New analytical expression for the rotational factor in Raman transitions

1995 ◽  
Vol 73 (9-10) ◽  
pp. 559-565 ◽  
Author(s):  
M. Korek ◽  
H. Kobeissi
Keyword(s):  
Direct Calculation ◽  
Matrix Elements ◽  
Numerical Application ◽  
Raman Transitions ◽  
The Matrix ◽  
Present Formalism ◽  
Vibration Wave ◽  
Schrödinger Perturbation ◽  
Analytical Expressions ◽  
Rotational Factor

The matrix elements of the polarizability anisotropy γ in the Raman spectra of diatomic molecules are investigated. These matrix elements are given by [Formula: see text] where Gνν′(m) is the rotational factor with m = [(J′(J′ + 1) − J(J + 1)]/2 and J′ − J = ±2. By using a nonconventional approach to the Rayleigh–Schrödinger perturbation theory the rotational factor can be written as Gνν′(m) = A0 + A1m + A2m2 where the coefficients A0, A1, and A2 are given by simple analytical expressions in terms of the integrals [Formula: see text] and [Formula: see text] where Y stands for Ψ(0) (the pure vibration wave function), or Ψ(0) (the first rotational perturbative correction to Ψ(0), or Ψ(2) (the second correction). A numerical application is presented for the ground states of CO and H2 molecules. A comparison with a numerical and direct calculation of the rotational factor Gνν′(m) shows the accuracy of the present formalism.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

ALGEBRAIC APPROACH TO THE PSEUDOHARMONIC OSCILLATOR IN 2D

2002 ◽  
Vol 11 (02) ◽  
pp. 155-160 ◽  
Author(s):  
SHI-HAI DONG ◽  
ZHONG-QI MA
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Algebraic Approach ◽  
Matrix Elements ◽  
Ladder Operators ◽  
Commutation Relations ◽  
The Matrix ◽  
Pseudoharmonic Oscillator ◽  
Analytical Expressions

A realization of the ladder operators for the solutions to the Schrödinger equation with a pseudoharmonic oscillator in 2D is presented. It is shown that those operators satisfy the commutation relations of an SU(1, 1) algebra. Closed analytical expressions are evaluated for the matrix elements of some operators r2 and r∂/∂ r

A REALIZATION OF DYNAMIC GROUP FOR AN ELECTRON IN A UNIFORM MAGNETIC FIELD

2002 ◽  
Vol 11 (04) ◽  
pp. 265-271 ◽  
Author(s):  
SHISHAN DONG ◽  
SHI-HAI DONG
Keyword(s):  
Magnetic Field ◽  
Relativistic Electron ◽  
Uniform Magnetic Field ◽  
Wave Functions ◽  
Matrix Elements ◽  
Eigenvalues And Eigenfunctions ◽  
Radial Wave ◽  
Creation And Annihilation Operators ◽  
The Matrix ◽  
Analytical Expressions

The eigenvalues and eigenfunctions of the Schrödinger equation with a non-relativistic electron in a uniform magnetic field are presented. A realization of the creation and annihilation operators for the radial wave-functions is carried out. It is shown that these operators satisfy the commutation relations of an SU(1,1) group. Closed analytical expressions are evaluated for the matrix elements of different functions ρ2 and [Formula: see text].

scholarly journals SPIN-POLARIZED PHOTOCURRENT THROUGH QUANTUM DOT PHOTODETECTOR

2017 ◽  
Vol 24 (1&2) ◽  
pp. 7-13
Author(s):  
Nguyen Van Hieu ◽  
Nguyen Bich Ha
Keyword(s):  
Quantum Dot ◽  
Electron Tunneling ◽  
Electronic Transitions ◽  
Matrix Elements ◽  
Semiconductor Quantum Dot ◽  
Spin Polarized ◽  
The Matrix ◽  
Electron Photon ◽  
Analytical Expressions ◽  
Polarized Photons

The theory of the photocurrent through the photodetector based on a two-level semiconductor quantum dot (QD) is presented. The analytical expressions of the matrix elements of the electronic transitions generated by the absorption of the circularly polarized photons are derived in the lowest order of the perturbation theory with respect to the electron tunneling interaction as well as the electron-photon interaction. From these expressions the mechanism of the generation of the spin-polarized of electrons in the photocurrent is evident. It follows that the photodetector based on the two-level semiconductor QD can be used as the model of a source of highly spinpolarized electrons.

THE HIDDEN SYMMETRY FOR A QUANTUM SYSTEM WITH A PÖSCHL–TELLER-LIKE POTENTIAL

2003 ◽  
Vol 12 (06) ◽  
pp. 809-815 ◽  
Author(s):  
SHI-HAI DONG ◽  
GUO-HUA SUN ◽  
YU TANG
Keyword(s):  
Quantum System ◽  
Wave Functions ◽  
Matrix Elements ◽  
Eigenvalues And Eigenfunctions ◽  
Creation And Annihilation Operators ◽  
Commutation Relations ◽  
Hidden Symmetry ◽  
The Matrix ◽  
The Creation ◽  
Analytical Expressions

The eigenvalues and eigenfunctions of the Schrödinger equation with a Pöschl–Teller (PT)-like potential are presented. A realization of the creation and annihilation operators for the wave functions is carried out. It is shown that these operators satisfy the commutation relations of an SU(1,1) group. Closed analytical expressions are evaluated for the matrix elements of different functions, sin (ρ) and [Formula: see text] with ρ=πx/L.

Développement complet de l'hamiltonien de vibration–rotation adapté à l'étude des interactions dans les molécules toupies sphériques. Application aux bandes ν2 et ν4 de 12CH4

1977 ◽  
Vol 55 (20) ◽  
pp. 1802-1828 ◽  
Author(s):  
Jean-Paul Champion
Keyword(s):  
Ground State ◽  
Coupling Scheme ◽  
Order Of Approximation ◽  
Matrix Elements ◽  
Coupling Coefficients ◽  
Linear Combinations ◽  
Coriolis Interaction ◽  
Raman Transitions ◽  
The Matrix ◽  
Vibration Rotation

Using an unsymmetrized coupling scheme in the group Td, we determine all the vibration–rotation operators of the Hamiltonian of XY4 molecules, including all possible interactions, up to any order of approximation. We define a basis of Hamiltonian matrices of which the matrix elements are functions of coupling coefficients of the group chain [Formula: see text] only. Thus, we develop a general formalism available for any vibrational sublevels of whatever symmetry. The parameters relative to the different vibrational sublevels are known linear combinations of the coefficients of the Hamiltonian expansion. From these, we deduce simple relations between the parameters associated with the ground state, the fundamentals, and the harmonic and combination bands.We apply this formalism to the study of the Coriolis interaction between ν2 and ν4 of CH4. With only 21 parameters for the two bands, we obtain a standard deviation of 34 mK for 251 Raman transitions of ν2 and 20 mK for 243 ir transitions of ν4.

The SU(2) realization for the Morse potential and its coherent states

2002 ◽  
Vol 80 (2) ◽  
pp. 129-139 ◽  
Author(s):  
S -H Dong
Keyword(s):  
Coherent States ◽  
Morse Potential ◽  
Transition Probability ◽  
Matrix Elements ◽  
Commutation Relations ◽  
Raising And Lowering Operators ◽  
The Matrix ◽  
Harmonic Perturbation ◽  
Analytical Expressions ◽  
Lowering Operators

A realization of the raising and lowering operators for the Morse potential is presented. We show that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expressions are derived for the matrix elements of different functions such as 1/y and d/dy. The harmonic limit of the SU(2) operators is also studied. The transition probability between two eigenstates produced by a harmonic perturbation as a function of the operators [Formula: see text]±,0 is discussed. The average values of some observables in the coherent states |α > for the Morse potential are also calculated. PACS Nos.: 02.30+b, 03.65Fd, 42.50Ar, and 33.10Cs

"Nonintegral" expressions for the diatomic centrifugal distortion constants

1995 ◽  
Vol 73 (5-6) ◽  
pp. 339-343
Author(s):  
Hafez Kobeissi ◽  
Chafia H. Trad
Keyword(s):  
Model Potential ◽  
Centrifugal Distortion ◽  
Numerical Application ◽  
Lennard Jones ◽  
Particular Solution ◽  
New Formulation ◽  
Centrifugal Distortion Constants ◽  
Schrödinger Perturbation ◽  
Analytical Expressions ◽  
Nonhomogeneous Equation

The problem of the centrifugal distortion constants (CDC), Dν, Hν, … for a diatomic molecule is considered. It is shown that a new formulation of the standard Rayleigh–Schrödinger perturbation theory can give simple and compact analytical expressions of the CDC (up to any order). Thus, the constants e1 = Bν, e2 = −Dν, e3 = Hν,…, en,… are all of the form en = lim σn(r)/σ0(r) as r → ∞. σ0 is the particular solution of the nonhomogeneous equation y″ + k(Eν – U)y = s, with s = ψν, where (Eν, ψν) is the eigenvector corresponding to the rotationless potential U(r) and to the vibrational level ν; and where σ0(0) = σ′0(0) = 0. σn is the particular solution of the above equation, where s is known for each order of n. The numerical application to the standard Lennard–Jones model potential shows that good results are obtained for Dν, Hν, Lν,…,Oν, Pν, for ν = 0 to 22, which is only at 2 × 10−4 of the well depth. The program uses one routine (the integration of the equation y″ + fy = s) repeated for different s; it is quite simple and gives no difficulties at the boundaries and there is no need to use any mathematical or numerical artifices.

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.

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