Wave Functions Of Stationary States Research Articles

On the symmetry algebra of a one-dimensional quantum-mechanical oscillator on a hyperbola

The quantum-mechanical problem of a harmonic oscillator on a hyperbola as a one-dimensional space of constant negative curvature is considered in this article. A generalization to the singular oscillator model in the context of one-dimensional Cayley – Klein geometries is given by the factorization method. The energy spectrum and wave functions of stationary states are found having the curvature of space as a parameter. For the energy levels of the singular oscillator, the effect of non-zero curvature is clearly manifested through a positive or negative term, depending on the sign of the curvature, which is quadratic in the level number. The results obtained are consistent with those previously published. The dynamical symmetry of the problem is shown explicitly as a quadratic Hahn algebra QH(3) or its isomorphic Higgs algebra.

Tunneling dynamics of an oscillating universe model

Quasiclassical methods for non-adiabatic quantum dynamics can reveal new features of quantum effects, such as tunneling evolution, that are harder to analyze in standard treatments based on wave functions of stationary states. Here, these methods are applied to an oscillating universe model introduced recently. Our quasiclassical treatment correctly describes several expected features of tunneling states, in particular just before and after tunneling into a trapped region where a model universe may oscillate through many cycles of collapse and expansion. As a new result, the oscillating dynamics is found to be much less regular than in the classical description, revealing a succession of cycles with varying maximal volume even when the matter ingredients and their parameters do not change.

Investigating and improving student understanding of the basics for a system of identical particles

We discuss an investigation of students' difficulties with systems of non-interacting identical particles. The investigation was carried out in advanced quantum mechanics courses at both the undergraduate and graduate levels by administering free-response and multiple-choice questions and conducting individual interviews with students. This research on student difficulties was used as a guide to develop and evaluate a Quantum Interactive Learning Tutorial (QuILT) that helps students develop a functional understanding of the concepts necessary for determining the many-particle Hamiltonian and stationary state wavefunctions for a system of non-interacting identical particles.

Positrons vs electrons channeling in silicon crystal: energy levels, wave functions and quantum chaos manifestations

The motion of fast electrons through the crystal during axial channeling could be regular and chaotic. The dynamical chaos in quantum systems manifests itself in both statistical properties of energy spectra and morphology of wave functions of the individual stationary states. In this report, we investigate the axial channeling of high and low energy electrons and positrons near [100] direction of a silicon crystal. This case is particularly interesting because of the fact that the chaotic motion domain occupies only a small part of the phase space for the channeling electrons whereas the motion of the channeling positrons is substantially chaotic for the almost all initial conditions. The energy levels of transverse motion, as well as the wave functions of the stationary states, have been computed numerically. The group theory methods had been used for classification of the computed eigenfunctions and identification of the non-degenerate and doubly degenerate energy levels. The channeling radiation spectrum for the low energy electrons has been also computed.

The “Forgotten” Pseudomomenta and Gauge Changes in Generalized Landau Level Problems: Spatially Nonuniform Magnetic and Temporally Varying Electric Fields

By perceiving gauge invariance as an analytical tool in order to get insight into the states of the "generalized Landau problem" (a charged quantum particle moving inside a magnetic, and possibly electric field), and motivated by an early article that correctly warns against a naive use of gauge transformation procedures in the usual Landau problem (i.e. with the magnetic field being static and uniform), we first show how to bypass the complications pointed out in that article by solving the problem in full generality through gauge transformation techniques in a more appropriate manner. Our solution provides in simple and closed analytical forms all Landau Level-wavefunctions without the need to specify a particular vector potential. This we do by proper handling of the so-called pseudomomentum K (or of a quantity that we term pseudoangular momentum Lz), a method that is crucially different from the old warning argument, but also from standard treatments in textbooks and in research literature (where the usual Landau wavefunctions are employed - labeled with canonical momenta quantum numbers). Most importantly, we go further by showing that a similar procedure can be followed in the more difficult case of spatially-nonuniform magnetic fields: in such case we define K and Lz as plausible generalizations of the previous ordinary case, namely as appropriate line integrals of the inhomogeneous magnetic field - our method providing closed analytical expressions for all stationary state wavefunctions.. (see paper) ... also provide explicit Berry's phase calculations and their connection to probability currents and ...issues in elementary Quantum Mechanics and Condensed Matter Physics. As an added feature, we also show how the possible presence of an additional electric field can be treated through a further generalization of pseudomomenta and their proper handling.

Construction of a neuroanatomical shape complex atlas from 3D MRI brain structures

Brain atlas construction has attracted significant attention lately in the neuroimaging community due to its application to the characterization of neuroanatomical shape abnormalities associated with various neurodegenerative diseases or neuropsychiatric disorders. Existing shape atlas construction techniques usually focus on the analysis of a single anatomical structure in which the important inter-structural information is lost. This paper proposes a novel technique for constructing a neuroanatomical shape complex atlas based on an information geometry framework. A shape complex is a collection of neighboring shapes - for example, the thalamus, amygdala and the hippocampus circuit - which may exhibit changes in shape across multiple structures during the progression of a disease. In this paper, we represent the boundaries of the entire shape complex using the zero level set of a distance transform function S(x). We then re-derive the relationship between the stationary state wave function ψ(x) of the Schrödinger equation [formula in text] and the eikonal equation [formula in text] satisfied by any distance function. This leads to a one-to-one map (up to scale) between ψ(x) and S(x) via an explicit relationship. We further exploit this relationship by mapping ψ(x) to a unit hypersphere whose Riemannian structure is fully known, thus effectively turn ψ(x) into the square-root of a probability density function. This allows us to make comparisons - using elegant, closed-form analytic expressions - between shape complexes represented as square-root densities. A shape complex atlas is constructed by computing the Karcher mean ψ¯(x) in the space of square-root densities and then inversely mapping it back to the space of distance transforms in order to realize the atlas shape. We demonstrate the shape complex atlas computation technique via a set of experiments on a population of brain MRI scans including controls and epilepsy patients with either right anterior medial temporal or left anterior medial temporal lobectomies.

Resonant coherent excitation of hydrogen-like ions planar channeled in a crystal; Transition into the first excited state

The presented program is designed to simulate the characteristics of resonant coherent excitation of hydrogen-like ions planar-channeled in a crystal. The program realizes the numerical algorithm to solve the Schrödinger equation for the ion-bound electron at a special resonance excitation condition. The calculated wave function of the bound electron defines probabilities for the ion to be in the either ground or first excited state, or to be ionized. Finally, in the outgoing beam the fractions of ions in the ground state, in the first excited state, and ionized by collisions with target electrons, are defined. The program code is written on C++ and is designed for multiprocessing systems (clusters). The output data are presented in the table. Program summary Program title: RCE_H-like_1 Catalogue identifier: AEKX_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEKX_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 2813 No. of bytes in distributed program, including test data, etc.: 34 667 Distribution format: tar.gz Programming language: C++ (g++, icc compilers) Computer: Multiprocessor systems (clusters) Operating system: Any OS based on LINUX; program was tested under Novell SLES 10 Has the code been vectorized or parallelized?: Yes. Contains MPI directives RAM: <1 MB per processor Classification: 2.1, 2.6, 7.10 External routines: MPI library for GNU C++, Intel C++ compilers Nature of problem: When relativistic hydrogen-like ion moves in the crystal in the planar channeling regime, in the ion rest frame the time-periodic electric field acts on the bound electron. If the frequency of this field matches the transition frequency between electronic energy levels, the resonant coherent excitation can take place. Therefore, ions in the different states may be observed in the outgoing beam behind the crystal. To get the probabilities for the ion to be in the ground state or in the first excited state, or to be ionized, the Schrödinger equation is solved for the electron of ion. The numerical solving of the Schrödinger equation is carried out taking into account the fine structure of electronic energy levels, the Stark effect due to the influence of the crystal electric field on electronic energy levels and the ionization of ion due to the collisions with crystal electrons. Solution method: The wave function of the electron of ion is the superposition of the wave functions of stationary states with time-dependent coefficients. These stationary wave functions and corresponding energies are defined from the stationary Schrödinger equation. The equation is reduced to the problem of the eigen values and vectors of Hermitian matrix. The corresponding matrix equation is considered as the linear equation system. Then the time-dependent coefficients of the electron wave function are defined from the Schrödinger equation, with a time-periodic crystal field. The time-periodic field is responsible for the transitions between the stationary states. The final time-dependent Schrödinger equation represents the matrix equation which has been solved by means of the QR-algorithm. Restrictions: As expected the program gives the correct results for relativistic hydrogen-like ions with the kinetic energies up to 1 GeV/u and at the crystal thicknesses of 1–100 μm. The restrictions are: first, the program might give inadequate results, when the ion kinetic energy is too large (>10 GeV/u); second, the unaccounted physical factors may be significant at specific conditions. For example, the spontaneous emission by exited highly charged ions, as well as both energy and angular spread of the incident beam, could lead to additional broadening of the resonance. The medium polarization by the electric field of ion can influence the electronic energy levels of the ion in the non-relativistic case. The role of these factors was discussed in the references. Also, the large crystal thickness may require large computational time. Running time: In general, the running time depends on the number of processors. In our tests we used the crystal thickness up to 100 μm and the number of 2.66 GHz processors was up to 100. The running time was about 1 hour in these conditions.

Bipolar Reaction Path Hamiltonian Approach for Reactive Scattering Problems.

In this work we present a method for calculating the stationary state wave functions and reaction probabilities of a multidimensional reactive scattering system. Our approach builds upon the counter-propagating wave methodology (CPWM) developed by Poirier and co-workers for calculating one-dimensional stationary state wave functions. The method involves the formulation of a bipolar decomposition for multidimensional stationary scattering wave functions within the context of a reaction path Hamiltonian, so we refer to this work as the bipolar reaction path Hamiltonian (BRPH) approach. Benchmark calculations are presented for several 2D model scattering systems with linear reaction coordinates. We show that the BRPH approach is competitive with conventional calculations based on discrete variable representation (DVR) methods.

Effect of interdiffusion on band structure and intersubband absorption coefficient of GaAs/GaAlAs double quantum well

The influence of Ga and Al atoms interdiffusion on the band structure and absorption coefficient of coupled double quantum wells are investigated on the basis of a potential composed of four modified Wood–Saxon potentials. Wave functions of the real potential are expressed as a superposition of the stationary state wave functions of the “basal” potential. It is shown that interdiffusion leads to the smearing of the rectangular profile of the potential and to the disappearance of the barrier between the wells, as a result of which, the degradation of doublet character of the electron energetic spectrum is observed. It is also shown that interdiffusion leads to the significant change in oscillator strength and in the enhancement of the thermal stability of the absorption spectrum of the considered heterostructure.

Construction of neuroanatomical shape complex atlas from 3D brain MRI.

This paper proposes a novel technique for constructing a neuroanatomical shape complex atlas using an information geometry framework. A shape complex is a collection of shapes in a local neighborhood. We represent the boundary of the entire shape complex using the zero level set of a distance function S(x). The spatial relations between the different anatomical structures constituting the shape complex are captured via the distance transform. We then leverage the well known relationship between the stationary state wave function psi(x) of the Schrödinger equation -h2nabla2 psi + psi = 0 and the eikonal equation //nablaS// = 1 satisfied by any distance function S(x). This leads to a one-to-one map between psi(x) and S(x) and allows for recovery of S(x) from psi(x) through an explicit mathematical relationship. Since the wave function can be regarded as a square-root density function, we are able to exploit this connection and convert shape complex distance transforms into probability density functions. Furthermore, square-root density functions can be seen as points on a unit hypersphere whose Riemannian structure is fully known. A shape complex atlas is constructed by first computing the Karcher mean psi(x) of the wave functions, followed by an inverse mapping of the estimated mean back to the space of distance transforms in order to realize the atlas. We demonstrate the shape complex atlas computation via a set of experiments on a population of brain MRI scans. We also present modes of variation from the computed atlas for the control population to demonstrate the shape complex variability.

The Analytic Solution of Schrödinger Equation with Potential Function Superposed by Six Terms with Positive-power and Inverse-power Potentials

The analytic solution of the radial Schrödinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely analytic solutions and the conditions that decide the existence of analytic solution have been searched when the potential of the radial Schrödinger equation is V(r) = α1r8 + α2r3 + α3r2 + β3r−1 + β2r−3 + β1r−4. Generally speaking, there is only an approximate solution, but not analytic solution for Schrödinger equation with several potentials' superposition. However, the conditions that decide the existence of analytic solution have been found and the analytic solution and its energy level structure are obtained for the Schrödinger equation with the potential which is motioned above in this paper. According to the single-value, finite and continuous standard of wave function in a quantum system, the authors firstly solve the asymptotic solution through the radial coordinate r → and r → 0; secondly, they make the asymptotic solutions combining with the series solutions nearby the neighborhood of irregular singularities; and then they compare the power series coefficients, deduce a series of analytic solutions of the stationary state wave function and corresponding energy level structure by tight coupling among the coefficients of potential functions for the radial Schrödinger equation; and lastly, they discuss the solutions and make conclusions.

The analytic solution of the radial Schr?dinger equation for the superposed potentials

The method of solving the radial Schrdinger equation is studied by under the tight coupling condition of several positivepower and inversepower potential functions. The precise analytic solutions and the conditions that determine the existence of analytic solution are searched when the potential of the radial Schrdinger equation is V（r）=α1r8＋α2r3+α3r2+β3r-1＋β2r-3＋β1r-4. According to the single valued, bounded and continuous stipulations of wave function in a quantum system, firstly, the asymptotic solution is solved when the radial coordinate r→∞ and r→0; secondly, the asymptotic solutions are combined with the series solutions in the neighborhood of irregular singularities; and then the power series coefficients are compared. A series of analytic solutions of the stationary state wave function and the corresponding energy level structure are deduced by tight coupling between the coefficients of potential functions for the radial Schrdinger equation. And the solutions are discussed and the conclusions are made.

Canonical transformations and non-unitary evolution

We test the idea that transformations which, at the classical level, can be interpreted as evolutions are represented within quantum mechanics by unitary operators. To this end, we consider non-trivial canonical transformations which leave invariant the form of the Hamilton function of a system. We demonstrate that infinite families of such transformations exist for a variety of familiar conservative systems of one degree of freedom. We show how the precise form of integral equations for the stationary state wavefunctions implied by the existence of these canonical transformations can be pinned down by exploiting the algebra of the transformations and a symmetry of their generating functions. We recover several integral equations found in the literature on standard special functions of mathematical physics. We find that when one of the classical canonical transformations we consider is non-linear, its quantum implementation is non-unitary. We end with some comments on the implications of our findings for semiclassical studies and a brief discussion relevant to string theory of the generalization to scalar field theories in 1 + 1 dimensions.

Influence of acceleration on the evolution of a quantum mechanical harmonical oscillator

Wave functions of stationary states for a uniformly accelerated harmonical oscillator and its energy spectrum are obtained. The oscillator excitation due to the acceleration is studied. Based on the obtained results a descriptive interpretation of fulling- Unruh effect is given.

Two-electron states in a double quantum dot in a constant electric field

The wave functions of stationary states and the spectrum of two-electron system are analytically determined in a symmetric double quantum dot. It is shown that in the ground state when the external electric field is absent, electrons cannot reside in the same quantum dot due to the Coulomb blockade. This situation changes in an external electric field. At a critical field strength, the probability of finding both electrons in the same quantum dot jumpwise increases from zero to unity.

Quantum chaos and nuclear structure

The quantum level statistics were studied for the first time in nuclear spectra 60 years ago [I.I. Gurevich, JEPT 9 (1939) 1283]. Now, we know that atomic nuclei and complex atoms are good examples of mesoscopic quantum systems where the effects of quantum many-body chaos are clearly seen both in spectra and in the properties of complicated wave functions of stationary states. We present a short and subjective review of those ideas, applications to nuclear structure, and open problems.

Letter: Quantum Mechanics Forbids Singularities in a Closed Radiation-dominated FRW Universe

The existence of singularities in a closed FRWuniverse depends on the assumption that generalrelativity is valid for distances less than the Plancklength. However, the stationary state wave functions of the Schrodinger equation for a closedradiation-dominated FRW universe derived by Elbaz et al.are zero at zero radius of curvature. Thus, even ifgeneral relativity is assumed valid at distances less than the Planck length, quantum mechanics seems to forbid singularities in a closed radiation-dominatedFRW universe.

New exact solutions of the Schrödinger equation with potentials of spin and soliton origin

New classes of exact solutions of the Schrödinger equation with simple explicit analytic expressions for potential fields, energy levels, and wave functions of stationary states are considered. The solutions are discovered with the help of new original methods elaborated in the quantum theory of spin systems. The corresponding effective potentials are compared to similar models of soliton origin. The main attention is paid to peculiar phenomena such as quasi-exact solvability, potentials with multiple and flexible profiles, fourth-order extrema, finite-band spectra and structural transformations in energy bands, and the spin–soliton analogy.

Behaviour of an Electron Wave in Electron-Focusing Geometry

In electron-focusing geometry in the presence of a magnetic field, numerical calculations have been made for the wave functions of stationary states and for transmission probability of the electron wave from the injector to the collector. The pattern behaviour of transmission probability is demonstrated as a function of the external magnetic field. The results obtained here reproduce the experimental focusing spectra qualitatively.

The time-dependent variational description of nuclear transition probabilities

Abstract A time-averaged gauge invariant wavefunction is considered as a representative for the stationary-state wavefunction of the nuclear collective motion. To verify its accuracy the diagonal and off-diagonal transition probabilities are numerically calculated in three closed-shell nuclei and compared to the corresponding generator coordinate method results. The agreement, even for strongly anharmonic motion, provides strong support for the time-dependent variational description. Thus, this method emerges as an interesting alternative for calculating the nuclear transition probabilities.