Hamiltonian Operator Research Articles

Unitary evolution for a two-level quantum system in fractional-time scenario.

The time-evolution operator obtained from the fractional-time Schrödinger equation(FTSE) is said to be nonunitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the nonunitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.

The NOSL-296 high resolution 14N216O line list for atmospheric applications

The Nitrous Oxide Spectroscopic Line list (NOSL-296) presented in this work is an empirical line list for the main isotopologue of nitrous oxide, 14N216O, mainly dedicated to atmospheric applications. It contains line positions and intensities at a reference temperature of 296 K calculated using the effective operator approach. The intensity cutoff is 10−30 cm/molecule. Almost 900,000 lines covering the 0.02–13,378 cm−1 spectral range are included in NOSL-296. The line list is based on the fitted non-polyad models of the effective Hamiltonian and effective dipole moment operator describing globally an exhaustive review of experimental data collected from the literature (more than 51,500 line intensities). The non-polyad model of effective Hamiltonian has been elaborated in [Tashkun SA, J Quant Spectrosc Radiat Transfer 2019;231:88–101, doi:10.1016/j.jqsrt.2019.04.023] while the non-polyad model of the effective dipole moment operator is developed in the present work. The fitted models are able to reproduce measured positions and intensities with accuracies compatible with measurement uncertainties. Comparison of NOSL-296 with HITRAN2020 and Ames-296 line lists are discussed. The NOSL-296 line list is provided as a Supplementary Material and freely accessible at https://ftp.iao.ru/pub/LTS/NOSL-296/.

Time-dependent Schrieffer-Wolff-Lindblad perturbation theory: Measurement-induced dephasing and second-order Stark shift in dispersive readout

We develop a time-dependent Schrieffer-Wolff-Lindblad perturbation theory to study effective interactions for driven open quantum systems. The starting point of our analysis is a given Lindblad equation, based on which we obtain an effective (averaged) map that describes the renormalization of both the Hamiltonian and collapse operators due to the drive. As a case study, we apply this method to the dispersive readout of a transmon qubit and derive an effective disperive map that describes measurement-induced dephasing and Stark shift for the transmon. The effective map we derive is completely positive and trace-preserving under adiabatic resonator response. To benchmark our method, we demonstrate good agreement with a numerical computation of the effective rates via the Lindbladian spectrum. Our results are also in agreement with, and extend upon, an earlier derivation of such effects by Gambetta et al. (Phys. Rev. A 74, 042318 (2006)) using the positive P-representation for the resonator field.

Existence of the transfer matrix for a class of nonlocal potentials in two dimensions

Evanescent waves are waves that decay or grow exponentially in regions of the space void of interaction. In potential scattering defined by the Schrödinger equation, for a local potential v, they arise in dimensions greater than one and are generally present regardless of the details of v. The approximation in which one ignores the contributions of the evanescent waves to the scattering process corresponds to replacing v with a certain energy-dependent nonlocal potential . We present a dynamical formulation of the stationary scattering for in two dimensions, where the scattering data are related to the dynamics of a quantum system having a non-self-adjoint, unbounded, and nonstationary Hamiltonian operator. The evolution operator for this system determines a two-dimensional analog of the transfer matrix of stationary scattering in one dimension which contains the information about the scattering properties of the potential. Under rather general conditions on v, we establish the strong convergence of the Dyson series expansion of the evolution operator and prove the existence of the transfer matrix for as a densely-defined operator acting in .

On New Hamiltonian Structures of Two Integrable Couplings

In this paper, we present new Hamiltonian operators for the integrable couplings of the Ablowitz–Kaup–Newell–Segur hierarchy and the Kaup–Newell hierarchy. The corresponding Hamiltonians allow nontrivial degeneration. Multi-Hamiltonian structures are investigated. The involutive property is proven for the new and known Hamiltonians with respect to the two Poisson brackets defined by the new and known Hamiltonian operators.

Quasilinear Systems of First Order PDEs with Nonlocal Hamiltonian Structures

In this paper we wonder whether a quasilinear system of PDEs of first order admits Hamiltonian formulation with local and nonlocal operators. By using the theory of differential coverings, we find differential-geometric conditions necessary to write a given system with one of the three Hamiltonian operators investigated.

The Disappearance and Reappearance of Potential Energy in Classical and Quantum Electrodynamics

In electrostatics, we can use either potential energy or field energy to ensure conservation of energy. In electrodynamics, the former option is unavailable. To ensure conservation of energy, we must attribute energy to the electromagnetic field and, in particular, to electromagnetic radiation. If we adopt the standard energy density for the electromagnetic field, then potential energy seems to disappear. However, a closer look at electrodynamics shows that this conclusion actually depends on the kind of matter being considered. Although we cannot get by without attributing energy to the electromagnetic field, matter may still have electromagnetic potential energy. Indeed, if we take the matter to be represented by the Dirac field (in a classical precursor to quantum electrodynamics), then it will possess potential energy (as can be seen by examining the symmetric energy-momentum tensor of the Dirac field). Thus, potential energy reappears. Upon field quantization, the potential energy of the Dirac field becomes an interaction term in the Hamiltonian operator of quantum electrodynamics.

Pointwise Weyl laws for Schrödinger operators with singular potentials

We consider the Schrödinger operators HV=−Δg+V with singular potentials V on general n-dimensional Riemannian manifolds and study the eigenvalues and eigenfunctions under this perturbation. These singular potentials appear naturally in physics, most notably the Coulomb potential |x|−1. Sogge and the first author [14] proved the sharp Weyl laws for these HV with potentials in the Kato class, which is the minimal assumption to ensure that HV is essentially self-adjoint and bounded from below and the eigenfunctions of HV are bounded. Later, Frank-Sabin [9] studied the problem on the pointwise Weyl laws for these HV in three dimensions by extending the method of Avakumović [2], while it is unknown how to reconstruct this argument in other dimensions. In this paper, we completely solve this problem in any dimensions by using a different argument. First, we establish the pointwise Weyl law for potentials in the Kato class on any n-dimensional manifolds. This extends the 3-dimensional results of Frank-Sabin [9] by a different method. Second, we prove that the pointwise Weyl law with the standard sharp error term O(λn−1) holds for potentials in Ln(M). This extends the classical results for smooth potentials by Avakumović [2], Levitan [18] and Hörmander [12] to critically singular potentials. In three dimensions, this L3 condition also naturally appears in Boccato-Brennecke-Cenatiempo-Schlein [6] on the ground state energy of the Hamilton operator in the Gross-Pitaevskii regime. These two results are sharp, and our proof exploits Li-Yau's heat kernel bounds and Blair-Sire-Sogge's eigenfunction estimates.

Tensor product scheme for computing bound states of the quantum mechanical three-body problem

We develop a computationally and numerically efficient method to calculate binding energies and corresponding wave functions of quantum mechanical three-body problems in low dimensions. Our approach exploits the tensor structure of the multidimensional stationary Schrödinger equation, being expressed as a discretized linear eigenvalue problem. In one spatial dimension, we solve the three-body problem with the help of iterative methods. Here the application of the Hamiltonian operator is represented by dense matrix–matrix products. In combination with a newly-designed preconditioner for the Jacobi–Davidson QR, our highly accurate tensor method offers a significantly faster computation of three-body energies and bound states than other existing approaches. For the two-dimensional case, we additionally make use of a hybrid distributed/shared memory parallel implementation to calculate the corresponding three-body energies. Our novel method is of high relevance for the analysis of few-body systems and their universal behavior, which is only governed by the particle masses, overall symmetries, and the spatial dimensionality. Our results have straightforward applications for ultracold atomic gases that are widespread and nowadays utilized in quantum sensors.

Coherent robust [formula omitted] control of uncertain linear quantum systems with direct and indirect couplings

In this paper, a robust H∞ analysis method is presented for a class of uncertain quantum systems where uncertainties may exist in the system and interaction Hamiltonians, and the coupling operators. We provide a sufficient condition to guarantee this class of uncertain systems robust strictly bounded real with respect to a given disturbance attenuation. Moreover, we propose a robust H∞ control method for quantum systems with uncertain system Hamiltonian and uncertain coupling operators. The controller and the quantum plant are connected via direct and indirect couplings. This robust H∞ control problem is shown to have a connection with a scaled H∞ problem. We propose a numerical procedure to find the corresponding coefficients of the desired H∞ controller by using an LMI formulation and multi-step optimization.

Time Fisher information associated with fluctuations in quantum geometry

As time is not an observable, we use Fisher information (FI) to address the problem of time. We demonstrate that although the Hamiltonian operator is the generator of time, the Hamiltonian constraint operator cannot observe the change that arises through the passage of time. This is because the Hamiltonian constraint does not contain time FI. As we also identify an alternative operator on the world-sheet which does contain time FI, we argue that the timelessness of quantum gravity occurs due to the use of an inappropriate operator. We propose that a criterion for a physically meaningful operator in any quantum gravitational process is that it should contain non-vanishing time FI. Although we explicitly derive these results on the world-sheet of bosonic strings, we argue that it holds for any quantum geometry associated with time-reparametrization–invariant classical geometry.

Simulation of the dispersed fluorescence spectrum of the NO[formula omitted][formula omitted] origin vibronic band

A vibronic Hamiltonian and transition dipole operator parametrized by high-level equation-of-motion coupled-cluster calculations are used as a basis for predicting the dispersed fluorescence spectrum of the B̃−X̃ origin band in the nitrate radical (NO3). The position and intensities of transitions to ground state vibrational levels observed in the photodetachment spectrum of the NO3 anion have previously been predicted to high fidelity with this Hamiltonian; it is expected that it also offers a basis for accurate prediction of the dispersed fluorescence spectrum. Comparison to experimental spectra provides considerable insight into various assignments. Significantly, this work joins a growing body of research that supports reassignment of the ground state ν3 vibrational fundamental to the region between 1050–1060 cm−1, which is more than 400 cm−1 below the feature to which it has traditionally been assigned. The in-plane vibrational levels below 2000 cm−1 are reviewed, with theory and experiment agreeing to within 25 cm−1 for all that have been observed experimentally. Finally, the various vibronic coupling mechanisms that affect the ground state energy levels of NO3 are summarized, and a discussion is provided about how the Lanczos algorithm can be used to efficiently calculate the dispersed fluorescence spectrum of the B̃−X̃ origin band.

Invariant quadratic operators associated with linear canonical transformations and their eigenstates

The main purpose of this work is to identify invariant quadratic operators associated with Linear Canonical Transformations (LCTs) which could play important roles in physics. In quantum physics, LCTs are the linear transformations which keep invariant the Canonical Commutation Relations (CCRs). In this work, LCTs corresponding to a general pseudo-Euclidian space are considered and related to a phase space representation of quantum theory. Explicit calculations are firstly performed for the monodimensional case to identify the corresponding LCT-invariant quadratic operators then multidimensional generalizations of the obtained results are deduced. The eigenstates of these operators are also identified. A first kind of LCT-invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of the coordinates and momenta operators themselves. It is shown that these statistical variances-covariances can be related with thermodynamic variables. In this context, new quantum corrections to the ideal gas state equation are deduced from correction to the Hamiltonian operator of non-relativistic free quantum particles that is suggested by LCT-covariance. Two other LCT-invariant quadratic operators, which can be considered as the number operators of some quasiparticles, are also identified: the first one is a number operator of bosonic type quasiparticles and the second one corresponds to fermionic type. This fermionic LCT-invariant quadratic operator is directly related to a spin representation of LCTs. It is shown explicitly, in the case of a relativistic pentadimensional theory, that the eigenstates of this operator can be considered as basic quantum states of elementary fermions. A classification of the fundamental fermions, compatible with the Standard Model of particle physics, is established from a classification of these states.

Quantum cosmology of a Friedmann–Lemaitre–Robertson–Walker universe with radiation and its tomographic properties

We introduce in this paper a tomographic analysis of the properties of a Friedmann–Lemaitre–Robertson–Walker (FLRW) universe with a perfect fluid. We first review previous works where the Schutz’s parametrization in terms of Clebsch potentials was adopted to describe the perfect fluid. This approach allows to introduce a fiducial time in the Wheeler–De Witt equation. We revisit the properties of the perfect fluid and the introduced Clebsch potentials. In particular, we see that thermasy plays an extremely important role in the definition of fiducial time. The definition of a time and the condition [Formula: see text] for the expansion factor imply that the Hamiltonian operator must be self-adjoint which implies a restriction on the initial conditions for the wave packet. We show that these allow to obtain well-defined tomograms. Tomograms are marginal functions which incorporate all the information contained in the wave function of the universe, but have the properties of classical probability functions. They can be defined for classical distributions on the phase space as well, enabling us to describe quantum and classical states with the same family of functions. The aim of this paper is to compare the difference between classical tomograms where the Hawking and Penrose theorems imply the inevitability of an initial singular state and the well-defined initial quantum states found in literature. Finally the introduction of a time in the Wheeler–DeWitt allows us to consider the evolution of the classical and quantum initial states of the universe which can be accomplished by introducing a transition probability function for tomograms.

Gauge Theory and the Analytic Form of the Geometric Langlands Program

We present a gauge-theoretic interpretation of the “analytic” version of the geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete operators acting on a Hilbert space of quantum states. The gauge theory ingredients required to understand this construction—such as electric–magnetic duality between Wilson and ’t Hooft line operators in four-dimensional gauge theory—are the same ones that enter in understanding via gauge theory the more familiar formulation of geometric Langlands, but now these ingredients are organized and applied in a novel fashion.

Opposing magnetic communications in two dinuclear Ni(II) complexes: Ferromagnetic Ni-N-Ni and antiferromagnetic Ni-O-Ni moiety

Two dinuclear Ni(II) complexes [Ni2(L)2(N3)2(H2O)]·2H2O (1) and [Ni2(L)2(NCS)2(H2O)] (2) [HL = 2-methoxy-6-[(2-morpholin-4-yl-ethylimino)-methyl]-phenol] have been synthesized and characterized by single crystal X-ray crystallography and FT-IR spectroscopy. The complexes 1 and 2 are dinuclear and structurally close comparable with metals doubly bridged by a pseudohalide anion (μ1,1-azido and N,N-thiocyanato, respectively) and a μ-(phenoxido). The two HL ligands behave differently with N2O2 and N2O donor set towards the octahedral metal centres. Magnetic studies indicate an overall ferromagnetic interaction between nickel(II) centers in both the complexes. The experimental data can be simulated using the isotropic exchange Hamilton operator Ĥ = −2JS1S2 with following parameters J = +7.9(4) cm−1, g1,2 = 2.246(5), D1,2 = 2.35(10) cm−1 for complex 1 and J = +13.3(5) cm−1, g1,2 = 2.102(4), D1,2 = 4.18(12) cm−1, for complex 2. The structures of the complexes were theoretically optimized in their ground state and the experimental UV–vis electronic absorption spectral data have been compared with the theoretical results obtained from TD-DFT calculation.

The Impact of deformed space-phase parameters into HAs and HLM systems with the improved Hulthen plus Hellmann potentials model in the presence of temperature-dependent confined Coulomb potential within the framework of DSE

In this work, the improved Hulthen plus Hellmann potentials model in the presence of temperature-dependent confined Coulomb potential ´ IHHPTd model is adopted as the quark-antiquark interaction potential for studying the mass spectra of heavy mesons in the three-dimensional nonrelativistic quantum mechanics noncommutative phase space (3DNRQm-NCSP) symmetries. In addition, we found another application for this potential through its description for hydrogen atoms He+, Li+2 and Be+. We solved the deformed Schrödinger equation analytically using the generalized Bopp’s shift method and standard perturbation theory. The new energy eigenvalues E (u,d)hy nc−n and (E hlm n−g, E hlm n−mand E hlm n−l ) for hydrogen atoms and heavy mesons such as charmonium cc and bottomonium cb and corresponding deformed Hamiltonian operators H nc hhp(r, Θ, θ, λ, λ, σ, σ, ²) and H nc hhp(r, Θ, θ, λ, λ, σ, σ, gs) were obtained, respectively. The present results are applied for calculating the new mass of heavy mesons. Four special cases were considered when some of the improved potential parameters were set to zero, resulting into improved Hellmann potential, improved Yukawa potential, improved Coulomb potential, and improved Hulthen potential, in ´ (3DNRQm-NCSP) symmetries. The limiting cases are analyzed for Θ, σ and χ −→ 0 are compared with those of literature.

Unified one-electron Hamiltonian formalism of spin-orbit Jahn-Teller and pseudo-Jahn-Teller problems in tetrahedral and octahedral symmetries.

Heavy element compounds with high symmetries often feature both spin-orbit coupling and vibronic coupling. This is especially true for systems with tetrahedral and octahedral symmetries, whose electronic states may be threefold degenerate and experience complicated Jahn-Teller and pseudo-Jahn-Teller interactions. To accurately describe these interactions, high quality spin-orbit vibronic Hamiltonian operators are needed. In this study, we present a unified one-electron Hamiltonian formalism for spin-orbit vibronic interactions for systems in all tetrahedral and octahedral symmetries. The formalism covers all spin-orbit Jahn-Teller and pseudo-Jahn-Teller problems in the symmetries with arbitrary types and arbitrary numbers of vibrational modes and generates Hamiltonian expansion formulas of arbitrarily high order.

Absence of ground states in the renormalized massless translation-invariant Nelson model

We consider a model for a massive uncharged non-relativistic particle interacting with a massless bosonic field, widely referred to as the Nelson model. It is well known that an ultraviolet renormalized Hamilton operator exists in this case. Further, due to translation-invariance, it decomposes into fiber operators. In this paper, we treat the renormalized fiber operators. We give a description of the operator and form domains and prove that the fiber operators do not have a ground state. Our results hold for any non-zero coupling constant and arbitrary total momentum. Our proof for the absence of ground states is a new generalization of methods recently applied to related models. A major enhancement we provide is that the method can be applied to models with degenerate ground state eigenspaces.

A Unified Quantization of Gravity and Other Fundamental Forces of Nature

We quantized the interaction of gravity with Yang–Mills and spinor fields; hence, offering a quantum theory incorporating all four fundamental forces of nature. Let us abbreviate the spatial Hamilton functions of the standard model by HSM and the Hamilton function of gravity by HG. Working in a fiber bundle E with base space S0=Rn, where the fiber elements are Riemannian metrics, we can express the Hamilton functions in the form HG+HSM=HG+t−23H˜SM, if n=3, where H˜SM depends on metrics σij satisfying detσij=1. In the quantization process, we quantize HG for general σij but H˜SM only for σij=δij by the usual methods of QFT. Let v resp. ψ be the spatial eigendistributions of the respective Hamilton operators, then, the solutions u of the Wheeler–DeWitt equation are given by u=wvψ, where w satisfies an ODE and u is evaluated at (t,δij) in the fibers.