Stationary Perturbations

The physical reality shows that systems, even at microscopic level, are not isolated, but subjects of various perturbations. While temporal perturbations were treated within causal motion by the preceding postulate, the stationary ones are the subject of the present discourse  

The stationary perturbations are those that do not affect the system structure but only its eigen-spectrum (eigen-energies and eigen-junctions). 

Analytically, this postulate may be formulated through the coupling Hamiltonian  

However, the problem is to express the corrected terms of perturbation is terms of eigen-energies and wave functions of the non-perturbed (isolated) system, assumed with the complete determined solution. Yet, note that the perturbed eigen-function are not per se normalized, the normalization procedure being reload for each order the preservative approximation is considered. 

The general algorithm of finding eigen-energies and states as well as some basic atomic, molecular and free solid states applications follows. 

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Kutzelnigg, W., Stationary perturbation theory. Theor. Chim. Acta (1992) 83 263-312. 

We have derived Eq. (11-36) with mixing of the singlet and triplet states of a radical pair in mind, but it is quite general for the time evolution of two levels separated by an energy under a stationary perturbation. In order to get the dynamic behavior of a radical pair, we should add a diffusion operator D and an operator K for the chemical reaction to Eq. (11-29),... 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

Since in the Floquet representation the Hamiltonian K defined on the enlarged Hilbert space is time-independent, the analysis of the effect of perturbations (like, e.g., transition probabilities) can be done by stationary perturbation theory, instead of the usual time-dependent one. Here we will present a formulation of stationary perturbation theory based on the iteration of unitary transformations (called contact transformations or KAM transformations) constructed such that the form of the Hamiltonian gets simplified. It is referred to as the KAM technique. The results are not very different from the ones of Rayleigh-Schrodinger perturbation theory, but conceptually and in terms of speed of convergence they have some advantages. 

For very small field amplitudes, the multiphoton resonances can be treated by time-dependent perturbation theory combined with the rotating wave approximation (RWA) [10]. In a strong field, all types of resonances can be treated by the concept of the rotating wave transformation, combined with an additional stationary perturbation theory (such as the KAM techniques explained above). It will allow us to construct an effective Hamiltonian in a subspace spanned by the resonant dressed states, degenerate at zero field. 

The mechanism of the TP transition, which is principally a second-order process, involves an ED transition from the initial state to an intermediate state (with or without resonance), and thence another from the intermediate state to the final state [221]. However, the states and energies in Eqs. (50)-(52) are only eigenstates and eigenvalues of the zero-order Hamiltonian H0 in a stationary perturbation treatment, in which the complete Hamiltonian is... 

A compact formulation of stationary perturbation theory in the non-relativistic theory has been given [72] in a Lie-algebraic language. One of the essential messages of stationary perturbation theory is that under a certain condition the essential theorems of exact perturbation theory, (e.g. that the first order energy correction is equal to the expectation value of the perturbation with the unperturbed wave function) remain valid. The condition is that all perturbation corrections are formulated in terms of a variational group , with respect to which the unperturbed energy expectation value is stationary. 

Figure 5-44. Energy efficiency of plasma-chemical CO2 dissociation (without cost of compression) as a function of specific energy input at different Mach numbers (1) M

Last case here is to consider also a stationary perturbation, i.e.,... 

Other special appearances of the stationary perturbations are to be exposed in what following for some paradigmatic physical situations. 

This results says that from above two free wave fimetions that onee considered previously as sin plays the role of the lower ground state, while that with eos is the excited one in the erystal configuration moreover, under the potential perturbation they separate in what is usually known in solid state theory as being the valence and conduction bands. We success therefore to construct this more realistic picture of the solid state crystals with the useful tool as stationary perturbation algorithm is. 

In the limit where H o can be treated as a stationary perturbation, the energy corrected to first order becomes... 

For the sake of completeness of our discussion, the perturbation on matrix A will next be briefly consid< red. The perturbation may be due to the deviation from the perfect absorption at the N - - I)th well. It may also be due to the fact that A has non-zero next-nearest-neighbor transition probabilities, or the transition probabilities may be dependent on concentration and time in such a way that the additional elements in a perturbed matrix are small compared with the corresponding unperturbed matrix. Only the stationary perturbation will be considered here. The analogous case for time-dependent perturbation follows in a manner similar to that used in quantum mechanics. 

The following stationary perturbation theory on the matrix A is also derived from the direct analogy of the present case to that of quantum mechanics. ... 

The M0ller-Plesset perturbation theory [26] corresponds to the application of the stationary perturbation theory to the calculation of the correlation energy using the Hartree-Fock Slater determinant as the zeroth order wavefunction. These methods are denoted MPn where n is the order of the perturbative corrections included. In the M0ller-Plesset method, the unperturbed Hamiltonian operator is chosen as a sum of Fock operators... 

A perturbation calculation starting from an approximate wave function (such as T hf) is susceptible to ambiguities and large errors (see [39, p. 18]). In order to obtain reliable results, it is necessary to stabilize the solutions with respect to a certain class of variations. This approach is called stationary perturbation theory its Hartree-Fock level version is called Coupled Hartree-Fock (CHF). 

Perturbation theory in general is a very useful method in quantum mechanics it allows us to find approximate solutions to problems that do not have simple analytic solutions. In stationary perturbation theory (SPT), we assume that we can view the problem at hand as a slight change from another problem, called the unperturbed case, which we can solve exactly. The basic idea is that we wish to find the eigenvalues and eigenfunctions of a hamiltonian H which can be written as two parts ... 

Problem about disintegration of streams by means of consideration of stability of the given current of a liquid. Mathematical research of stability of motion in relation to perturbations can be solved by means of the motion equations (Vitman, 1961 and Pirumov, 1961). With that end in view on stationary main current the non-stationary perturbation is imposed so that resulting motion fulfilled to the motion equations. At an outflow velocity having practical interest, gravity effects on liquid motion can to be considered. In this case on a liquid stream forces of viscosity, a superficial tension and a seepage force act [29-35]. 

The excited atom is assumed to emit a wave shifted in frequency by equation (8.17) and the lineshape for the sample as a whole is obtained by averaging over the probability distribution of the stationary perturbers. This approximation is used to calculate the Stark broadening produced by ions in a plasma and for neutral atom broadening at pressures s 100 Torr. From equations (8.19) and (8.20) we see that the quasi-static approximation is likely to be better at high densities, where T is short, and at low temperatures. 

We consider first the interaction of an excited atom at r=0 with a single stationary perturber. The probability that the perturber lies in the spherical shell between r and r+dr is given by... 

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