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Journal of Physics B Atomic and Molecular Physics

An alternative extension of Hartree theory to time dependent problems

R Phythian

Published Sep 1, 1970• 0 citations
Publisher

Abstract

The probability amplitude for the transition of a many particle system between given initial and final states in some finite time interval is considered where these states are products of single particle functions. It is pointed out that, if this amplitude is calculated by approximating the intermediate states by means of the usual time dependent Hartree theory, a spurious time dependence of the amplitude is obtained. An alternative time dependent extension of Hartree theory, more suited to this problem, is obtained by employing suitable trial functions in a variational principle for the transition amplitude. Some simple properties of the approximation are discussed.

DOI
10.1088/0022-3700/3/9/013

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The probability amplitude for the transition of a many particle system between given initial and final states in some finite time interval is considered where these states are products of single particle functions. It is pointed out that, if this amplitude is calculated by approximating the intermediate states by means of the usual time dependent Hartree theory, a spurious time dependence of the amplitude is obtained. An alternative time dependent extension of Hartree theory, more suited to this problem, is obtained by employing suitable trial functions in a variational principle for the transition amplitude. Some simple properties of the approximation are discussed.
An exactly solvable time-dependent quantum mechanical problem is employed to study the convergence properties of transition amplitudes calculated by using the Schwinger variational principle. A detailed comparison between the amplitudes approximated by the perturbative series and by their associated Schwinger variational principles is performed. The much better performance obtained by the variational principle is documented through different case studies. For a given order of the Schwinger principle, it is observed that the transition amplitudes do not converge to the exact one for large perturbations. The latter is true even though large combinations of unperturbed states with constant coefficients are taken as trial wave functions. As a matter of fact, it is shown that the improvement of the method comes from using better trial wave functions and increasing the order of the Schwinger principle employed.
The exact study of many-body microscopic systems is impossible when the number of particles is large (N ≥ 3). Approximate methods are then used. The time-independent mean-field (TIMF) approximation has been proposed for the description of collisions in many-body systems. Collision amplitudes are derived by the use of a variational principle and the choice of trial functions as products of single-particle orbitals. Resulting mean-field equations with a nonvanishing right-hand side turn out to be a generalization of the traditional Hartree or Hatree–Fock type equations. These TIMF equations are successfully solved numerically for the case of short-range forces. In this paper, we test the validity of this theory for the Coulomb interaction between two particles, that is, a long-range interaction. A numerical comparison between the exact and the mean-field solutions is conducted PACS Nos.: 31.15.Ne, 31.15.Pf, 21.45.+v,25.10.-i
Working within a time-dependent Hartree-Fock framework, one develops a semiclassical approximation appropriate for large systems. It is demonstrated that the standard semiclassical approach, the Thomas-Fermi approximation, is inconsistent with Hartree-Fock theory when the basic two-body interaction is short-ranged (as in nuclear systems, for example). However, by introducing a simple extension of the Thomas-Fermi approximation, one overcomes this problem. One also discusses the infinite nuclear matter problem and point out that time-dependent Hartree-Fock theory yields collective modes of the zero sound variety instead of ordinary hydrodynamic (first) sound. One thus emphasizes that one should be extremely circumspect when attempting to cast the equations of motion of time-dependent Hartree-Fock theory into a hydrodynamic-like form.
If a particle which can decay spontaneously is subjected to collisions with other particles the transition probability may be changed. The calculation of such an effect by the usual time-dependent perturbation methods is complicated by the circumstance that energy can be conserved in some intermediate states, because of the possibility of elastic scattering. The perturbation calculation is here modified in such a way that the resulting expression for the whole transition probability does not contain any divergences or spurious effects. By suitable approximations the expression is separated into two parts, one of which gives the probability of elastic scattering followed by a spontaneous transition, and the other the true stimulated transition probability which causes a change in the lifetime.
We derive a multiconfigurational time-dependent Hartree theory for systems with particle conversion. In such systems particles of one kind can convert to another kind and the total number of particles varies in time. The theory thus extends the scope of the available and successful multiconfigurational time-dependent Hartree methods---which were solely formulated for and applied to systems with a fixed number of particles---to a broader class of physical systems and problems. As a guiding example we treat explicitly a system where bosonic atoms can combine to form bosonic molecules and vice versa. In the theory for particle conversion, the time-dependent many-particle wave function is written as a sum of configurations made of a different number of particles, and assembled from sets of atomic and molecular orbitals. Both the expansion coefficients and the orbitals forming the configurations are time-dependent quantities that are fully determined according to the Dirac-Frenkel time-dependent variational principle. By employing the Lagrangian formulation of the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the atomic and molecular orbitals and one for the expansion coefficients. The first set is comprised of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with coefficients forming a time-dependent Hermitian matrix. Particular attention is paid to the reduced density matrices of the many-particle wave function that appear in the theory and enter the equations of motion. There are two kinds of reduced density matrices: particle-conserving reduced density matrices which directly only couple configurations with the same number of atoms and molecules, and particle nonconserving reduced density matrices which couple configurations with a different number of atoms and molecules. Closed-form and compact equations of motion are derived for contact as well as general two-body interactions, and their properties are analyzed and discussed.