Application of group symmetry to spectral problems of quantum mechanics

1988 ◽  
Vol 31 (5) ◽  
pp. 420-423
Author(s):  
V. G. Bagrov ◽  
A. S. Vshivtsev ◽  
V. N. Chekalin
Keyword(s):  
Quantum Mechanics ◽  
Group Symmetry ◽  
Spectral Problems

scholarly journals The intersection between dual potential and sl(2) algebraic spectral problems

2020 ◽  
Vol 35 (32) ◽  
pp. 2050208
Author(s):  
William H. Pannell
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Transformation Formula ◽  
Algebraic Construction ◽  
Spectral Problems ◽  
Partner Potential

The relation between certain Hamiltonians, known as dual, or partner Hamiltonians, under the transformation [Formula: see text] has long been used as a method of simplifying spectral problems in quantum mechanics. This paper seeks to examine this further by expressing such Hamiltonians in terms of the generators of sl(2) algebra, which provides another method of solving spectral problems. It appears that doing so greatly restricts the set of allowable potentials, with the only nontrivial potentials allowed being the Coulomb [Formula: see text] potential and the harmonic oscillator [Formula: see text] potential, for both of which the sl(2) expression is already known. It also appears that, by utilizing both the partner potential transformation and the formalism of the Lie-algebraic construction of quantum mechanics, it may be possible to construct part of a Hamiltonian’s spectrum from the quasi-solvability of its partner Hamiltonian.

scholarly journals On the Relativistic Quantum Mechanics of a Particle in Space with Minimal Length

2012 ◽  
Vol 57 (9) ◽  
pp. 942
Author(s):  
Ch.M. Scherbakov
Keyword(s):  
Quantum Mechanics ◽  
Commutation Relation ◽  
Relativistic Quantum ◽  
Heisenberg Algebra ◽  
Minimal Length ◽  
Group Symmetry ◽  
Relativistic Quantum Mechanics ◽  
Noncommutative Space ◽  
Quantum Mechanical

A noncommutative space and the deformed Heisenberg algebra [X,P] = iħ{1 – βP2}1/2 are investigated. The quantum mechanical structures underlying this commutation relation are studied. The rotational group symmetry is discussed in detail.

scholarly journals TBA equations and quantization conditions

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Yoan Emery
Keyword(s):  
Quantum Mechanics ◽  
Quantum Corrections ◽  
Wkb Method ◽  
Spectral Problems ◽  
Wall Crossing ◽  
Polynomial Potentials

Abstract It has been recently realized that, in the case of polynomial potentials, the exact WKB method can be reformulated in terms of a system of TBA equations. In this paper we study this method in various examples. We develop a graphical procedure due to Toledo, which provides a fast and simple way to study the wall-crossing behavior of the TBA equations. When complemented with exact quantization conditions, the TBA equations can be used to solve spectral problems exactly in Quantum Mechanics. We compute the quantum corrections to the all-order WKB periods in many examples, as well as the exact spectrum for many potentials. In particular, we show how this method can be used to determine resonances in unbounded potentials.

scholarly journals Spectral Problems for Quasinormal Modes of Black Holes

Universe ◽  
2021 ◽  
Vol 7 (12) ◽  
pp. 476
Author(s):  
Yasuyuki Hatsuda ◽  
Masashi Kimura
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Perturbation Theory ◽  
Quantum Mechanics ◽  
Quasinormal Modes ◽  
Review Article ◽  
Wkb Method ◽  
Spectral Problems ◽  
Points Of View ◽  
Numerical Treatments

This is an unconventional review article on spectral problems in black hole perturbation theory. Our purpose is to explain how to apply various known techniques in quantum mechanics to such spectral problems. The article includes analytical/numerical treatments, semiclassical perturbation theory, the (uniform) WKB method and useful mathematical tools: Borel summations, Padé approximants, and so forth. The article is not comprehensive, but rather looks into a few examples from various points of view. The techniques in this article are widely applicable to many other examples.

scholarly journals Uncertainty relation in quantum mechanics with quantum group symmetry

1994 ◽  
Vol 35 (9) ◽  
pp. 4483-4496 ◽  
Author(s):  
Achim Kempf
Keyword(s):  
Quantum Mechanics ◽  
Quantum Group ◽  
Uncertainty Relation ◽  
Group Symmetry

Artefact in 20Å-Resolution Electron Images of Negatively Stained Twodimensional Membrane Protein Crystals

1982 ◽  
Vol 40 ◽  
pp. 92-93
Author(s):  
Douglas L. Dorset ◽  
Barbara Moss
Keyword(s):  
Electron Scattering ◽  
Cross Correlation ◽  
Transmission Function ◽  
Group Symmetry ◽  
Asymmetric Structure ◽  
Object Model ◽  
Phase Object ◽  
Computing Systems ◽  
Resolution Electron ◽  
Plane Group

A number of computing systems devoted to the averaging of electron images of two-dimensional macromolecular crystalline arrays have facilitated the visualization of negatively-stained biological structures. Either by simulation of optical filtering techniques or, in more refined treatments, by cross-correlation averaging, an idealized representation of the repeating asymmetric structure unit is constructed, eliminating image distortions due to radiation damage, stain irregularities and, in the latter approach, imperfections and distortions in the unit cell repeat. In these analyses it is generally assumed that the electron scattering from the thin negativelystained object is well-approximated by a phase object model. Even when absorption effects are considered (i.e. “amplitude contrast“), the expansion of the transmission function, q(x,y)=exp (iσɸ (x,y)), does not exceed the first (kinematical) term. Furthermore, in reconstruction of electron images, kinematical phases are applied to diffraction amplitudes and obey the constraints of the plane group symmetry.

Structure of stacking faults in pyrite using TEM techniques

1992 ◽  
Vol 50 (1) ◽  
pp. 358-359
Author(s):  
Raja Subramanian ◽  
Kenneth S. Vecchio
Keyword(s):  
Lattice Structure ◽  
Stacking Faults ◽  
Covalent Bond ◽  
Group Symmetry ◽  
Iron Pyrite ◽  
Dislocation Glide ◽  
Partial Dislocations ◽  
Transmission Electron ◽  
Contrast Analysis ◽  
Chlorine Ions

The structure of stacking faults and partial dislocations in iron pyrite (FeS2) have been studied using transmission electron microscopy. Pyrite has the NaCl structure in which the sodium ions are replaced by iron and chlorine ions by covalently-bonded pairs of sulfur ions. These sulfur pairs are oriented along the <111> direction. This covalent bond between sulfur atoms is the strongest bond in pyrite with Pa3 space group symmetry. These sulfur pairs are believed to move as a whole during dislocation glide. The lattice structure across these stacking faults is of interest as the presence of these stacking faults has been preliminarily linked to a higher sulfur reactivity in pyrite. Conventional TEM contrast analysis and high resolution lattice imaging of the faulted area in the TEM specimen has been carried out.

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