C. K. Au

The dispersion theory of dispersion forces

The long-range forces that act between neutral atoms and molecules have been known as dispersion forces since the work of London, who was the first to make manifest the connection between these forces and the dispersion of light by atoms, already guessed at by Newton. The analysis and application of the analyticity properties of scattering amplitudes, developed several decades ago in the context of quantum field theory, is known as dispersion theory. In this article we review the approach to dispersion forces based on dispersion theory. We give a general discussion of the concept of potential in quantum field theory and then show how a study of two-photon exchange amplitudes leads to a model-independent derivation and generalization of the formulas describing retarded dispersion forces for the case of two neutral atoms, first obtained by Casimir and Polder. We then review later extentions to the interaction of a charged and a neutral system and recent work on the case where both systems are charged. The connection between the dispersion-theory approach and more conventional methods is described. We also illustrate the use of dispersion-theory techniques to study forces arising from two-neutrino exchange and two-meson exchange. The effect of dispersion forces on the energy levels of the Rydberg states of hilium is briefly sketched. Finally, some open questions are mentioned.

Excited bound state logarithmic perturbation theory without nodes

The logarithmic perturbation theory is modified slightly in order to deal with excited states. Instead of considering a real wavefunction describing the physical stationary state, the authors consider a complex wavefunction at the same energy, by mixing in the ghost state. For excited bound states, the former has nodes, while the latter is guaranteed not to have any nodes, and can be represented simply as exp(-G), to which the logarithmic perturbation method can be applied in a straightforward manner. The physical entities (the energy corrections) are independent of the amount of mixing of the ghost state. The connection to the Green function method is also shown. The freedom to mix in the ghost state allows the authors to justify an ad hoc approach whereby the simple version of the logarithmic perturbation theory is applied to excited bound states. The formalism is illustrated with simple examples.

The question of gauge dependence of transition probabilities in quantum mechanics: Facts, myths and misunderstandings

Abstract We show that, in terms of physical observables, transition probabilities in quantum mechanics can be calculated in a truly gauge invariant way, i.e., independent of the choice of gauge and that recent arguments for the preference of a particular gauge are due to misconceptions.

Dynamic multipole polarisability of hydrogen

A closed-form expression is derived for the dynamic multipole polarisability of hydrogen in an arbitrary S state. This expression is analytic around omega =0 and can thus be used to generate the negative-order multipole sum rules in hydrogenic atoms. The residues and discontinuities across the branch cut in this expression lead to closed-form expressions for the multipole transition matrix elements in hydrogenic atoms.

Logarithmic perturbation method for the scattering phase shift

Abstract A hierarchical scheme expresses the phase shift, order by order, in terms of quadrature. An important advantage is the freedom to choose the unperturbed system, for example, as a piece-wise constant potential, making the remaining correction small, especially if the unperturbed potential has the right number of bound states. Unitary is maintained order by order.

A quantum-field theory approach to the calculation of energy levels in helium-like Rydberg atoms

We discuss the fine structure splitting of the energy levels in Rydberg states of helium or helium-like ions on the basis of quantum electrodynamics, using time-independent perturbation theory and the radiation gauge. For the zero-order description of the states we use products of Dirac-type wavefunctios with shielding for the outer electron. The perturbing interaction includes the residual electrostatic potential, the interaction coming from the exchange of virtual photons, and the creation of virtual electron-positron pairs. It is shown that the level shifts for low-Z ions are given to high accuracy by a procedure followed previously. This consists of adding the expectation value of a retardation-correction potential V/sup corr//sub 2//sub ..gamma../(r/sub 2/), arising from two-photon exchange, to the result obtained from relativistic multiple-Coulomb exchange. For principal quantum number of the Rydberg electron of the order of ten, we describe the range of values of Z for which this procedure gives the splitting to one-percent accuracy. Copyright 1987 Academic Nress, Inc.

A new algorithm for one-dimensional problems

Abstract A new computation scheme in perturbation theory is developed for one-dimensional nonrelativistic problems with static potentials. If λ is a measure of the strength of the perturbation, after n steps in this scheme, the energy and wave function corrections are known to order λ 2 n −1 , whereas the same are known only to order λ n in the ordinary perturbation expansion.

`A new efficient method for calculating perturbative energies using functions which are not square integrable': Regularization and justification

The method recently proposed by Skala and Cizek for calculating perturbation energies in a strict sense is ambiguous because it is expressed as a ratio of two quantities which are separately divergent. Even though this ratio comes out finite and gives the correct perturbation energies, the calculational process must be regularized to be justified. We examine one possible method of regularization and show that the proposed method gives traditional quantum mechanical results.

Consistency of the Aharonov-Bohm effect with quantum theory

SummaryA recent claiming the inconsistency of the Aharonov-Bohm effect with the principles of quantum theory is shown to contain incorrect arguments.

CONVERGENT PERTURBATION EXPANSION FOR THE ANHARMONIC OSCILLATOR

Abstract We study the ground state as well as the first three excited states of the anharmonic oscillator with anharmonicity λx4 for a range of λ = (0, 10) with the first-order logarithmic perturbation iteration method (FOLPIM). This leads to convergent results. The initial choice of the wave function seems only to affect the rate of convergence in the case of the ground state but may critically affect the convergence for the excited states. For large values of λ, convergence is best obtained by choosing the asymptotic solution as the initial “unperturbed” wave function.