The Zeeman Effect in Triplet States of Rotating Asymmetric Molecules

1975 ◽  
Vol 53 (19) ◽  
pp. 1853-1860 ◽  
Author(s):  
J. C. D. Brand ◽  
C. di Lauro ◽  
D. S. Liu
Keyword(s):  
Magnetic Field ◽  
Field Theory ◽  
Energy Levels ◽  
Zeeman Effect ◽  
Homogeneous Magnetic Field ◽  
Triplet States ◽  
Matrix Elements ◽  
The Matrix ◽  
Asymmetric Rotor ◽  
Magnetic Tuning

Intermediate field theory is used to obtain the matrix elements which determine the action of a homogeneous magnetic field on the energy levels of triplet states of asymmetric rotor molecules. Applications of these formulas are discussed (i) in relation to the Zeeman effect on the rotational fine structure of triplet–singlet transitions, where conditions are identified under which individual lines remain unbroadened by the field, and (ii) in connection with the magnetic tuning of singlet–triplet resonance.

Die Übertragung der LCAO-Interpolationsmethode auf Gitterelektronen im Magnetfeld / The LCAO Interpolation Method Applied to Crystal Electrons in a Magnetic Field

1975 ◽  
Vol 30 (11) ◽  
pp. 1378-1384 ◽  
Author(s):  
H. G. Müller
Keyword(s):  
Magnetic Field ◽  
Interpolation Method ◽  
Energy Levels ◽  
Recursion Formula ◽  
Point Group ◽  
Energy Structure ◽  
Matrix Elements ◽  
Linear Difference Equations ◽  
The Matrix ◽  
Crystal Electron

The splitting of the energy levels of crystal electrons in homogeneous magnetic fields of moderate strength is described in terms of the matrix elements of the LCAO method. The matrix elements are regarded as adjustable parameters connected with the energy structure of the crystal electron without magnetic field. The translational symmetry of the crystal in a magnetic field is exactly taken into account, and the matrix elements contain the symmetry of the point group of the material.The resulting eigenvalue equation is a system of linear difference equations. In analogy to the treatment of a similar one-dimensional difference equation for the differential equation of Mathieu type the eigenvalues result from an investigation of a vector recursion formula. In connection with this method the limiting cases magnetic field and crystal potential approaching zero are discussed in short

STUDY OF A PROPOSAL FOR DETERMINING THE g-FACTOR ANOMALY FOR ELECTRONS BY RESONANCE EXCITATION IN A MAGNETIC FIELD

1963 ◽  
Vol 41 (10) ◽  
pp. 1571-1579 ◽  
Author(s):  
J. Byrne
Keyword(s):  
Magnetic Field ◽  
Electromagnetic Field ◽  
Experimental Determination ◽  
Homogeneous Magnetic Field ◽  
Resonance Excitation ◽  
Matrix Elements ◽  
G Factor ◽  
The Matrix

Following recent proposals for the experimental determination of the electron g-factor anomaly, a study has been made of the properties required of a perturbing electromagnetic field for the production of particular transitions between the states of an electron moving in a constant homogeneous magnetic field. The matrix elements have been calculated for the transitions of interest and some suggestions have been advanced for the solution of certain experimental problems indicated by the results of the study.

Matrix Representation for Calculation of the Second-Order Aberration in Ion Optics

1966 ◽  
Vol 21 (1-2) ◽  
pp. 9-14
Author(s):  
Isao Takeshita
Keyword(s):  
Magnetic Field ◽  
Matrix Method ◽  
Arbitrary Shape ◽  
Matrix Representation ◽  
Homogeneous Magnetic Field ◽  
Second Order ◽  
Matrix Elements ◽  
Ion Trajectory ◽  
Ion Optics ◽  
The Matrix

The matrix representation for calculating the ion trajectory is extended to include the secondorder aberration terms.Properties of the matrix and relations between its elements are discussed.As an example of the application of the matrix method, the second-order coefficients of a homogeneous magnetic field of arbitrary shape are derived and the relations between the matrix elements and Hinterberger's notations are shown.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

Klein–Gordon oscillator with position-dependent mass in the rotating cosmic string spacetime

2018 ◽  
Vol 33 (04) ◽  
pp. 1850025 ◽  
Author(s):  
Bing-Qian Wang ◽  
Zheng-Wen Long ◽  
Chao-Yun Long ◽  
Shu-Rui Wu
Keyword(s):  
Magnetic Field ◽  
Cosmic String ◽  
Energy Levels ◽  
Homogeneous Magnetic Field ◽  
Spinless Particle ◽  
Klein Gordon ◽  
Coulomb Type ◽  
Dependent Mass ◽  
String Background ◽  
Position Dependent Mass

A spinless particle coupled covariantly to a uniform magnetic field parallel to the string in the background of the rotating cosmic string is studied. The energy levels of the electrically charged particle subject to the Klein–Gordon oscillator are analyzed. Afterwards, we consider the case of the position-dependent mass and show how these energy levels depend on the parameters in the problem. Remarkably, it shows that for the special case, the Klein–Gordon oscillator coupled covariantly to a homogeneous magnetic field with the position-dependent mass in the rotating cosmic string background has the similar behaviors to the Klein–Gordon equation with a Coulomb-type configuration in a rotating cosmic string background in the presence of an external magnetic field.

Causal amplitudes and the Yang-Feldman formalism

1957 ◽  
Vol 53 (4) ◽  
pp. 843-847 ◽  
Author(s):  
J. C. Polkinghorne
Keyword(s):  
Field Theory ◽  
Green’S Functions ◽  
Free Field ◽  
Dispersion Relations ◽  
Matrix Elements ◽  
Field Equations ◽  
Commutation Relations ◽  
The Matrix ◽  
Free Fields ◽  
Set Up

ABSTRACTThe Yang-Feldman formalism vising the Feynman-like Green's functions is set up. The corresponding free fields have non-trivial commutation relations and contain information about the scattering. S-matrix elements are simply the matrix elements of anti-normal products of the field φF′(x). These are evaluated, and they give directly expressions used in the theory of causality and dispersion relations. It is possible to formulate field theory in a form in which the fields obey free field equations and the effects of interaction are contained in their commutation relations.

scholarly journals Interação quadrupolar e sua correlação com o efeito Mössbauer

1982 ◽  
Vol 4 (4) ◽  
pp. 01
Author(s):  
Sylvestre Schneider
Keyword(s):  
Moment Tensor ◽  
Energy Levels ◽  
Multipole Expansion ◽  
Multipole Moment ◽  
Quadrupolar Interaction ◽  
Matrix Elements ◽  
Multipole Moments ◽  
Classical Charge ◽  
The Matrix ◽  
Mechanical Expression

By means multipole expansion, we guess the concept of "multipole moment". They are given examples of multipole moments. We start of them to define the "quadruploe moment tensor". By starting from the classical chargedents distribution of loads ρ (r-> ') and from the potencial expanded in a Taylor series, in which one one of the terms allows us to see the quadrupole moment, we construct a classical Hamiltonian in terms of quadrupole. The quantum-mechanical expression ĤQ por HQ is given, by substitution of the classical charge density ρ (r->) by an operator ρ (op), which describes adequately the real configuration in a non-continuous charge distribution. By use of the Clebsh-Gordan tecnology-coefficients with the irredutible tensors, the matrix-elements of ĤQ was calculated. The relation develloped, which allows the calcule of the matrix-elements of ĤQ, this is used for an application to a specific case, of a strong mafnetic field exerted on an atom. It is obtained a particular relation, for calculating the energy levels of quadrupolar interaction. The sequente purpose was to work out an application for an atom or ion in the fundamental staton 2S1/2, and nuclear spin 3/2 in a strong magnetic field.The results was a splitting in energy levels for the quadrupolar interaction. The quadrupolar interaction was examined in correlation of the Mössbauer-Effect in different examples.

scholarly journals Hyperfine structure of muonic mesomolecules tdµ, tpµ, dpµ in variational method

2019 ◽  
Vol 222 ◽  
pp. 03011
Author(s):  
A.V. Eskin ◽  
V.I. Korobov ◽  
A.P. Martynenko ◽  
V.V. Sorokin
Keyword(s):  
Variational Method ◽  
Hyperfine Structure ◽  
Vacuum Polarization ◽  
Energy Levels ◽  
Computer Code ◽  
Wave Functions ◽  
Matrix Elements ◽  
Gaussian Form ◽  
Stochastic Variational Method ◽  
The Matrix

The hyperfine structure of energy levels of muonic molecules tdµ, tpµ and dpµ is calculated on the basis of stochastic variational method. The basis wave functions are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. Vacuum polarization, relativistic and nuclear structure corrections are taken into account to increase the accuracy. For numerical calculation, a computer code is written in the MATLAB system. Numerical values of energy levels of hyperfine structure in muonic molecules tdµ, tpµ and dpµ are obtained.

scholarly journals Energy levels in muonic helium

2019 ◽  
Vol 222 ◽  
pp. 03009
Author(s):  
A.V. Eskin ◽  
V.I. Korobov ◽  
A.P. Martynenko ◽  
V.V. Sorokin
Keyword(s):  
Hyperfine Structure ◽  
Bound States ◽  
Energy Levels ◽  
Computer Code ◽  
Bound State ◽  
Relativistic Correction ◽  
Matrix Elements ◽  
Gaussian Form ◽  
Stochastic Variational Method ◽  
The Matrix

The energy spectrum of bound states and hyperfine structure of muonic helium is calculated on the basis of stochastic variational method. The basis wave functions of muonic helium are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. For numerical calculation a computer code is written in the MATLAB system. As a result, numerical values of bound state energies and hyperfine structure are obtained. We calculate also correction to the structure of the nucleus, vacuum polarization and relativistic correction.

The magnetic rotation spectrum of formaldehyde: singlet–triplet perturbations in the 2141, 2143, 2243, and 2341 levels of the à lA2 state of H2CO

1983 ◽  
Vol 61 (1) ◽  
pp. 6-14 ◽  
Author(s):  
C. M. L. Kerr ◽  
D. C. Moule ◽  
D. A. Ramsay
Keyword(s):  
Magnetic Activity ◽  
Triplet States ◽  
Magnetic Rotation ◽  
Matrix Elements ◽  
Molecular Constants ◽  
Absorption Bands ◽  
Vibrational Assignments ◽  
The Matrix ◽  
Upper Level ◽  
Rotation Spectrum

The absorption spectra and magnetic rotation spectra of the [Formula: see text],[Formula: see text],[Formula: see text], and [Formula: see text] bands of the [Formula: see text] system of formaldehyde have been studied at high resolution. The earlier analyses of the absorption bands have been refined and improved molecular constants obtained. Several singlet–singlet perturbations have been noted and possible mechanisms discussed.The magnetic activity falls into two classes. Three of the bands show limited singlet–singlet activity with characteristics similar to those found earlier for the [Formula: see text] and [Formula: see text] bands. All the bands show singlet–triplet activity which is characterized by the appearance of rR, rP, pR, and PP transitions associated with a common upper level. Tables of perturbed excited-state singlet levels are given. No appreciable shifts of the perturbed levels have been found indicating that the matrix elements are small. For the [Formula: see text] band two perturbation mechanisms are proposed and the vibrational symmetries and energies of the perturbing triplet states are determined. The energies are too high to permit unambiguous vibrational assignments.

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