<i>Applications of Group Theory in Quantum Mechanics</i>

Petrashen, M. I.; Trifonov, E. D.; Lichtenberg, D. B.

doi:10.1063/1.3022793

Cited by 24 publications

(39 citation statements)

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“…The Jahn-Teller distortion of a finite system necessarily results in its shape change, usually accompanied by a change of shape symmetry. The application of the Jahn-Teller effect to describing the shape and symmetry changes in molecules has been expounded in many textbooks, for instance, in [87,88]. And only later the Jahn-Teller effect started being considered for infinite crystals.…”

Section: Geometric Shape Transitionsmentioning

confidence: 99%

Effects of symmetry breaking in finite quantum systems

2013

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The review considers the peculiarities of symmetry breaking and symmetry transformations and the related physical effects in finite quantum systems. Some types of symmetry in finite systems can be broken only asymptotically. However, with a sufficiently large number of particles, crossover transitions become sharp, so that symmetry breaking happens similarly to that in macroscopic systems. This concerns, in particular, global gauge symmetry breaking, related to Bose-Einstein condensation and superconductivity, or isotropy breaking, related to the generation of quantum vortices, and the stratification in multicomponent mixtures. A special type of symmetry transformation, characteristic only for finite systems, is the change of shape symmetry. These phenomena are illustrated by the examples of several typical mesoscopic systems, such as trapped atoms, quantum dots, atomic nuclei, and metallic grains. The specific features of the review are: (i) the emphasis on the peculiarities of the symmetry breaking in finite mesoscopic systems; (ii) the analysis of common properties of physically different finite quantum systems; (iii) the manifestations of symmetry breaking in the spectra of collective excitations in finite quantum systems. The analysis of these features allows for the better understanding of the intimate relation between the type of symmetry and other physical properties of quantum systems. This also makes it possible to predict new effects by employing the analogies between finite quantum systems of different physical nature.

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Section: Geometric Shape Transitionsmentioning

confidence: 99%

Effects of symmetry breaking in finite quantum systems

2013

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“…Indeed, let T be the matrix simultaneously D'(g) = TD(g) T-1. Therefore, by Wigner's theorem [27] one obtains that B' has also quasidiagonal form, i.e. the general set (38) is reduced to a number of independent sets of equations of essentially lower order.…”

Section: Search For Dipole Momentsmentioning

confidence: 98%

Multipole Theory of the Polarization of Solids by Point Defects. I. Dipole Approximation

Kantorovich¹

1983

Physica Status Solidi (b)

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The electronic structure of solids is investigated where the electronic density is localized on structure elements (atoms, molecules, bonds). For these solids containing point defects the Schrödinger equation is obtained in the framework of a molecular cluster model taking self‐consistently into account the polarization of the rest of the solid outside the cluster. The polarization is considered in terms of multipole moments on structure elements induced by the defects neglecting exchange interactions. In the first part of the paper the general theory developed is applied to ionic crystals in the dipole approximation. The way to use the cluster symmetry in polarization calculations is also analyzed.

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“…Let us formulate our quantum problem in repre sentation (6), where the φ(t) classical oscillator phase is selected as α, (8) Let us write the Schrödinger equation for the Φ(x, t) function, or, according to (7),…”

Section: Problem Statementmentioning

confidence: 99%

Quantum-mechanical oscillator in a uniform force field

Trifonov

2012

Russ. J. Phys. Chem. B

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The symmetry group of an isotropic oscillator explaining additional degeneracy of its energy levels was for the first time considered by Demkov [1-3]. He directed attention to the fact that harmonic oscillator Hamiltonian symmetry was beyond the scope of usual geometric concepts and could be expressed using canonical transformations relating coordinates and momenta. We show in this paper how the symmetry prop erties of the harmonic oscillator Hamiltonian can be used to solve the dynamic problem of an oscillator in a uniform force field.

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Applications of Group Theory in Quantum Mechanics

Cited by 24 publications

References 0 publications

Effects of symmetry breaking in finite quantum systems

Effects of symmetry breaking in finite quantum systems

Multipole Theory of the Polarization of Solids by Point Defects. I. Dipole Approximation

Quantum-mechanical oscillator in a uniform force field

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