B. L. Burrows

Lower bounds for quartic anharmonic and double‐well potentials

Rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the Schrodinger equation with quartic anharmonic and symmetric double‐well potentials of the form V(A,B)=Ax2/2+Bx4(B≥0) are presented. This procedure exploits some exactly soluble model potentials and appears to be of quite general utility.

Optimization and Control of a Pendulum-driven Cart-pole System

This paper investigates the motion generation for a pendulum-driven cart-pole system. The dynamic model of this system is developed by using the Newtons law. A six-step motion strategy is simulated by using MATLAB/SIMULINK. The pendulum is driven by an open-loop controller to track the desired pendulum angular velocity profile. The optimal configuration which based on the profile is addressed under some specified conditions.

Exact solutions for shell-confined hydrogen-like atoms: polarisabilities and Shannon entropies

An idealised model to treat the effect of spherically confining the electron in a hydrogen-like atom is studied, where the potential is infinite in all space except for a spherical shell. The exact solution of the Schrödinger equation is obtained in terms of two independent solutions of the Kummer equations. It is found that, in some cases, it is necessary to use the standard Kummer M function and a non-standard second solution. In other cases we may use the Kummer U function and in a limiting case the two standard solutions of Bessels equation. The effect of an imposed dipole field on the shell is treated using the first-order perturbation equation from which the polarisability can be calculated. In addition, the exact wavefunction is used to calculate the Shannon entropies of both position and momentum and it is shown that these measures give insight into the form of the wavefunction.

Confined quantal oscillator with time-dependent boundaries

A Lie algebraic treatment is described for a generic model of a quantum mechanical oscillator whose range is restricted, and with time-dependent boundary conditions. The model provides a general framework for describing molecular confinement, and we obtain formally exact analytic solutions.

Exact Solutions for Confined Model Systems Using Kummer Functions

We treat model systems where an electron is confined in a region of space. The particular models considered have solutions which may be expressed in terms of the Kummer functions. Both standard and non-standard Kummer functions are used in these models and a comprehensive summary of the usual and exceptional Kummer functions is given. The definition of confinement is widened to treat radial confinement in any spherical shell, including the asymptotic region and cases where the electron is confined to a lower dimension. Initially we consider the theory in K dimensional space and then give particular examples in 1, 2, and 3 dimensions. A commonly treated model is the radially confined hydrogen atom in 3 dimensions with an infinite barrier on a confining sphere so that the wavefunction is identically zero on this sphere. We have extended this model to treat a more general model of spherical confinement where the derivative of the charge density is zero on the confining sphere. It is shown that the analogous models for the radial harmonic oscillator and radial constant potentials may be treated using a generic technique.

Quasi exact solutions for an asymmetric double well potential

Algebraic methods are used to derive approximate quasi exact solutions for a parametric model potential having the functional form V(x) = V 0 – Mβx + ½x 2(α + βx)2 with M a positive integer and β > 0. This yields an asymmetric double well potential provided that the parameters (M, α, β) satisfy the conditions – 1/12√3 < ω = – Mβ2/2α3 < 1/12√3. If k = α2/β is sufficiently large there is a high barrier between the two wells, and the quasi exact spectrum is essentially harmonic. More generally, each quasi exact solution is the product of a finite polynomial and a universal asymptotic factor, exp[–(½αx 2 + ⅓βx 3)], and is mainly localized in the deeper well, even when the spectrum is significantly non-harmonic. For ω sufficiently small, both the quasi exact spectrum and the lower excited bound state spectrum are determined quite accurately by low order Rayleigh–Schrodinger perturbation theory following a suitable (but non-standard) canonical transformation of the reduced Hamiltonian.

Stark energy levels of symmetric-top molecules: an elementary algebraic treatment

We derive simple semi-analytical expressions for the Stark energy levels of a general symmetric-top molecule placed in a uniform electric field of arbitrary strength. These are obtained by means of third-order perturbation theory following a suitable canonical transformation of the system Hamiltonian, which is conveniently written in terms of the generators of either of the Lie algebras SO(3) or SO(2,1).

Response to ‘Comment on Exact solutions for shell-confined hydrogen-like atoms: polarisabilities and Shannon entropies’

Aquino [1] has presented some results on the model problem in our paper [2] that differ from our values. These calculations have been done using a variety of methods and some of them have been checked by a referee. We thank both of them for their careful calculations for which we are grateful. The problem has arisen, not as the author speculated from the method that we used, but from a subtle error in the simplification of our solution. In this response we explain the source of the error and argue that our method has many advantages, not least the fact that we obtain essentially analytical solutions for the problem.

Effects of Orbital Overlap on Calculations of Charge Exchange in Atom-Surface Scattering

Calculations of ionization (or neutralization) probabilities when an atom (or ion) is scattered off a solid surface usually neglect the nonorthogonality of the valence orbitals on the atom (or ion) and the atoms in the solid. In order to investigate the validity of this, the more complicated equations, which arise when orbital overlap is included, are derived. They are used to determine how far the probabilities change from their values when overlap is ignored. It is found that there can be a significant change which suggests that it is advisable, where possible, to take explicit account of the non-orthogonality in the calculation of ionization (or neutralization) probabilitiies.