Quantum Mechanical Wave Function Research Articles

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.

Characterization of nuclear wave packets describing molecular photodissociation

A bound-to-free transition initiated by femtosecond excitation of diatomic molecules results in photofragments with a distribution of kinetic energies. A measurement of the kinetic-energy distribution yields the modulus squared of the asymptotic momentum-space wave packet prepared in the laser excitation process. On the other hand, the coordinate-space density of the wave packet entering the interaction-free region can be determined from pump–probe integrated fluorescence spectroscopy. We provide several numerical examples to show that this information can be used to determine the phase of the asymptotic wave packet so that this particular quantum-mechanical wave function can be characterized completely. To achieve this aim we use an iteration scheme (Gerchberg–Saxton algorithm) which does not require any further information about the system or the laser pulses.

God's Action in the World: The Relevance of Quantum Mechanics

It has been suggested that God can act on the world by operating within the limits set by Heisenberg's uncertainty principle (HUP) without violating the laws of nature. This requires nature to be intrinsically indeterministic. However, according to the statistical interpretation the quantum mechanical wavefunction represents the average behavior of an ensemble of similar systems and not that of a single system. The HUP thus refers to a relation between the spreads of possible values of position and momentum and so is consistent with a fully deterministic world. This statistical interpretation of quantum mechanics is supported by reference to actual measurements, resolves the quantum paradoxes, and stimulates further research. If this interpretation is accepted, quantum mechanics is irrelevant to the question of God's action in the world.

Measurement of Helmholtz wave fields

A simple formalism is found for the measurement of wave fields that satisfy the Helmholtz equation in free space. This formalism turns out to be analogous to the well-known theory of measurements for quantum-mechanical wave functions: A measurement corresponds to the squared magnitude of the inner product (in a suitable Hilbert space) of the wave field and a field that is associated with the detector. The measurement can also be expressed as an overlap in phase space of a special form of the Wigner function that is tailored for Helmholtz wave fields.

A quantum molecular dynamics study of exciton self-trapping in conjugated polymers: Temperature dependence and spectroscopy

We examine the dynamics of exciton self-trapping in conjugated polymer systems using mixed quantum-classical molecular dynamics. The model treats the exciton as a two-dimensional quantum mechanical wave function representing a particle/hole quasiparticle interacting with a classical vibrational lattice [M. N. Kobrak and E. R. Bittner, J. Chem. Phys. 112, 5399 (2000)]. We show that the dynamics are influenced strongly by thermal disorder in the lattice, and that there is a dramatic change in the self-trapping mechanism as temperature increases. At low temperatures, the rate of localization is limited by the time required for the vibrational lattice to respond to the creation of the particle–hole pair, while at higher temperatures thermal disorder permits localization on time scales limited primarily by electronic response. We simulate the time-resolved fluorescence spectrum for the model system, and compare the temperature dependence of the spectrum to recent time-resolved fluorescence upconversion studies on polydiacetylene derivatives.

Optical Solitons with Gaussian Tails**The project supported by National Natural Science Foundation of China, the Development Foundation for Science and Technology of ECNU, the Hong Kong Research Grant Council and the Hong Kong Baptist University Faculty Research Grant

Based on an extended method of multiple scales, an analytical description of the optical solitons in a model for the propagation of nonlinear pulses in dispersion-managed optical fibers is presented. The results show that the optical pulses can be expressed by quantummechanical wavefunctions of a linear oscillator, which have tails of Gaussian type, modulated by hyperbolic secant solitons of the nonlinear Schrödinger equation.