On-the-energy shell

It should be noted that the integral equations [2] determining the elements Kp B. aE derived as an energy-variational problem, correspond also to the stationary condition of the variational functional proposed by Newton (9). Thus the K-matrix elements obeying equation [2] guarantee a stationary value for the K-matrix on the energy shell. 

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

From this starting point, the authors develop equations leading to the evaluation of the real symmetric K matrix to specify the scattering process on the repulsive surface and propose its determination by a variational method. Furthermore, they show explicitly the conditions under which their rigorous equations reduce to the half-collision approximation. A noteworthy result of their approach which results because of the exact treatment of interchannel coupling is that only on-the-energy-shell contributions appear in the partial linewidth. Half-collision partial linewidths are found not to be exact unless off-the-shell contributions are accidentally zero or (equivalently) unless the interchannel coupling is zero. The extension of the approach to indirect photodissociation has also been presented. The method has been applied to direct dissociation of HCN, DCN, and TCN and to predissociation of HCN and DCN (21b). 

We mention that TKK and fKK are equivalent on the energy shell. The Mpller operator may be determined from the Lippman-Schwinger equation... 

This paper presents an account of the dynamics of electric charges coupled to electromagnetic fields. The main approximation is to use non-relativistic forms for the charge and current density. A quantum theory requires either a Lagrangian or a Hamiltonian formulation of the dynamics in atomic and molecular physics the latter is almost universal so the main thrust of the paper is the development of a general Hamiltonian. It is this Hamiltonian that provides the basis for a recent demonstration that the S-matrix on the energy shell is gauge-invariant to all orders of perturbation theory. 

Here the differential d with d = d J = d b indicates that the requirement of energy conservation always has to be fulfilled (the matrix element is taken on-the-energy-shell ). 

X S(Eph - kinl - kin2 - ,+ +) d ph, (10.52a) where the deferential cross section which is a smooth function of the kinetic energies can be taken out of the integral as an on-the-energy-shell value. Integration over the photon energies then leads to... 

The matrix element is understood to be on-the-energy-shell i.e., the energy e of the photoelectron has to be calculated according to equ. (1.29a). Due to the different binding energies of electrons ejected from different shells of the atom, it is therefore possible to restrict the calculation of the matrix element to the selected process in the present example to photoionization in the Is shell only. As a consequence, the matrix element factorizes into two contributions, a matrix element for the two electrons in the Is shell where one electron takes part in the photon interaction, and an overlap matrix element for the other electrons which do not take part in the photon interaction (passive electrons). The overlap matrix element is given by... 

The energy of a time-independent Hamiltonian system is a conserved quantity. In this case an invariant measure can be constructed on the energy shell //(q, p) = E, that is. 

In order to obtain a statistical measure for the ionization process we have calculated for an ensemble of trajectories the fraction of ionized orbits as a function of time. The initial internal energy was chosen to correspond to a completely chaotic phase space of the He -ion if the nuclear mass were infinite. The initial conditions for the internal motion have been selected randomly on the energy shell. In Fig. 11 we have illustrated the fraction of ionized orbits as a function of time up to T = 10 a.u. for a series of different CM energies and for a very strong laboratory field strength of B = 10 a.u.. For an initial CM energy of Ec = 0.053 a.u. which corresponds to an initial CM... 

LRMT provides useful information on how the transition is approached when T(E) is less than 1 and energy is localized to a finite number of states on the energy shell. The extent of localization of molecular vibrations can be determined spectroscopically by the dilution factor, which is proportional to the inverse participation ratio for state n,... 

Expression (5.15) has a remarkable property. To find its behaviour at short distances (near zone) we expand and go to the small kR limit. There is no kR-independent term the leading term is in k2R2< giving for the short-range result, with k given its value on the energy shell... 

The first matrix element in this product is calculated on the energy shell. It could be directly obtained from a study of scattering by the potential (6.3.1). It will be admitted, in the static approximation, that the nucleus is at the origin of coordinates (rA = 0). Then the second matrix element in the product reduces to the quantity (y a) = dy[l. The collision amplitude which appears in (6.3.8) can be written... 

Thus, preparing a microcanonical ensemble involves choosing points at random on the energy shell. 

We have determined the direction of g, but not its magnitude. To find the point along the direction of h that lies exactly on the energy shell we are interested in, we let be parameterized by its magnitude s ... 

It should be noted that in the T-noninvariant Lagrangian (4.3) there are no terms with ir-meson fields. This is a consequence of the fact that the P-conserving coupling of pseudoscalar particles to nucleons on the energy shell is T-invariant [64,65]. 

Quantitatively, in the framework of many-body theory, the damping rate Fe e, that is, lifetime broadening contribution for an excited electron with energy i> Ef, can be obtained on the energy-shell approximation in terms of the imaginary part of the complex nonlocal self-energy operator E [2, 5] as... 

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