Nonlinear Approximations in the Theory of Electron Detachment

  • Yu. N. Demkov
  • V. N. Ostrovskii
Part of the Physics of Atoms and Molecules book series (PAMO)


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
$$\rm \Big[H_0+\mid \phi >\gamma\ t^{-1}<\phi \mid \Big]\ \ \mid \psi >\ =\ i\ {\partial\over \partial t}\mid \psi >$$
has a solution of the form
$$\rm \mid \psi >\ =\ N{\int\limits_C}\ G(E)\ \mid \phi >\ exp\ \Big\{i\ {\int\limits^E}\ <\phi\ \mid G({E^\prime})\mid \phi >\ d{E^\prime}\ -\ Et\Big\}\ dE,$$
which appears to be even simpler than the form of equation (9.1.4). Unfortunately, the perturbation term in (10.1.2) is singular at t = 0, which makes it difficult to apply this equation to real systems. The general behavior of the terms for a simplified problem which has only three discrete energy levels is illustrated in Fig. 10.1. In certain cases when the terms which go to infinity may be ignored, this model can still be used. In the particular case of the time-dependent equation
$$\rm \Big(H_0\ e^{at}\ +\ \mid \phi >\ b\ <\ \phi \mid\ \Big)\ \mid\ \psi >\ =\ i\ {\partial\over \partial t}\ \mid\ \psi\ >,$$
there is a system of exponentially diverging terms which interact with a single horizontal term. Making the substitution exp(at) = s, we transform equation (10.1.3) to the form (10.1.1); hence, the exact solution of equation (10.1.3) can be obtained making use of equation (10.1.2). A two-state system of this type was considered by Nikitin and also by Demkov (for references see (26) ), and it was applied to problems of non-resonant charge exchange and to the calculation of the fine-structure transitions in alkali metals caused by impact.


Total Cross Section Momentum Distribution Quadratic Approximation Nonlinear Approximation Hankel Function 
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Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • Yu. N. Demkov
    • 1
  • V. N. Ostrovskii
    • 1
  1. 1.Leningrad State UniversityLeningradUSSR

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