Quantum Mechanical Perturbation Theory Research Articles

Análise da convergência na Teoria da Perturbação Estacionária

Neste trabalho, apresentamos uma revisão da teoria de perturbação na Mecânica Quântica e mostramos que a série perturbativa de Rayleigh-Schrödinger não converge para o resultado exato obtido por integração numérica para o caso do oscilador anarmônico do tipo x4 e do efeito Stark no átomo de hidrogênio. Abordamos um método de renormalização da energia perturbativa, denominado aproximante de Padé, para contornar o problema da convergência das séries de potências. Como ilustração, tratamos o caso da teoria de perturbação de quarta ordem no oscilador anarmônico lambdax4 e verificamos a eficácia do aproximante de Padé em comparação à solução numérica.

Electronic structure of piezoelectric double-barrier InAs/InP/InAs/InP/InAs (111) nanowires

We present a theoretical study of an n-type InAs nanowire with built-in InAs/InP heterojunctions in the effective-mass approximation via self-consistent Poisson–Schrödinger calculations in cylindrical coordinates. Rapid convergence and efficiency are achieved by (i) a suitable transformation of the radial part of the Hamiltonian matrix thereby maintaining symmetry (ii) using quantum mechanical perturbation theory to derive an expression for the change in electron density with electrostatic potential. We calculate the energy levels in a 150 Å long InAs quantum dot surrounded by 50 Å long InP barriers within an InAs quantum wire of radius 200 Å, having a doping level of 3×1016 cm−3 and conduction-band discontinuities of ΔECB=0.6 eV. In equilibrium, the lowest quantum dot state is at 15 meV above the Fermi level and we find that upon variation of the applied collector–emitter voltage VCE, resonance occurs at VCE=88 mV. This is in good agreement with an experimental study of resonant tunneling in a nominally undoped InAs/InP nanowire of similar dimensions grown in the [111] direction, where resonance was detected at VCE=80 mV, and a small shift (<5 mV) in its position occurred upon inverting the voltage polarity. We rule out barrier asymmetry, bandbending due to impurities or defects, and contact effects as being the origin of the resonant-voltage shift, and attribute it to the strain-induced charges at the InP/InAs interfaces. Both InAs and InP segments are shown to be under in-plane compression giving a piezoelectric field of 0.155 meV/Å in the InAs quantum dot while resonant tunneling, as calculated, occurs at 84 mV for VCE<0 and at 87 mV for VCE>0. This is in contrast to two-dimensional pseudomorphic heteroepitaxy, where the InP is under in-plane tensile strain yielding a very strong resonance-voltage shift (≫5 mV). The small magnitude of the measured shift indicates that in nanowires any strain at the heterointerfaces relaxes within a few atomic layers.

Nonlocal dielectric functions on the nanoscale: Screened forces from unscreened potentials

In this work, we prove that an intramolecular dielectric model yields accurate results for the forces between nonoverlapping molecules, at first order in the intermolecular interaction. The analysis is valid within the Born–Oppenheimer approximation. Within any perturbed molecule, a nonlocal dielectric function εv−1(r,r′) describes the screening of external potentials due to the induced redistribution of electronic charge, i.e., this function acts as the integral kernel that determines the effective potential at point r (within linear response), when an external potential φex(r′) acts on the molecule, at other points r′. The dielectric function εv−1(r,r′) depends on the nonlocal charge-density susceptibility, which can be calculated ab initio or by density functional techniques. From quantum mechanical perturbation theory, at first order the interaction energy of two molecules is determined by the unscreened Coulomb interaction energy of the unperturbed molecular charge distributions. Yet the first-order forces on the nuclei include dielectric screening effects, due to the redistribution of the electronic charge density of each molecule in the presence of the other. This counterintuitive result follows from a relation between the charge-density susceptibility and the derivatives of the electronic charge density with respect to nuclear coordinates. The derivation provides a quantum mechanical validation for dielectric screening models on the nanoscale, when the dielectric function for electronic response is nonlocal.

Mean field and computer simulation study of a nematogenic lattice model including three-body interactions

By now, nematogenic lattice models have been extensively studied in the literature dealing with liquid crystals; they usually involve cylindrically symmetric (uniaxial) particles and pairwise additive interaction potentials. On the other hand, quantum mechanical perturbation theory shows that interatomic or intermolecular potentials are only approximately pairwise additive; the pairwise additivity approximation (PAA) of molecular interactions has been extensively and systematically used in the statistical mechanics of condensed matter, and has proven rather successful. As an attempt to move beyond the PAA in the simulation of mesogenic systems, we have considered here a nematogenic lattice model consisting of uniaxial particles, whose centres of mass are associated with a simple-cubic lattice, and whose interaction potentials consist of a pairwise additive dispersion (Nehring–Saupe) term, restricted to nearest neighbours (and already studied in the literature), plus a short-range triplet-additive one, i.e., the Kielich–Stogryn generalization of the Axilrod–Teller–Muto formula for three atoms. The model has been studied by Mean Field theory and Monte Carlo simulation; the three-body term was found to produce a recognizable quantitative effect on the nematic ordering transition.

Double perturbation theory: a powerful tool in computational coordination chemistry

Quantum mechanical double perturbation theory offers a route to access a large variety of important chemical and physical molecular properties. The properties that are considered here can be defined as the second derivative of the total energy of a molecule with respect to perturbation parameters arising from, e.g. frequency-dependent or -independent external electric and magnetic fields, nuclear and electronic magnetic moments, relativistic corrections, nuclear displacements, etc. The accessible properties cover, among others, the important fields of electronic and vibrational spectra, nuclear magnetic resonance (NMR), electron paramagnetic resonance (EPR) and optical activity. We will outline the methodology that gives access to these properties, and discuss a number of them in detail together with applications to transition metal compounds. All these properties are treated within the same theoretical framework in order to illustrate the similarities between them.

PP-wave string interactions from perturbative Yang-Mills theory

Recently, Berenstein et al. have proposed a duality between a sector of N=4 super-Yang-Mills theory with large R-charge J, and string theory in a pp-wave background. In the limit considered, the effective 't Hooft coupling has been argued to be lambda'=g_{YM}^2 N/J^2=1/(mu p^+ alpha')^2. We study Yang-Mills theory at small lambda' (large mu) with a view to reproducing string interactions. We demonstrate that the effective genus counting parameter of the Yang-Mills theory is g_2^2=J^4/N^2=(4 pi g_s)^2 (mu p^+ alpha')^4, the effective two-dimensional Newton constant for strings propagating on the pp-wave background. We identify g_2 sqrt{lambda'} as the effective coupling between a wide class of excited string states on the pp-wave background. We compute the anomalous dimensions of BMN operators at first order in g_2^2 and lambda' and interpret our result as the genus one mass renormalization of the corresponding string state. We postulate a relation between the three-string vertex function and the gauge theory three-point function and compare our proposal to string field theory. We utilize this proposal, together with quantum mechanical perturbation theory, to recompute the genus one energy shift of string states, and find precise agreement with our earlier computation.

시계열 섭동 모델링 알고리즘 : 운전자 프로그래밍과 양자역학 섭동이론의 통합

Genetic programming (GP) has been combined with quantum mechanical perturbation theory to make a new algorithm to construct mathematical models and perform predictions for chaotic time series from real world. Procedural similarities between time series modeling and perturbation theory to solve quantum mechanical wave equations are discussed, and the exemplary GP approach for implementing them is proposed. The approach is based on multiple populations and uses orthogonal functions for GP function set. GP is applied to original time series to get the first mathematical model. Numerical values of the model are subtracted from the original time series data to form a residual time series which is again subject to GP modeling procedure. The process is repeated until predetermined terminating conditions are met. The algorithm has been successfully applied to construct highly effective mathematical models for many real world chaotic time series. Comparisons with other methodologies and topics for further study are also introduced.

Quantum theory of the effect of post-collision interaction on the absorption of X-ray photons by a deep atomic shell

A fundamental problem for modern atomic physics, the theoretical description of the many-particle effects of post-collision interactions in the photoionization of deep atomic shells outside the framework of quantum-mechanical perturbation theory is considered. The analytical solution of the stationary many-particle Schrödinger equation for the non-relativistic wavefunction of the system of interacting continuous spectra of atomic states is obtained using a class of weight-functions of Cauchy–Lorentz type. States considered are those resulting from: (i) direct photoionization of a deep shell of a free atom and (ii) Auger-decay of a deep vacancy with two (photo- and Auger-) electrons in the continuous spectrum. The solution takes into account: (a) the rearrangement of the continuum photoelectron wavefunction in the changing field of the atomic residue, and (b) the electrostatic interaction of photo- and Auger-electrons resulting from the Auger-decay of the vacancy. The corresponding analytical structure of the photoionization cross section of the deep atomic shell is derived. A generalization is proposed for: (a) two or more channels of Auger-decay of the deep vacancy, and (b) direct photoexcitation of the deep atomic shell.

Stability Properties of Feynman's Operational Calculus for Exponential Functions of Noncommuting Operators

Stability properties of Feynman's operational calculus are addressed in the setting of exponential functions of noncommuting operators. Applications of some of the stability results are presented. In particular, the time-dependent perturbation theory of nonrelativistic quantum mechanics is presented in the setting of the operational calculus and application of the stability results of this paper to the perturbation theory are discussed.

Per‐Olov Löwdin curriculum vitae

Per‐Olov Löwdin curriculum vitae

Two-photon excitation and ionization of the 1s shell of the argon atom

The nonrelativistic absolute values and the form of the cross section for two-photon one-electron excitation and ionization of the 1s shell of the Ar atom are calculated in the second order of the quantum-mechanical perturbation theory by taking into account both the many-body effects of relaxation of the atomic core in the fields of arising virtual vacancies and the decay of vacancies via the autoionization Auger and (or) radiative channels. The appearance of the Cooper minimum in the two-photon absorption cross section is established. The calculations are carried out for the linear and circular polarizations of photons of an incident laser beam. The calculated results may be used for predictions.

Effects of classical noises on atom cooling and trapping

Abstract From a Fokker–Planck equation for the evolution of Wigner distribution function, the influence of classical noises, classified as additive noise and multiplicative noise, on the dynamics of atom in cooling and trapping processes is investigated. Our method permits an exact calculation for the case of harmonic trap which includes dissipation and noises simultaneously. It is found that the average energy heated by the noises does not increase simply in an exponential manner as predicted by Savard et al. (T.A. Savard, K.M. O'Hara, J.E. Thomas, Phys. Rev. A 56 (1997) R1095) based on the quantum mechanical first-order time-dependent perturbation theory. For some noise strengths, the average energy can increase much faster, especially at the beginning stage of evolution. We consider the disagreement between our results and theirs is due to the violation of the Virial theorem in the nonequilibrium process. The previous treatment actually implies the Virial theorem. On the other hand, an estimate shows that a nonharmonic trap is more sensitive to classical noises than a harmonic one. Furthermore, it is found that the atom momentum fluctuations in three directions can become correlated due to their coupling to the same noise reservoir.

Energy-averaged electron–ion momentum transport cross section in the Born approximation and Debye–Hückel potential: Comparison with the cut-off theory

An exact analytical expression for the energy-averaged electron–ion momentum transport cross section in the Born approximation and Debye–Hückel exponentially screened potential has been derived and compared with the formulae given by other authors. A quantitative comparison between cut-off theory and quantum mechanical perturbation theory has been presented. Based on results from the Born approximation and Spitzer's formula, a new approximate formula for the quantum Coulomb logarithm has been derived and shown to be more accurate than previous expressions.

Projection operator approach to time-independent perturbation theory in quantum mechanics

The time-independent perturbation theory in quantum mechanics is formulated using projection operator techniques. The determination of the perturbed eigenvalue can be decoupled from that of the perturbed eigenstate. Both the Brillouin–Wigner theory and the Rayleigh–Schrödinger theory come out straightforwardly. Both degenerate and nondegenerate cases can be treated in a unified way for arbitrarily high order perturbations.

Energy-averaged electron–ion momentum transport cross section in the Born Approximation and Debye–Hückel potential: Comparison with the cut-off theory

An exact analytical expression for the energy-averaged electron–ion momentum transport cross section in the Born approximation and Debye–Hückel exponentially screened potential has been derived and compared with the formulae given by other authors. A quantitative comparison between cut-off theory and quantum mechanical perturbation theory has been presented. Based on results from the Born approximation and Spitzer's formula, a new approximate formula for the quantum Coulomb logarithm has been derived and shown to be more accurate than previous expressions.

Zel'dovich's method of perturbation theory in quantum mechanics

Many years ago Zel'dovich showed how the Lagrange condition in the theory of differential equations can be utilized in the perturbation theory of quantum mechanics. Zel'dovich's method enables us to circumvent the summation over intermediate states. As compared with other similar methods, in particular the logarithmic perturbation expansion method, we emphasize that this relatively unknown method of Zel'dovich has a remarkable advantage in dealing with excited states. That is, the ground and excited states can all be treated in the same way. The nodes of the unperturbed wavefunction do not give rise to any complication.

Semiclassical quantization and resonance in spin tunnelling

We derive a semiclassical quantization for a spin, study it for not too small a spin quantum number , and compute the 2S+1 eigenvalues of a Hamiltonian exhibiting resonant tunnelling as the magnetic field parallel to the anisotropy axis is increased. Special attention is paid to the resonance condition. As a corollary we prove that semiclassical quantization and quantum-mechanical perturbation theory agree there where they should.

An ab initio pair potential for Ne2 and the equilibrium properties of neon

Fourth-order Möller—Plesset quantum mechanical perturbation theory with all excitations up to quadruple included, MP4(SDTQ), is used to calculate an ab initio pair potential energy function for neon. This potential is in reasonable agreement with that obtained from experiment, having a well depth some 8.5% smaller but with a well position which agrees within experimental error. This pair potential is used to calculate the second virial coefficient, the room temperature isotherm and the compressibility factor for bulk neon. There is reasonable agreement with experimental values.

Dispersive interparticle forces -pharmaceutical aerosols

METHOD In commercially available dry powder inhalers the proportion of the drug in a dose which reaches the lungs is low. Formulations contain the active drug usually blended with lactose which acts as an inert carrier. Various interparticle forces play a role in both the adhesion and subsequent redispersion of the drug from the lactose particle surface. In previous papers we have reported on the electrostatic (Colbeck and Amass, 1997a) and polarisation (Colbeck and Amass, 1998) forces. Here we consider the interaction between two non-polar particles. Three equations for the dispersive energy were differentiated to obtain expressions for the interaction forces between salbutamol and lactose particles. The first equation was derived from that given by London (1936). The expression given by London is based on quantum mechanical perturbation theory and it determines instantaneous and fluctuating dipole moments of the molecules although the ensuing interaction is of an electrostatic nature. The second expression for the dispersive force was derived from the energy of attraction between two spherical particles which was given by Hamaker (1937).The final equation is based on the work of Davis (1984) who developed the Hamaker expression into one for the interaction of spheres of unequal size. The interaction constants for each of the three equations depends upon all, or a combination, of the polarisability, the ionisation energy and the density of the interacting particles. The polarisability was calculated from the molar refractivity of the molecule and the number of molecules in a particle. The ionisation energy of a lactose or salbutamol particle was assumed to be correlated to that of its molecule and the number of molecules in a particle. RESULTS The dispersion force calculated from the theories of London, Hamaker and Davis are shown in figures 1,2 and 3 respectively. Values were calculated for salbutamol sulphate and four samples of lactose, each recrystallised by different methods (Colbeck and Amass, 1997b). From figure 1 it is seen that the London dispersion forces between drug and lactose are negative and hence attractive. The force of interaction increased with an increase in the size of the lactose. This can be explained by the assumption of the direct correlation of the molecular properties to the particle properties. For the Hamaker force the magnitude is greatly reduced, yet still attractive. Again the force between the salbutamol and lactose increased with the increase in the size of the lactose, for the same reason as that mentioned above. From figure 3 it can be seen that the Davis dispersion force may be both attractive and repulsive. This behaviour may be explained by the repulsive effects of the two electron clouds in close proximity dominating the force at small separations.

Theoretical model for intravalley and intervalley free-carrier absorption in semiconductor lasers: beyond the classical Drude model

Free-carrier absorption is calculated from the second-order perturbation theory of quantum mechanics by considering the interactions between carriers and polar optical phonons, deformation potential optical phonons, deformation potential acoustic phonons, piezoelectric acoustic phonons, and charged impurities in the intravalley transition and the intervalley transition. A formula is derived from our theoretical model for the coefficient of free-carrier absorption by incorporating the state-filling effect and the degenerate carrier distribution. Our results indicate that the classical Drude model is inadequate to describe many features of the free-carrier absorption. Alternatively, our theoretical model may provide an efficient method for investigating the effect of free-carrier absorption on the functionality or performance of the related optoelectronic device.