Coupled-channel Schrodinger equation

Johnson, B.R., Renormalized Numerov method applied to calculating bound states of coupled-channel Schrodinger equation, J. Chem. Phys., 69, 4678 688, 1978. 

Typically, however, the more channels there are that are strongly coupled in an inelastic collision the better it is to approximate the dynamics by classical mechanics i.e., there are more channels the heavier the particles, but this is also the limit in which classical mechanics is a better approximation. Thus there have been many classical trajectory simulations of inelastic collision processes [71]. These have the advantage that no approximations other than the use of classical mechanics need be made, and the number of classical equations of motion to be solved (Hamiltonian s equations) grows linearly with the number of particles, while the numbers of coupled channels in the coupled-channel Schrodinger equation grows exponentially with this number. 

A little more complicated system is the de-excitation of He(2 P) by Ne, where the deexcitation is dominated by the excitation transfer and only a minor contribution from the Penning ionization is involved. The experimental cross section obtained by the pulse radiolysis method, together with the numerical calculation for the coupled-channel radial Schrodinger equation, has clearly provided the major contribution of the following excitation transfer processes to the absolute de-excitation cross sections [151] (Fig. 15) ... 

To capture the essence of the Feshbach resonance phenomenon, we will need to understand what happens to the ground vibrational state 4>o(R) of the ground electronic state, also depicted in Figure 1.13, because of the interaction with the continuum of states excited electronic state. The physical process described above can be formulated as a two coupled channels problem where the solution irg(R) in the closed channel (the ground state) depends on the solution ire(R) in the open channel (the excited state) and vice-versa. The coupled Schrodinger equations read... 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

Using the orthogonality of the number states ]N) along with Eqs. (12.15), (12.33), and (12.34), we transform the Schrodinger equation [Eq. (12.24)] into at set of coupled differential equations, the so-called coupled-channels equations. ... 

Carrington, Teach, Marr, Shaw, Viant, Hutson and Taw [211] used a coupled-channel theory, described by Hutson [217], in order to solve the Schrodinger equation, written in the form... 

If we substitute the right-hand side of (10.164) into the total Schrodinger equation (10.162) and project onto a single case (e) basis function VtJ/V we obtain a set of coupled differential equations for the channel wave fimctions XaJLSjaR W-As we ave seen above, for each J and s there are three coupled equations corresponding to the allowed values of Ja and R. These equations may be written in matrix notation,... 

A review is given on the application of the coupled-channel method for the calculation of the electronic energy loss of ions as well as ionization in matter. This first principle calculation, based on the solution of the time-dependent Schrodinger equation, has been apphed to evaluate the impact parameter and angular dependence of the electronic and nuclear energy losses of ions as well as the ionization due to high-power short laser pulses. The results are compared to experimental data as well as to other current theoretical models. 

Virtually all non-trivial collision theories are based on the impact-parameter method and on the independent-electron model, where one active electron moves under the influence of the combined field of the nuclei and the remaining electrons frozen in their initial state. Most theories additionally rely on much more serious assumptions as, e.g., adiabatic or sudden electronic transitions, perturbative or even classical projectile/electron interactions. All these assumptions are circumvented in this work by solving the time-dependent Schrodinger equation numerically exact using the atomic-orbital coupled-channel (AO) method. This non-perturbative method provides full information of the basic single-electron mechanisms such as target excitation and ionization, electron capture and projectile excitation and ionization. Since the complex populations amplitudes are available for all important states as a function of time at any given impact parameter, practically all experimentally observable quantities may be computed. 

Although the Schrodinger equation associated with the A -i- BC reactive collision has the same form as for the nonreactive scattering problem that we considered previously, it cannot be solved by the coupled-channel expansion used then, as the reagent vibrational basis functions cannot directly describe the product region (for an expansion in a finite number of terms). So instead we need to use alternative schemes of which there are many. 

To develop coupled-channel methods to solve the Schrodinger equation, we first transform the Hamiltonian (A3.11.81) to hyperspherical coordinates, yielding ... 

Substituting into the Schrodinger equation, multiplying by " " and integrating over r, shows that the unknown functions F (R) for the relative motion in channel n satisfy the infinite set of coupled equations... 

Such quantum capture theories in three dimensions have been developed to solve the Schrodinger equation for the long-range attractive entrance channel potential in a coupled rotational-states formalism. A rotationally adiabatic approximation to this theory has been developed by constructing potential curves that describe the evolution of... 

Substituting the expansion (1.36) into the total Schrodinger equation and projecting onto a single channel function ct) (t) yields a set of coupled equations. 

Hutson, J.M., Coupled-channel methods for solving the hound-state Schrodinger equation, Comput. Phys. Commun., 84, 1-18,1994. 

Cold alkali metal atoms have a variety of magnetically tunable resonances that have been exploited in a number of experiments to control the properties of ultracold quantum gases or to make cold molecules. For the most part, experiments have succeeded with species that either do not have inelastic loss channels, or, if they do, the loss rates are very small. Thus, for practical purposes, we can set the resonance decay rate yc = 0 in examining a wide class of magnetically tunable resonances. While general coupled channel methods can be setup to solve the multichannel Schrodinger equation [1], we will use simpler models to explain the basic features of tunable Feshbach resonance states. 

Several other approaches have been explored for collisions involving polyatomics. In particular we want to mention reviews emphasizing a quasiclassical treatment of energy transfer into polyatomics [41], a semiclassical coupled-channels approach for polyatomics [42,43], quantal treatments where slow (usually rotational) degrees of freedom are treated in a sudden-collision approximation [44], and approaches based on the solution of the time-dependent Schrodinger equation for scattering wave-packets [45-50], No attempt will however be made to review the extensive literature on molecular collisions. This has been periodically done in publications of reviews and workshop lectures [51-55]. 

The APH surface functions axe also sector adiabatic . When the APH wavefunc-tion of Eq. (37) is substituted into the Schrodinger equation, the resulting exact Coupled Channel or Close Coupling (CC) equations axe of the form... 

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