Hermitian operators description

Heisenberg-type descriptions for two observers, 667, 668 Heitler, W., 723 Helicity operator, 529 Hermitian operator, 393 Hermitian operator Q describing electric charge properties of particles, 513... 

The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

Unlike classical quantum mechanics, the spontaneous processes of the damped oscillator are irreversible, so its quantum mechanical description needs changes to some instruments of classical quantum mechanics. To do this, we use the Heisenberg picture of quantum processes. In this picture, the observables are time-dependent linear Hermitian operators, and the state vector of the system is time independent. Using the terminology introduced in the first part, the infinitesimal time transformation of the Hermitian operator could happen in two ways ... 

The origin of the tensorial description of molecular sets has been proved appearing as a consequence of quantum mechanical expectation value computation, via approximate QSPR operators. TDQS measures constitute a second order building block set for such QSPR operators. TDQS measures provide, at this second order approximation level, QSPR operator unconstrained optimal structures, which comply with the usual quantum mechanical form of the Hermitian operators, associated to the observables for every molecular stmcture. 

A -representability conditions [28]. Let us start this description by focusing on the RDM s properties, which may be deduced from their definition as expectation values of density fermion operators. Thus the ROMs are Hermitian, are positive semidefinite, and contract to finite values that depend on the number of electrons, N, and in the case of the HRDMs on the size of the one-electron basis of representation, 2/C. Thus... 

The operators F corresponding to all physically measurable quantities are Hermitian this means that their matrix representations obey (see Appendix C for a description of the bra I > and kef < I notation used below) ... 

A unitary transformation is one which, when applied to both the state function and the observables of a system, leaves the description of the system unchanged. Denote by U an operator with the property that its Hermitian conjugate or adjoint is equal to its inverse, that is... 

An important feature is a certain flexibility in the choice of the Hermi-tian potential V. It was pointed out by Kelly that any term of the form (1 — P)Sl(l — P), where P = 1 — J2iLl I >< is a projection operator and fl is an arbitrary Hermitian function, can be added in to give K.8 This has the advantage that Q can be varied to give the best description of the physical situation. 

The application of Eq. (221) to the description of the time evolution of a quantum system requires us to speedy the initial state of the system. Generally, there is some statistical indeterminancy in its initial preparatioa Thrrs, we adopt a statistical description that employs the derrsity operator /o(0) to specify the initial state. As witheqrrilibrirrm density operators, the density operator p(0) is assumed to have the following properties (i) Trp(0) = 1, (ii) p(0) is Hermitian, and (iii) the diagonal matrix elemerrts of p(0)... 

---