Quantum Mechanical Perturbation Theory Research Articles

New quantum mechanical perturbation technique using an ‘‘electronic scratchpad’’ on an inexpensive computer

The authors have developed a new method for doing numerical quantum mechanical perturbation theory. It has the flavor of Rayleigh–Schrödinger perturbation theory (division of the Hamiltonian into an unperturbed Hamiltonian and a perturbing term, use of the basis formed by the eigenfunctions of the unperturbed Hamiltonian) while turning out to be a variational technique. Furthermore, it is easily implemented by means of the widely used ‘‘electronic scratchpad,’’ MathCAD 2.0, using an inexpensive computer. As an example of the method, the problem of a harmonic oscillator with a quartic perturbing term is examined.

Perturbation theory in high-energy transmission electron diffraction

A perturbation theory for many-beam high-energy transmission electron diffraction in noncentrosymmetric crystals is described for both the nondegenerate and degenerate cases. This perturbation theory differs from the conventional quantum-mechanical perturbation theory by perturbing the electron wavevectors instead of the total electron energy, which is constant for elastically scattered electrons. The relations between the perturbation theory and some other approximations commonly used in electron diffraction are discussed. It is shown that the few-beam approximation and the Kambe approximation are both applications of degenerate perturbation theory. Finally, as an example, this degenerate perturbation theory is applied to obtain an analytical solution to a four-beam case with two systematic (0 and g) and two nonsystematic (h and l) beams. This four-beam solution shows that the intensity of a four-beam interaction depends on all the four three-phase invariants involved, and also shows that the effects of the g beam on the three-beam interaction of 0, h and l are localized to the region near the Bragg condition of g. This may serve as a guide for future experiments using three-beam interactions for the measurement of structure-factor phases of an unknown structure.

Large order perturbation theory and summation methods in quantum mechanics

Large order perturbation theory and summation methods in quantum mechanics

A note on Cooley's energy correction formula

A simple alternative derivation based on standard first-order perturbation theory in quantum mechanics is given for Cooley's energy correction formula which is employed in the popular iterative Numerov-Cooley method for numerically solving the one-dimensional eigenvalue Schrödinger equation.

Quasiparticle treatment of quantum-mechanical perturbation theory in the free electron gas

The correlation energy of the free electron gas is investigated. The point of departure is Wigner's idea of selective summation of the perturbation-theory series for the ground-state energy. The introduction of a quasiparticle operator (with a quasiparticle understood as representing momentum propagation in the system) makes possible the resummation of the perturbation-theory series in such a way that the order of the terms is determined, not by the number of different particles that participate in an interaction, but by the number of different lines (momentum transfers). The matrix elements of the quasiparticle operator are obtained as the solution of an integral equation determined in the Macke-Tamm-Dankoff approximation and the RPA. The first order of the quasiparticle perturbationtheory series (it includes the Hartree-Fock exchange energy) is simply the sum of the energies of the quasiparticles. The obtained expressions are identical to the results of other theories. The second order corresponds to exchange interaction of the quasiparticles. For it, numerical calculations are made in the range of r S from 0.I to i0 a.u. Comparison with the analogous results obtained in the e S method has shown the inadequacy of the e S method, in which the best successes have been achieved in the investigation of the correlation energy of the free electron gas. In the case of high densities (small r S) the quasiparticle exchange interaction tends to the particle exchange interaction, which was first calculated by Gell-Mann and Brueckner and for which an exact value was obtained by Onsager and collaborators. The results for the total correlation energy (the sum of the first and second terms but with the Hartree-Fock exchange energy subtracted) agrees better with the results of Hubbard and Nozi~res and Pines than with other results.

COLLISION-INDUCED DISSOCIATION (CID) (Ⅰ)——TIME-DEPENDENT QUANTUM THEORY

Collision-induced dissociation dynamic processes have been studied in this paper by using Time-dependnet quantum mechanical perturbation theory. The molecular systems are of the type of the Morse oscillator and the intermolecular exponential repulsive interaction potential. The analytical formula established can be conveniently applied to analyse a variety of physical and chemical problems relevent to CID processes. The numerical calculation for the (I2, He) system shows that vibrational energy is more effective than translational energy in enhancing the rate of the CID processes, as the total energy of the system is fixed. In addition, except the energy barrier, there are no extra dynamical restrictions for CID processes.

The wake potential and dynamic screening in a hydrogen system

Abstract The wake potential of a fast charged particle is calculated from first-order quantum-mechanical perturbation theory for hydrogen 1s → 2s and 1s → 2p transitions without any cutoff parameter. In the first-order approximation, the wake charge distribution is a linear charge along the fast charged particle path, with a charge distribution radius of a few A around the fast charged particle path. The screening of the charged parade by its own wake potential is also calculated and found to be important.

Sticking probability of light atoms on the graphite (0001) surface

Sticking of light gas atoms (He,D,H) on the (0001) surface of graphite is studied theoretically in the framework of the quantum mechanical perturbation theory. The formulas include a detailed microscopic description of the solid structure and vibrations. The calculations are employed in conjunction with the published experimental data to investigate the properties of the inelastic gas–surface coupling in the He/graphite system. The inelastic coupling is suggested to have a significant attractive component, and to be confined to highly localized spots on the surface. The low energy sticking probability is shown to be very sensitive to the weak coupling between the graphite layers. The high energy sticking via excitation of the optical out-of-plane phonons is found to be very inefficient. The isotopic effect is investigated in the H,D/graphite system. While the heavy isotope sticks about two times more efficiently in the low energy limit, the decay of the sticking probability with the energy is slower for the lighter isotope. Therefore the difference in the sticking probability of the two isotopes decreases strongly towards higher energies.

Dynamics of continuous-time random walk, fractal time dispersion, and fractional exponential time relaxation.

This paper attempts to establish the dynamics of a microscopic model for a continuous-time random walk. The waiting-time distribution Q(t) is derived from the time-dependent perturbation theory of quantum mechanics for the walker's motion coupled with the media. The walker's motion includes the hopping of a localized particle and a spin (or dipole) flip. The medium is modeled as a harmonic heat bath. The walker moves among a set of degenerate localized states. The scaling behavior of the effective spectrum at low frequency with index \ensuremath{\beta} is modeled by using stochastic theory. It is found that Q(t)=exp(-${\mathrm{at}}^{(2\mathrm{\ensuremath{-}}\ensuremath{\beta})}$) for 0\ensuremath{\le}\ensuremath{\beta}2 and Q(t)\ensuremath{\sim}${t}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$ for \ensuremath{\beta}=2. The applications of our theory include dispersion diffusion, the transient drift of hopping control light excitation in a-Si:H, and thermoremanent magnetization relaxation in spin glasses.

Some Remarks on Critical Phenomena of Fluid Metals

Critical phenomena of fluid metals are studied by observing the potential range of effective ion-ion interaction in their liquid phases. This is done by two different methods. First, the Fourier transform of March's potential with a cutoff is calculated. Secondly, by means of quantum mechanical perturbation theory, the effective potential is derived and discussed within the lowest order approximation for the electronic dielectric function. Both methods yield the result that the effective potential is of short-range from a viewpoint of critical phenomena. On the basis of ion-ion interaction obtained by the second method, a scaling form for the critical temperature is derived and examined by the use of existent experimental data.

The induction energy of an assembly of polarizable molecules

An expression is given for the induction energy of an arbitrary assembly of polarizable molecules, each described by distributed multipoles and polarizabilities up to arbitrary order. In a self-consistent treatment, the induced moments of each molecule are determined by the field arising from the polarized moments of the others, but the induction energy is obtained as the interaction between the induced moment of each molecule and the unpolarized moments of the others. Comparison of the classical result with quantum-mechanical perturbation theory shows that the self-consistent induction energy corresponds to the inclusion of some, but not all, of the terms in a perturbation expansion to infinite order.

The anionic group theory of the non-linear optical effect and its applications in the development of new high-quality NLO crystals in the borate series

Abstract Starting from a general quantum-mechanical perturbation theory on the nonlinear optical (NLO) effect in crystals, this review gives a systematic presentation of the basic concepts and calculation methods of the ‘anionic group theory for the NLO effect of crystals’ and a brief discussion of the approximations involved. Calculations have been made for the second harmonic generation (SHG) coefficients of a few typical NLO crystals. Comparisions between these theoretical values and the experimental values made both on powdered crystals and on single crystals suffice to show the feasibility of the theoretical treatment and calculation methods. On this basis, borate ions of various structure types are classified and systematic calculations are carried out for the NLO susceptibilities of some typical borate crystals with good prospects of applications in opto-electronics. Through these calculations, a series of structural criteria serving as useful guidelines for searching and developing new NLO crystals in borate series are presented. These structural criteria have good prospects of wider applications in searching and developing for other new types of NLO crystal materials. The criteria are as follows. (1) The planar six-membered ring (B3O6)3- and the planar trigonal (BO3)3 group, each possessing a conjugated 7t-orbital system, are far more favourable for producing larger SHG coefficients than the non-planar tetrahedral (BO4)5- group. Moveover, in the planar group, the larger the electronic population in the conjugated 7c-orbital system, the greater the SHG effects will be. As a result, the SHG effects will decrease in the order (2) The SHG coefficients can be adjusted to a certain extent by suitable arrangement of the three-and four-coordinated B atoms, such as (BO3)3- and (BO4)5-, (B3O6)3- as opposed to(B3O7)5- and (B3O8)7-. On the basis of these structural criteria, we have been successful in developing some new high-quality NLO materials, including the LiB3O5 crystal as an excellent NLO material. Suggestions are put forward for searching for and developing other promising new NLO materials in the borate series. The recent development of two new-type NLO borate crystals, β-BaB2O4 and LiB3O5, is described.

Change of the hydrogen excitation cross sections due to the wake potential of a fast charged particle

The wake potential of a fast charged particle is derived for an atomic system using first-order quantum-mechanical perturbation theory. The Coulomb excitation cross sections of the target atoms by this wake potential are then derived and found to be considerable. Numerical results on a model system of condensed atomic hydrogen show that the wake potential is very important and must be included in calculating the integrated Coulomb excitation cross sections in condensed matter.

Quantum black holes and Planck's constant

The author shows that the Planck-scale black holes of quantum gravity must obey a consistency condition relating Planck's constant to the integral of the mass of the black holes over time, if the usual path integral formulation of quantum mechanics is to make sense on physical spacetime. The author also shows, using time-dependent perturbation theory in ordinary quantum mechanics, that a massless particle will not propagate on physical spacetime with the black holes present, unless the same condition is met.

PERTURB: A special‐purpose algebraic manipulation program for classical perturbation theory

AbstractClassical perturbation theory provides a particularly promising route to EBK quantization of nonseparable systems. However, the number of terms generated when implementing perturbation theory for systems with more than two degrees of freedom can prove too large for general purpose symbolic manipulators to handle. We describe PERTURB, a specialized algebraic manipulation program written in C for quantization of multidimensional systems. A review of operator based classical perturbation theory is given, and the relationship between this type of perturbation theory and quantum mechanical Van Vleck perturbation theory discussed. The relative performance of the Dragt–Finn and Lie transform algorithms is assessed.

Formation of the H-beta line in solar chromosphere magnetic field

We consider the effects of microscopic electric field, Doppler broadening and radiation damping in the solar atmosphere magnetic field and use the perturbation theory of quantum mechanics to determine the broadening mechanisms and the absorption coefficient of the Hβ line. We then use the VAL quiet chromosphere model to solve the Unno-Beckers equations of radiative transfer for the Hβ line and obtain a numerical solution of the Stokes parameters of this line.

Spectroscopic constants of alkali halides and hydrides using modified T-Rittner model

The T-Rittner model derived from quantum mechanical exchange perturbation theory is modified by incorporating (i) new values of molecular state polarizabilities, (ii) the recent values of van der Waals potentials estimated from the Kirkwood–Muller (KM) formula, and (iii) the repulsive hardness parameters evaluated from the overlap integrals using the exchange charge model (ECM). With the help of this modified potential model we calculate values of binding energy W, dipole moment μ, vibrational frequency ωe, vibrational anharmonicity constant ωeXe, and rotational constant αe for diatomic molecules of alkali halides and hydrides. The results obtained in the present study are compared with the experimental data. A good agreement between calculated and experimental values supports the validity of the modified T-Rittner model used in the present study.

On the perturbative analysis of the dynamics of reactive collisions

This paper explores the application of quantum mechanical perturbation theory to the determination of reactive transition probabilities. The very simple problem of particle reflection and transmission at a one-dimensional potential barrier is examined over a wide range of energies and for alternate perturbative schemes. The first scheme is conventional; the second is based on a quasi-adiabatic (QA) description of the dynamics. We find that for a choice of model parameter values, loosely based on the H+H2 system, the first (conventional) scheme provides quantitatively accurate results. However, at higher system masses, the first scheme results are in only moderate agreement with exact results and are highly sensitive to the variation of distortion potential parameters. For the high system mass case, the QA results are in much better agreement with the exact results. However, the QA results are also highly sensitive to the variation of QA potential parameters. We discuss finally a number of avenues for further investigation.

On summation of perturbation series for screened coulomb potentials

Higher orders of perturbation theory (PT) in quantum mechanics and field theory rise factorially at k → ∞ ( k is the PT order). We discuss the computation results of higher orders of PT and summation methods for divergent perturbation series (PS) for quantum mechanical problems: the Yukawa and Hulthén potentials, the funnel-shaped potential, etc. Criteria of PS summability by Padé approximants are examined.

An alternative to the magnus expansion in time-dependent perturbation theory

An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.