Hermitian equations

Equation 4.6 can be simplified by making the following substitutions. First we note that the Hamiltonians are Hermitian (equation A.7)P Then we use the abbreviations and make the definitions shown in equations 4.8... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

VIII. Hermitian Matrices and The Turnover Rule The eigenvalue equation ... 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

The equation defines V(u), of course, since H is understood to be H(u). The matrix V is unitary. To see this, multiply (7-53) by l/t on the left then multiply the associate (or adjunct) of (7-53), —MVt — Wffi, by U on the right and note that H is hermitian W = H. Subtracting one of the resulting forms from the other gives the further result... 

Upon taking the hermitian adjoint of this equation, we obtain... 

The isomorphism between the tilde operation and hermitian conjugation, implies that upon performing this -operation on the Dirac equation we find that [Pg.524]

Thus the current operator indeed transforms like a vector. This must be the case in order that the equation Qdu(x) = ju(x) transform properly, assuming the transformation property (11-267) for Au(x). We now inquire briefly into tike question of the uniqueness of the U(ia) operator, in particular into the question of the phase associated with the fermion field operator. Note that the phase of the photon field operator is uniquely determined (Eq. (11-267)) by the fact that An is a hermitian field which commutes with the total charge operator Q. The negaton-positon field operator on the other hand does not commute with the total charge operator, in fact... 

If we restrict ourselves to the case of a hermitian U(ia), the vanishing of this commutator implies that the /S-matrix element between any two states characterized by two different eigenvalues of the (hermitian) operator U(ia) must vanish. Thus, for example, positronium in a triplet 8 state cannot decay into two photons. (Note that since U(it) anticommutes with P, the total momentum of the states under consideration must vanish.) Equation (11-294) when written in the form... 

The density operator r(<) is a Hermitian and positive function of time, and satisfies the generalized Liouville-von Neumann (LvN) equation(47, 45)... 

For illustration, we consider some examples involving only one variable, namely, the cartesian coordinate x, for which w x) = 1. An operator that results in multiplying by a real function /(x) is hermitian, since in this case fix) = fix) and equation (3.8) is an identity. Likewise, the momentum operator p = (i)/i)(d/dx), which was introduced in Section 2.3, is hermitian since... 

By setting B equal to A in the product AB in equation (3.11), we see that the square of a hermitian operator is hermitian. This result can be generalized to... 

The eigenvalues of a hermitian operator are real. To prove this statement, we consider the eigenvalue equation... 

Because A is hermitian, the left-hand sides of equations (3.14) and (3.15) are equal, so that... 

The orthogonality theorem can also be extended to cover a somewhat more general form of the eigenvalue equation. For the sake of convenience, we present in detail the case of a single variable, although the treatment can be generalized to any number of variables. Suppose that instead of the eigenvalue equation (3.5), we have for a hermitian operator 4 of one variable... 

Now, the operator A is hermitian with respect to the functions 0, with a weighting function equaling unity, so that the integral on the left-hand side of equation (3.20) becomes... 

A comparison with equation (3.8) shows that if A is hermitian, then we have = A and A is said to be self-adjoint. The two terms, hermitian and self-adjoint, are synonymous. To find the adjoint of a non-hermitian operator, we apply equations (3.33). For example, we see from equation (3.10) that the adjoint of the operator d/dx is —d/dx. 

Since TI = 1, we see that A = 1 and that the eigenvalues A, which must be real because IT is hermitian, are equal to either -hi or —1. To find the eigenfunetions >p i(q), we note that equation (3.65) now becomes... 

Applying the hermitian property of H X) to the third integral on the right-hand side of equation (3.69) and then applying (3.66) to the second and third terms, we obtain... 

Application of equation (3.33) reveals that the operator is the adjoint of a, which explains the notation. Since the operator d is not equal to its adjoint d neither d nor d is hermitian. (We follow here the common practice of using a lower case letter for the harmonic-oscillator ladder operators rather than our usual convention of using capital letters for operators.) We readily observe that... 

Neither nor J is hermitian. Application of equation (3.33) shows that they are adjoints of each other. Using the definitions (5.18) and (5.14) and the commutation relations (5.13) and (5.15), we can readily prove the following relationships... 

Show explicitly by means of integration by parts that the operator Hi in equation (6.18) is hermitian for a weighting function equal to... 

The eigenvalues of a hermitian matrix are real. To prove this statement, we take the adjoint of each side of equation (1.47), apply equation (1.10), and note that A = A ... 

We shall now use the theorem previously mentioned. Since is hermitian during this rank computation, so too are D and X. The above constraint on, Equation (20), along with its hermiticity property, leads to the following number of conditions on its elements, and therefore ultimately on the P elements ... 

This number is the answer to the question originally posed. This is the number of real conditions required to fix experimentally a complex, normalized, hermitian, projection matrix. For example, this number of experimental structure factors, Equation (1), would suffice to fix P Equation (6). 

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.,... 

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