Plane wavefunction of an electron

In order to understand the wavefunction of an electron emitted from an atom by a certain ionization process, the wavefunction of a free particle with wavenumber k travelling along the positive z-axis will first be considered. The space and time dependence of this wavefunction follows from the time-dependent Schrodinger equation with zero potential1- and is given by 

The expansion into partial waves is of importance for the desired wavefunction of an emitted electron, because it provides a classification into individual angular momenta ( which refer to the centre of mass of the atom. As a starting point, the expansion of a plane wave will be considered. If a certain origin is selected and the direction of k is chosen to agree with the z-axis (the quantization axis see 

In Fig. 7.3 the individual terms of the given expansion are shown in Fig. 7.3(a) for ( = even which, because of the phase (F) in equ. (7.11), gives real functions in Fig. 7.3(h) for ( = odd which gives imaginary functions. Each individual contribution, shown on the left-hand side, is centred at the common origin and displays certain symmetry properties with respect to this origin. (This aspect becomes important if the emission of electrons is considered, for example, photoionization of an s-electron leads to only one partial wave with ( = 1, i.e., this pattern provides an approximate view for such an emission process (for the 

In a next step, the plane wave, travelling in the direction of the quantization axis z, is generalized to a plane wave moving along the direction k which differs from the z-direction (see Fig. 7.2(b)). From equ. (7.8) 

If such a wave also contains the electron spin with the projection ms along the 

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