Hamiltonian continuum eigenstates

Figure 1.8 The probability density of several continuum eigenstates of the Hamiltonian in Eq. (24) plotted on the baseline of their corresponding energy. The potential is also plotted for convenience. Note that most continuum states (dashed lines) are delocalized and have a very small amplitude inside the potential well between the two barriers, whereas there are continuum functions that are localized inside the well. The localized eigenstate (solid line) is the same as shown in Figure 1.1.

Let us denote the excited state after initial excitation by x,) with quantum numbers denoted by the composite index i, which is assumed to be in a particular vibrational state v,. The continuum eigenstate of the full Hamiltonian H is denoted as whose quantum numbers are given by the composite index /. The probability per unit energy that the system will end up in the continuous state / is given by... 

A continuum eigenstate of the total Hamiltonian h with eigenvalue E and degeneracy index jS will then be written as a superposition of the unperturbed eigenstates... 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

Let us start with zero approximation states of H0 consisting of the discrete states (Xx, X2), 2(Xls X2),..., n( -i, X2) and continuum states time-independent) eigenstates of the physical system are obtained by diagonalizing the total Hamiltonian in this representation, and can be displayed as a superposition of these zero-order states. For the sake of simplicity we consider just one zero-order... 

Complex rotation can be usefully applied also to the case of the interaction of an atom with a time-dependent perturbation. With the Floquet formalism by Shirley [41], it was shown that, for a time-periodic field, the dressed states of the combined atom-field system can be characterized non-perturbatively by the eigenstates of a time-independent, infinite-dimensional matrix. The combination of the Floquet approach with complex rotation, proposed by Chu, Reinhardt, and coworkers [37, 42, 43], permits to account for the field-induced coupling to the continuum in an efficient way. As in the time-independent case, this results in complex eigenvalues (this time to the Floquet Hamiltonian matrix) and again the imaginary parts give the transition rate to the continuum. This combination has since then been successfully used to examine various strong field phenomena a review can be found in Ref. [44]. 

The diagonalisation of the hydrogen Hamiltonian in a Slater-function (4.38) basis has been reviewed by Callaway (1978) in the context of variational solutions of the integrodifferential equations. This basis has useful features. The inclusion of all the Slater functions necessary for the radial eigenstate u r) produces exact eigenstates for principal quantum numbers up to n in the / manifold. The remainder are pseudostates, which represent the higher discrete states and the continuum. Since Slater functions are not orthonormal there are linear-dependence difficulties that severely limit the size of the basis for which the diagonalisation is numerically feasible. 

R continuum), and in the intennediate well. For example, for energies lower than the top of the bander these can be taken as eigenstates of Hamiltonians defined with the potentials (a), (b), and (c) shown on the right of Fig. 9.5. The results obtained above then correspond to the case where direct coupling between the L and R states can be disregarded and where it is assumed that the intermediate well between the two barriers can support only one state 11). Jq r, Eq. (9.87), and. /o > / (Eq. (9.87) with L replacing R everywhere) are then the transmitted and reflected fluxes, respectively, associated with one state 0) in the L continuum. 

Of course, the Dirac operator for H-like ions has continuum states as well, including ultrarelativistic ones. One can therefore neither expect that all eigenstates are analytic in c , nor that the entire Dirac operator allows an expansion in powers of c. This can at best be the case for the projection of D to positive-energy non-ultrarelativistic states. The paradigm, on which the Foldy-Wouthuysen transformation is based, to construct a Hamiltonian, related to the Dirac operator by a unitary transformation, in an expansion in c ... 

The experimentally achievable localized excitations are typically described by one of the zero-order basis states (see Section 3.2), which are eigenstates of a part of the total molecular Hamiltonian. Localization can be in a part of the molecule or, more abstractly, in state space . The localized excitations are often described by extremely bad quantum numbers. The evolution of initially localized excitations is often more complex and fascinating than an exponential decay into a nondescript bath or continuum in which all memory of the nature of the initial excitation is monotonically lost. The terms in the effective Hamiltonian that give birth to esoteric details of a spectrum, such as fine structure, lambda doubling, quantum interference effects (both lineshapes and transition intensity patterns), and spectroscopic perturbations, are the factors that control the evolution of an initially localized excitation. These factors convey causality and mechanism rather than mere spectral complexity. 

Later, when making comparisons with nonrelativistic calculations, we subtract the electron rest energy mc firom e. ) The choice of the potential U r) is more or less arbitrary one important choice being the (Dirac) Hartree-Fock potential. Eigenstates of Eq. (1) fall into three classes bound states with —mc < k < rn< , continuum states with > m( , and negative energy (positron) states ej < —me . Since contributions from virtual electron-positron pairs are projected out of the no-pair Hamiltonian, we will be concerned primarily with bound and continuum electron states. 

One can use other general considerations to study the problem of polarizabilities and there are two common methods for doing so. One is based on the sum over eigenstates to the unperturbed Hamiltonian and is usually slowly converging because of the contributions from the continuum [6,14]. The other one is based on operator inequalities and can yield upper and lower bounds to the polarizabilities [llj. From the theory of operator inequalities [11,16] for the dipole polarizability in three dimensions we can write the ot2)zz component of the polarizability tensor in the form... 

Expansion in a finite set of discrete pseudostates evidently converges much faster than expansion in true eigenstates of Hq (which would involve also continuum states). The same behaviour is observed for the expansion in eigenstates of Fq (the one-particle Fock Hamiltonian), so that convergence of conventional TDHF methods may prove rather slow, especially with large bases of atomic orbitals... 

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