Wave function scattering

The energy and state resolved tiansition probabilities are the ratio of two quantities obtained by projecting the initial wave function on incoming plane waves (/) and the scattered wave function on outgoing plane waves [F)... 

Section IIC showed how a scattering wave function could be computed via Fourier transformation of the iterates q k). Related arguments can be applied to detailed formulas for S matrix elements and reaction probabilities [1, 13]. For example, the total reaction probability out of some state consistent with some given set of initial quantum numbers, 1= j2,h), is [13, 17]... 

The complex scattering wave function can be specified by nodal points at which u = 0,v = 0. They have great physical significance since they are responsible for current vortices. We have calculated distribution functions for nearest distances between nodal points and found that there is a universal form for open chaotic billiards. The form coincides with the distribution for the Berry function and hence, it may be used as a signature of quantum chaos in open systems. All distributions agree well with numerically computed results for transmission through quantum chaotic billiards. 

The typical way to open a billiard is to attach some reservoirs with continuous energy spectrum, for example, the leads or microwave waveguides, as shown in fig. 3 below. Full information about the scattering properties of the billiard is given by the scattering wave function which is a solution of the Schrodinger equation Hip = Exp with the total Hamiltonian... 

The scattering wave function Hpj = Etp can be mapped onto the interior space of the billiard by the projection operator ipB = PbV where Pb = J2n n)(n Then this truncated scattering wave function can be expanded in the eigenfunctions of the closed billiard ipn(x,y) (A.F. Sadreev et.al., 2003)... 

Even for the resonant transmission through the Sinai billiard, computations show that many eigenfunctions contribute to the scattering wave function as shown in fig. 1. An assumption of a complex RGF for the scattering function (9) means that the joint probability density has the form... 

Using the relation, the binary density matrix in momentum representation may be expressed in terms of scattering wave functions. 

Asymptotically, the system consists of two isolated atomic systems. This suggests an expansion of the electronic scattering wave-function on a two-center atomic basis... 

The other source of a channel phase is the complex continuum wave function at the final energy E. At first it would appear from Eq. (15) that the phase of ESk) should cancel in the cross term. This conclusion is valid if the product continuum is not coupled either to some another continuum (i.e., if it is elastic) or to a resonance at energy E. If the continuum is coupled to some other continuum (i.e., if it is inelastic), the product scattering wave function can be expanded as a linear combination of continuum functions,... 

For an exact scattering wave function, the coefficients c/is in Eq. (8.1) would be determined as linear functions of the matrix elements otips, since the algebraic equations are linear. By factoring these coefficients... 

This is the reciprocal of the logarithmic derivative of the wave function Xdiq) in the dissociation channel, for q = qd. At given total energy E these R-matrix elements are matched to external scattering wave functions by linear equations that determine the full scattering matrix for all direct and inverse processes involving nuclear motion and vibrational excitation. Because the vibronic R-matrix is Hermitian by construction (real and symmetric by appropriate choice of basis functions), the vibronic 5-matrix is unitary. 

Molecular-orbital approaches to edge structures differ for semiconducting and isolating molecular complexes. The latter and transition-metal complexes allow one to minimize solid-state effects and to obtain molecular energy levels at various degrees of approximation (semiempirical, Xa, ab initio). The various MO frameworks, namely, multiple-scattered wave-function calculations (76, 79, 127, 155) and the many-body Hartree-Fock approach (13), describe states very close to threshold (bound levels) and continuum shape resonances. 

For a central potential it is natural to expand the scattering wave function X (k,r) in orbital-angular-momentum eigenstates. We first consider the case of zero potential where the wave function is a plane wave (4.42), which takes the form... 

The phase of the scattering wave function for a real potential is constant and equal to the phase shift. 

The scattering wave function in the outer region in a particular dissociation channel a consists of an incoming wave in this channel, and outgoing waves in all open channels, ... 

Equations (30) and (31) complete the construction of the scattering wave function in the continuum. According to Eq. (30), the expansion coefficients over the basis functions in the interaction region are uniquely determined by the boundary values of the normal derivatives f. On the other hand, in each dissociation channel (3 one has = (this is the matching condition). Thus, the derivatives / can be evaluated, using Eq. (22), from the solutions F in the outer region by exploiting the matrix Y from Eq.(31). ... 

In the time-independent scattering formalism, the partial cross section is proportional to the square of the overlap between the scattering wave function in channel j, n [cf. Eq. (17)] and the initial molecular state o-The distributions Pn E) and Pf E) can be estimated directly from the solutions of the time-independent Schrodinger equation described in 4.2. In general, one obtains different PSD s for different initial states fo of the molecule. 

The superposition of the outgoing and incoming waves alters the wave function of the photoelectron at the site of the given emitter. The final state wave function, /), becomes q> + (psc) where (pe and (pscSXQ the emitted and scattered wave functions. The photoelectron absorption process involves the final state wave function and the quantum mechanical expression for the X-ray absorption coefficient is given by. 

Table II illustrates another aspect of the inward propagation, with P, as the initial ratio. A longer path is needed to recover P,. Thus the initial value does not really matter. This is to be understood as due to the different behaviour of the ingoing and outgoing components of the scattering wave function. Even with an initial ratio contaminated by the ingoing component, the outgoing component which increases in an inward propagation will always dominate. For...

In scattering theory, it can be proven that the scattered wave function, outside the effective radius from the scattering potential, should look like [39] ... 

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