Scattering theory boundary conditions

The KR variational principle determines a wave function with correct boundary conditions at a specified energy, the typical conditions of scattering theory. Energy values are deduced from consistency conditions. 

In classical physics we are familiar with another kind of stationary states, so-called steady states, for which observables are still constant in time however fluxes do exist. A system can asymptotically reach such a state when the boundary conditions are not compatible with equilibrium, for example, when it is put in contact with two heat reservoirs at different temperatures or matter reservoirs with different chemical potentials. Classical kinetic theory and nonequilibrium statistical mechanics deal with the relationships between given boundary conditions and the resulting steady-state fluxes. The time-independent formulation of scattering theory is in fact a quantum theory of a similar nature (see Section 2.10). 

In the quantum mechanical treatment of stationary scattering theory the wave function pk(r) corresponding to the wave number k2 = l E/h2 must obey the boundary condition... 

Fig. 7a. Plot of light scattering intensity, ocS(q, t) vs wavenumbers q for different times after the quench from To = 75 °C to T — 49 °C, for the same mixture as in Fig. 6a). From Bates and Wilzius [36], b Structure factor S(q, t) vs wavenumber q for different times after a quench from infinite temperature (i.e. the Initial state is an ather-mal blend) to kBT/t = 1.0, for N = 32, <[>, = 0.6, eAB = e, eAA = cBB = 0, and averages over 40 quenches in a simple cubic lattice of size 40 x 40 x 40 (with periodic boundary conditions) are taken. Arrow shows the prediction for qm of the linearized theory of spinodal decomposition. Time t after the quench is measured in attempted moves per monomer. From Sariban and Binder [155]...

The Korringa, Kohn3 Rostoker (KKR) method [1.17,18] employs an expansion inside the MT spheres similar to the cellular and APW methods. In the interstitial region between the spheres, however, the potential must be flat and the wave functions are expanded in phase-shifted spherical waves. The boundary condition can then be expressed as the condition for self-consistent multiple scattering between the muffin-tin spheres, or alternatively as the condition for destructive interference of the tails of these waves in the core region (Sect.2.1). This is the other most widely used computational technique in band theory. 

In the next section (Sec. 2), we will develop the theory of the BCRLM. We discuss the solution of the coupled-channel equations in both natural collision coordinates " and hyperspherical coordinates. " Both coordinate systems are widely used to treat collinear reactive scattering processes. We will discuss the projection " of the hyperspherical equations on coordinate surfaces appropriate for applying scattering boundary conditions and review the definition of integral and differential scattering cross sections in this model. 

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