Energy Levels Of Bound States Research Articles

On the energy levels of a finite square-well potential

The theory of Cauchy integrals and the Riemann problem is used to derive an explicit formula for the bound-state energy levels of the finite square-well potential problem and is used to obtain a simple asymptotic expression for them.

A study of the predissociation of NaK molecules in the 6 1Σ+ state by optical–optical double resonance spectroscopy

Predissociation of a high-lying Σ+1 state of NaK is studied using the optical–optical double resonance technique. A single-mode ring dye laser is set to a particular 2(A)1Σ+(v,J)←1(X)1Σ+(v′′,J′′) transition. Another single-mode laser (Ti–sapphire) is then used to excite the molecule from the 2(A)1Σ+(v,J) level, to rovibrational levels of a higher predissociating electronic state, which we identify as 6 1Σ+. The predissociation is monitored by the atomic potassium emission on the 3 2D3/2→4 2P1/2 transition at 1.17 μm, while bound state radiative processes are monitored by total violet fluorescence from the upper state to the various rovibrational levels of the ground 1(X)1Σ+ state. By scanning the Ti–sapphire laser, different rovibrational levels of the 6 1Σ+ state can be excited. The vibrational levels probed range from v=13 to 20 with rotational states ranging from 9 to 99. The bound state energy level positions are measured from the center frequencies of lines recorded with the Ti–sapphire laser excitation scans. The 6 1Σ+ state is then described by the following molecular constants which are calculated from the experimental values of the level energies: Te=25 560.373 cm−1, ωe=89.179 26 cm−1, ωexe=0.730 691 cm−1, Be=0.067 327 0 cm−1, αe=0.000 675 35 cm−1, De=−3.298 31×10−8 cm−1, βel=1.518 17×10−8 cm−1. The potential well depth is De=4416.0 cm−1, if we assume the most likely asymptotic limit of Na(3 2S1/2)+K(5 2P1/2). The equilibrium separation is Re=4.158 Å. We also report measured and calculated intensities (Franck–Condon factors) for the 6 1Σ+→1(X)1Σ+ violet band. The absolute predissociation rates of 6 1Σ+ levels are directly measured from the linewidths recorded on the Ti–sapphire laser excitation scans. We measure predissociation rates ranging up to 9.4×109 s−1. The dependence of the absolute predissociation rates on rovibrational quantum numbers is studied with an attempt to predict the shape of the repulsive potential curve causing the predissociation, its crossing point with the bound state, and the type of perturbative interaction leading to the predissociation. The state causing the predissociation is determined from correlation diagrams to be the continuum of either the 3 3Π, the 3 1Π, or the 5 3Σ+ state with Na(3S)+K(3D) dissociation limit. We measure the collisional broadening rate coefficients of some 6 1Σ+←2(A)1Σ+ lines due to both argon and potassium perturbers, and obtain the average values, kbrAr=(1.1±0.2)×10−8 cm3 s−1 and kbrK=(1.1±0.6)×10−8 cm3 s−1. Velocity-changing collisions and collisional excitation transfer between individual rotational levels of the 2(A)1Σ+ state are also investigated.

Quantum black hole model and Hawking’s radiation

The black hole model with a self-gravitating charged spherical symmetric dust thin shell as a source is considered. The Schroedinger-type equation for such a model is derived. This equation appeared to be a finite differences equation. A theory of such an equation is developed and general solution is found and investigated in details. The discrete spectrum of the bound state energy levels is obtained. All the eigenvalues appeared to be infinitely degenerate. The ground state wave functions are evaluated explicitly. The quantum black hole states are selected and investigated. It is shown that the obtained black hole mass spectrum is compatible with the existence of Hawking's radiation in the limit of low temperatures both for large and nearly extreme Reissner-Nordstrom black holes. The above mentioned infinite degeneracy of the mass (energy) eigenvalues may appeared helpful in resolving the well known information paradox in the black hole physics.

QUANTUM BLACK HOLE MODEL AND HAWKING’S RADIATION

The black hole model with a self-gravitating charged spherical symmetric dust thin shell as a source is considered. The Schroedinger-type equation for such a model is derived. This equation appeared to be a finite differences equation. A theory of such an equation is developed and general solution is found and investigated in details. The discrete spectrum of the bound state energy levels is obtained. All the eigenvalues appeared to be infinitely degenerate. The ground state wave functions are evaluated explicitly. The quantum black hole states are selected and investigated. It is shown that the obtained black hole mass spectrum is compatible with the existence of Hawking’s radiation in the limit of low temperatures both for large and nearly extreme Reissner-Nordstrom black holes. The above mentioned infinite degeneracy of the mass (energy) eigenvalues may appeared helpful in resolving the well known information paradox in the black hole physics.

The long-range interaction between He and H+2: an ab initio and dynamical study

A multi-configuration self-consistent field, multi-reference configuration interaction (MCSCF-MRCI) calculation with extended atomic basis sets has been carried out on the electronic ground state of HeH2 + for intermediate-to-large intermolecular distances R between He and H2 + for both C∞v and C2v geometries at a fixed H2 + bond length of 2·0 a 0. All energies were corrected for basis set superposition error using the counterpoise method. The most stable geometry occurs in C∞v with a well depth of 7·16 kcal mol-1 (2504 cm-1) at an intermolecular distance of 2·98 a0 (1·576 Å), in good agreement with three previous high level calculations. The anisotropy of the interaction is strong, and is accounted for largely by the induced-dipole–quadrupole long-range force. The computed energy points were fitted to an analytical two-term Legendre expansion incorporating the appropriate long-range behaviour. This potential function was used to compute bound-state rovibrational energy levels of the HeH2 +complex variationally using a basis set method. The intermolecular zero-point energy of the complex was found to be 1020 cm-1, yielding a dissociation energy of 1484 cm-1 (4·24 kcal mol-1). The influence of the anisotropy on complex formation and ion-molecule reactions at thermal energies was assessed using close-coupling optical potential calculations of capture cross-sections; significant positive deviations from Langevin behaviour were found at sub-eV energies. The surface should be of value in interpreting the thermal energy Penning ionization of H2 by excited He.

Interface-related defects in a Ge0.4Si0.6/Si single well

Deep-level transient spectroscopy measurements were performed on a Ge0.4Si0.6/Si single quantum well grown by molecular beam epitaxy. The results show hole traps labelled H1 and H2. H1 is a deep level in Si whose activation energy measured from the valence band and capture cross section are 0.3 eV and 3*10-22 cm2 respectively. When an electric field was applied to the quantum well, an anomalous behaviour was observed for H2 only. The activation energy of H2 increased and then saturated as the applied electric field increased. Such behaviour can be explained by identifying H2 as an emission from interface states to a bound state in a quantum well. The depth profile of H2 also showed a spatially confined behaviour which strongly supports the idea that H2 is an interface state. Calculations using a variational method were performed to determine the bound state energy levels of the quantum well in a constant electric field. Comparing the results of experiments with those of the calculations, H2 corresponds to an interface state whose energy level lies at 0.45 eV above the valence band edge of the well; the capture cross section of H2 is 2.1*10-18 cm2.

An improved potential energy surface of Ar–C2H2

The intermolecular interaction of the Ar–C2H2 system was investigated using the coupled-states approximation to analyze the scattering process at a collision energy of 649.5 cm−1 (80.5 meV) and a close-coupling method to predict bound state energy levels. In both cases, the acetylene molecule is treated as a rigid rotor. A primary rainbow peak observed in the total differential scattering experiment provides useful information on the well depth while dimer rotational frequencies give accurate information on the shape around the minimum of the ground state potential. Four existing potentials for the system were tested against the total differential scattering cross section and rotational spectroscopic measurements. No one potential showed good agreement with both measurements. An improved potential energy surface was obtained by fitting simultaneously both scattering and spectroscopic data.

Photonic bound states in two-dimensional photonic crystals probed by coherent-microwave transient spectroscopy

The photonic analog of a semiconductor quantum well (QW) is constructed for the first time to our knowledge from two-dimensional periodic dielectric arrays. A sharp transmission peak is observed in the photonic band-gap spectral region of the barrier and is attributed to resonant transmission of an electromagnetic wave through a photonic bound state in this structure. In contrast with bound-state energy levels in semiconductor QW’s, the photonic bound state shifts to a higher frequency as the well width is increased, and a second bound state appears on the lower-frequency side.

Excitation and ionization of hydrogen atoms by ultra-intense fields

The Volkov wavepacket is used to trace the evolution of a ground-state hydrogen atom in an ultra-intense field, where the peak electric field strength is larger than the mean Coulomb binding field. We find the probabilities for excitation and ionization by projecting the time-dependent wavefunction onto the field-free atomic states. The ground-state atom is drastically shaken up into highly excited Rydberg states and continuum states within a very short time. Results are given for linearly and circularly polarized oscillating fields and for static fields. The dominant mechanism is field excitation and emission as a result of the instantaneous binding potential being suppressed below all bound-state energy levels. Our results are quite different from those of several other authors, some of whom claim that bound-state stabilization occurs in the high-field limit. We see no stabilization and we comment on possible reasons for the diverse results.

Hamilton-Jacobi theory and the relativistic action variable

The relativistic quantum Hamilton-Jacobi equations, which are physically equivalent to the relativistic Schrödinger and the Dirac equations, are postulated by extending the recent Leacock-Padgett theory to the relativistic case. Then the relativistic quantum action variables are obtained to show that the exact bound state energy levels of a relativistic system may be determined more conveniently by these variables without solving the dynamical equations.

Anharmonic corrections to variational transition state theory calculations of rate constants for a model activated reaction in solution

Previous errors in harmonic variational transition state theory calculations of rate constants for a model of an activated reaction in solution are corrected by including anharmonicities in the calculations of bound state energy levels. Harmonic variational transition state theory including multidimensional quantum mechanical corrections for reaction coordinate motion has previously been used to calculate the rate constant for a reaction model corresponding to a single solute coordinate coupled to a harmonic bath. Comparison of these rate constants with accurate quantum mechanical rate constants showed good agreement, in general, with the worst discrepancy to be a factor of 3.6. By including anharmonicities in the bound vibrational potentials to infinite order and calculating the energy levels by a uniform semiclassical prescription, the maximum error is reduced to 42%.

Mutual friction in superfluid 3He: Effects of bound states in the vortex core.

The motion of singular quantized vortex lines in superfluid $^{3}\mathrm{He}$ is considered for the A and B phases. Mutual friction is calculated within a microscopic quantum-mechanical Green's-function formalism, valid for dynamical processes. This enables us to include all the different physical phenomena in a unified approach. We consider axisymmetric vortices for temperatures considerably lower than ${\mathit{T}}_{\mathit{c}}$. In this regime, the main contribution to the force exerted on a moving vortex originates from the localized Fermi excitations occupying quantized energy eigenstates in the vortex core. These $^{3}\mathrm{He}$ quasiparticle states are similar to the quantized motion of charge in a magnetic field; thus vortex motion in $^{3}\mathrm{He}$ resembles the Hall phenomenon in metals. The outcome is that the viscous drag cannot simply be expressed through the cross sections for $^{3}\mathrm{He}$ quasiparticles scattering off the vortex, but is rather due to the mutual interactions between the localized quasiparticles and the normal excitations. Our calculations conform with the experimental values for the mutual-friction parameters. We also discuss vortex oscillations, and predict that strong dissipation should be observed at a resonant frequency of about 10 kHz, owing to transitions between the bound-state energy levels. This effect could be used for detecting and measuring the quantization of the bound-state spectrum for superfluid $^{3}\mathrm{He}$ in the vortex-core matter.

The calculation of highly excited bound-state energy levels for a triatomic molecule by using three-arrangement basis sets and contracted basis functions

The rovibrational energy levels of the nonrotating H+3 molecular ion have been calculated using a basis set defined in three-arrangement Jacobi coordinates. Energy levels corresponding to the A′1 symmetry are reported up to 22 600 cm−1 above the potential energy minimum and are converged to within 1.8 cm−1, energy levels in the A2 symmetry are reported up to 28 800 cm−1 and are converged to within 0.7 cm−1, and energy levels in the E′ symmetry are reported up to 24 300 cm−1 and are converged to within 1.8 cm−1. The maximum deviation of the reported results from the results of Carter and Meyer [J. Chem. Phys. 93, 8902 (1990)] is about 2 cm−1, and most of the results agree to within 1 cm−1. An advantage to our basis set is that it can represent the full D3h symmetry of this system, and thus the energy levels are assigned to the correct symmetry group unambiguously. In order to improve computational efficiency, contracted basis functions were formed using a successive diagonalization-truncation procedure.

Wave Function Properties and Shifted Energy Levels of Bound States in a Plasma

On the basis of earlier work we show a simple way to estimate the properties of bound states in a plasma. The Bethe-Salpeter equation is approximated by an effective Schrodinger equation. The energy eigenvalues are found via a variation procedure. The treatment is applicated to helium-like bound states and excited hydrogen-like states. The effect of the new energy eigenvalues on the plasma composition is discussed for the symmetrical electron-positron plasma.

Hyperspherical coordinates method in the theory of exciton-ionized donor complex of semiconductors

The hyperspherical coordinates method is used to study the problem of exciton-ionized donor complex in semiconductors. The critical mass ratio σ c = m e/m h) for the existence of such a complex is found to be 0.34. As a concrete example, we calculate the low-lying bound state energy levels of the complex in GaAs. Two bound states are found, one level lies between −1.0576 and −1.0471, the first excited state bounded loosely between −1.0235 and −1.0138 in unit of isolated donor binding energy.

Application of a generalized Feynman-Hellmann theorem to bound-state energy levels

We derive a generalization of the Feynman-Hellmann theorem and use it to describe how the energy of a bound state changes with the change in mass of one of the constituent particles.

High-energy atomic photoelectric effect and bremsstrahlung.

The relativistic photoionization cross sections from ${\mathrm{ns}}_{1/2}$,${\mathrm{np}}_{1/2}$,${\mathrm{np}}_{3/2}$ subshells, in the high-energy limit, have been analytically obtained. The main effects of screening enter through their effects on bound-state normalizations and energy levels. To estimate further effects, the screened potential at small distances is analytically described by a power-series expansion in the small distance r. For this high-energy-limit situation, both bound and continuum wave functions are expanded in power series for small r. The bound-free transition cross sections are then calculated analytically. Our results show that, beyond the known screening effect described by normalization factors, the screening effect enters the cross sections primarily through the change in bound-state energy and is otherwise not too sensitive to the expansion coefficients of the potential. The formulas contain no explicit n dependence. Comparisons with existing finite-energy numerical results indicate that ratios, though not absolute values, of cross sections attain their high-energy limits relatively early. Using Poincar\'e's theorem, all the photoionization results may be analytically continued to the high-frequency region of the spectrum of electron bremsstrahlung. Screening is dominated by the ``extra'' screening (namely, the screening which is not described by the known ``normalization screening'') except in the very tip region (a few eV above threshold), where normalization screening is much more important. The extension to positron bremsstrahlung is also discussed.

Quantum action-angle-variable analysis of basic systems

Quantum action-angle variables are used to describe and analyze a number of familiar systems. For a given system, the quantum canonical transformation from the old coordinates, e.g., linear or polar, to the new coordinates, action-angle variables, is found by generalizing the corresponding classical transformation using a method based upon the correspondence principle, the Hermiticity and canonical nature of the old coordinates, and the requirement that the Hamiltonian be independent of the quantum angle variable. The bound-state energy levels and other important system properties follow immediately from the canonical transformation. Harmonic oscillators of various dimensions and the three-dimensional angular momentum system are used as illustrations; these illustrations provide interesting alternatives to the usual quantum treatments.

A lower bound for ground-state energy by Steiner symmetrisation of the potential

It is shown that Steiner symmetrisation of the potential leads to a lowering of the ground-state energy of the Hamiltonian. Some special cases in which Steiner symmetrisation of the potential leads to the lowering of all the bound-state energy levels of the system are presented. A connection between certain exactly solvable potentials via Steiner symmetrisation is also brought out. Counterexamples are given to show that, in general, the lowering of energy eigenvalues on Steiner symmetrisation does not occur for all levels.

1/N expansion for Gaussian potential

The method of large-N expansion has been applied to an attractive radial Gaussian potential to obtain its bound state energy levels. It has been shown that the shifted 1/N expansion technique provides a better approximation than the method of unshifted 1/N expansion and yields energy values which are in good agreement with the accurate numerical and analytic results.