Quantum Mechanical Wave Function Research Articles

Quasiperiodic orbit analysis of nonadiabatic cis–trans photoisomerization dynamics

Adopting a multidimensional model of nonadiabatic cis–trans photoisomerization, quantum-mechanical and classical simulations of the ultrafast wave-packet dynamics associated with this photoreaction are presented. The quantum calculations demonstrate that nonadiabatic photoisomerization typically leads to a largely delocalized and diffuse wave function, which hampers an intuitive understanding of the dynamics in terms of specific nuclear motion. To facilitate a classical description, a recently proposed theoretical formulation is employed that affords an exact mapping of discrete electronic states onto continuous degrees of freedom and therefore provides a well-defined classical limit of a nonadiabatically coupled system. It is shown that a simple quasiclassical implementation of the mapping formulation is able to reproduce at least qualitatively the complex quantum dynamics of the system. In addition, the classical description allows us to characterize the nonadiabatic photoisomerization dynamics in terms of a few “quasiperiodic orbits.” These orbits are close to a true unstable periodic orbit but are exactly periodic only with respect to the slow reaction coordinate of the system. Various types of quasiperiodic orbits of nonadiabatic photoisomerization are identified and analyzed. It is shown that the diffuse appearance of the quantum-mechanical wave function can be directly connected to irregular classical orbits propagating on vibronically coupled potential-energy surfaces. The chaotic behavior of the system is mainly caused by the relatively high energy corresponding to photoexcitation, the large anharmonicity of the isomerization potentials, and the reflection of the trajectory at surface crossings. The results demonstrate that quasiperiodic orbits represent a concept well suited to analyze the quantum dynamics of complex systems in terms of classical trajectories without the cumbersome search for periodic orbits.

On the meaning of the string-inspired noncommutativity and its implications

We propose an alternative interpretation for the meaning of noncommutativity of the string-inspired field theories and quantum mechanics. Arguments are presented to show that the noncommutativity generated in the stringy context should be assumed to be only between the particle coordinate observables, and not of the spacetime coordinates. Some implications of this fact for noncomutative field theories and quantum mechanics are discussed. In particular, a consistent interpretation is given for the wavefunction in quantum mechanics. An analysis of the noncommutative theories in the Schr\"odinger formulation is performed employing a generalized quantum Hamilton-Jacobi formalism. A formal structure for noncommutative quantum mechanics, richer than the one of noncommutative quantum field theory, comes out. Conditions for the classical and commutative limits of these theories have also been determined and applied in some examples.

A Wavefunction-Based Criterion for the Detection of Intermolecular Interactions in Molecular Dynamics Simulations

The availability of quantum mechanical wave functions in molecular dynamics simulations for example in those of the Car−Parrinello type offers the ability to analyze intermolecular interactions of a system in terms of quantum chemical descriptors. We demonstrate how standard population analyses can be utilized for a semiquantitative analysis of intermolecular interactions. The approach is therefore of particular value for the study of those systems for which geometric criteria and predefined interaction potentials have not yet been obtained. This is demonstrated for a DMSO−water mixture, for which the population-analysis criterion provides a simple measure for different interaction types, e.g., between water−oxygen and methyl−hydrogen atoms. In the case of a polypeptide, it is shown that the wave-function-based criterion provides insight into hydrogen bonding of the CO groups with a hydrogen atom attached to a carbon atom from the peptide's backbone.

Fractional Fourier Transformation for Quantum Mechanical Wave Functions Studied by Virtue of IWOP Technique**The project supported by National Natural Science Foundation of China under Grant No. 10175057, The President Foundation of the Chinese Academy of Sciences, and Shanghai Jiao Tong University

Starting from the optical fractional Fourier transform (FFT) and using the technique of integration within an ordered product of operators we establish a formalism of FFT for quantum mechanical wave functions. In doing so, the essence of FFT can be seen more clearly, and the FFT of some wave functions can be derived more directly and concisely. We also point out that different FFT integral kernels correspond to different quantum mechanical representations. They are generalized FFT. The relationship between the FFT and the rotated Wigner operator is studied by virtue of the Weyl ordered form of the Wigner operator.

On the Concept of Information and Its Role in Nature

In this article we address some fundamental questions concerning information: Can the existing laws of physics adequately deal with the most striking property of information, namely to cause specific changes in the structure and energy flows of a complex system, without the information in itself representing fields, forces or energy in any of their characteristic forms? Or is information irreducible to the laws of physics and chemistry? Are information and complexity related concepts? Does the Universe, in its evolution, constantly generate new information? Or are information and information-processing exclusive attributes of living systems, related to the very definition of life? If that were the case, what happens with the physical meanings of entropy in statistical mechanics or wave function in quantum mechanics? How many distinct classes of information and information processing do exist in the biological world? How does information appear in Darwinian evolution? Does the human brain have unique properties or capabilities in terms of information processing? In what ways does information processing bring about human self-consciousness? We shall introduce the meaning of "information" in a way that is detached from human technological systems and related algorithms and semantics, and that is not based on any mathematical formula. To accomplish this we turn to the concept of interaction as the basic departing point, and identify two fundamentally different classes, with information and information-processing appearing as the key discriminator: force-field driven interactions between elementary particles and ensembles of particles in the macroscopic physical domain, and information-based interactions between certain kinds of complex systems that form the biological domain. We shall show that in an abiotic world, information plays no role; physical interactions just happen, they are driven by energy exchange between the interacting parts and do not require any operations of information processing. Information only enters the non-living physical world when a living thing interacts with it-and when a scientist extracts information through observation and measurement. But for living organisms, information is the very essence of their existence: to maintain a long-term state of unstable thermodynamic equilibrium with its surroundings, consistently increase its organization and reproduce, an organism has to rely on information-based interactions in which form or pattern, not energy, is the controlling factor. This latter class comprises biomolecular information processes controlling the metabolism, growth, multiplication and differentiation of cells, and neural information processes controlling animal behavior and intelligence. The only way new information can appear is through the process of biological evolution and, in the short term, through sensory acquisition and the manipulation of images in the nervous system. Non-living informational systems such as books, computers, AI systems and other artifacts, as well as living organisms that are the result of breeding or cloning, are planned by human beings and will not be considered here.

EPR entangled states and complex fractional Fourier transformation

Starting from the Einstein-Podolsky-Rosen entangled state representations of continuous variables we derive a new formulation of complex fractional Fourier transformation (CFFT). We find that two-variable Hermite polynomials are just the eigenmodes of the CFFT. In this way the CFFT is linked to the appropriate operator transformation between two kinds of entangled states in the context of quantum mechanics. In so doing, the CFFT of quantum mechanical wave functions can be derived more directly and concisely.

Multidimensional classical Liouville dynamics with quantum initial conditions

A simple and numerically efficient approach to Wigner transforms and classical Liouville dynamics in phase space is presented. (1) The Wigner transform can be obtained with a given accuracy by optimal decomposition of an initial quantum-mechanical wave function in terms of a minimal set of Gaussian wave packets. (2) The solution of the classical Liouville equation within the locally quadratic approximation of the potential energy function requires a representation of the density in terms of an ensemble of narrow Gaussian phase-space packets. The corresponding equations of motion can be efficiently solved by a modified leap-frog integrator. For both problems the use of Monte Carlo based techniques allows numerical calculation in multidimensional cases where grid-based methods such as fast Fourier transforms are prohibitive. In total, the proposed strategy provides a practical and efficient tool for classical Liouville dynamics with quantum-mechanical initial states.

Proton-Coupled Electron Transfer in DNA−Acrylamide Complexes

A theoretical study of proton-coupled electron transfer (PCET) in the radical anionic thymine−acrylamide complex is presented. This study is based on a multistate continuum theory, in which the solute is represented by a multistate valence bond model, the solvent is described by a dielectric continuum, and the transferring hydrogen nucleus is represented by a quantum mechanical wave function. In this application, the ground and excited electronic states are calculated with the complete active space self-consistent-field (CASSCF) method, the electronic coupling for the electron transfer reaction is calculated with the generalized Mulliken−Hush method, and the solvation properties are calculated with the frequency-resolved cavity model. The influence of neighboring DNA base pairs is determined by studying solvated DNA−acrylamide models in addition to the solvated thymine−acrylamide complex. The calculations indicate that the final product corresponds to single electron transfer (ET) for the solvated thymine...

Neutrino oscillations.

The wave theory of light, and in particular the principle of interference, was formulated by Thomas Young in 1801. In the 20th century, the principle of interference was extended to the quantum mechanical wave functions describing matter. The phenomenon of quantum mechanical interference of different neutrino states, neutrino oscillations, has provided one of the most exciting developments in high energy particle physics of the last decade. Observations of the flavour oscillations of neutrinos produced by distant sources, such as from the core of the Sun, provide compelling evidence that neutrinos have mass. This article describes the main features and the most significant experimental observations of this unusual application of the principle of interference.

Plasmon-assisted transmission of entangled photons.

The state of a two-particle system is said to be entangled when its quantum-mechanical wavefunction cannot be factorized into two single-particle wavefunctions. This leads to one of the strongest counter-intuitive features of quantum mechanics, namely non-locality. Experimental realization of quantum entanglement is relatively easy for photons; a starting photon can spontaneously split into a pair of entangled photons inside a nonlinear crystal. Here we investigate the effects of nanostructured metal optical elements on the properties of entangled photons. To this end, we place optically thick metal films perforated with a periodic array of subwavelength holes in the paths of the two entangled photons. Such arrays convert photons into surface-plasmon waves--optically excited compressive charge density waves--which tunnel through the holes before reradiating as photons at the far side. We address the question of whether the entanglement survives such a conversion process. Our coincidence counting measurements show that it does, so demonstrating that the surface plasmons have a true quantum nature. Focusing one of the photon beams on its array reduces the quality of the entanglement. The propagation of the surface plasmons makes the array effectively act as a 'which way' detector.

Atomic Structure Determination Using the Focal-Series Reconstruction Technique

During the last five years the technique focal-series reconstruction has evolved to a powerful tool for investigating materials science problems in high-resolution transmission electron microscopy. Compared to the conventional interpretation of one single high-resolution image, the quantum mechanical electron wave function at the exit plane of the object (exit-plane wave function, EPW) is in many cases a better starting point for the materials analysis. The retrieval of the EPW is achieved on a routine basis by applying automated numerical procedures to a series of images taken from the same specimen area at different objective lens defocus values. The application of the reconstruction procedure allows one to remove numerically all the well known instrumental artifacts, such as nonlinear contrast formation or delocalisation effects due to spherical aberration and other parasitic aberrations. The reconstructed EPW gives thus direct insight into the atomic structure in the case of sufficiently thin objects and renders tedious image simulations of complicated defects unnecessary in many cases.

Absorption of solar radiation by charged water droplets

We consider the absorption of solar radiation by charge accumulated on the surface of an otherwise neutral water droplet. We estimate the order of magnitude of the additional contribution from the surface charges to the photoabsorption cross-section. The approximate location and strength of major spectral lines is indicated. This spectral information requires determination of the quantum mechanical energy levels and wave functions of the absorbing system. An approximate calculation is carried out using a simple model for electrons on the surface of the droplet. The implications for atmospheric applications are considered.

Diffraction in time with dissipation

The more usual problems discussed in classical mechanics in elementary textbooks are the non-dissipative ones in which there is a Hamiltonian representing the energy as a constant of motion. The translation of this type of problem to quantum mechanics is very well known. Conversely, there are very simple classical mechanical problems that involve dissipation, but whose translation to quantum mechanics on the basis of a corresponding Hamiltonian is sometimes misinterpreted, since the underlying classical formalism involves non-canonical transformations that lead to non-unitary transformations of the quantum mechanical wavefunctions. In this paper we shall discuss the problem of a beam of particles of given momentum incident from the left on a shutter that is opened at time t = 0. The solution has the well known properties of diffraction in time, but we will analyse it quantum mechanically both when dissipation is and is not present. The objective is to get a better understanding of the effect of dissipation in the quantum mechanical picture and to estimate the influence of the above mentioned non-unitary transformations on this particular problem.

From Quantum Motion to Classical Motion—Seeking the Lost Reality

We show that the natural motion of particles in continuous space-time (CSTM) is not classical continuous motion (CCM), but one kind of essentially discontinuous motion, the wave function in quantum mechanics is the very mathematical complex describing this kind of motion, and Schrodinger equation is just its simplest nonrelativistic motion equation, we call such motion quantum discontinuous motion or quantum motion; furthermore, we show that, when considering gravity the space-time will be essentially discrete, and the motion in discrete space-time (DSTM) will naturally result in the collapse process of the wave function, this finally brings about the appearance of classical continuous motion (CCM) in macroscopic world.

Polaron recombination dynamics in linear polyenes

We examine the dynamics of polaron recombination in conjugated polymer systems using mixed quantum-classical molecular dynamics. The model treats the particle-hole pair as a two-dimensional quantum mechanical wavefunction interacting with a classical vibrational lattice. This description allows a natural evolution of the particle-hole wavefunction from the polaron limit to the exciton limit. We have performed real-time simulations of the coupled nuclear and electronic dynamics associated with the scattering of polarons into exciton states. We use these simulations to calculate cross-sections for exciton formation as a function of spin state, and explore the variation of these cross-sections with respect to changes in the magnitude of the particle-hole Coulomb interaction and the effective masses of the quasiparticles. Our results indicate that for an optimal choice of parameters, the electro-luminescence quantum yields as high as 60% can be achieved, substantially greater than the 25% predicted by simple spin statistics.

Exchange energy in a double-well potential profile from fluctuation theory

This article shows that the asymptotic quantum mechanical exchange energies and wave functions in a bistable potential can be obtained by computing the complex-valued first-passage time across the potential barrier from a transient state endowed with Smoluchowski boundary conditions, whose probability density is concentrated on one side of the barrier. This interpretation is validated by one of the authors’ previous work on diffusion in a random field, showing that this model yields a diffusion equation equivalent to quantum mechanical equations.

Gate tunneling currents in ultrathin oxide metal–oxide–silicon transistors

Carrier tunneling through ultrathin (1–3 nm) SiO2 layers in MOS (metal–oxide–silicon) structures is investigated using the Bardeen–Harrison transition probability method. Quantum mechanical wave function matching at the two abrupt potential boundaries of a trapezoidal Si/SiO2/Si barrier gives an electric-field dependent preexponential factor in the Wentzel–Kramers–Brillouin tunneling probability, which significantly affects the current–voltage characteristic at low fields. An analytical theory is employed to predict the relative importance of three elastic tunneling pathways (electrons, valence electrons, and holes) and two geometrical tunneling locations (channel region and source or drain overlap regions) in MOS transistors (MOSTs), showing (1) hole tunneling dominant in p+gate pMOST (p-channel MOST) at low gate voltages, and (2) overlap regions dominant prior to base-region inversion in both p+gate pMOST and n+gate nMOST (n-channel MOST). The analytic theory is used to analyze the experimental tunneling currents measured at the gate, source, well, and substrate terminals of sourced MOS capacitors to give the oxide thickness and impurity doping concentrations in the base and source regions.

The influence of strain on confined electronic states in semiconductor quantum structures

A continuum finite element technique is adopted to study electronic properties of submicron electronic devices where function hinges on quantum mechanical effects. Of particular interest is the influence of mechanical strain on confined electronic states. The steady state Schrödinger equation, which governs the electronic behavior of such devices, is modified to include the potential induced by a strain field which is present as a consequence of the fabrication. The governing equation is cast in a variational form, and it is discretized on a standard finite element mesh which is more refined in regions where large quantum mechanical wave function gradients are expected. Multiple energy bands and three-dimensional structures can be considered, and effects including strain enhanced charge confinement and strain induced energy band mixing are studied. As examples, a Ge [5 0 1] faceted island, or quantum dot, on a Si substrate and a Ge v-groove quantum wire on a Si substrate are considered. The technique is used to determine size ranges in which these devices are expected to be most useful. The nonuniform mismatch strain field in the structures is found to affect the energies of experimentally accessible confined states and in some cases to enhance quantum mechanical confinement.

Orbit Sum Rules for the Quantum Wave Functions of the Strongly Chaotic Hadamard Billiard in Arbitrary Dimensions

Sum rules are derived for the quantum wave functions of the Hadamard billiard in arbitrary dimensions. This billiard is a strongly chaotic (Anosov) system which consists of a point particle moving freely on a D-dimensional compact manifold (orbifold) of constant negative curvature. The sum rules express a general (two-point)correlation function of the quantum mechanical wave functions in terms of a sum over the orbits of the corresponding classical system. By taking the trace of the orbit sum rule or pre-trace formula, one obtains the Selberg trace formula. The sum rules are applied in two dimensions to a compact Riemann surface of genus two, and in three dimensions to the only non-arithmetic tetrahedron existing in hyperbolic 3-space. It is shown that the quantum wave functions can be computed from classical orbits. Conversely, we demonstrate that the structure of classical orbits can be extracted from the quantum mechanical energy levels and wave functions (inverse quantum chaology).

Quantum molecular dynamics study of polaron recombination in conjugated polymers

We examine the dynamics of polaron recombination in conjugated polymer systems using mixed quantum classical molecular dynamics. The model treats the particle-hole pair as a fully correlated two-particle quantum mechanical wave function interacting with a one-dimensional classical vibrational lattice. This description allows a natural evolution of the particle-hole wave function from the polaron limit to the exciton limit, and we have performed real-time simulations of the coupled nuclear and electronic dynamics associated with the scattering of polarons into exciton states. We use these simulations to calculate cross sections for exciton formation as a function of spin state, and explore the variation of these cross sections with respect to changes in the magnitude of the particle-hole Coulomb interaction and the effective masses of the quasiparticles. Our results indicate that for an optimal choice of parameters the electroluminescence quantum yield may be as high as 59%, substantially greater than the 25% predicted by simple spin statistics. We interpret these results in a diabatic framework, and suggest strategies for the design of organic systems for use in electroluminescent devices.