Incoming boundary condition

The above equations have to be complemented with rules for dealing with the singularities emerging when PHP has eigenvalues at energy E, which is the case primarily when the Wp) states are continuum states. In such a case, we denote /p) = c,E 1) and ) = Ic.ii"), with c standing for all quantum numbers related to a continuum channel (the channel index). The notation E = E - iO, serves to remind of the incoming boundary conditions used for example, for the c,E ) states one has... 

The states correspond to wave packet controlled in the far past and in the far future, respectively. Let us see what this means. In the absence of external time-dependent fields, the scattering component of the time-dependent wave function i/r(f) can be expanded in terms of either of the two sets of scattering states for example, those with incoming boundary conditions... 

The scattering states fulfilling outgoing (incoming) boundary conditions, i.e., which correspond to wave packets controlled in the past (future), are given by... 

To conclude, the scattering states with incoming boundary conditions, which correspond, in the far future, to Coulomb plane waves, are a linear combination of the states... 

As detailed in Section 5.1.2, the photoelectron distribution in a channel, identified by a parent ion a = Is, 2s, 2p is obtained by projecting the propagating wavefunction l (t) onto the helium scattering states, which satisfy incoming boundary conditions... 

Thus A + ) has the outgoing boundary condition while A ) has the incoming boundary condition and they are called scattered waves. From Eqs. (63-67), the asymptotic boundary conditions for (A ) are... 

Combining Eqs. (44 and 48) we express the incoming boundary condition wavefunction in terms of the S-matrix... 

Here, C is a constant, is the scattering wavefunction on the upper electronic potential energy surface which satisfies incoming boundary conditions, i is the transition dipole operator and is the... 

The partial wave basis functions with which the radial dipole matrix elements fLv constructed (see Appendix A) are S-matrix normalized continuum functions obeying incoming wave boundary conditions. 

Besides the resuspension of particles, the perfect sink model also neglects the effect of deposited particles on incoming particles. To overcome these limitations, recent models [72, 97-99] assume that particles accumulate within a thin adsorption layer adjacent to the collector surface, and replace the perfect sink conditions with the boundary condition that particles cannot penetrate the collector. General continuity equations are formulated both for the mobile phase and for the immobilized particles in which the immobilization reaction term is decomposed in an accumulation and a removal term, respectively. Through such equations, one can keep track of the particles which arrive at the primary minimum distance and account for their normal and tangential motion. These equations were solved both approximately, and by numerical integration of the governing non-stationary transport equations. 

The superscripts plus and minus refer to outgoing and incoming Coulomb boundary conditions respectively. Of course xt Xk exact solu-... 

Equ. (7.20) describes for an out -state the asymptotic behaviour of the stationary wavefunction. As discussed above, the characteristic property of this state is that the incoming spherical waves e Kr/r have the scattering amplitude /(-)( ). It is this minus sign in the exponential term of the incoming spherical waves which is kept as a superscript to characterize the out -state, and the relation described by equ. (7.20) is frequently called the incoming spherical waves boundary condition. Hence, one should not mix up the state with the waves. 

This expression differs from the expansion of a plane wave as given in equ. (7.14) in three respects. First, a different overall normalization is used (normalization in K-space, see equ. (7.28f)). Second, the radial functions RK( r) are different from the spherical Bessel functions j( Kr). Third, the incoming spherical wave boundary condition leads to an additional factor, b( K). 

Remembering the phase factor e ia>t, this relation expresses quantitatively the result which was used above, i.e. a plane wave can be described asymptotically as a sum of undisturbed outgoing and incoming spherical waves, which differ only in their relative phases. The boundary condition of equ. (7.20) can then be expressed as... 

This form is similar to the one presented in equ. (7.18) for a free plane wave, except for the incorporation of the incoming spherical wave boundary condition, the separate treatment of the radial function, and the normalization of these radial functions on the energy scale. Furthermore, it should be noted that equ. (7.29a) contains a j dependence of the phases and the radial function which can be understood only within a relativistic treatment. 

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