Generalized Gaussian wave packet dynamics, Schrödinger equation, and stationary phase approximation

Gaussian wave packet dynamics and scattering in the interaction picture

Using semiclassical methods, it is possible to construct very accurate approximations in the short-wavelength limit of quantum dynamics that rely exclusively on classical dynamical input. For systems whose classical realization is strongly chaotic, there is an exceedingly short logarithmic Ehrenfest time scale, beyond which the quantum and classical dynamics of a system necessarily diverge, and yet the semiclassical construction remains valid far beyond that time. This fact leads to a paradox if one ponders the reversibility and predictability properties of quantum and classical mechanics. They behave very differently relative to each other, with classical dynamics being essentially irreversible/unpredictable, whereas quantum dynamics is reversible/stable. This begs the question: 'How can an accurate approximation to a reversible/stable dynamics be constructed from an irreversible/unpredictable one?' The resolution of this incongruity depends on a couple of key ingredients: a well-known, inherent, one-way structural stability of chaotic systems; and an overlap integral not being amenable to the saddle point method.

The charge-transfer processes ${\mathrm{O}}^{+}(^{4}S^{0},^{2}D^{0},^{2}P^{0})+\mathrm{He}\ensuremath{\rightarrow}\mathrm{O}(^{3}P)+{\mathrm{He}}^{+}$ have been investigated by using a fully quantal molecular-orbital close-coupling (QMOCC) approach. Cross sections are presented for ion energies from 0.5 to 10 keV and compared with those from recent experiments and semiclassical theory. Good agreement is found between the QMOCC results and the measurements. Particular attention is given to addressing the metastable component of the experimental ion beams. We further argue that the so-called ``suppressed electron-capture effect'' for metastable ions proposed by Wolfrum et al. is not a viable mechanism to explain their measurements. However, the current QMOCC calculations were found to reproduce neither the ground-state nor metastable-state cross sections predicted by the semiclassical method.

This work is concerned with the development of approximate methods to describe relatively large chemical systems. This effort has been divided into two primary directions: First, we have extended and applied a semiclassical transition state theory (SCTST) originally proposed by Miller to obtain microcanonical and canonical (thermal) rates for chemical reactions described by a nonseparable Hamiltonian, i.e. most reactions. Second, we have developed a method to describe the fluctuations of decay rates of individual energy states from the average RRKM rate in systems where the direct calculation of individual rates would be impossible. Combined with the semiclassical theory this latter effort has provided a direct comparison to the experimental results of Moore and coworkers. In SCTST, the Hamiltonian is expanded about the barrier and the ``good`` action-angle variables are obtained perturbatively; a WKB analysis of the effectively one-dimensional reactive direction then provides the transmission probabilities. The advantages of this local approximate treatment are that it includes tunneling effects and anharmonicity, and it systematically provides a multi-dimensional dividing surface in phase space. The SCTST thermal rate expression has been reformulated providing increased numerical efficiency (as compared to a naive Boltzmann average), an appealing link to conventional transition state theory (involving a ``prereactive`` partition function depending on the action of the reactive mode), and the ability to go beyond the perturbative approximation.

Semiclassical quantization of complex Henon–Heiles systems

An accurate description of quantum dynamics is challenging from both a theoretical and computational point of view. We illustrate some representative calculations based on a handful of trajectories whereby the semiclassical initial value representation theory is able to produce approximate but accurate quantum dynamical results. The range of applicability of SCIVR is vast and includes, among others, spectroscopy in gas and condensed phase, wavefunction calculations, and non-adiabatic dynamics. Here, we present a few SCIVR applications of different complexity. They involve a vibrational study of an N2 molecule adsorbed on a TiO2 surface, the IR spectroscopy of a methane molecule, and the determination of the vibrational ground state wavefunction of the formic acid dimer. These calculations demonstrate that our semiclassical approaches are able not only to provide an effective approximation to nuclear quantum dynamics but also to slash computational costs, which is mandatory when dealing with large-dimensional systems or complex problems. Finally, as a perspective, we move to a non-adiabatic framework and, by means of a linearized semiclassical approach, we compute the electronic state populations in a non-adiabatic dynamical process integrating 10,000 trajectories. The very good agreement with split-operator results motivates us to perform future research in the field aiming at few-trajectory-based calculations of comparable accuracy.

In this research paper, quantum particle swarm optimization (QPSO) algorithm, an algorithm founded on the basic theory of particle cluster and properties of quantum mechanics, has been introduced for design of uniformly spaced scanned linear array antennas with definite value of sidelobe level (SLL), first null beam width (FNBW), and broad null depth. An effort is made such that these parameters are made equal to their related specific values. This is usually done by changing excitation current amplitude of the elements. The expression of array factor is derived by using the property of linear array antennas. One example has been presented with 35 isotropic antennas. Generated Pattern is scanned to an angle of 30°. The obtained results have shown that this technique is able to find required value of SLL, FNBW, and wide null depth. Although, the proposed method is developed and applied to a linear array of isotropic antennas; however, the principle can easily be applied to other types of arrays.

Results coming from the study of relativistic vector mesons interacting with a constant magnetic field are examined through Johnson-Lippmann implications on one-dimensional oscillatorlike systems. We obtain specific nonrelativistic Hamiltonians showing new properties in quantum mechanics and leading to superpositions of bosons and pseudofermions. Moreover, two “potentials” are introduced and discussed in comparison with recent developments usually obtained in p=2 parasupersymmetric quantum mechanics. Pseudofermions are also examined, particularly with respect to orthofermions.

Non-local games are widely studied as a model to investigate the properties of quantum mechanics as opposed to classical mechanics. In this paper, we consider a subset of non-local games: symmetric XOR games of n players with 0-1 valued questions. For this class of games, each player receives an input bit and responds with an output bit without communicating to the other players. The winning condition only depends on XOR of output bits and is constant w.r.t. permutation of players. We prove that for almost any n-player symmetric XOR game the entangled value of the game is Theta((sqrt(ln(n)))/(n^{1/4})) adapting an old result by Salem and Zygmund on the asymptotics of random trigonometric polynomials. Consequently, we show that the classical-quantum gap is Theta(sqrt(ln(n))) for almost any symmetric XOR game.

In this work we explore the implications of applying the formalism of symplectic geometry to quantum mechanics, particularly focusing on many-particle systems. We extend the concept of a symplectic indicator of entanglement, originally introduced by Sawicki et al., Commun. Math. Phys. 305, 441 (2011), to these complex systems. Specifically, we demonstrate that the restriction of the symplectic structure to manifolds comprising all states characterized by isospectral reduced one-particle density matrices ${M}_{\ensuremath{\mu}(\ensuremath{\psi})}^{0}$ exhibits degeneracy for nonseparable states. We prove that the degree of degeneracy at any given state $|\ensuremath{\varphi}\ensuremath{\rangle}\ensuremath{\in}{M}_{\ensuremath{\mu}(\ensuremath{\psi})}^{0}$ corresponds to the degree of degeneracy of the symplectic form $\ensuremath{\omega}$ when restricted to the manifold of states that are locally unitary equivalent with $|\ensuremath{\varphi}\ensuremath{\rangle}$. Additionally, we provide a physical interpretation of this symplectic indicator of entanglement, articulating it as an inherent ambiguity within the associated classical dynamical framework. Our findings underscore the pivotal role of symplectic geometry in elucidating entanglement properties in quantum mechanics and suggest avenues for further exploration into the geometric structures underlying quantum state spaces.

Properties of quantum mechanics have enabled the emergence of quantum cryptographic protocols achieving important goals which are proven to be impossible classically. Unfortunately, this usually comes at the cost of needing quantum power from every party in the protocol, while arguably a more realistic scenario would be a network of classical clients, classically interacting with a quantum server. In this paper, we focus on copy-protection, which is a quantum primitive that allows a program to be evaluated, but not copied, and has shown interest especially due to its links to other unclonable cryptographic primitives. Our main contribution is to show how to dequantize quantum copy-protection schemes constructed from hidden coset states, by giving a construction for classically-instructed remote state preparation for coset states, which preserves hardness properties of hidden coset states. We then apply this dequantizer to obtain semi-quantum cryptographic protocols for copy-protection and tokenized signatures with strong unforgeability. In the process, we present the first secure copy-protection scheme for point functions in the plain model and a new direct product hardness property of coset states which immediately implies a strongly unforgeable tokenized signature scheme. ⋆ Work done while at CRED and DIENS. 1 This is called hybrid quantum cryptography in [AGKZ20]. Technical Overview Copy-Protection In this section, we give an overview of our remote coset state preparation protocol and its proof of soundness. To give the reader a glimpse of the functionality of our protocol and how it can be used as a generic compiler to obtain semi-quantum copy-protection, we first start by analysing security requirements for several existing quantum copy-protection schemes based on coset states. Security Requirements. For our discussion, we focus on the copy-protection of pseudorandom functions scheme in the plain model and the single-decryptor scheme in the plain model presented in [CLLZ21] . The common point is that security of these constructions all reduce to a monogamy-ofentanglement property of coset states [CLLZ21, CV22]. Informally, this property states that a triple of quantum algorithms Alice, Bob and Charlie cannot cooperatively win the following monogamy game with a challenger, except with negligible probability. The challenger first prepares a uniformly random coset state |A s,s ′ ⟩ and gives the state to Alice. Alice outputs two (possibly entangled) quantum states and sends them to Bob and Charlie respectively. No communication is allowed between Bob and Charlie. Finally, Bob and Charlie both get the description of the subspace A. The game is won if Bob outputs a vector in A + s and Charlie outputs a vector in A ⊥ + s ′ , where A ⊥ denote the dual subspace of A. If our goal is to design a semi-quantum protocol for preparing coset states such that it can be used in a plug-and-play manner for the aforementioned protocols, our protocol needs to have the following properties: • Correctness. If the prover is honest, at the end of the protocol execution, the prover must have a hidden coset state |A s,s ′ ⟩ in its registers.

We define and characterize a new abstract notion of “quasiBoolean algebra,” intermediate in nature between an (ortho)lattice and a Boolean algebra. It will turn out that such algebras are natural candidates for representing the simultaneously definite properties of a quantum system. We then prove a general theorem about maximal quasiBoolean subalgebras of an ortholattice which we use to derive a number of different proposals in the literature for what properties of a quantum system should be regarded as simultaneously definite.

In this study, the mechanical and thermoelectric properties of Mg2Si doped with In were successfully calculated using DFT package CASTEP, and semiclassical Boltztmann transport theory in relaxation time approximation BoltzTraP and Phono3py packages. The mechanical property calculations show that the doped compound is softer, less rigidity, and less brittle. The band structure and density of states calculations show that the doped compound showing metallic behavior with increased density of states at the Fermi level due to Si 3p states, Mg 2p states, and In 5p states. The thermoelectric property calculations show decreased magnitude of Seebeck coefficient, and conduction type changes from n to p type. The conductivity of In doped increases as well as the thermal conductivity, leading to reduced figure of merit at high temperature from 0.43 in undoped compound to 0.15 in In-doped compound. In conclusion, the In dopant has improved the mechanical properties of Mg2Si, but at the same time reduced the thermoelectric properties of the compound.KeywordsDensity functional theoryThermoelectric materialsFirst principle study

Forward-backward trajectory formulations of time correlation functions are reviewed. Combination of the forward and reverse time evolution operators within the time-dependent semiclassical approximation minimizes phase cancellation, giving rise to an efficient methodology for simulating the dynamics of low-temperature fluids. A quantum mechanical version of the forward-backward formulation, based on the hydrodynamic formulation of time-dependent quantum mechanics, is also available but is practical only for small systems.

A semiclassical reactive flux method for the calculation of condensed phase activated rate constants