Discrete Energy Eigenvalues Research Articles

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

Single-Shot Factorization Approach to Bound States in Quantum Mechanics

Closed form expressions for the Green’s function of a quantum graph—a scattering approach

In this work we present a three step procedure for generating a closed form expression of the Green’s function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard (2001 Phys. Rev. E 65 016205) and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate. The derivation of the Green’s function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green’s function is derived. The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green’s functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.

Isospectral mapping for quantum systems with energy point spectra to polynomial quantum harmonic oscillators

We show that a polynomial Hˆ(N) of degree N of a harmonic oscillator hamiltonian allows us to devise a fully solvable continuous quantum system for which the first N discrete energy eigenvalues can be chosen at will. In general such a choice leads to a re-ordering of the associated energy eigenfunctions of Hˆ such that the number of their nodes does not increase monotonically with increasing level number. Systems Hˆ have certain ‘universal’ features, we study their basic behaviours.

Analytical solutions for the radial Scarf II potential

The real Scarf II potential is discussed as a radial problem. This potential has been studied extensively as a one-dimensional problem, and now these results are used to construct its bound and resonance solutions for l=0 by setting the origin at some arbitrary value of the coordinate. The solutions with appropriate boundary conditions are composed as the linear combination of the two independent solutions of the Schrödinger equation. The asymptotic expression of these solutions is used to construct the S0(k)s-wave S-matrix, the poles of which supply the k values corresponding to the bound, resonance and anti-bound solutions. The location of the discrete energy eigenvalues is analyzed, and the relation of the solutions of the radial and one-dimensional Scarf II potentials is discussed. It is shown that the generalized Woods–Saxon potential can be generated from the Rosen–Morse II potential in the same way as the radial Scarf II potential is obtained from its one-dimensional correspondent. Based on this analogy, possible applications are also pointed out.

Continuity of the entropy of macroscopic quantum systems.

The proper definition of entropy is fundamental to the relationship between statistical mechanics and thermodynamics. It also plays a major role in the recent debate about the validity of the concept of negative temperature. In this paper, I analyze and calculate the thermodynamic entropy for large but finite quantum mechanical systems. A special feature of this analysis is that the thermodynamic energy of a quantum system is shown to be a continuous variable, rather than being associated with discrete energy eigenvalues. Calculations of the entropy as a function of energy can be carried out with a Legendre transform of thermodynamic potentials obtained from a canonical ensemble. The resultant expressions for the entropy are able to describe equilibrium between quantum systems having incommensurate energy-level spacings. This definition of entropy preserves all required thermodynamic properties, including satisfaction of all postulates and laws of thermodynamics. It demonstrates the consistency of the concept of negative temperature with the principles of thermodynamics.

Coherent States of Quantum Free Particle on the Spherical Space

In this paper, we study the quantum free particle on the spherical space by applying da costa approach for quantum particle on the curved space. We obtain the discrete energy eigenvalues and associated normalized eigenfunctions of the free particle on the sphere. In addition, we introduce the Gazeau-Klauder coherent states of free particle on the sphere. Then, the Gaussian coherent states is defined, which is used to describe the localized particle on the spherical space. Finally, we study the relation between the f-deformed coherent states and Gazeau-Klauder ones for this system.

OptaDOS: A tool for obtaining density of states, core-level and optical spectra from electronic structure codes

We present OptaDOS, a program for calculating core-electron and low-loss electron energy loss spectra (EELS) and optical spectra along with total-, projected- and joint-density of electronic states (DOS) from single-particle eigenenergies and dipole transition coefficients. Energy-loss spectroscopy is an important tool for probing bonding within a material. Interpreting these spectra can be aided by first principles calculations. The spectra are generated from the eigenenergies through integration over the Brillouin zone. An important feature of this code is that this integration is performed using a choice of adaptive or linear extrapolation broadening methods which we show produces higher accuracy spectra than standard fixed-width Gaussian broadening. OptaDOS may be straightforwardly interfaced to any electronic structure code. OptaDOS is freely available under the GNU General Public licence from http://www.optados.org. Program summaryProgram title:OptaDOSCatalogue identifier: AESK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicencing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 110299No. of bytes in distributed program, including test data, etc.: 1889705Distribution format: tar.gzProgramming language: Fortran 95.Computer: Any architecture with a Fortran 95 compiler.Operating system: Linux, Mac OS X.Has the code been vectorised or parallelised?: Yes, using MPIRAM: 10 MBWord size: 32 or 64Classification: 7.2, 7.3.External routines: MPI to run in parallel, CASTEP or any other electronic structure code capable of generating single-point eigenenergies and dipole transition coefficients.Nature of problem:Many properties of materials can be described using integration over the Brillouin zone such as core-level and low-loss EELS and optical spectra. This integration is performed computationally using a grid of k-points. The discrete energy eigenvalues must be interpolated into a continuous spectra. The most common method broadens the eigenvalues using a Gaussian function. Gaussian broadening suffers from slow convergence with number of k-points and a difficulty in resolving fine spectral features.Solution method:OptaDOS improves the underlying Brillouin zone integration beyond fixed-width Gaussian broadening by using band gradients to perform adaptive and linearly extrapolated broadening. This increases the resolution of the predicted spectra.Unusual features:Simple and user-friendly input system. Along with the usual band energies, band gradients are used to generate the linear extrapolation and adaptive broadening schemes producing a superior output able to represent both dispersive and localised bands concurrently.Additional comments:The input data to OptaDOS are single-particle eigenenergies and dipole transition coefficients. OptaDOS has an interface to obtain these from the CASTEP plane wave density-functional theory (DFT) code. The interfacing of OptaDOS with other electronic structure codes, which are also capable of generating such inputs, is currently being undertakenRunning time:A few seconds to ∼10 min.

Eigenvalue spectra of a PT-symmetric coupled quartic potential in two dimensions

The Schrodinger equation was solved for a generalized PT-symmetric quartic potential in two dimensions. It was found that, under a suitable ansatz for the wave function, the system possessed real and discrete energy eigenvalues. Analytic expressions for the energy eigenvalues and the eigenfunctions for the first four states were obtained. Some constraining relations among the wave function parameters rendered the problem quasi-solvable.

Fractional recurrence in discrete-time quantum walk

AbstractQuantum recurrence theorem holds for quantum systems with discrete energy eigenvalues and fails to hold in general for systems with continuous energy. We show that during quantum walk process dominated by interference of amplitude corresponding to different paths fail to satisfy the complete quantum recurrence theorem. Due to the revival of the fractional wave packet, a fractional recurrence characterized using quantum Pólya number can be seen.

EIGENVALUES AND EIGENFUNCTIONS OF WOODS–SAXON POTENTIAL IN PT-SYMMETRIC QUANTUM MECHANICS

Using the Nikiforov–Uvarov method which is based on solving the second-order differential equations, we firstly analyzed the energy spectra and eigenfunctions of the Woods–Saxon potential. In the framework of the PT-symmetric quantum mechanics, we secondly solved the time-independent Schrödinger equation for the PT and non-PT-symmetric version of the potential. It is shown that the discrete energy eigenvalues of the non-PT-symmetric potential consist of the real and imaginary parts, but the PT-symmetric one has a real spectrum. Results are obtained for s-states only.

Complexified Pöschl–Teller II potential model

With the help of complexifying a five-parameter exponential-type potential model, we obtain a general complex version of the Pöschl–Teller II potential, V(x)=−V 1 q c q 0 sech q c 2λx+V 2 q c q 0 cosech q c 2λx , where q c = q 0 e 2 iαε , real V 1>0, q 0>0 and 0<λε< π 2 . It has been shown that this complex potential is P-pseudo-Hermitian and PT-symmetric, where the parity operator P acts on the position operator as PxP −1= lnq 0 λ −x . The discrete energy eigenvalues are shown to be real when V 2⩾− q 0λ 2 4 while they are complex conjugate pairs if V 2<− q 0λ 2 4 .

PT symmetry and shape invariance for a potential well with a barrier

We construct an exponential-type PT-symmetric potential, which includes the PT-symmetric versions of the Rosen–Morse well and Scarf potential and the complex PT-invariant potential well V( x)= q 2tanh 2 αx+ i( q 1/2)sech αxtanh αx+ q 0, q 2>0. The discrete energy eigenvalues of the latter complex potential are shown to be real when | q 1|⩽ α 2/2+2 q 2, while they are complex conjugate pairs if | q 1|> α 2/2+2 q 2. The PT symmetry is unbroken in the former case and spontaneously broken in the latter case.

PT-symmetric sextic potentials

The family of complex PT-symmetric sextic potentials is studied to show that for various cases the system is essentially quasi-solvable and possesses real, discrete energy eigenvalues. For a particular choice of parameters, we find that under supersymmetric transformations the underlying potential picks up a reflectionless part.

Observation of shell structure in sodium nanowires

The fact that the rare gas atoms at the end of each row in the periodic table of the elements are exceptionally stable is explained by the closed-shell configuration of their electronic structure. In general, the quantum states of a system of particles in a finite spacial domain form a set of discrete energy eigenvalues, which are usually grouped into bunches of degenerate or close-lying levels, called shells [1]. For fermions this gives rise to a local minimum in the total energy of the system when all states of a given shell are occupied. Shell effects have previously been observed for protons and neutrons in nuclei and for clusters of metal atoms [2], [3], [4]. Here, we report the first observation of a shell effect in an open system, a metal nanowire. When recording the statistical distribution of conductance values observed while pulling off the contact between two bulk sodium metal electrodes, the histogram shows oscillations up to contacts larger than 100 atoms in cross section. The period follows the law expected for the electronic shell-closing effect similar to that determining the “magic numbers” in metal clusters [3], [4]

Smooth transformations of Kratzer’s potential in N dimensions

We study smooth transformations V(r)=g(−1/r)+f(1/r2) of Kratzer’s potential −a/r+b/r2 in N⩾2 spatial dimensions. Eigenvalue approximation formulas are obtained which provide lower or upper energy bounds for all the discrete energy eigenvalues Enl and all N⩾2, corresponding, respectively, to the two cases that the transformation functions g and f are either both convex (g″⩾0) and f″⩾0) or both concave (g″⩽0 and f″⩽0). Detailed results are presented for V(r)=−a/r+b/rβ and V(r)=−(v/r)[1−ar/(1+r)]+b/r2.

Eigenvalue bounds for a class of singular potentials

We study smooth transformations of the solvable potentials . Eigenvalue approximation formulae are obtained which provide lower or upper energy bounds for all the discrete energy eigenvalues , , accordingly as the transformation functions g and f are both convex or both concave. Detailed results are presented for the special case of two-term singular potentials of the form , and also for the potentials and , , , for .

Application of Prony’s inversion method in determining the lifetime of a resonance

Certain multi-channel (or Feshbach) resonances are associated with systems with internal degree(s) of freedom. For these systems, in general, the probability of decay is not a simple exponential function, rather it involves the sum of a number of terms each with oscillatory as well as exponential dependence on time. It is shown how from the observed time-dependence of the decay probability one can, using Prony’s method, obtain the number of quasi-stationary states, the imaginary part of the complex discrete energy eigenvalues and the level spacings. These quantities enable one to determine, in an approximate way, some properties of the interaction.

Level curvatures and many-spin quantum systems

For quantum systems depending on several parameters, we define the tensor of level curvatures as the second derivatives of the discrete energy eigenvalues with respect to the parameters: ∂ 2E n ∂x ξ∂x n . For quantum systems with irregular spectra of random-matrix type, we show that both the diagonal and the off-diagonal components of the tensor of level curvatures obey a similar statistical distribution with a long tail of universal character within the orthogonal, the unitary, and the symplectic ensembles. The curvature distribution is numerically calculated for a finite and planar square cluster of spins in dipolar interaction like in nuclear magnetism. The tail of the curvature distribution is shown to agree with the theoretical prediction. Moreover, we give a partial list of possible quantities which could behave like level curvatures.

Equivalent Hamiltonians with additional discrete states.

Given a particular Hamiltonian {ital H}, we present a method to generate a new Hamiltonian {ital {tilde H}}, which has the same discrete energy eigenvalues and the same continuum phase shifts as {ital H}, but which also has additional given discrete eigenstates. This method is used to generate a Hamiltonian {ital h}{sub 1}, which gives rise to a complete orthonormal set of basis states, which contain a given set of biorthonormal discrete states, the continuum states of which are asymptotic to plane waves (have zero phase shifts). Such a set of states may be helpful in representing the medium modification of the Green's function due to the Pauli principle, as well as including Pauli exclusion effects into scattering calculations.

Continuum and discrete s states for the exponential cosine screened coulomb potential by the Ecker-Weizel approximation

We show that the Ecker-Weizel approximation corresponds to approximating the exponential cosine screened Coulomb potential by a Hulthen potential with an energy dependent strength parameter and use this fact to obtain the S-matrix, discrete energies and wave functions for the s states. The discrete energy eigenvalues are in excellent agreement with those of variational calculations.