Wavefunction of an emitted electron

In the remote past, before the photon interaction took place, the state vP[c-)(r, t) 

Instead of the time-dependent description of the photoelectron wavefunction it is easier and correct to use stationary wavefunctions P(K )(r). The boundary condition for the wave packets then transforms into a boundary condition in the distance coordinate r and requires (for details see [Sta82a]) 

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. 

The boundary condition formulated in equ. (7.20) for the asymptotic behaviour of the photoelectron wavefunction has implications which not only are asymptotic, but also apply to the wavefunction F K )(r) represented in full space. In order to 

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). 

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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 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... 

This process is referred to as auto-ionization (AI). It does not follow that all AI transitions lead to completely uncorrelated carrier pairs. Most of the emitted electrons will thermalize within the Coulomb capture radius of the geminate positive ion, forming a transient charge-transfer state that can either decay to the ground state or dissociate by the absorption of ambient energy. Following Jortner (3), the wavefunction of such an AI state can be written as... 

Such fluctuations of the photon flux, emitted from a molecule, have been predicted to be due to cooperative effects (21.22). The theory is based on an idea of Prigogine and coworkers (s. e.g.(22)) who treated the irreversible part of a physical process by transforming the wavefunctions of a dissipative system into another space using a "dynamical" non-unitary representation D = exp(-iVT /fI) with a "star-Hermitian" time operator 3 and V describing the interaction of a relevant local system Hq, e.g. the complex chromophore, and the total system H, i.e. our crystal. In the new representation y>=D Y no additional time dependence is introduced, dD/dt = 0, any expectation value of an operator M=DMD should be unchanged = M> and the total Hamiltonian is transformed by 1T=DHD 1 = Hq to the local system Hamiltonian (21.22). To describe the time development in the new representation, the electron density... 

Here, n is an integer, is the frequency of the vibration, and h is known as Planck s constant This quantization of energy, as it is known, was first postulated by Max Planck in 1900 as a key part of his theory to explain the frequency distribution of radiation emitted by a black body. It is found that energy is quantized whenever a particle is confined to a small space because of the need to match the wavefunction of the particle to the space available. This applies just as much to electrons travelling around an atomic nucleus as it does to atoms vibrating in a solid. 

The absorption of a photon creates an electron-hole pair whose wavefunctions initially overlap. After the absorption process, the electron and hole thermalize to the band edge and diffuse apart. The different thermalization mechanisms in extended and localized states are reflected in the diffusion properties. In the extended states, the thermalization time, required to emit the excess energy AE as n phonons is... 

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