Hermitian character

The imaginary unit i contained in the operator p is essential for the hermitian character of that operator. The operator L>x = A/Ax is not hermitian because... 

The hermitian character of an operator depends not only on the operator itself, but also on the functions on which it acts and on the range of integration. An operator may be hermitian with respect to one set of functions, but not with respect to another set. It may be hermitian with respect to a set of functions defined over one range of variables, but not with respect to the same set over a different range. For example, the hermiticity of the momentum operator p is dependent on the vanishing of the functions ipi at infinity. 

Due to the hermitian character of the dynamical matrix, the eigenvalues are real and the eigenvector satisfies the orthonormality and closure conditions. The coupling coefficients are given by... 

In order to prove the Hermitian character of the superoperator X it is sufficient to show that ... 

Equations (129) and (130) prove that equation (128) and, hence equation (127), are valid. This shows the Hermitian character of the... 

As mentioned in the Introduction, one of our purposes is reduction of the non-Hermitian character of the ordinary EOMCC formalism (or, in some variants of the new theory, its complete elimination). Unlike the original Hamiltonian Hk, the transformed Hamiltonian H used in the EOMCC method is no longer Hermitian and the non-Hermiticity of H measured by a difference H — (H) shows up already in the first-order of MBPT, i.e. 

Again, both the electric and magnetic fields are obviously of Hermitian character. What also emerges from the route of their derivation through Eqs. (3), (6), and... 

It immediately follows from the hermitian character of the respective subsystem hamiltonians and their eigenvalue equations that these energy functions satisfy the following differential Hellmann-Feynman theorems ... 

The notion of complex energies in quantum physics was introduced ad hoc, via a simple model, by Gamow [25], soon after the establishment of fhe well-known Hilbert-space QM developed for the discrete spectra. This notion was in radical departure from fhe Hermitian character of QM, whose normal... 

In the above fonnula, integrals with the imaginary unit i equal zero because the integrand is an odd function. After using the Hermitian character of the momentum operator we obtain 1 = (I5 p ls) = The last equality follows from Appendix H available at... 

In the second row, the scalar product of spinors is used, while in the third row, the Hermitian character of the operator p. Further,... 

The first of Eqs. (5.18) is evident (the unperturbed Schrodinger equation does not contain any unknowns). The second equation involves two unknowns, and E There is a way to eliminate by using the Hermitian character of the operators. Indeed, by making the scalar... 

This can always be assured (by suitable orthogonalization and normalization) and follows from the Hermitian character of the operator . 

---