Hamiltonian channel

N pW0c) the channel hamiltonian Hp — QpHQp,i.e. the hamiltonian projected... 

Since these formal bases, which are supposed to describe the true continuum background, will be represented upon finite sets, all the qnantities which must be interpolated from these representations (i.e. matrix elements and phaseshifts) must be smooth functions of the energy index this reqnires a snitable redefinition of the channel hamiltonian Hp if this supports narrow shape resonances. 

Another approach widely used for nonadiabatic reactions is the diabatic one. The channel Hamiltonians Hex and H determining the zeroth-order Born-Oppenheimer electron states of the donor A and acceptor B and the perturbations Vt and Vf leading to the forward and reverse electron transitions, respectively, are separated... 

Long after the scattering several arrangement channels can be populated, with channel Hamiltonians H13 (/ = 1,2,...), and... 

The total Hamiltonian H of the projectile—target system is partitioned into a projectile—target potential V, whose range must be short compared with the Coulomb potential, and channel Hamiltonian K. In the case of a charged target there is a residual Coulomb potential Vc, which is subtracted from V and added to K. 

The channel Hamiltonian governs the system at macroscopic separation distances, in particular the injection and detection distances of the experiment. It consists of the kinetic-energy operator Kq of the electron and the Hamiltonian Hr of the target. 

The differential cross section for ionisation is given by (6.60). To formulate the T-matrix element we partition the total Hamiltonian H into a channel Hamiltonian K and a short-range potential V and use the distorted-wave representation (6.77). The three-body model is defined as follows. 

The channel Hamiltonian K (10.7) is separable in the electron coordinates. We define the following one-electron states. 

We return now to the coupled-channels approach based on operator equations. The formalism is adequately covered in several books (see e.g. Goldberger and Watson, 1964 Newton, 1966 Levine, 1969) and we shall only present the main equations. Assume for simplicity that only two reaction channels a (for A + BC)and 6(AB + C) exist. The total Hamiltonian H may be split into two terms, a channel Hamiltonian Hc for free motion and a channel interaction Vc, with c a, b. If a is the initial free state and we want the scattering states in channel a, i.e. those in the absence of rearrangement, then the Lippmann-Schwinger equation gives... 

Here //bg is referred to as the bare entrance-channel Hamiltonian describing, in the hypothetical absence of interchannel coupling, the configuration of Zeeman states in which the individual atoms of -wave interacting pairs in an ultracold gas are prepared. The potential V),g(r) thus mimics a bare entrance-channel interaction. 

Two-channel approaches can be used when the entrance channel is coupled significantly only to a single Zeeman-state configuration of atom pairs that belongs to an energetically closed channel. Accordingly, the bare closed-channel Hamiltonian, describes such atom pairs in the hypothetical absence of interchannel... 

Given the assumption that the interaction between the channels is caused mainly by (t)res). the closed-channel Hamiltonian might be replaced simply by the following one-dimensional term ... 

Here the resonance State I ( )res> is assumed to be unit normalized, that is ((t)res ( )res) = 1. Such a replacement in a two-channel Hamiltonian shall be referred to in the following as a single-resonance or configuration-interaction [3] model. Contrary to the two-channel two-potential method, Equation 11.18 always allows the stationary Schrodinger equation associated with the Hamiltonian of Equation 11.14 to be solved analytically in terms of the bare bound and continuum states of Hbg [3,59]. 

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