Hermitian assumption

The first-order state /i) is therefore obtained simply by applying Fq to the zeroth-order state, with the caution of rejecting /o)(/ol Fq j/g), because the contribution coming from the state /q) is already includ in the basis set. We could go on that way, as this is just how Mori proceeds in his celebrated papers. However, to avoid the Hermitian assumption made by Mori, we build up a biorthogonal basis set. This means that the state /i) has to be associated with the corresponding left state (we assume (/q = (/o )... 

However, if one were to exactly follow what seem to be Pecora s assumptions about the scalar product being hermitian, one would get a different result from Pecora when counting the number of real conditions on the complex P matrix, arising from the constraint + = In In fact, when the + matrix is considered to be hermitian, the normalization condition on the N complex diagonal elements of QQ+ yields N real conditions and not 2N as Pecora seemed to tacitly suppose. This is due to the fact that the diagonal elements are already known to be real since + is hermitian, and hence, Im = 0 is not a separate constraint. 

The basic assumption made by Schwinger is that the infinitesimal generating operator 3iP i2 is obtained by a variation of the quantities contained in a Hermitian operator 12 which, because of the additive requirement given in eqn (8.76), must have the general form... 

In addition to the equidistant distribution of the nodal points the Lagrange interpolation polynomial are defined via the wave-function not taking into account its derivative. Hermitian interpolation polynomials are in addition defined by the assumption that value and derivation of the wave-function are correct at the nodal points. Hence we get the following ansatz... 

Multiphoton processes taking place in atoms in strong laser fields can be investigated by the non-Hermitian Floquet formalism (69-71,12). This time-independent theory is based on the equivalence of the time-dependent Schrodin-ger description to a time-independent field-dressed-atom picture, under assumption of monochromaticity, periodicity and adiabaticity (69,72). Implementation of complex coordinates within the Floquet formalism allows direct determination of the complex energy associated with the decaying state. The... 

Tlie 4m + 2 rule actually does not depend on the Huckel assumptions (16.18) to (16.21). The C H tt MOs (16.51) were derived solely by symmetry considerations and are the correct SCF minimal-basis-set tt MOs. With k = 0, we have an MO with all plus signs in front of the AOs. Clearly this MO has a lower energy than any of the others. For the remaining MOs, the pair with k=j and k = nc j ate complex conjugates of each other and must have the same energy. (Since H is Hermitian, we have... 

The presented algorithm works if and only if the pivot entry k, k) is zero in none step k,k =, ..., n-, or, equivalently, if and only if for none fc the (n - k) x ( - k) principal submatrix of the coefficient matrix A is singular. (Apx p submatrix of A is said to be the principal if it is formed by the first p rows and by the first p columns of A.) That assumption always holds (so the validity of the above algorithm is assured) in the two important cases where A is row or column diagonally dominant and where A is Hermitian (or real symmetric) positive definite (compare the list of special matrices in Section II.D). (For example, the system derived in Section II.A is simultaneously row and column diagonally dominant and real symmetric multiplication of all the inputs by -1 makes that system also positive definite. The product V V for a nonsingular matrix y is a Hermitian positive definite matrix, which is real symmetric if V is real.)... 

We have used non-Hermitian scattering theory to calculate the transition probability amphtude, within the framework of the complex adiabatic approach. It should be stressed that non-Hermitian quantum mechanics allows us to use complex adiabatic potential energy surfaces in cases where one has to go beyond the adiabatic approximation in Hermitian quantum mechanics [1,2]. In the adiabatic approximation, we assume that the motion in the y direction is much slower than in the x direction. This assumption is based on the geometry of the two-dimensional potential surface (see Fig. 5). 

Next, we consider the resonance integrals or bond integrals H12, H23, and //13. (The requirement that be hermitian plus the fact that the x s and Hjt are real suffices to make these equal to i/21, 7/32, and //31, respectively.) The interpretation consistent with these integrals is that H 2, for instance, is the energy of the overlap charge between XI and X2- Symmetry requires that Hn = H23 in the allyl systena. However, even when symmetry does not require it, the assumption is made that all Hj are equal to the same quantity (called P) when i and j refer to neighbors (i.e., atoms connected by a bond). It is further assumed that Htj = 0 when i and j are not neighbors. Therefore, in the allyl case,... 

On the assumption that quantities A are real (as is the case when L is a Hermitian operator) and that A is real, it follows from equation (4.10) that... 

The matrix of the operator of the perturbation energy is therefore not Hermitian. The secular problem (34) has previously been investigated in detail — however, with the tacit assumption that is Hermitian, as is usually the case. But we... 

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