Unperturbed eigenvector

The next piece of evidence we have to consider is the almost universal insensitivity of calculated reaction rates when the transition probabilities in the model are varied this can be seen in diatomic dissociation [75.P1], chemical activation [72.R 77.Q], and in thermal unimolecular reactions [79.T2]. The reason for this is as follows. Since measurements are most often made at times long after the internal relaxation has ceased, the (normalised) steady distribution during the reaction is (SolilVoli, see equation (3.9). Moreover, the perturbed eigenvector To is rather similar to the unperturbed eigenvector Sq, with the dominant terms in the perturbation arising from the decay terms In fact, Tq=(1 -5)So, where... 

The eigenvalues and eigenvectors of the unperturbed hamiltonian are assumed to be known ... 

However, it is easy to verify that neither %a nor %b is an eigenvector of this unperturbed Hamiltonian, and neither are ea and eb its eigenvalues (see Example 3.18). More generally, since H/(0) is clearly Hermitian, it cannot have any non-orthogonal eigenvectors, by virtue of the theorem (note 79) quoted above. 

What is the unperturbed Hamiltonian (H//(0)) whose eigenvectors are xa and Xb an(l whose eigenvalues are ea and eb We can construct this operator explicitly from the set of /U-orthogonal vectors xa and xb ... 

Thus We are lead to extend the meaning of p.m. a second time p.m. should be regarded as still valid if we can show that the number of the almost stationary states is just equal to the number of the corresponding unperturbed eigenvalues, and that the pseudo-eigenvalues and pseudo-eigenvectors belonging to these almost... 

Explicit calculation of the coefficients. The calculation of the coefficients qf V)( t ), etG is in general very complicated, though possible by making use of the spectral formula of If. In what follows we shall restrict ourselves to the case where the initial state [Pg.72]

This is a system of inhomogeneous linear equations for the functions (vectors) T m ) (the mixed notation for the perturbation corrections to eigenvalues and eigenvectors is used above). The 0-th order in A yields the unperturbed problem and thus is satisfied automatically. The others can be solved one by one. For this end we multiply the equation for the first order function by the zeroth-order wave function and integrate which yields ... 

The problem of finding a vector is usually solved by representing the required vector as an expansion with respect to some natural set of basis vectors. Following this method one can expand the vector of the n-th order correction to the k-th unperturbed vector- 44 n terms of the solutions b p (eigenvectors) of the unperturbed problem eq. (1.51) ... 

By this, the expansion coefficients uffl are themselves of the 0-eth order in A. The restriction l / k indicates that the correction is orthogonal to the unperturbed vector. In order to get the corrections to the /c-th vector, we find the scalar product of the perturbed Schrodinger equation for it written with explicit powers of A with one of the eigenvectors of the unperturbed problem p (j k). For the first order in A we get ... 

In the previous section we described the result of turning on a perturbation on the wave functions (eigenvectors) of the unperturbed Hamilton operator with nondegenerate spectrum in the lowest order when this effect takes place. In quantum mechanics the wave function is an intermediate tool, not an observable quantity. The general requirement of the theory is, however, to represent the interrelations between the observables. For this we give here the formulae describing the effect of a perturbation upon an observable. Let us assume that in one of its unperturbed states the system is characterized by the expectation value of an observable A ... 

Let the operator p(°) project to some subspace spanned by several eigenvectors of the unperturbed Hamiltonian iP°). It is known that a set of operators projecting to a subspace of the same dimensionality and including p(°) can be parametrized in the following form [29-31] ... 

The eigenvectors of K are the same ones as those for K, and the eigenvalues just have to be multiplied by Ha. The difference with the preceding discussion is that here Ho and V are both of order e. Thus we take as the unperturbed Floquet Hamiltonian just... 

We remark that in the approach by adiabatic elimination a further approximation is implicitly made, since the eigenvectors (or the initial conditions) when it is applied to dynamics are not transformed with e W. This amounts to the approximation esWl = 11 + eW + 1. This does not produce a big difference when adiabatic elimination is applied to adiabatic processes with laser pulses, since, as we will see in Section V, the initial and final eigenvectors of the perturbed Hamiltonian coincide with those of the unperturbed one. 

The coefficients are the eigenvector components of the Fock operator of the unperturbed (Est = — 0) polymer). The new matrices A OT are also cyclic hypermatrices and can be block-diagonalized in the same way as before. Therefore, one obtains finally again equation (86) but on the r.h.s. we now have... 

We can now insert this expansion into the first-order equation and multiply the resulting equation by each c in turn. The orthonormality of the c° and the fact that they are eigenvectors of the original unperturbed equation gives an expression for each d, ... 

Again writing the expansion of the first-order correction in terms of the eigenvectors of the unperturbed Hartree-Fock matrix ... 

Let eA be the perturbation on the unperturbed matrix A q, where e is some parameter. Expanding the perturbed eigenvectors and eigenvalues in power series in s, one obtains ... 

---