Hermitian conjugates systems

The polarized-Iight and spin examples have shown that, even though a quantum system may be in a definite state, as established by an exhaustive measurement, a subsequent observation does not necessarily yield a definite result. Knowing the result of an observation therefore does not reveal the state, the system was in at the time of the measurement, and neither does knowing the state of a system predict the exact outcome of any observation. Quantum theory only predicts the statistical outcome of many measurements of some property. To achieve this, a physical state is represented by a column vector or (equivalently) by the Hermitian conjugate row vector ... 

Some care has to be exercised when demonstrating an expansion theorem in terms of Eq. (A.58), because the differential operator (A.52) is not Hermitian. It is, however, very easy to find a conjugate system of eigenfunctions 24 they are obtained by substituting —km for kx in Eq. (A.58). We then have for an arbitrary function ... 

The operator with dagger implies Hermitian conjugate of the operator, as usual. We put the system is in a one-dimensional box of size L with periodic boundaries. As a result, the wave numbers are discrete. We have k = Inn/L with n integer. The spectrum of frequencies (s>k is discrete as well. 

From the definition of 6k it should be noted that Bk and k are Hermitian conjugates. According to the Golden Rule of time-dependent perturbation theory the probability per unit time, Wt f k, co), that the field (k, to) induces a transition in the system from the initial state f> to the final state / ) is given by... 

For Kramers (e.g. one-electron) states where the eigenvalue of T2, A = — 1, the metric gAA is antisymmetric, and so relating to symplectic algebras (in relativistic terms to pure torsion rather than to curvature), rather than symmetric as for non-Kramers systems. The joint action of Hermitian conjugation and time reversal (which is not commutative) is summarized with the above results for these individual operations in Table 1. 

Equations (5.6) and (5.7), together with the normalization condition (5.1), form the self-consistent system of two equations and one additional condition for the determination of two unknown functions n and ip and one unknown number A (in Eq. (5.7) ip+ is the function Hermitian-conjugated to ip), i.e. for the complete description of the equilibrium spatial structure of the globule. 

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... 

In the equation above, the symbol q designates determinants that lie outside the principal projection space used in the CC equations [Eq. (7)]—that is, triples, quadruples, and so on, for CCSD. Note that the first two terms strongly resemble those in the exact expression [Eq. (10)]. In fact, to the extent that the right and left CC wave functions approximate the exact wave function and its Hermitian conjugate and that the excited states of the system are represented by the... 

M is the mass diagonal matrix, its inverse is the row vector obtained by hermitian conjugation of the column vector P, The potential V(R ) represents the force field associated to the solute system. 

It seems the key step in this derivation, which differs from the analysis of CGM, is the following. In the system of equations resulting from the constraint C C+ = Ijv, Pecora considers that N(N - 1) of [them] are simply complex conjugates of each other , yielding a total number of complex conditions equal to N(N + l)/2. This is, in fact, equivalent to considering theCC1 matrix as hermitian, i.e.,... 

---