Nonrelativistic Quantum Mechanics Research Articles

Quantum Mechanics Based on an Extended Least Action Principle and Information Metrics of Vacuum Fluctuations

We show that the formulations of non-relativistic quantum mechanics can be derived from an extended least action principle. The principle can be considered as an extension of the least action principle from classical mechanics by factoring in two assumptions. First, the Planck constant defines the minimal amount of action a physical system needs to exhibit during its dynamics in order to be observable. Second, there is constant vacuum fluctuation along a classical trajectory. A novel method is introduced to define the information metrics to measure additional observability due to vacuum fluctuations, which is then converted to an additional action through the first assumption. Applying the variational principle to minimize the total actions allows us to recover the basic quantum formulations including the uncertainty relation and the Schrödinger equation in the position representation. In the momentum representation, the same method can be applied to obtain the Schrödinger equation for a free particle while further investigation is still needed for a particle with an external potential. Furthermore, the principle brings in new results on two fronts. At the conceptual level, we find that the information metrics for vacuum fluctuations are responsible for the origin of the Bohm quantum potential. Even though the Bohm potential for a bipartite system is inseparable, the underlying vacuum fluctuations are local. Thus, inseparability of the Bohm potential does not justify a non-local causal relation between the two subsystems.responsible for manifesting entanglement effects without underlying physical interactions, implying that entanglement effects are non-causal. At the mathematical level, quantifying the information metrics for vacuum fluctuations using more general definitions of relative entropy results in a generalized Schrödinger equation that depends on the order of relative entropy. The extended least action principle is a new mathematical tool. It can be applied to derive other quantum formalisms such as quantum scalar field theory.

Retraction: Non-Hermiticity of tunneling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics [Phys. Rev. A 107 , 052220 (2023)

Retraction: Non-Hermiticity of tunneling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics [Phys. Rev. A <b>107</b> , 052220 (2023)

Quantum ontology without textbooks. Nor overlapping

In this paper, I critically assess two recent proposals for an interpretation-independent understanding of non-relativistic quantum mechanics: the overlap strategy (Fraser & Vickers, 2022) and the textbook account (Egg, 2021). My argument has three steps. I first argue that they presume a Quinean-Carnapian meta-ontological framework that yields flat, structureless ontologies. Second, such ontologies are unable to solve the problems that quantum ontologists want to solve. Finally, only structured ontologies are capable of solving the problems that quantum ontologists want to solve. But they require some dose of speculation. In the end, I defend the conservative way to do quantum ontology, which is (and must be) speculative and non-neutral.

Neutrino oscillations as a single Feynman diagram

We propose an approach to neutrino oscillations in vacuum, based on quantum field theory (QFT). The neutrino emission and detection are identified with the charged-current vertices of a single second-order Feynman diagram for the underlying process, enclosing neutrino propagation between these two points. The key point of the presented approach is the definition of the space-time setup typical for neutrino oscillation experiments, implying macroscopically large but finite volumes of the source and detector separated by a sufficiently large distance L. We derive an L-dependent master formula for the charged lepton production rate, which provides the QFT basis for the analysis of neutrino oscillations. This formula depends on the underlying process and is not reducible to the conventional approach resorting to the concept of neutrino oscillation probability, which originates from non-relativistic quantum mechanics. We demonstrate that for some particular choice of the underlying process the derived master formula approximately coincides with the conventional one under some assumptions. In support to presented approach we show that it provides the QFT framework not only for neutrino–neutrino but also neutrino-antineutrino oscillations. It is also argued that the proposed formalism allows us to consistently incorporate medium effects, when neutrinos oscillate in dense matter.

Diffeomorphism invariance and quantum mechanical paradoxes

Paradoxes in gravitational physics, such as grandfather paradoxes with closed timelike curves or the AMPS paradox, are often constructed in a weak gravity regime but upon further examination engender strong gravitational responses such as unstable Cauchy horizons or firewalls. In contrast, some proposed paradoxes in nonrelativistic quantum mechanics ignore gravity completely. Such nonrelativistic proposed paradoxes are often not gauge invariant — they use local operators which should be forbidden by diffeomorphism-invariant quantum gravity. Ignoring this complication is reasonable if there is a consistent weak gravity regime where the paradox can be formulated with gauge invariant observables up to some order in Newton’s constant. We show that this approach remains inconsistent due to a lack of analyticity of the necessary solutions. As a consequence, there is no consistent weak gravity, diffeomorphism invariant embedding of such paradoxes—just as in other gravitational paradoxes they generate a strong gravitational response and are in principle sensitive to quantum gravity.

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&lt;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.

Quantum Scalar Field Theory Based on an Extended Least Action Principle

Recently it is shown that the non-relativistic quantum formulations can be derived from an extended least action principle Yang (2023). In this paper, we apply the principle to massive scalar fields, and derive the Schrödinger equation of the wave functional for the scalar fields. The principle extends the least action principle in classical field theory by factoring in two assumptions. First, the Planck constant defines the minimal amount of action a field needs to exhibit in order to be observable. Second, there are constant random field fluctuations. A novel method is introduced to define the information metrics to measure additional observable information due to the field fluctuations, which is then converted to the additional action through the first assumption. Applying the variation principle to minimize the total actions allows us to elegantly derive the transition probability of field fluctuations, the uncertainty relation, and the Schrödinger equation of the wave functional. Furthermore, by defining the information metrics for field fluctuations using general definitions of relative entropy, we obtain a generalized Schrödinger equation of the wave functional that depends on the order of relative entropy. Our results demonstrate that the extended least action principle can be applied to derive both non-relativistic quantum mechanics and relativistic quantum scalar field theory. We expect it can be further used to obtain quantum theory for non-scalar fields.

Earthquake Quantization

In this homage to Einstein&amp;apos;s 144th birthday we propose a novel quantization prescription, where the paths of a path-integral are not random, but rather solutions of a geodesic equation in a random background. We show that this change of perspective can be made mathematically equivalent to the usual formulations of non-relativistic quantum mechanics. To conclude, we comment on conceptual issues, such as quantum gravity coupled to matter and the quantum equivalence principle.

Quantum Fisher information in moving reference frame

In the field of quantum metrology, an important application is quantum parameter estimation. As the fundamental theory of quantum parameter estimation, quantum Cramér-Rao inequality shows that the variance of parameter estimation is determined by the inverse of quantum Fisher information. Higher quantum Fisher information corresponds to a lower variance, thereby improving the precision of parameter estimation. Quantum Fisher information has been extensively investigated in many aspects of non-relativistic quantum mechanics, including entanglement structure detection, quantum teleportation, quantum phase transition, quantum chaos, and quantum computation. However, there are few researches considering the influence of relativistic effect on quantum Fisher information, and therefore, we attempt to investigate this topic in this work. The relativistic transformation of particle states is employed, and the quantum Fisher information about amplitude parameter &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \theta $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and phase parameter &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$\varphi $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; are investigated in moving reference frame. In this work, the parameters to be estimated are encoded into the spin degree of freedom, and the pure single-qubit state and the pure two-qubit state are both considered. The quantum Fisher information about &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \theta $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M5.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M5.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$\varphi $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M6.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M6.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; of single-qubit state and two-qubit state in moving reference frame are numerically calculated, respectively. It can be observed that the quantum Fisher information is associated with rapidity, amplitude parameter, and the ratio of the width to the particle mass &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M7.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M7.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. The quantum Fisher information of the estimated parameters decreases with rapidity increasing for both single-qubit state and two-qubit state. As rapidity approaches infinity, i.e. increases to the speed of light, the quantum Fisher information reaches to a constant which decreases as the ratio &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M8.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M8.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; increases. More importantly, for the phase parameter &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ \varphi $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M9.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M9.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, it is observed that the quantum Fisher information of two-qubit state reduces more significantly than that of single-qubit state. While, for the amplitude parameter &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$\theta $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M10.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M10.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the quantum Fisher information of two-qubit state is greater than that of single-qubit state. These results are useful and valuable for improving the precision of parameter estimation under the influence of relativistic effect.

New spin and p-spin symmetries of the deformed Dirac equation for the IMRQHPs including the improved YLT interaction in three-dimensional extended relativistic quantum mechanics symmetries

In this paper, the modified Dirac equation (MDE) has been solved approximately for improved Manning–Rosen plus quasi-Hellman potentials (IMRQHPs), including the improved Yukawa tensor interaction under new spin symmetry and pseudospin symmetry for arbitrary spin–orbit number [Formula: see text], an appropriate approximation employed on the centrifugal terms in the symmetries of three-dimensional extended relativistic quantum mechanics (3D-ERQM). This potential is a superposition of Manning–Rosen plus quasi-Hellman potentials and some other exponential terms. Through the application of both new approximations to deal with the centrifugal term, Bopp’s shift method and the independent-time perturbation theory method, the recent modified approximate energy corrections under IMRQHPs are obtained for nuclei [Formula: see text]O and [Formula: see text]F and (O2, I2, N2, H2, LiH, CO and NO) diatomic molecules in 3D-ERQM and three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries. The recent values that we get appear sensitive to [Formula: see text], which are known as the discrete atomic quantum numbers, the mixed potential depths ([Formula: see text], [Formula: see text]) and [Formula: see text], the range of the potential [Formula: see text] and non-commutativity parameters [Formula: see text]. In both 3D-ERQM and 3D-ENRQM symmetries, we show that the recent corrections obtained on the new bound states under IMRQHPs are infinitesimal to the principal part of energy in the ordinary cases of 3D-RQM and 3D-NRQM symmetries. In the new symmetries of 3D-ERQM symmetries, it is not possible to get the exact analytical solutions [Formula: see text] and [Formula: see text], only the approximate solutions are possible. In addition, we discussed in detail the non-relativistic study of the studied mixed potentials. Four special cases, i.e. we investigated some special cases in the context of modified Schrödinger theories. In the framework of known relativistic quantum mechanics, we have clearly demonstrated that the modified Schrödinger equation can be represented as the Dirac equation under the influence of IMRQHPs.

The connection of mathematics and logic in the structure of axiomatized and formalized theories

The aim of the research is to study the relationship between logic and mathematics in the structure of axiomatized and formalized scientific theories. The object of the study is the explication of this connection and its explanation. The subject of the study is syntactic and semantic views on the structure of scientific theories, the relationship between logic and mathematics has not been studied in detail in them. In the syntactic view, the structure of the theory is understood as a linguistic construct build from various logical propositions of the theoretical level, correspondence propositions and observation propositions. The structure of the theory does not take into account the variety of model representations of the theory that generate a variety of language constructs. The semantic view overcomes this disadvantage, and in it the structure of the theory is presented as a hierarchy of models: from axioms to theoretical-level models, experimental models and data models. The structure of the theory, the connection of logic and mathematics were studied using comparative analysis, methods of interpretive analysis and reconstruction of scientific theories. The methods made it possible to explicate mathematical concepts in the structure of the theory and correlate them with logic and natural language. Comparative analysis has shown that in the syntactic view, the connection between logic and mathematics lies in the fact that mathematical concepts of physics are interpreted in the language of logic of first-order predicates with equality. The connection between mathematical concepts is provided by the axiomatic method, which serves as a means of formalizing concepts. Mathematics comes down to logic. In the semantic approach, in order to identify the connection between mathematics and logic, it was necessary to reconstruct the structure of non-relativistic quantum mechanics. With the help of the set-theoretic predicate of Suppes, its axioms were determined, the connection between mathematical structures, postulates of the theory, axioms, and observable quantities was established. Logic and mathematics are related to each other in such a way that metamathematics or linguistics is a part of mathematics. Mathematics includes set theory and model theory, i.e. mathematical logic. The connection of mathematical formalisms with phenomena and with natural language remains problematic, and there is this drawback in the syntactic approach. The novelty lies in the fact that the research contributes to the methodology and logic of science, to the explanation of the connection between logic and mathematics in scientific theory, which was illustrated by various examples from various fields of physics.

New Eigensolution of the Klein--Gordon and Shrödinger equations for improved modified Yukawa-Kratzer potential and its applications using Boop's Shift method and standard perturbation theory in the 3D-ERQM and 3D-ENRQM symmetries

The deformed Klein-Gordonequation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the improved modified Yukawa-Kratzer potential (IMYKP) model under the influence of the deformation space-space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H2) molecules under the IMYKP model it was shown to be sensitive to the atomic quantum numbers (j, l, s, m), mixed potential depths (V0, De, re), the screening parameter’s inverse α and noncommutativity parameters (Θ,τ ,χ). In addition, we analyzed the nonrelativistic energy values by applying the well-known transmission rules known in the literature. In addition, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the improved modified Kratzer potential, the improved generalized Kratzer potential, the improved Kratzer potential, the improved modified Kratzer plus screened Coulomb potential, the improved Hellmann potential, the improved Yukawa potential, and improved inversely square Yukawa potential. We noticed that these particular results are identical to our previous work and other known works in the literature. The study is further extended to calculate the mass spectra of mesons of charmonium (cc) and bottomonium (bb) within the framework of the IMYKP model in three-dimensional extended non-relativistic quantum mechanics (3D-ENRQM) symmetries.

The role of the pion in the lineshape of the X(3872)

We determine the contribution of long-range pion interactions to the X(3872) dynamics, assuming it is a loosely bound D0D¯⁎0 molecule. Our result is based on the distorted wave Born approximation in non-relativistic quantum mechanics. Despite their long-range nature, we find that pion interactions cannot produce a large and negative effective range. Nonetheless, they introduce imaginary parts. In particular, they contribute to the total decay width of the X(3872) a term associated with, but not precisely corresponding to, the D⁎ width. Our approach can also be applied to the recently discovered Tcc+ state.

The use of Laplace transform in the inverse problem from bound states

"In a previous work dedicated to non-relativistic quantum mechanics in the three-dimensional space with spherical symmetry, we have shown the possibility of reconstructing the ground state density from its moments in the case of bound states. The method is based on relationships connecting these moments to the lower energy of each state of angular momentum `. The ground state density yields the ground state wave function, which allows us to calculate the equivalent local potential by inverting the Schrodinger equation. This calculus uses the Fourier transform of the ground state density ρ(r). The aim of this work is to check how to improve the results by replacing the Fourier transform by the Laplace transform of the quantity r2 ρ(r). The method is first presented. Then examples and results are compared to those obtained previously. "

Adiabatic Shortcuts Completion in Quantum Field Theory: Annihilation of Created Particles.

Shortcuts to adiabaticity (STA) are relevant in the context of quantum systems, particularly regarding their control when they are subjected to time-dependent external conditions. In this paper, we investigate the completion of a nonadiabatic evolution into a shortcut to adiabaticity for a quantum field confined within a one-dimensional cavity containing two movable mirrors. Expanding upon our prior research, we characterize the field's state using two Moore functions that enables us to apply reverse engineering techniques in constructing the STA. Regardless of the initial evolution, we achieve a smooth extension of the Moore functions that implements the STA. This extension facilitates the computation of the mirrors' trajectories based on the aforementioned functions. Additionally, we draw attention to the existence of a comparable problem within nonrelativistic quantum mechanics.

Relativistic approach to scattering

The phenomenon of scattering is an important tool to understand the structure of particles and forces between them. Many of the properties like charge, spin, size etc. of microscopic particles are better understood with the help of scattering experiments. Essentially, the scattering theory in non-relativistic quantum mechanics is concerned with the solution of Schrödinger wave equation for variety of potentials with appropriate boundary conditions. Common approaches to scattering problems in quantum mechanics are the method of partial waves and Born approximation methods. In this paper we review the general features of potential scattering. Then we extend the formalism of the method of partial waves for relativistic case and apply the method to study scattering by few well known potentials.

Deriving the generalized momentum operators from covariant derivatives

The generalized momentum operators are derived in the framework of non-relativistic quantum mechanics, taking into account an especial assumption on the covariant derivative. According to this assumption, a scalar density doesn’t change under the action of covariant derivative. The Hermitian form of the covariant derivative is discussed. It is shown that, with the help of the adjoint of covariant derivatives, generalized momentum operators can be derived. The inverse of covariant derivative is calculated. It is shown that the inverse of generalized momentum operators can be deduced from the inverse of covariant derivatives.

The Influence of Deformation Space-Space on High and Low Energy Spectra of Fermionic Particles and Spectra of Heavy Quarkonia with Improved Hulthén Plus Hyperbolic Exponential Inversely Quadratic Potential

In this work, the modified approximation to the centrifugal barrier term is applied to find approximate bound-state solutions of the deformed Dirac equation for the spin and pseudospin symmetries in a model with the improved hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials (IHHEIQPs) using the parametric method of Bopp’s shift and the standard perturbation theory in the extended relativistic quantum mechanics (ERQM). Our results indicate that the new energy eigenvalues are highly sensitive to the potential parameters (ν1, A) and to the values of quantum atomicnumbers (j, k, l, m,̃︀ l, ̃︀ m, s, ̃︀s), range of the potential ν, and noncommutativity parameters (θ, σ, β). We found that the effect of a space-space deformation gives a correction in the energy spectrum, where the main energy term remains due to the effect of the hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials known in the literature. The new nonrelativistic energies are obtained by applying the nonrelativistic limit to the relativistic spin-energy equation in the extended nonrelativistic quantum mechanics (ENRQM). The proposed potential model reduces to the improved Hulth´en and exponential inversely quadratic potentials as special cases in ERQM. The present results are applied for calculating the new mass spectra Mhiqpnc−nl of heavy mesons such as cc, bb, bc, bs, cs, and bq,q = (u, d) in ENRQM. It turns out that the values of masses come from the contribution of the mass spectra Mhiqpnl in NRQM, while the effect of a space-space deformation δMhiqpnc−nl is an infinitesimal correction as compared with Mhiqpnl. Our results seem to be significant and agree perfectly with the ones in the literature.

RETRACTED: Non-Hermiticity of tunneling time in a spacetime-symmetric extension of nonrelativistic quantum mechanics

Time continues to be an intriguing physical property in the modern era. On the one hand, we have the classical and relativistic notions of time, where space and time have the same hierarchy, which is essential in describing events in spacetime. On the other hand, in quantum mechanics time appears as a classical parameter, meaning that it does not have an uncertainty relation with its canonical conjugate. In this work, we use a recent spacetime-symmetric proposal [Phys. Rev. A 95, 032133 (2017)] that tries to solve the unbalance in nonrelativistic quantum mechanics by extending the usual Hilbert space, having the time parameter $t$ and the position operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{X}$ in one subspace and the position parameter $x$ and time operator $\mathbb{T}$ in the other subspace. Time as an operator is better suitable for describing tunneling processes. We then solve the $1/2$-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunneling time through a rectangular barrier. Using a Gaussian energy distribution, we demonstrate that, for wavepackets well resolved in time, the expectation value of the operator $\mathbb{T}$ is the energy average of the classical time ${T}_{\text{class}}=\ensuremath{\partial}S/\ensuremath{\partial}E$, where $S$ is the classical action, which can be real or imaginary. For wavepackets not well resolved in time, the contribution of ${T}_{\text{class}}$ consistently vanishes, and solely properties of the energy distribution contribute to $\mathbb{T}$. We show that the time of travel for nontunneling particles is purely real. When tunneling is involved, complex arrival times emerge, becoming a signature of tunneling. Furthermore, we apply our results to a constant energy distribution, obtaining pure imaginary times for energies below the barrier while obtaining complex times for particles with a wavepacket spreading energies below and above the barrier, and show a comparison to previous works.

Spectrum of the wave equation with Dirac damping on a non-compact star graph

We consider the wave equation on non-compact star graphs, subject to a distributional damping defined through a Robin-type vertex condition with complex coupling. It is shown that the non-self-adjoint generator of the evolution problem admits an abrupt change in its spectral properties for a special coupling related to the number of graph edges. As an application, we show that the evolution problem is highly unstable for the critical couplings. The relationship with the Dirac equation in non-relativistic quantum mechanics is also mentioned.