Semiclassical Wave Functions Research Articles

Semiclassical Wave Functions and Energy Spectra in Polygon Billiards

A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism is presented. It is argued that it is in the spirit of the semiclassical wave function formalism to make necessary rationalization of respective quantities accompanied the procedure of the semiclassical quantization in polygon billiards. First the rational polygon billiards and their unfoldings into corresponding Riemann surfaces are considered which show periodic structures with 2g independent periods. The space of the real linear combinations of these periods is however two dimensional and just these space can be rationalized consistently approximating real periods and leading to discrete lattices built of all such approximated periods. It is then shown that semiclassical quantization of both the classical momenta and energy spectra are determined completely by periodic structure of the rationalized lattices. Each Riemann surface can be then reduced to elementary polygon patterns as its basic periodic elements which built it. Each such elementary polygon pattern can be glued to a torus of genus g. Semiclassical wave functions are then constructed on such elementary polygon patterns. These semiclassical wave functions have forms of coherent sums of plane waves and satisfy on the billiards boundaries well defined conditions - the Dirichlet, the Neumann or the mixed ones. Not every mixing of such conditions is allowed however and the respective limitations can ignore some semiclassical states in the presented formalism. A relation between the superscar solutions and the semiclassical wave functions constructed in the paper are discussed. Finally an extension of the formalism on irrational polygons is described shortly as well.

Semiclassical Monte Carlo: a first principles approach to non-adiabatic molecular dynamics.

Modeling the dynamics of photophysical and (photo)chemical reactions in extended molecular systems is a new frontier for quantum chemistry. Many dynamical phenomena, such as intersystem crossing, non-radiative relaxation, and charge and energy transfer, require a non-adiabatic description which incorporate transitions between electronic states. Additionally, these dynamics are often highly sensitive to quantum coherences and interference effects. Several methods exist to simulate non-adiabatic dynamics; however, they are typically either too expensive to be applied to large molecular systems (10's-100's of atoms), or they are based on ad hoc schemes which may include severe approximations due to inconsistencies in classical and quantum mechanics. We present, in detail, an algorithm based on Monte Carlo sampling of the semiclassical time-dependent wavefunction that involves running simple surface hopping dynamics, followed by a post-processing step which adds little cost. The method requires only a few quantities from quantum chemistry calculations, can systematically be improved, and provides excellent agreement with exact quantum mechanical results. Here we show excellent agreement with exact solutions for scattering results of standard test problems. Additionally, we find that convergence of the wavefunction is controlled by complex valued phase factors, the size of the non-adiabatic coupling region, and the choice of sampling function. These results help in determining the range of applicability of the method, and provide a starting point for further improvement.

Escape of photo-detached electron from an annular nanomicrocavity

The escaped probability density of the photo-detached electron in an annular nanomicrocavity shows strong oscillations as a function of the length of the escape orbits. We present a semiclassical theory that describes theses oscillations in terms of bundles of escape orbits. Due to the interference effects of the electron waves travelling along different escaped orbits, oscillatory structures appear in the escaped probability density. In addition, the calculation results suggest that the escaped probability density of the photo-detached electron is not only related to the inner radius of the annular microcavity, but also related to the laser polarization. In order to show the correspondence between the escaped probability density and the detached electron’s escaped orbits clearly, we calculate the Fourier transformed semiclassical wave function and find that the peak positions agree well with the length of the detached electron’s orbits. We hope that our results will be useful in understanding the escape and propagation process of particles through semiconductor microjunctions or ballistic microstructure.

Maslov indices, Poisson brackets, and singular differential forms

Maslov indices are integers that appear in semiclassical wave functions and quantization conditions. They are often notoriously difficult to compute. We present methods of computing the Maslov index that rely only on typically elementary Poisson brackets and simple linear algebra. We also present a singular differential form, whose integral along a curve gives the Maslov index of that curve. The form is closed but not exact, and transforms by an exact differential under canonical transformations. We illustrate the method with the 6j-symbol, which is important in angular-momentum theory and in quantum gravity.

Phonon-induced superconductivity at high temperatures in electrical graphene superlattices

Due to the chiral nature of the Dirac equation, overlying of an electrical superlattice (SL) can open new Dirac points on the Fermi-surface of the energy spectrum. These lead to novel low-excitation physical phenomena. A typical example for such a system is neutral graphene with a symmetrical unidirectional SL. We show here that in smooth SLs, a semiclassical approximation provides a good mathematical description for particles. Due to the one-dimensional nature of the unidirectional potential, a wavefunction description leads to a generalized Bohr-Sommerfeld quantization condition for the energy eigenvalues. In order to pave the way for the application of semiclassical methods to two dimensional SLs in general, we compare these energy eigenvalues with those obtained from numerical calculations, and with the results from a semiclassical Gutzwiller trace formula via the beam-splitting technique. Finally, we calculate ballistic conductivities in general point-symmetric unidirectional SLs with one electron and one hole region in the fundamental cell showing only Klein scattering of the semiclassical wavefunctions.

Semiclassical wave functions and semiclassical dynamics for the Kepler/Coulomb problem

We investigate the semiclassical Kepler/Coulomb problem using the classical constants of the motion in the framework of Nelson’s stochastic mechanics. This is done by considering the eigenvalue relations for a family of coherent states (known as the atomic elliptic states) whose wave functions are concentrated on the elliptical orbit corresponding to the associated classical problem. We show that these eigenvalue relations lead to identities for the semiclassical energy, angular momentum and Hamilton–Lenz–Runge vectors in the elliptical case. These identities are then extended to include the cases of circular, parabolic and hyperbolic motions. We show that in all cases the semiclassical wave function is determined by our identities and so our identities can be seen as defining a semiclassical Kepler/Coulomb problem. The results are interpreted in terms of two dynamical systems: one a complex valued solution to the classical mechanics for a Coulomb potential and the other the drift field for a semiclassical Nelson diffusion.

Spin effects and the Pauli principle in semiclassical electron dynamics

Several approaches to the semiclassical dynamics of fermions have been proposed in the past. The main subject under discussion was the inclusion of the Pauli principle, i.e., the fact that two electrons with parallel spins must be in orthogonal states. In the past, this was sometimes achieved by adding repulsive Pauli potentials or by using antisymmetric trial states. In this article we show that (a) the use of semiclassical propagators based on classical trajectories is sufficient to account for the Pauli principle, but (b) a semiclassical wave-function approach is not satisfactory.

Vector fields in holographic cosmology

We extend the holographic formulation of the semiclassical no-boundary wave function (NBWF) to models with Maxwell vector fields. It is shown that the familiar saddle points of the NBWF have a representation in which a regular, Euclidean asymptotic AdS geometry smoothly joins onto a Lorentzian asymptotically de Sitter universe through a complex transition region. The tree level probabilities of Lorentzian histories are fully specified by the action of the AdS region of the saddle points. The scalar and vector matter profiles in this region are complex from an AdS viewpoint, with universal asymptotic phases. The dual description of the semiclassical NBWF thus involves complex deformations of Euclidean CFTs.

Escape of photodetached electrons from a nanocircular microcavity

This paper investigates the escape of photodetached electron from a nanocircular microcavity on the basis of the semiclassical theory. The escaped orbit of the photodetached electron is investigated and the escaped probability density of this system is derived and calculated. The calculation results suggest that oscillating structures appear in the escaped probability density, which are caused by the interference effects of the electron waves traveling along different escaped trajectories. Our study also suggests that the escaped probability density of the photodetached electron depends on the laser polarization sensitively. To show the correspondence between the escaped probability density and the detached electron’s escaped orbits clearly, we calculate the Fourier transform of the semiclassical wave function and find each peak corresponds to the length of one detached electron’s orbit. We hope that our results will be useful in understanding the escape and transport process of particles through semiconductor microjunctions.

Semiclassical wave functions for open quantum billiards

We present a semiclassical approximation to the scattering wave function Ψ(r,k) for an open quantum billiard, which is based on the reconstruction of the Feynman path integral. We demonstrate its remarkable numerical accuracy for the open rectangular billiard and show that the convergence of the semiclassical wave function to the full quantum state is controlled by the mean path length or equivalently the dwell time for a given scattering state. In the numerical implementation a cutoff length in the maximum path length or, equivalently, a maximum dwell time τ(max) included implies a finite energy resolution ΔE~τ(max)(-1). Possible applications include leaky billiards and systems with decoherence present.

Semiclassical approach to the physics of smooth superlattice potentials in graphene

Due to the chiral nature of the Dirac equation, governing the dynamics of electrons in graphene, overlying of an electrical superlattice (SL) can open new Dirac points on the Fermi surface of the energy spectrum. These lead to novel low-excitation physical phenomena. A typical example for such a system is neutral graphene with a symmetrical unidirectional SL. We show here that in smooth SLs, a semiclassical approximation provides a good mathematical description for particles. Due to the one-dimensional nature of the unidirectional potential, a wave-function description leads to a generalized Bohr-Sommerfeld quantization condition for the energy eigenvalues. In order to pave the way for the application of semiclassical methods to two-dimensional SLs in general, we compare these energy eigenvalues with those obtained from numerical calculations and with the results from a semiclassical Gutzwiller trace formula via the beam-splitting technique. Finally, we calculate ballistic conductivities in general point-symmetric unidirectional SLs with one electron and one hole region in the fundamental cell showing only Klein scattering of the semiclassical wave functions.

Escape of Particles from an Open Square-Shaped Cavity

The escape of particles in an open square-shaped cavity has been examined. We consider a family of trajectories launched from the left bottom lead of the square cavity and escaped from the right boundary. For each escaping trajectories, we record the propagation time and the detector position. We find that the escape time graph exhibits a regular sawtooth structure. For a set of detector points, we search for the classical trajectories from the source point to the detector points. Then we use semiclassical theory to construct the wave function at different given points. The calculation results suggest that the escape probability density depends on the detector position and the momentum of the particle sensitively. The Fourier transform of the semiclassical wave function gives the path length spectrum. Each peak in the path length spectrum corresponds to the length of one escape trajectory of the particle. We hope that our results will be useful in understanding the escape and transport process of particles inside a microcavity.

Dominant-interaction Hamiltonians for high-order-harmonic generation in laser-assisted collisions

We formulate the concept of dominant interaction Hamiltonians to obtain an integrable approximation to the dynamics of an electron exposed to a strong laser field and an atomic potential leading to high-order-harmonic generation. The concept relies on local information in phase space to switch between the interactions. This information is provided by classical integrable trajectories from which we construct a semiclassical wave function. The high-order-harmonic spectrum obtained is in excellent agreement with the accurate quantum spectrum. The separation in the atomic potential and laser coupling interactions should facilitate the calculation of high-order-harmonic spectra in complex systems.

Semiclassical initial value representation with complex trajectories

Semiclassical time-dependent solutions to the Schroedinger equation can be obtained by use of integral formulas known as initial value representations (IVRs). Over the last few decades IVR formulas have established a sucessful reputation, most notably in the field of physical chemistry, for it is often found that the quantum behavior of molecular systems is very well approximated by semiclassical wavefunctions. The IVR input consists in a swarm of independent classical trajectories, each of which obeys the dynamics dictated by the system’s unquantized Hamiltonian. In a recent work [Aguiar et al, Chem. Phys. 370 (2010)] an IVR expression based on complex classical trajectories was formulated. The complex nature of the classical trajectories in this particular formula is a consequence of employing minimum uncertainty wavepackets (coherent states) to represent the time evolution operator upon which the formula is constructed. Here, an improved and simpler version of this complex IVR formula is presented. In order to test its accuracy, we apply it to a simple model system.

Exponentially accurate semiclassical tunneling wavefunctions in one dimension

We study the time behavior of wavefunctions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit. We determine the leading-order component of the wavefunction that tunnels. It is exponentially small in 1/ℏ. For a wide variety of incoming wave packets, the leading-order tunneling component is Gaussian for sufficiently small ℏ. We prove this for both the large time asymptotics and for moderately large values of the time variable.

The calculation of multidimensional semiclassical wave functions in the forbidden region using real valued coordinates.

A method that uses only real valued coordinates is presented for integrating the many dimensional semiclassical wave function into the forbidden region. The procedure first determines a surface of caustic points by running the set of trajectories that define the wave function in the allowed region. In the forbidden region, the momentum and the action integral are both complex functions of position, and their imaginary parts vanish on the caustic surface. The direction of the imaginary part of the momentum p(I) can be chosen to the perpendicular to the caustic surface at all points on that surface. Equations are derived for integrating the values of the real and imaginary parts of the momentum along the curves that follow the direction of p(I). The equations for the change in the action integral and the prefactor for the semiclassical wave function along these curves are also obtained, allowing for the determination of the semiclassical wave function in the forbidden region. Calculations are performed for a two dimensional problem, and the semiclassical wave function is found to be is excellent agreement with the results of exact quantum calculations.

Q-Legendre transformation: partition functions and quantization of the Boltzmann constant

In this paper we construct a q-analogue of the Legendre transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables). Our approach is based on an analogy between the semiclassical wavefunctions in quantum mechanics and the quasithermodynamic partition functions in statistical physics. The basic idea is to go from the q-Hamilton–Jacobi equation in mechanics to the q-Legendre transformation in thermodynamics. It is shown that this requires a non-commutative analogue of the Planck–Boltzmann constants (ℏ and kB) to be introduced back into the classical formulae. Being applied to statistical physics, this naturally leads to an idea to go further and to replace the quasithermodynamic parameter corresponding to the Boltzmann constant with an infinite collection of generators of the so-called epoché (bracketing) algebra. The latter is an infinite-dimensional non-commutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of ‘deformations of deformations’ of the Weyl algebra. The generators mentioned are naturally indexed by planar binary leaf-labelled trees in such a way that the trees with a single leaf correspond to the observables of the limiting thermodynamic system.

Semiclassical initial value treatment of wave functions

A semiclassical initial value approximation for time-independent wave functions, previously derived for integrable systems, is rederived in a form which allows it to be applied to more general systems. The wave function is expressed as an integral over a Lagrangian manifold that is constructed by propagating trajectories from an initial manifold formed on a Poincaré surface. Even in the case of bound, integrable systems, it is unnecessary to identify action-angle variables or construct quantizing tori. The approximation is numerically tested for separable and highly chaotic two-dimensional quartic oscillator systems. For the separable (but highly anharmonic) system, the accuracy of the approximation is found to be excellent: overlaps of the semiclassical wave functions with the corresponding quantum wave functions exceed 0.999. For the chaotic system, semiclassical-quantum overlaps are found to range from 0.989 to 0.994, indicating accuracy that is still very good, despite the short classical trajectories used in the calculations.

A singularity free surface hopping expansion for the multistate wave function

A version of a surface hopping wave function for nonadiabatic multistate problems, which is free of turning point singularities, is derived and tested. The primitive semiclassical form of the particular surface hopping method considered has been shown to be highly accurate, even for classically forbidden processes. However, this semiclassical wave function displays the usual singular behavior at turning points and caustics in the classical motion. Numerical data has shown that this somewhat reduces its accuracy when the energy is near the crossing energy of the diabatic electronic surfaces. The singularity free version of this surface hopping wave function is derived by partitioning the x-axis into a large number of small steps for one dimensional problems. The adiabatic electronic energy surfaces are approximated to be linear functions within each step. The matching conditions required by the continuity of the wave function and its derivative at each step boundary provide the needed conditions to obtain the amplitudes for changes in electronic state and/or reflection of the trajectory for the motion of the nuclei. This leads to a form of the surface hopping wave function that is free of turning point singularities. The method is tested for a one dimensional model problem, and it is found to be highly accurate at all energies considered, even when the energy is near the crossing energy.

Modeling of ionization composition in argon plasma with fast electrons

Discharge and laser produced plasmas often have non-equilibrium electron distributions containing fast electrons that may have a profound effect on the ionization balance. The influence of high energy electrons on the ionization balance in the collisional-radiative equilibrium model for optically thin plasma is considered forthwith. A nonmaxwellian electron distribution with fast electrons at energies and concentrations is used for calculating collisional rates; this according to the Hartree-Fock-Slater (HFS) quantum-statistical model in the average atom approximation. The calculative approach is based on the distorted wave approximation along with numerical and semiclassical wave functions calculated in the self-consistent HFS potential. The results from ionization equilibrium modeling in argon plasma at temperatures T ∼ 5–1000 eV with fast electron energies ranging from 1 to 10 keV and their different relative concentrations (from 0.1–10%) are analyzed. It is shown that fast electrons increase the impact ionization rates and significantly change the ionization stage as compared with equilibrium plasma.