A quasi classical approach to fully differential ionization cross sections

A classical approximation to time dependent quantum mechanical scattering in the Møller formalism is presented. Numerically, our approach is similar to a standard Classical–Trajectory–Monte–Carlo calculation. Conceptually, however, our formulation allows one to release the restriction to stationary initial distributions. This is achieved by a classical forward–backward propagation technique. As a first application and for comparison with experiment we present fully differential cross sections for electron impact ionization of atomic hydrogen in the Erhardt geometry. PACS numbers: 34.80D, 03.65.Sq, 34.10+x Classical models and approximations are frequently used for atomic and molecular problems, despite their inherent quantum nature. One of the reasons is that our intuitive understanding is mainly based on classical terms and pictures by which we are surrounded in every day life. Another reason for not doing (fully) quantum mechanical calculations is the complexity of a problem: fully differential cross sections in higher dimensional atomic systems, e.g., often require a numerical effort still beyond present computing power. Only recently the quantum mechanical Coulomb three– body scattering problem was solved numerically [1, 2]. On the other hand, a remarkably successful classical approach to collisional atomic problems has been developed over the years, the so called Classical–Trajectory–Monte– Carlo method (CTMC). It was introduced as a purely classical model based on a “planetary atom” with a major axis of two meters (!) [3]. This model has produced reasonable results for total or energy differential ionization cross sections on atomic hydrogen [4, 5] and for other few–body Coulomb collision processes [6]. Attempts to reduce the limitation of this classical model aimed at changing the description of the initial state for the hydrogen atom from a microcanonical distribution to one, that is closer to the quantum density [7, 8]. Another idea was to introduce additional ad hoc stabilisation potentials in order to be able to treat multi–electron targets [9]. However, all attempts took as a starting point not the quantum problem but the previously formulated classical model. Hence, the proposed amendments were accompanied by inconsistencies or the need of ”fit parameters” determined from cross sections. In the end, one must say that it is up to now not possible to describe higher differential cross sections or targets with more than one active electron consistently in a classical collision framework. Letter to the Editor 2 To achieve progress in this situation we decided to go one step backwards and start with a time dependent quantum mechanical scattering formalism. By following the approximations which lead to the classical description, i.e. the CTMC method, we can identify the source and nature of deviations between the classical and the quantum result which serve as a guide to improve the classical description to a quasi–classical approximation. We divide the problem into three logically separate steps: (1) preparation of the initial state before the collision, (2) propagation in time, and (3), extraction of the cross section. For a consistent quasiclassical picture each of these steps has to be approximated in the same way. To keep the derivation transparent, we will concentrate on electron impact ionization of one active target electron in the following. Step (1): The initial wave function for the collision problem is translated first into a quantum phase space distribution by the Wigner transformation [10]. The resulting Wigner distribution is reduced to a classical distribution w(p, q) which can be propagated classically in phase space by taking the usual limit h̄ → 0. The difference to an a priori classical approach is the use and interpretation of negative parts of the distribution: Viewed as the h̄ → 0 limit of a quantum problem they contribute to the observables in the same way as the positive parts since they do not need to be interpreted as weights for real paths of classical particles. Yet, there arise additional problems when using this type of general initial phase space distributions in the usual classical framework: Most of them are not stationary under classical propagation, their Poisson bracket with the Hamilton function does not vanish, {H,w} 6= 0. Hence, the initial target distribution will look very different at the time the projectile has approached and the collision actually happens. Step (2): The formulation of the propagation is crucial since it must resolve the problem of the non–stationary classical initial distribution, as described above. Traditionally, the time dependent scattering is described by calculating the transition amplitude between initial and final state through the S–matrix, which is in turn related to the t–matrix describing directly the cross section, see, e.g., [11]. In a simplified version where the asymptotic initial and final states are eigenstates of the asymptotic Hamiltonians H (i) 0 and H (f) 0 one normally writes for the transition amplitude Sfi = lim t→∞ 〈f |U(t)|i〉 (1) where U(t) = exp[−iHt] denotes propagation with the full Hamiltonian. By a Wigner transform the quantum time evolution operator U(t) can be directly transformed with the help of the quantum Liouville operator Lq, which reduces to the classical Liouville operator Lc in the limit h̄ → 0 [12]. The latter describes the evolution of a phase space distribution w(r, p, t) according to the Poisson bracket ∂tw = {H,w} ≡ −iLcw (2) in analogy to the quantum evolution of the density matrix ρ generated by the commutator,