Eigenvalue equations relation

That relation follows from the fact that the operator X2 + Y2 has positive eigenvalues. To see this, write the eigenvalue equations for X, Y, and X2 + Y2 ... 

As one can see, for each set of parameters Z, k, L and d, one can select a in such a way that Eqs. (21) and (22) are fulfilled. Then, the resulting expectation values correspond to the variational minima and are equal to the appropriate exact eigenvalues of either Dirac or Schrodinger (or rather Levy-Leblond) Hamiltonian. However the corresponding functions do not fulfil the pertinent eigenvalue equations they are not eigenfunctions of these Hamiltonians. This example demonstrates that the value of the variational energy cannot be taken as a measure of the quality of the wavefunction, unless the appropriate relation between the components of the wavefunction is fulfilled [2]. 

Here, the Hamiltonian HXot is given by Eq. (19) or (20). Next, it is suitable to introduce twice a closeness relation using the eigenstates of the Hamiltonians (19) or (20), which are both obeying the eigenvalue equation ... 

Moreover, using the eigenvalue equation (A.l) of the molecular Hamiltonian, and then suppressing the closeness relation between the dipole moment operator, the SD is transformed into... 

Quite often chemical engineering systems are encountered with widely different time constants, which give rise to both long-term and short-term effects. The corresponding ordinary differential equations have widely different eigenvalues. Differential equations of this type are known as stiff systems. Seader and Henley (1988) derived the expressions for maximum and minimum eigenvalues for the differential component mass balance equations related to intermediate plates and reboiler respectively. 

This is known as the Jaffe series [117]. The important recurrence relations (6.468) can be represented by the matrix eigenvalue equation [118]... 

In all cases the vector operator and its components are related to the quantum numbers by eigenvalue equations analogous to ... 

Very recently Sinha et. al /13 6/ have analytically shown that the CC—based LRT for IP is algebraically equivalent to the CC formalism of Haque and Mukherjee/69/ for one-hole valence problem. This indicates that the CC equations for determining the IP s can be cast into an equivalent eigenvalue equation. In fact Sinha et. aj /136/ have demonstrated that the relation between the CC-LRT for IP and the errresponding CC theory is the same as that between a Cl problem and the associated partitioned problem as obtained by the Soliverez transformation/137/. For open-shells containing more than one valence, the correspondence between the CC-LRT and the CC equations no longer holds, but Sinha et. al showed the CC... 

Substituting the state vector f) in (20), the above equations provide the relations among Cj f) and Cj (/). There are two ways to couple the two components Cj f) and Cj (/). If we couple the X component to the nearest-neighbor Y component and vice versa, we obtain a set of eigenvalue equations given by... 

Although for the sake of clarity the previous discussion was limited to the case of a binary mixture, these results are easily generalized to the study of an n-component mixture. Because of the coupling between the mobile phase components, the velocity eigenvalues are related to the slopes of the tangents to the n-dimensional isotherm surface, in the n composition path directions. These slopes can be calculated when the isotherm surface is known. Conversely, systematic measurement of the retention times of very small vacancy pulses for various compositions of the mobile phase may permit the determination of competitive equilibrium isotherms, but only if a proper isotherm model is available. Least-squares fitting of the set of slope data to the isotherm equations allows the calculation of the isotherm parameters. If an isotherm model, i.e., a set of competitive isotherm equations, is not available, the experimental data cannot be used to derive an empirical isotherm (see Chapter 4). 

The relation between the wavefunction corrections determined by the WSCI and PSCI methods and the corresponding Newton-Raphson methods may be determined using the matrix partitioning approach. For example, the solution to the PCI matrix eigenvalue equation is equivalent to the solution of the linear equation... 

Substitutions of Eqs. (78-85) in Eq. (73) lead to three-term recurrence relations of the expansion coefficients contained in Eqs. (32-39) in Ref. [6] and not reproduced here for the sake of space. The three-term recurrence relations can be cast onto the form of matrix eigenvalue equations with eigenvalues li and the eigenvectors, just like in the case of Section 2.3. 

An equation closely related to the first order response equation at the Hartree-Fock level (223) is the eigenvalue equation... 

Thus far, we have shown how one can obtain eigenvalue equations, in which the energy eigenvalues correspond to the intensive EAs (or IPs), by postulating that the anion (or cation) wave function can be related to the neutral molecule wave function through an operator. We have also shown how the EA and IP-EOM can be combined to generate... 

Solving the 2n dimensional generalized eigenvalue equation, one obtains 2n real eigenvalues and 2n eigenvectors v. The eigenvectors satisfy the orthogonality relations. 

In the q-coordinate system, the vibrational normal coordinates, the SA atom-dimensional Schrodinger equation can be separated into SA atom one-dimensional Schrodinger equations, which are just in the form of a standard harmonic oscillator, with the solutions being Hermite polynomials in the q-coordinates. The eigenvectors of the F G matrix are the (mass-weighted) vibrational normal coordinates, and the eigenvalues ( are related to the vibrational frequencies as shown in eq. (16.42) (analogous to eq. (13.31)). 

As an example, in Fig. 5.1 we return to our favored ammonia molecule and list all nuclear permutations, with and without the all-particle inversion operator, that leave the full Hamiltonian invariant. Nuclear permutations are defined here in the same way as in Sect. 3.3. A permutation such as (ABC) means that the letters A, B, and C are replaced by B, C, and A, respectively. The inversion operator, E, inverts the positions of all particles through a common inversion center, which can be conveniently chosen in the mass origin. In total, 12 combinations of such operations are found, which together form a group that is isomorphic to Ds. How is this related to our previous point group At this point it is very important to recall that the state of a molecule is not only determined by its Hamiltonian but also, and to an equal extent, by the boundary conditions. The eigenvalue equation is a differential equation that has a very extensive set of mathematical solutions, but not all these solutions are also acceptable states of the physical system. The role of the boundary conditions is to define constraints that Alter out physically unacceptable states of the system. In most cases these constraints also lead to the quantization of the energies. 

The matrix representation of a Hermitian operator 0 in an arbitrary basis i> is generally not diagonal. However, its matrix representation in the basis formed by its eigenvectors is diagonal. To show this we multiply the eigenvalue equation (1.72) by [Pg.16]

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