–Tamm method, TEUR has recently been derived for neutrino oscillations by connecting the uncertainty in neutrino energy with the characteristic timescale of oscillations. Interestingly, the suggested interpretation of neutrinos as unstable-like particles has proved to naturally emerge in this context. Further aspects were later discussed in semiclassical gravity theory, by computing corrections to the neutrino energy uncertainty in a generic stationary curved spacetime, and in quantum field theory, where the clock observable turns out to be identified with the non-conserved flavor charge operator. In the present work, we give an overview on the above achievements. In particular, we analyze the implications of TEUR and explore the impact of gravitational and non-relativistic effects on the standard condition for neutrino oscillations.

Bound-state formation in time-dependent potentials

We study the temporal formation of quantum-mechanical bound states within a one-dimensional attractive square-well potential by first solving the time-independent Schr\"odinger equation and then studying a time-dependent system with an external time-dependent potential. For this we introduce Gaussian potentials with different spatial and temporal extensions and generalize this description also for subsequent pulses and for random noisy potentials. Our main goal is to study the timescales in which the bound state is populated and depopulated. Particularly, we clarify a likely connection between the uncertainty relation for energy and time and the transition time between different energy eigenstates. We demonstrate that the formation of states is not delayed due to the uncertainty relation but follows the pulse shape of the perturbation. In addition we investigate the (non-)applicability of first-order perturbation theory on the considered quantum system.

Exploring the implications of the uncertainty relationships in quantum mechanics

Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.

Transactional Interpretation and the Generalized Poisson Distribution

The aim of this paper is to study the quantum-like approach to the description of the market in the context of the principle of minimum Fisher information. We wish to investigate the validity of using squeezed coherent states as market strategies. For this purpose, we focus on the representation of any squeezed coherent state with respect to the basis of the eigenvectors of the observable of market risk. We derive a formula for the probability of being the squeezed coherent state in one of these states. The distribution that we call generalized Poisson establishes the relation between the squeezed coherent states and their description in the language of risk in quantum terms. We provide a formula specifying the total risk of squeezed coherent strategy. Then, we propose a risk of risk concept that is in fact the second central moment of the generalized Poisson distribution. This is an important numerical characterization of squeezed coherent strategies. We provide its interpretations on the basis of the uncertainty relation for time and energy.

Generalized Heisenberg uncertainty relation in spherical coordinates

Following to the Weil method, we generalize the Heisenberg–Robertson uncertainty relation for arbitrary two operators. Consideration is made in spherical coordinates, where the distant variable [Formula: see text] is restricted from one side, [Formula: see text]. By this reason, accounting of suitable boundary condition at the origin for radial wavefunctions and operators is necessary. Therefore, there arise extra surface terms in comparison with traditional approaches. These extra terms are calculated for various solvable potentials and their influence is investigated. At last, the time–energy uncertainty relations are also analyzed. Some differences between our approach and that, in which a direct product for separate variances were considered, are discussed.

A time–energy uncertainty relation in chemical thermodynamics

An analogy between the thermodynamic inequalities presented by Nicholson et al. [Nat. Phys. 16, 1211 (2020)] and by Yoshimura and Ito [Phys. Rev. Res. 3, 013175 (2021)] is discussed. As a result, a time–energy uncertainty relation in chemical thermodynamics in terms of Gibbs free energy and chemical potential is derived. It is numerically demonstrated that the uncertainly relation holds in a model system of oscillatory Brusselator reactions. Our result bridges the thermodynamic time–information uncertainty relation and free energy evolution in chemical reactions.

Neutrino Mixing and Oscillations in Quantum Field Theory: A Comprehensive Introduction

We review some of the main results of the quantum field theoretical approach to neutrino mixing and oscillations. We show that the quantum field theoretical framework, where flavor vacuum is defined, permits giving a precise definition of flavor states as eigenstates of (non-conserved) lepton charges. We obtain the exact oscillation formula, which in the relativistic limit reproduces the Pontecorvo oscillation formula and illustrates some of the contradictions arising in the quantum mechanics approximation. We show that the gauge theory structure underlies the neutrino mixing phenomenon and that there exists entanglement between mixed neutrinos. The flavor vacuum is found to be an entangled generalized coherent state of SU(2). We also discuss flavor energy uncertainty relations, which impose a lower bound on the precision of neutrino energy measurements, and we show that the flavor vacuum inescapably emerges in certain classes of models with dynamical symmetry breaking.

Thermodynamic origin of quantum time–energy uncertainty relation

The problem of time is a notable obstacle towards the recognition of quantum theory as the ultimate fundamental description of nature. Quantum theory may not be complete if founded upon classical notions. Louis de Broglie, seeming to be more or less convinced about the ontology of his proposed matter waves, tried to develop a theory of sub-quantum degrees of freedom relying on statistical thermodynamics. He realized a quantum particle as a fluctuating dense corpuscle formed via non-linear effects from a sub-quantum medium. A wave on the medium guides the vibrating corpuscle. He argued that an intrinsic clock of a quantum particle is related to its Brownian motion at the sub-quantum level. This led him to conjecture a relation between the de Broglie clock frequency mc 2/h and its implicit temperature, which equals that of the surrounding sub-quantum medium. About the same time, Mandelbrot was the first to derive in a classical setting a thermodynamic uncertainty relation between energy and temperature, that was, coincidentally or not, anticipated by Bohr and Heisenberg in the first years of development of quantum theory. We show here that, when the de Broglie temperature–time conjecture is assumed, the thermodynamic temperature–energy uncertainty relation leads to the quantum time–energy uncertainty relation.

On the Speed Limit for Imaginary-Time Schrödinger Equation with Application to Quantum Searches

Recently, Okuyama and Ohzek [1] derived a speed limit for the imaginary-time Schrödinger equation, which is inspired by the prior work of Kieu, who had shown a new class of time–energy uncertainty relations suitable for actually evaluating the speed limit of quantum dynamics. In this paper, we apply the result of Okuyama and Ohzek to obtain a generalized speed limit for Grover’s search in imaginary-time quantum annealing. An estimate of the lower bound on the computational time is shown, from which the role of the coefficient function corresponding to the final Hamiltonian played in the quantum dynamics for the problem is sticking out. However, when trying to apply the speed limit to the analogue of Grover’s problem, we find that not only the coefficient of the target Hamiltonian is related to the time complexity of the algorithm, but also the coefficient of the initial Hamiltonian is crucial for determining the time complexity. This is new and generalizes one of the results in our previous work.

Time–information uncertainty relations in thermodynamics

Physical systems powering motion and creating structure in a fixed amount of time dissipate energy and produce entropy. Whether living, synthetic or engineered, systems performing these dynamic functions must balance dissipation and speed. Here, we show that rates of energy and entropy exchange are subject to a speed limit—a time–information uncertainty relation—imposed by the rates of change in the information content of the system. This uncertainty relation bounds the time that elapses before the change in a thermodynamic quantity has the same magnitude as its s.d. From this general bound, we establish a family of speed limits for heat, dissipated/chemical work and entropy depending on the experimental constraints on the system and its environment. In all of these inequalities, the timescale of transient dynamical fluctuations is universally bounded by the Fisher information. Moreover, they all have a mathematical form that mirrors the Mandelstam–Tamm version of the time–energy uncertainty relation in quantum mechanics. These bounds on the speed of arbitrary observables apply to transient systems away from thermodynamic equilibrium, independent of the physical constraints on the stochastic dynamics or their function. A time–information uncertainty relation in thermodynamics has been derived, analogous to the time–energy uncertainty relation in quantum mechanics, imposing limits on the speed of energy and entropy exchange between a system and external reservoirs.

On the time–energy uncertainty relation of Kieu applied to quantum evolutions of Grover’s search

Recently, Kieu derived a new class of time–energy uncertainty relations, which is not only formal but also suitable for evaluating the speed limit of quantum dynamics. However, when this time–energy uncertainty relation was applied to study the quantum adiabatic Grover’s search, some approximation had been done for reaching the right conclusion. That is, the final state being orthogonal to the initial state in this problem was assumed which may cause confusion, while the truth is that the overlap between these two states is indeed very small when the problem size is very big but not equal to zero exactly. In this paper, we modify the time–energy uncertainty relation of Kieu applied to two quantum evolutions of Grover’s problem, and find that the resulting computational complexities match well with the previous works on this topic but without the need of making the assumption like that in the work of Kieu.

Entropic time–energy uncertainty relations: an algebraic approach

We address entropic uncertainty relations between time and energy or, more precisely, between measurements of an observable G and the displacement r of the G-generated evolution e−irG. We derive lower bounds on the entropic uncertainty in two frequently considered scenarios, which can be illustrated as two different guessing games in which the role of the guessers are fixed or not. In particular, our bound for the first game improves the previous result by Coles et al [Phys. Rev. Lett. 122 100401 (2019)]. To derive our bounds, we extend a recently proposed novel algebraic method by Gao et al [arXiv:1710.10038 [quant-ph]] which was used to derive both strong subadditivity and entropic uncertainty relations for measurements.

Time-energy uncertainty relation for neutrino oscillations in curved spacetime

We derive the Mandelstam–Tamm time–energy uncertainty relation for neutrino oscillations in a generic stationary curved spacetime. In particular, by resorting to Stodolsky covariant formula of the quantum mechanical phase, we estimate gravity effects on the neutrino energy uncertainty. Deviations from the standard Minkowski result are explicitly evaluated in Schwarzschild, Lense–Thirring and Rindler (uniformly accelerated) geometries. Finally, we discuss how spacetime could affect the characteristic neutrino oscillation length in connection with the recent view of flavor neutrinos as unstable particles.

Cosmological Inflation Based on the Uncertainty-Mediated Dark Energy

It was suggested in our previous paper [Yu.V. Dumin, Grav. Cosmol. 25, 169 (2019)] that the cosmological Dark Energy might be mediated by the time–energy uncertainty relation in the Mandelstam–Tamm form, which is appropriate for the long-term evolution of quantum systems. The amount of such Dark Energy gradually decays with time, and the corresponding scale factor of the Universe increases by a “quasi-exponential” law (namely, the exponent is proportional to of the square root of time) throughout the entire cosmological evolution. While such a universal behavior looks quite appealing, an important question arises: Does the quasi-exponential expansion resolve the major problems of the early Universe in the same way as the standard inflationary scenario? In the present paper, we elucidate this issue by analyzing the causal structure of the space–time following from this model. It is found that the observed region of the Universe (the past light cone) covers a single causally connected domain developing from the Planck times (the future light cone). Consequently, there should be no appreciable inhomogeneity and anisotropy in the early Universe, creation of the topological defects will be suppressed, etc. From this point of view, the uncertainty-mediated Dark Energy can serve as a reasonable alternative to the standard (exponential) inflationary scenario.

Remarks on the uncertainty relations

We analyze general uncertainty relations and we show that there can exist such pairs of non-commuting observables [Formula: see text] and [Formula: see text] and such vectors that the lower bound for the product of standard deviations [Formula: see text] and [Formula: see text] calculated for these vectors is zero: [Formula: see text]. We also show that for some pairs of non-commuting observables the sets of vectors for which [Formula: see text] can be complete (total). The Heisenberg, [Formula: see text], and Mandelstam–Tamm (MT), [Formula: see text], time–energy uncertainty relations ([Formula: see text] is the characteristic time for the observable [Formula: see text]) are analyzed too. We show that the interpretation [Formula: see text] for eigenvectors of a Hamiltonian [Formula: see text] does not follow from the rigorous analysis of MT relation. We show also that contrary to the position–momentum uncertainty relation, the validity of the MT relation is limited: It does not hold on complete sets of eigenvectors of [Formula: see text] and [Formula: see text].

Critical Look at the Time--Energy Uncertainty Relations

The Heisenberg and Mandelstam-Tamm time-energy uncertainty relations are analyzed. The conlusion resulting from this analysis is that within the Quantum Mechanics of Schr\"{o}dinger and von Neumann, the status of these relations can not be considered as the same as the status of the position-momentum uncertainty relations, which are rigorous. The conclusion is that the time--energy uncertainty relations can not be considered as universally valid.

Time-Energy and Time-Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations.

In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation provides a general lower bound to the characteristic time with which the mean value of a generic quantum observable F can change with respect to the width of its uncertainty distribution (square root of F fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty (square root of energy fluctuations). Here we show that when unitary evolution is complemented with a steepest-entropy-ascent model of dissipation, the resulting nonlinear master equation entails that these lower bounds get modified and depend also on the entropy uncertainty (square root of entropy fluctuations). For example, we obtain the time–energy-and–time–entropy uncertainty relation where is a characteristic dissipation time functional that for each given state defines the strength of the nonunitary, steepest-entropy-ascent part of the assumed master equation. For purely dissipative dynamics this reduces to the time–entropy uncertainty relation , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty .

Flavor-energy uncertainty relations for neutrino oscillations in quantum field theory

In the context of quantum field theory, we derive flavor energy uncertainty relations for neutrino oscillations. By identifying the non conserved flavor charges with the clock observables, we arrive at the Mandelstam Tamm version of time energy uncertainty relations. In the ultrarelativistic limit these relations yield the well known condition for neutrino oscillations. Ensuing non relativistic corrections to the latter are explicitly evaluated. The analogy among flavor states and unstable particles and a novel interpretation of our uncertainty relations, based on the unitary inequivalence of Fock spaces for flavor and massive neutrinos, are also discussed.

On the problem of time in quantum mechanics

The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.