V. N. Ostrovskii

Electron Detachment in Slow Collisions between a Negative Ion and an Atom

Time-dependent problems with slowly changing parameters of the system, where two discrete energy levels (terms) come close together with a subsequent transition occuring between the two states, have been extensively investigated in quantum mechanics. For calculations of the transition probability the formulae of Landau-Zener or Landau-Teller or Zener-Rosen or some others (26) can be used, depending on the particular conditions. These theories are not applicable, however, to the case where an isolated energy level approaches the continuum boundary. Then the term interacts with an infinite number of energy levels rather than with only one, and the above theories are no longer applicable.

Zero-Range Potentials in Multi-Channel Problems

The theory developed in the previous chapter can be directly applied to describe electron scattering by molecules. Subramanyan (89) considered e + H2 scattering in the approximation of a two-center ZRP model. In the simplest case, the parameter a for an isolated hydrogen atom was determined from the scattering length computed in the static approximation without exchange. We have already pointed out in Chapter 1 that in the approximation without exchange, the hydrogen atom has two different (singlet and triplet) scattering lengths (a similar situation arises in all atoms with non-zero total spin). Therefore the spin state of the system may influence scattering considerably. Within the ZRP model, an account of spin was first made by Smirnov and Firsov (l7). Further development of the theory was reported in (90,91) and also in (64,86,87).

Zero-Range Potentials for Molecular Systems. Bound States

The determination of the bound-state energy levels of an electron moving in the potential field of a complicated molecule is a typical problem in quantum chemistry. The resulting field is a superposition of several spherically symmetrical potentials centered on the nuclei of the molecule. The solution of such problems with real potentials encounters great numerical difficulties. Therefore it is important to see whether simpler models can be developed to give a qualitatively correct description of the molecules. In this chapter, we shall consider a superposition of several ZRPs forming a many-center potential field. The corresponding Schrodinger equation for an arbitrary number of force centers admits an analytical solution. This model can be applied to study quantum-mechanically the motion of outer electrons in negative molecular ions. With a suitable choice of parameters, it can also be applied to the inner electrons of these molecular systems.

Time-Independent Quantum Mechanical Problems

The Hamiltonian of the problems considered in Chapters 8–10 depended explicitly upon time. In other words, there was an external parameter of the system whose dependence upon time was fixed and did not alter during the course of the collision. In reactions where colliding systems are ions and atoms, the nuclear separation R is such a parameter. If the energy of the nuclear motion is low and comparable with that of the electronic transitions, R must be included in the list of the dynamical variables of the system, i.e., the nuclear motion must be treated quantally. This approach is definitely required if, for instance, we deal with reactions near thresholds.

Time-Dependent Quantum Mechanical Problems Solvable by Contour Integration

Only a few quantum mechanical problems are known with a time-dependent Hamiltonian H(t) whose solutions can be expressed in a closed analytical form, and an extension of the list of such problems is of great interest to many particular branches of physics. In this chapter, we shall show that a rather wide class of time-dependent quantum mechanical problems can be solved in a closed form if the method of contour integration is applied to construct the solution.

Basic Principles of the Zero-Range Potential Method

It is known that the Schrodinger equation admits solutions expressible in a closed analytic form in only a few cases. The stationary Schrodinger equation is solvable analytically for a harmonic oscillator, for a particle moving in a Coulomb field or in a rectangular potential well, and in some other cases. Even the one-particle Schrodinger equation cannot be solved exactly for the great majority of potentials. In the case of the many-particle and non-stationary problems the situation becomes even more intractable.

Weakly Bound Systems in Electric and Magnetic Fields

In this chapter we shall consider the motion of an electrically charged particle in a combined field of a ZRP and an electric or magnetic field. In contrast to the approach used in Chapter 3, the external (electric or magnetic) field is not assumed here to be weak so that perturbation theory will not be used. This allows the inclusion of some new and interesting phenomena which could not be treated within the perturbational framework. The results derived in this chapter will be applicable to weakly bound systems such as negative ions or analogous systems in solids.

Motion of a Particle in a Periodic Field of Zero-Range Potentials

In all applications considered in the previous chapters of this monograph, it was assumed that the particle was in a field of a finite number of ZRPs. Now we shall consider an infinite number of potential wells. This will require the calculation of infinite sums extended over all force centers. A physically important model is an infinite number of identical wells forming a regular (periodic) structure similar to that of a crystal. The regular feature of this model facilitates the calculation of the sums. It can be considered as a generalization of the well-known Kronig-Penney model (81) . In the latter, a particle moves in the field of a one-dimensional periodic lattice in a one-dimensional space. In this and the next section, we shall consider a one-dimensional lattice in a three-dimensional space (a model of an electron moving in the field of a polymer molecule), where the result of the summation can be written in an analytical form. In Sec. 6.3 and Sec. 6.4 we shall study the case of two- and three-dimensional lattices and discuss various approximate methods of calculating the sums.

Nonlinear Approximations in the Theory of Electron Detachment

In Section 9.1 we considered a Hamiltonian with a separable potential multiplied by a coefficient linearly dependent on time, and showed that the solution of the corresponding time-dependent equation could be obtained with the help of contour integration. An equation where the coefficient of the separable potential is a linear function of time can be solved in a similar way. In the particular case when the coefficient of the separable potential is inversely proportional to time, the equation