Simple Energy Spectrum Research Articles

Spectral Problem of the Hamiltonian in Quantum Mechanics without Reference to a Potential Function

Following the celebrated postulates of quantum mechanics, we write the quantum mechanical wavefunction as a convergent series of suitably selected complete square-integrable basis functions in configuration space. The expansion coefficients of the series are energy orthogonal polynomials that contain all spectral information about the system. We exploit the properties of these polynomials to introduce physical systems with rich and highly nontrivial energy spectra. In this approach, no reference is made at all to the usual potential energy function. We consider, in this new approach, a few representative problems at the level of undergraduate students who took at least two courses in quantum mechanics and are familiar with the basics of orthogonal polynomials. Our aim is to expose students to quantum systems with rich energy spectra that goes beyond the very limited textbook examples of systems with very simple energy spectra (e.g., the harmonic oscillator, Coulomb, Morse, Pöschl–Teller, etc.) illustrating the physical significance of these energy polynomials in the description of a quantum system. To assist students, partial solutions are given in an appendix as tables and figures.

Analysis of destruction term in transport equation for turbulent energy dissipation rate

The K–\(\varepsilon \) model used in turbulence simulations involves the transport equations for the turbulent kinetic energy and its dissipation rate. In contrast to the turbulent energy equation derived from the Navier–Stokes equation, the transport equation for the dissipation rate has been modeled empirically and has less of a theoretical grounding. An analysis of the dependence of terms in the exact transport equation on the Reynolds number \(\textit{Re}\) has suggested that the two dominant \(O(\textit{Re}^{1/2})\) terms cancel out at the leading order and their O(1) difference yields the terms in the model equation. The two-scale direct interaction approximation (TSDIA) can be used to derive the model equation for inhomogeneous turbulence. In this study, as a first step toward deriving all the terms in the model equation, the two dominant terms in the exact transport equation were investigated theoretically and numerically for the case of homogeneous isotropic turbulence, in order to derive the destruction term. The two terms were first analyzed by using the TSDIA under the assumption of a simple energy spectrum form. However, the finite-width effect of the inertial range of the energy spectrum did not give the expected \(O(\textit{Re}^{-1/2})\) corrections to the leading-order terms. Then, the model equations of the Markovianized Lagrangian renormalized approximation (MLRA) were numerically solved to obtain the accurate energy spectrum profile of a decaying homogeneous isotropic turbulence. Thereby, it was shown that the deviation of the energy spectrum from the \(-5/3\) law was responsible for the \(O(\textit{Re}^{-1/2})\) corrections. By assuming the energy spectrum suggested by the MLRA results, the two dominant terms were analyzed again by using the TSDIA to successfully derive the destruction term in the model equation.

Multi-azimuthal-angle effects in self-induced supernova neutrino flavor conversions without axial symmetry

The flavor evolution of neutrinos emitted by a supernova (SN) core is strongly affected by the refractive effects associated with the neutrino-neutrino interactions in the deepest stellar regions. Till now, all numerical studies have assumed the axial symmetry for the "multi-angle effects" associated with the neutrino-neutrino interactions. Recently, it has been pointed out in \cite{Raffelt:2013rqa} that removing this assumption, a new multi-azimuthal-angle (MAA) instability would emerge in the flavor evolution of the dense SN neutrino gas in addition to the one caused by multi-zenith-angle (MZA) effects. Inspired by this result, for the first time we numerically solve the non-linear neutrino propagation equations in SN, introducing the azimuthal angle as angular variable in addition to the usual zenith angle. We consider simple energy spectra with an excess of \nu_e over anti-\nu_e. In these cases, we find that even starting with a complete axial symmetric neutrino emission, the MAA effects would lead to significant flavor conversions in normal mass hierarchy, in cases otherwise stable under the only MZA effects. The final outcome of the flavor conversions, triggered by the MAA instability, depends on the initial asymmetry between \nu_e and anti-\nu_e spectra. If it is sufficiently large, final spectra would show an ordered behavior with spectral swaps and splits. Conversely, for small flavor asymmetries flavor decoherence among angular modes develops, affecting the flavor evolution also in inverted mass hierarchy.

Anisotropic conductivity and weak localization in HgTe quantum wells with a normal energy spectrum

The results of experimental study of interference induced magnetoconductivity in narrow quantum well HgTe with a normal energy spectrum are presented. Analysis is performed by taking into account the conductivity anisotropy. It is shown that the fitting parameter ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ corresponding to the phase relaxation time increases in magnitude with the increasing conductivity ($\ensuremath{\sigma}$) and decreasing temperature following the $1/T$ law. Such a behavior is analogous to that observed in the usual two-dimensional systems with a simple energy spectrum and corresponds to the inelasticity of electron-electron interaction as the main mechanism of the phase relaxation. However, it drastically differs from that observed in the wide HgTe quantum wells with the inverted spectrum, in which ${\ensuremath{\tau}}_{\ensuremath{\phi}}$, being obtained by the same way, is practically independent of $\ensuremath{\sigma}$. It is presumed that a different structure of the electron multicomponent wave function for the inverted and normal quantum wells could be the reason for such a discrepancy.

Weak antilocalization in HgTe quantum wells with inverted energy spectra

The results of experimental study of the magnetoconductivity of 2D electron gas caused by suppression of the interference quantum correction in HgTe single quantum well heterostructure with the inverted energy spectrum are presented. It is shown that only the antilocalization magnetoconductivity is observed at the relatively high conductivity $\sigma>(20-30)G_0$, where $G_0= e^2/2\pi^2\hbar$. The antilocalization correction demonstrates a crossover from $0.5\ln{(\tau_\phi/\tau)}$ to $1.0\ln{(\tau_\phi/\tau)}$ behavior with the increasing conductivity or decreasing temperature (here $\tau_\phi$ and $\tau$ are the phase relaxation and transport relaxation times, respectively). It is interpreted as a result of crossover to the regime when the two chiral branches of the electron energy spectrum contribute to the weak antilocalization independently. At lower conductivity $\sigma<(20-30)G_0$, the magnetoconductivity behaves itself analogously to that in usual 2D systems with the fast spin relaxation: being negative in low magnetic field it becomes positive in higher one. We have found that the temperature dependences of the fitting parameter $\tau_\phi$ corresponding to the phase relaxation time demonstrate reasonable behavior, close to 1/T, over the whole conductivity range from $5G_0$ up to $130G_0$. However, the $\tau_\phi$ value remains practically independent of the conductivity in distinction to the conventional 2D systems with the simple energy spectrum, in which $\tau_\phi$ is enhanced with the conductivity.

Coronal Loops Heated by Magnetohydrodynamic Turbulence. I. A Model of Isobaric Quiet‐Sun Loops with Constant Cross Sections

Several recent papers have presented new observational results indicating that many coronal loops in active regions are nearly isothermal. It is expected that quiet-Sun loops may have similar thermal structures, since quiet-Sun differential emission measures look similar to those in active regions. In the quiet Sun, it is well known from observations that the nonthermal velocity inferred from the excess broadening of a line over thermal broadening reaches a peak of about 30 km s-1 around 3 × 105 K and then decreases with temperature, having a value of about 20 km s-1 at 1 × 106 K. In the present work, we make the assumption that the observed nonthermal velocities are a manifestation of magnetohydrodynamic (MHD) turbulence and present a model of static, isobaric coronal loops heated by turbulence. Instead of solving the MHD equations, we adopt simple energy spectra in MHD turbulence and infer the heating rate as a function of temperature from the observed nonthermal velocities. By solving the steady state energy equation of a loop in which temperature monotonically increases with height, we obtain the following results: (1) The heating rate is predominantly near the footpoints and decreases with the loop arc length. (2) There is a critical temperature above which the loop cannot be maintained in a steady state. (3) The loop is denser and is more isothermal than uniformly heated loops, being compatible with recent observations. (4) The theoretical differential emission measures are in good agreement with the empirical values at temperatures above 105 K. Below this temperature, we still have a large discrepancy. (5) It is possible to explain the observed strong correlation between intensity and nonthermal velocity of a spectral line in the quiet Sun. Our results support the idea that quasi-statically driven MHD turbulence of the direct current (DC) type in the stratified medium (transition region and corona) is a viable mechanism for coronal heating.

An analytical model of velocity-derivative skewness of rotating homogeneous turbulence

The velocity-derivative skewness of rotating homogeneous turbulence is analytically derived from the turbulent kinetic energy dissipation rate equation by assuming a simple energy spectrum function. It predicts the cut-off of spectral energy at large rotation rate, and the predicted reduction of skewness due to rotation agrees well with the results of direct numerical simulation and eddy-damped quasi-normal Markovian closure.

Electron correlation, geometry, and energy spectrum of quadruply excited states.

There is a class of quadruply excited states of $^{5}$${\mathit{S}}^{\mathit{o}}$ symmetry where electronic motion is highly correlated and where the electrons tend to form a tetrahedron as the excitation energy increases toward the four-electron ionization threshold. This conclusion has been reached following ab initio state-specific calculations in Be, for the lowest energy state of each intrashall manifold n, of the energies, the average radii ${\mathit{r}}_{\mathit{n}}$, and the average interelectronic angle ${\mathrm{\ensuremath{\theta}}}_{12}$. In order to calculate ${\mathrm{\ensuremath{\theta}}}_{12}$, a general theory is developed, applicable to arbitrary N-electron atomic states. The value of ${\mathrm{\ensuremath{\theta}}}_{12}$ is straightforward to compute, and is given from a prescription transforming the expression for the two-electron interaction energy of the state to a formula for the probability density of cos${\mathrm{\ensuremath{\theta}}}_{12}$. The state-specific calculations for each n, up to n=6, were done by the multiconfigurational Hartree-Fock method where all configurations with ${\mathit{n}}_{1}$=${\mathit{n}}_{2}$=${\mathit{n}}_{3}$=${\mathit{n}}_{4}$ are included. For n=3, the main configuration 3s3${\mathit{p}}^{3}$ has a weight of 0.90 while ${\mathrm{\ensuremath{\theta}}}_{12}$=103.3\ifmmode^\circ\else\textdegree\fi{}. As n increases, electron correlation increases relative to the Coulomb nuclear attraction. With increasing degeneracy, many configurations with high orbital angular momenta mix heavily, and ${\mathrm{\ensuremath{\theta}}}_{12}$ increases.For example, for n=6, the 6s6${\mathit{p}}^{3}$ configuration has a weight of only 0.59 and ${\mathrm{\ensuremath{\theta}}}_{12}$=106\ifmmode^\circ\else\textdegree\fi{}. In this case, doubly, triply, as well as quadruply excited configurations with respect to ${\mathit{nsnp}}^{3}$ contribute to the wave function significantly. Finally, these four-electron ionization ladder states have a simple energy spectrum, given to a very good approximation by ${\mathit{E}}_{\mathit{n}}$=-A'/${\mathit{n}}^{2}$ (${\mathit{n}}^{1/2}$\ensuremath{\sim}${\mathit{r}}_{\mathit{n}}$), where A' is a constant. In conjunction with our earlier results on the geometry and the spectra of special classes of doubly and triply excited states, this finding leads to the conclusion that for highly correlated electronic motion the spectrum is dictated essentially by one dynamical variable, the average radius from the nucleus.