Formulation Of Perturbation Theory Research Articles

Quadrupole perturbation of the harmonic oscillator potential

TheSU(3) symmetry of the harmonic oscillator potential is used to calculate exactly the second order energy correction, caused by quadrupole deformations. This can be done in a purely algebraic manner, using the Dalgarno-Schwartz formulation of perturbation theory. Some connections of this method with group theory are discussed and an extension to more general situations is proposed.

The calculation of isotopic partition function ratios by a perturbation theory technique

The vibrational Hamiltonian of a molecule in the harmonic approximation, H = (1/2) 𝒥 (gijpipj + fijqiqj), has been divided into a diagonal part (terms with i=j) and an off-diagonal part (i≠j), which is regarded as the perturbation. The vibrational partition function of the molecule is then calculated by Schwinger perturbation theory as the partition function of the unperturbed problem, corresponding to a collection of oscillators with frequencies 2πνi′ = (fiigii)1/2, plus perturbation correction terms which are calculated to second order. With the usual assumptions of isotope effect calculations that the molecular translations and rotations are classical and separable from the vibrations, the perturbation formulation of the vibrational partition function is easily transformed into a perturbation theory formulation of (reduced) isotopic partition function ratios. If, for example, the molecular potential function is expressed in terms of the displacements of bond stretches and bond angle bends from their respective equilibrium values, the unperturbed partition function ratio corresponds to the isotope effect expected for noninteracting bond-stretch and bond-angle-bend oscillators. Detailed comparison is made for a number of molecular systems of perturbation theory calculations of partition functions and isotopic partition function ratios with exact calculations carried out by actually obtaining the normal mode vibrational frequencies of the vibrational Hamiltonian. Good agreement is found. The utility of the perturbation theory formulation resides in the fact that it permits one to look at isotope effects in a very simple manner; some demonstrations are given.

Covariant perturbation theory and chiral superpropagators

We use a covariant formulation of perturbation theory for the non-linear chiral invariant pion model to define chiral superpropagators leading to S matrix elements which are independent of the choice of the pion field coordinates. The relation to the standard definition of chiral superpropagators is discussed.

A fluctuation-dissipation relation and its applications to thermodynamic fluctuations in superconducting cylinders

Modifying slightly Kubo's formulation of perturbation theory to take care of the fact that generalized susceptibility may not be zero at infinite frequency, we establish, in the classical limit, a general relation between covariance and generalized susceptibility. The relation is then applied to evaluate the covariances of the magnetic flux, currents, and magnetization of a superconducting cylinder. Expressions for the spectra of magnetic flux and magnetization are also obtained.

Quantum Theory of Transport in Narrow Channels

The electrical conductivity in a semiconductor surface channel or a thin film is written in terms of integrals over a retarded current-current correlation function and evaluated using a Green's-function formulation of perturbation theory. The perturbation theory exhibits four new features. (1) The boundary conditions at the surfaces of the channel are expressed in terms of a fluctuation potential rather than a Fuchs reflectivity parameter. (2) The quantization of the eigenvalues for motion normal to the channel is explicitly incorporated into the theory. (3) The averaging procedure used to obtain the diagrammatic definition of the propagators and correlation functions is extended to include the effects both of screening and of graded interface impurity doping by permitting summation of multiple-scattering effects within planes of impurities parallel to the surface prior to the consideration of interference between these planes. (4) The propagators and conductivity are evaluated at arbitrary temperatures, using the Matsubara formalism. The conductivity is calculated explicitly in the quantum limit that the energy spacings $\ensuremath{\Delta}E$ between the eigenvalues for motion normal to the surface satisfy $\ensuremath{\Delta}E\ensuremath{\gg}\mathrm{kT}$ for the occupied eigenstates. The approximations needed to reproduce the Boltzmann-equation analysis by Stern and Howard of the extreme quantum limit are delineated. The effects of dispersion and quantized-state mixing are examined for a $\ensuremath{\delta}$-function model of the fluctuation potential. They are found to be significant if a quantized eigenvalue is near the Fermi energy or if the doping in the channel is highly nonuniform.

Calculation of Natural Orbitals and Wavefunctions by Perturbation Theory

A computationally convenient formulation of perturbation theory is developed. In this formulation either the SCF wavefunction or the SCF function plus important corrections may be used as a trial function. Corrections to the trial function are expanded in powers of the residual error. The reduced density matrices may be expanded in a series in this error. If the SCF function is used as the trial function, parts of the density matrix are needed to second order in the error to determine the natural orbitals and geminals to zeroth order.

Molekularstrahlmessungen von Sto�querschnitten f�r �berg�nge zwischen definierten Rotationszust�nden zwei-atomiger Molek�le

Previously reported experimental results for inelastic cross sections for rotational excitation of TlF molecules in low-lying, well defined rotational states are interpreted in terms of a time dependent perturbation theory formulation of the high energy approximation. In order to calculate inelastic cross sections for the observed small angle scattering the Born approximation and the classical deflection function are shown to be applicable. In this approximation the ΔJ selection rules are characteristic of the individual terms in the expansion of the potential, whereas the ΔM selection rules depend on the orientation of the molecule with respect to the scattering trajectory. An approximation for dealing with a scattering gas consisting of molecules is introduced and the appropriate orientation averaging is carried out for the case of a generalized electrostatic potential. The measured results for the transition TlF(2.0)→ (3.0) in collisions with the rare gases and CH4 and SF6 are more than a factor three larger than calculated results for the induced dipole-quadrupole (α, μ, Q) interaction. Rough argeement is found between calculated results for a dipole-quadrupole interaction and the experimental results for the above mentioned transition produced by the scattering gases O2, N2 (air), N2O, and H2O. Finally, the dipole-dipole potential appears to provide an explanation for the large inelastic cross sections observed with NH3 and CF2Cl2. Calculated inelastic and total cross sections are however considerably larger (about a factor 2) than the measured results with NH3. Possible explanations are discussed.

Perturbation Theory and the Hellmann—Feynman Theorem

By using the Hellmann—Feynman theorem in conjunction with an operator formulation of perturbation theory it is shown that, while for a general Hamiltonian an energy derivative is determined only to order n by an nth-order wavefunction (Salem and Wilson), for a restricted class of systems it is determined to order n+1, and by restricting the class somewhat further one can obtain energy derivatives to order 2n+1. The example of the force between two atoms interacting through dispersion type forces is briefly discussed.

Sur la détermination de l'état fondamental d'un système de particules

The linked diagram expansions of the ground state energy and wave function of a system of particles are derived by a method involving the use of the time-dependent as well as the time-independent formulations of perturbation theory. The essential step is the derivation, in the time-dependent formulation, of a simple exponential formula obtained by summation of the disconnected vacuum-vacuum diagrams.