Time-independent Formalism Research Articles

A response-function approach to the direct calculation of the transition-energy in a multiple-cluster expansion formalism

In this paper, we have developed a response function approach to the direct determination of transition energy in a multiple-cluster expansion formalism. We adopt a time-independent formalism in a way reminiscent of the Fourier-transformed version of a response-function theory. The formalism has been used to describe specifically the linear response, and compact and usable formulae have been derived for the calculation of excitation energy and dynamic polarisability of closed-shell systems. Extension of the method for calculating higher order response functions and ionisation potentials is straight-forward.

An introductory guide to effective operators in nuclei

In this tutorial-style article we discuss the time-independent perturbation formalism of Brandow for effective interactions and operators in a nonrigorous, intuitive way. The simple example of a 2\ifmmode\times\else\texttimes\fi{}2 matrix is used to introduce the basic concepts, for instance the notion of folded diagrams. The same example is used subsequently in a discussion of the branch points arising from "level crossings" and the associated convergence difficulties with the perturbation series. Numerical calculations of the effective interaction and the effective charge are then reviewed, focussing, for the most part, on the ($1s0d$) shell. Whenever possible, simple physical models are used to illustrate the results. A fairly comprehensive summary of the calculations to date is given, and we attempt to synthesize and draw conclusions.

Distorted Cluster States in Scattering Theory

Difficulties connected with the asymptotic behavior of adiabatic wave functions in the scattering theory of slow collisions are discussed in the time-independent formalism. It is shown that the Born-Oppenheimer approximation, and its generalization involving a finite number of adiabatic states, cannot satisfy the correct asymptotic boundary conditions. A slightly different basis set of distorted cluster states is constructed which avoids this difficulty and yet fully contains the distortion effect. Symmetric treatment of the three-particle scattering system is given in the multidimensional form, and the nonadiabatic corrections and their approximations are discussed. Resonances as complex energy poles of the amplitudes are shown to be contained in the nonadiabatic correction terms and analyzed in terms of the effective potentials. An application of the method to high-energy fast collisions is briefly discussed.

The stationary states of interacting fields

As an application of a time-independent perturbation formalism developed earlier for systems with many degrees of freedom, we give in terms of diagrams the general perturbation expressions for the exact stationary states of interacting fields. The physical vacuum is obtained by applying to the bare vacuum the exponential of an operator simply expressed in terms of connected diagrams. For the states with one or more physical particles expressions are found in the form of suitably defined creation operators applied to the physical vacuum. The incoming and outgoing parts of a scattering state are analyzed in terms of products of physical particle creation operators, the connection being given by the S-matrix. Some remarks are made on renormalization. For electrodynamics in particular the ratio of renormalized to unrenormalized fine structure constants is shown to have a simple probabilistic interpretation: it is the probability to find the bare one-photon state in the physical one-photon state of vanishing momentum, divided by the probability to find the bare vacuum in the physical vacuum.