Quantization Of Energy Spectrum Research Articles

Classical and quantum-mechanical transition between regularity and irregularity in a Hamiltonian system.

We study the transition from regularity to irregularity for a physically real Hamiltonian system, viz., the hydrogen atom in a uniform magnetic field. We obtain quantitative correspondence between the transition in classical phase space and the change in the level statistics of the quantum energy spectra. Beyond the completely regular regime, the distributions of level spacings show a pronounced minimum at small spacings and cannot be accurately fitted by the semiclassical formula of Berry and Robnik [J. Phys. A 17, 2413 (1984)].

Quantum energy spectra and one-dimensional quasiperiodic systems.

Statistical properties of energy spectra in one-dimensional quasiperiodic systems are studied numerically. We find three distinctive level distributions: the Poisson, inverse-power-law (IPL), and cosine-band-like behaviors in the Harper model with an incommensurate potential. These depend on whether the electronic state is localized, critical, or extended, respectively. Energy spectra of electrons on the quasiperiodic Fibonacci lattice are also characterized by the IPL irrespective of the strength of the modulation, indicating that the state is always critical.

Monte Carlo Simulation of Electron–Phonon Scattering in the XUV‐Induced Electron Emission of NaCl

AbstractA Monte Carlo simulation of the electron emission induced in NaCl by 60 to 250 eV photons is carried out. The number as well as the initial energy and spatial distribution of conduction electrons, obtained in a previous paper, are used as the input. Electron–phonon scattering events are treated semiclassically. Optical as well as acoustical phonons are taken into account, so that with the increase of electron energy the scattering probability decreases for optical and increases for acoustical phonons. The parameters of the model are adjusted to fit the experimental quantum yield and energy spectra of emitted electrons. It appears that for high‐energy (>2 to 3 eV) electrons the electron–phonon scattering is mainly due to acoustical phonons.

Statistical properties of the quantized energy spectrum of a Hamiltonian system with classically regular and chaotic trajectories: A numerical study of level-spacing distributions for two-dimensional coupled Morse-oscillator systems.

A quantification of the degree of classical chaos manifested in the quantized energy spectra of two\char21{}degree-of-freedom coupled Morse-oscillator systems with sufficiently dense energy levels is attempted by use of Brody's repulsion parameter which characterizes his nearest-neighbor level-spacing distribution function. A close relationship is established numerically between the mass-ratio dependence of the Brody parameter and that of the relative area of the chaotic regions on the Poincar\'e surfaces of section in the corresponding classical system. It is shown that in the strong-coupling limit the distribution appears to tend to the Mehta-Gaudin distribution from the random matrix theory, suggesting that in this limit it is almost impossible to distinguish between the quantized version of the classical K system and those of other systems with a fairly small number of regular trajectories. The present analysis also demonstrates that the Brody parameter serves as a useful indicator for measuring the degree of mode coupling and for detecting an isolated local mode in the system.

A connection between classical chaos and the quantized energy spectrum: Level-spacing distributions in a kinetically coupled quantum morse system with two degrees of freedom

The effect of classical chaos on the quantized energy spectrum is investigated for a variety of kinetically coupled Morse systems with two degrees of freedom. It is shown that, for both the energy range and mass ratios generating a global chaos, the positions of spacing coordinates giving the peaks of the nearest-neighbor level-spacing histograms are sufficiently close to those giving the maxima of the Wigner distributions. The classical-to-quantum correspondence is conjectured to be rather mild for a three-atom system composed of ordinary light atoms.

Motion of wave packets in regular and chaotic systems

The quantum energy spectrum of a system which is classically integrable consists of families of nearly equidistant levels. There is no such regularity for a classically chaotic system. As a consequence, a small wave packet, initially centered in a regular region of phase space, will slowly disperse while following the (almost) periodic classical trajectory. A wave packet placed in a chaotic region will disperse much more rapidly. These predictions are illustrated by calculating the evolution of two wave packets with the same mean energy in the Hénon–Heiles model.

Nonlinear conductivity of thin semiconductor layers with a dimensionally quantized electron energy spectrum

UDC 537.311.33 The nonlinear transverse conductivity of thin semiconductor layers with a dimensionally quantized energy spectrum of the transverse motion of the electrons in the frequency region co, which is less than the characteristic distances between the dimensional quantization levels ~2 and arbitrary with respect to the reciprocal relaxation time of the population of these levels T(r), is obtained. The electron scattering is assumed to be quasielastic. The conductivity of these layers is mainly reactive, extremely nonlinear, and contains sections of negative dynamic differential conductivity. At low temperatures and frequencies (c0T(r) > 1 the current is extremely nonlinear and is reactive. When the frequency of the field is increased, the value of the current and capacitance jumps fails, and time oscillations of the current occur at frequencies ~tkn.

Solution of a One-Dimensional Three-Body Problem in Classical Mechanics

A one-dimensional problem of three equal particles interacting pairwise via a potential that is the sum of a quadratic and an inversely quadratic term is discussed from the point of view of classical mechanics. It is shown that the quantum energy spectrum can be obtained from the classical theory of action-angle variables and Bohr-Sommerfeld rules.

Tunneleffekt und quantenverteilung der energie bei der elektrochemischen entladung von protonen und deuteronen

The role of the tunnel effect in electrode processes has been studied in a series of earlier papers assuming a continuous energy distribution of the dissolved ions. In particular the probable shape and the permeability of the potential barrier for the discharge of protons and deuterons on a mercury cathode have been determined in this way on the basis of Post and Hiskey's experimental values.In the present paper the same problem is treated assuming a quantum distribution of the vibrational energy of the H+-OH2 bond. The calculations were carried out using a recently found expression for the permeability of a generalized Eckart barrier. It is found that the parameters of the barrier do not differ much from those determined previously. It is also shown that practically all particles penetrate this barrier at the quantum level n = 2 (for protons) or n = 3 (for deuterons). Actually these energy levels correspond approximately to the mean values of the energy of the protons or deuterons penetrating the barrier because the strictly quantized energy spectrum is strongly smeared out due to the interactions of the particles in the solution.