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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. [ABSTRACT FROM AUTHOR]

In this paper, we derive a quantum speed limit for unitary evolution for the case of mixed quantum states using the stronger uncertainty relation for mixed quantum states. This bound can be optimized over different choices of Hermitian operators for a better bound. We illustrate this with some examples and show its better performance with respect to three existing bounds for mixed quantum states. [ABSTRACT FROM AUTHOR]

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.

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.

In this paper, we derive a quantum speed limit for unitary evolution for the case of mixed quantum states using the stronger uncertainty relation for mixed quantum states. This bound can be optimized over different choices of Hermitian operators for a better bound. We illustrate this with some examples and show its better performance with respect to three existing bounds for mixed quantum states.

We revisit the arguments underlying two well-known arrival-time distributions in quantum mechanics, viz., the Aharonov–Bohm–Kijowski (ABK) distribution, applicable for freely moving particles, and the quantum flux (QF) distribution. An inconsistency in the original axiomatic derivation of Kijowski's result is pointed out, along with an inescapable consequence of the 'negative arrival times' inherent to this proposal (and generalizations thereof). The ABK free-particle restriction is lifted in a discussion of an explicit arrival-time set-up featuring a charged particle moving in a constant magnetic field. A natural generalization of the ABK distribution is in this case shown to be critically gauge-dependent. A direct comparison to the QF distribution, which does not exhibit this flaw, is drawn (its acknowledged drawback concerning the quantum backflow effect notwithstanding). [ABSTRACT FROM AUTHOR]

The measurement of the tunneling time delay in attosecond experiments, termed attoclock, sparked a heated debate about tunneling time and the role of time in quantum mechanics. We show a model that explains the experimental result of the attoclock in the nonadiabatic field calibration, where we find good agreement with the experimental data of Hofmann et al. (J. Mod. Opt. 66, 1052, 2019). We confirm our result with the numerical integration of the time-dependent Schrödinger equation. We also found that at a field strength F ≤ F a (F a is the atomic field strength), the model always predicts a time delay with respect to the quantum (lower) limit τ a at F = F a . For an adiabatic tunneling, the saturation at the limit (F = F a ) explains the well-known Hartman effect or Hartman paradox. • We show a model that explains the experimental result of the attoclock in the nonadiabatic field calibration. • A theory with a simple analytical expression for tunneling time delay composed of basic laser and atomic parameters. • The model provides simple physical picture for the tunnel-ionization time delay. • It sheds light on roles of time in quantum mechanics. [ABSTRACT FROM AUTHOR]

The Mandelstam–Tamm quantum speed limit (QSL) puts a bound on how fast a closed system in a pure state can evolve. In this paper, we derive several extensions of this QSL to closed systems in mixed states. We also compare the strengths of these extensions and examine their tightness. The most widely used extension of the Mandelstam–Tamm QSL originates in Uhlmann’s energy dispersion estimate. We carefully analyze the underlying geometry of this estimate, an analysis that makes apparent that the Bures metric, or equivalently the quantum Fisher information, will rarely give rise to tight extensions. This observation leads us to address whether there is a tightest general extension of the Mandelstam–Tamm QSL. Using a geometric construction similar to that developed by Uhlmann, we prove that this is indeed the case. In addition, we show that tight evolutions of mixed states are typically generated by time-varying Hamiltonians, which contrasts with the case for systems in pure states.

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. [ABSTRACT FROM AUTHOR]

The measurement of the tunneling time (T-time) in today’s attosecond and strong field (low-frequency) experiments, despite its controversial discussion, offers a fruitful opportunity to understand time measurement and the time in quantum mechanics. In addition, as we will see in this work, a related controversial issue is the particulate nature of the radiation. The T-time in attosecond experiment and its different aspects and models, is discussed in this work, especially in relation to my model of real T-time (Kullie, 2015), where a good agreement with the experiment and an intriguing similarity to the Bohr–Einstein photon box Gedanken experiment was found. The tunneling process itself is still not well understood, but I am arguing that a scattering mechanism (by the laser wave packet) offers a possibility to understand the tunneling process in the tunneling region. This is related to the question about the corpuscular nature of light which is widely discussed in modern quantum optics experiments. [ABSTRACT FROM AUTHOR]

In this article we consider linear operators satisfying a generalized commutation relation of a type of the Heisenberg-Lie algebra. It is proven that a generalized inequality of the Hardy's uncertainty principle lemma follows. Its applications to time operators and abstract Dirac operators are also investigated. [ABSTRACT FROM AUTHOR]

We develop a mathematical theory of time operators of a Hamiltonian with purely discrete spectrum. The main results include boundedness, unboundedness and spectral properties of them. In addition, possible connections of a time operator of H with regular perturbation theory are discussed. [ABSTRACT FROM AUTHOR]

Some results are briefly reviewed and developments are presented on the study of Time in quantum mechanics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the hermitian (more precisely, maximal hermitian, but non-selfadjoint) operator for Time which appears: (i) for particles, in ordinary non-relativistic quantum mechanics; and (ii) for photons (i.e., in first-quantization quantum electrodynamics). [ABSTRACT FROM AUTHOR]

Some new results are presented and recent developments are reviewed on the study of Time in quantum physics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the Hermitian (more precisely, maximal Hermitian, but non-self-adjoint) operator for Time which appears: (i) for particles, in ordinary nonrelativistic quantum mechanics; and (ii) for photons (i.e. in first-quantization quantum electrodynamics). In conclusion, various recent and possible future applications of the time quantum analysis for tunnelling processes, nuclear collisions and systems with (quasi)discrete energy spectra are indicated. [ABSTRACT FROM AUTHOR]

Let H be a self-adjoint operator on a Hilbert space $\mathcal{H}$, T be a symmetric operator on $\mathcal{H}$ and K(t)(t ∈ ℝ) be a bounded self-adjoint operator on $\mathcal{H}$. We say that (T, H, K) obeys the generalized weak Weyl relation (GWWR) if e-itHD(T) ⊂ D(T) for all t ∈ ℝ and Te-itHψ = e-itH(T+K(t))ψ, ∀ψ ∈ D(T) (D(T) denotes the domain of T). In the context of quantum mechanics where H is the Hamiltonian of a quantum system, we call T a generalized time operator of H. We first investigate, in an abstract framework, mathematical structures and properties of triples (T, H, K) obeying the GWWR. These include the absolute continuity of the spectrum of H restricted to a closed subspace of $\mathcal{H}$, an uncertainty relation between H and T (a "time-energy uncertainty relation"), the decay property of transition probabilities |〈ψ,e-itHϕ〉|2 as |t| → ∞ for all vectors ψ and ϕ in a subspace of $\mathcal{H}$, where 〈·,·〉 denotes the inner product of $\mathcal{H}$. We describe methods to construct various examples of triples (T, H, K) obeying the GWWR. In particular, we show that there exist generalized time operators of second quantization operators on Fock spaces (full Fock spaces, boson Fock spaces and fermion Fock spaces) which may have applications to quantum field models with interactions. [ABSTRACT FROM AUTHOR]