Hermitian product

Example 3.14 (Application to description given in Proposition 2.9). Let us consider Hermitian vector spaces V and W whose dimensions are n and 1 respectively. Then M = End(P)0End(l/) Hom(lT, P) Hom(P, W) becomes a vector space with a Hermitian product. We consider an action of G = U(P) on M given by... 

We remind the reader that all permutations are imitary operators. Since binary permutations are equal to dieir own inverses, they are also Hermitian. Products of commuting binaries are also Hermitian. 

This probability is determined by the amount to which vector E is contained in vector d. This quantity equals the hermitian product of the E and d [140]. The hermitian product can be written as the scalar product of E and d. As a result the probability Wabs of absorption of light, which is characterized by its E-vector, by a classical dipole is proportional to Ed 2. This probability can be divided into a dynamic part Tp... 

Eq. (3.22) previously given in Section 3.2. We will now analyze in greater detail the methods of solving this equation. As is known, the product E d has to be considered as a hermitian product of E and d [140]. It is just such a product which represents the amplitude with which the vector E is contained in the vector d. As a result we have for the matrix elements of the electric dipole transition... 

In accordance with the transformation properties (18), it follows that the Hermitian products... 

In order to obtain the theoretical expressions for the g- and /-tensor elements, we have to compare the theoretical Hamiltonian given in Eq. (IV.59) with the phenomenological Hamiltonian in Eq. (1.4). For this comparison we have to keep in mind that in the phenomenological expression corresponds to Pyjk in Eq. (IV.59) and that in order to make the phenomenological Hamiltonian Hermitian, products such as cos yZ Jy> must be symmetrisized to (cos yZJy- +Jy cos yZ)j2, etc. 

However, using the McWeeny approach [7], it is sufficient to calculate only the projection P >v/K on the subspace of virtual zero-order orbitals in order to get the second hyperpolarizability tensor. This projection is evaluated via a procedure similar to the one used in solving the first-order equation (21). Taking in (32) the Hermitian product with the unoccupied and using (19), one finds... 

This implies that the Hermitian conjugate of an antilinear operator is also antilinear. It should also be pointed out that the product of two antilinear... 

If Eq. (E.14) is satisfied for all elements of some point group G, A will be an invariant operator [13] (the Hermitian conjugate as well as the sum and/or product of two invariant operators are also invariant operators). Such an operator can be expanded in the form... 

One easily shows by differentiation with respect to t and using the hermitian property of H that this scalar product is independent of time, if and < ( )> are both solutions of (9-40). The probability... 

The operator H must be hermitian within this scalar product. One verifies that for this to be the case a and j8 must be hermitian matrices... 

The product of two hermitian operators may or may not be hermitian. Consider the product AB where A and B are separately hermitian with respect to a set of functions xpi, so that... 

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... 

If A and B are hermitian (self-adjoint), then we have (is)t = BA and further, if A and B commute, then the product AB is hermitian or self-adjoint. 

In counting the number of orthonormalization conditions on C, CGM apparently did not assume the hermiticity of the scalar product in the subspace, but rather chose to impose it. Their calculation of K ran along the following lines a complex projector, which is hermitian and normalized, may be factored into [13]... 

More fundamentally, what Pecora seems to assume - although never explicitly saying so - is the following property. Since the condition CC+ = In is actually the orthonormalization constraint on the scalar product between any two wavefunctions (ft is hermitian. That is to say, it is assumed that the subspace on which the projection is made is a Hilbert subspace. 

However, if one were to exactly follow what seem to be Pecora s assumptions about the scalar product being hermitian, one would get a different result from Pecora when counting the number of real conditions on the complex P matrix, arising from the constraint + = In In fact, when the + matrix is considered to be hermitian, the normalization condition on the N complex diagonal elements of QQ+ yields N real conditions and not 2N as Pecora seemed to tacitly suppose. This is due to the fact that the diagonal elements are already known to be real since + is hermitian, and hence, Im = 0 is not a separate constraint. 

Appendix Proof of the inner product hermiticity of a subspace of an hermitian space... 

Our first way of answering the last question will be based on the fundamental theorems on Hilbert space [14], Indeed, the theorem on separability tells us that any subspace of h is also a separable Hilbert space. As a consequence, the inner product defined on, say, the occupied subspace is hermitian irrespectively of the choice of the basis x f (/)], as long as this latter satisfies the fundamental requirements of Quantum Mechanics. One should therefore not have to impose this property as a constraint when counting the number of conditions arising from the constraint CC+ =1 but, on the contrary, can take it for granted. 

As a conclusion to this part, when counting the number of conditions arising from CC+ = kv, one does not have to impose the inner product to be hermitian but can take it for granted. 

If A is a square matrix and AT is a column matrix, the product AX is a so a column. Therefore, the product XAX is a number. This matrix expression, which is known as a quadratic form, arises often in both classical and quantum mechanics (Section 7.13). In the particular case in which A is Hermitian, the product XxAX is called a Hermitian form, where the elements of X may now be complex. 

The Hermitian conjugate c (dagger) of a column vector c, is a row vector, with the components c. The scalar product of the row vector w and a column vector, v is... 

A Hermitian operator p is a von Neumann density if it is nonnegative and has unit trace. In more concrete terms, if is the finite-dimensional Fock space for a quantum model where electrons are distributed over a finite number of states, then p is a von Neumann density if (i) v,pv) > 0 for all operators v on and (ii) l)g = 1. By the formula v,pv) we mean the trace scalar product of the operators v and pv, that is, v,pv) = traceg(u pu) since (p, l)g = tracegp = 1 we have used this scalar product to express the trace condition. More generally. 

Definition 15 A -body operator is a Hermitian operator that can be represented as a polynomial of degree 2 A in the annihilation and creation operators, and is of even degree in these operators. In addition, a A -body operator must be orthogonal to all k — l)-body operators, all k — 2)-body operators,. .., and all scalar operators, with respect to the trace scalar product. 

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