Tensor Hermitian

If the second-order Hermitian matrix follows the transformation rule for a (2,2) tensor, then this decomposition is the only possible manner of expressing these matrices as a sum of simpler parts so that the decomposition remains invariant under unitary tranformations of the basis [73]. 

Since any operator can be written as the sum of Hermitian and anti-Hermitian operators, we can restrict our discussion to these two types only. Further, any operator can be written as a linear combination of irreducible symmetry operators, so we can restrict ourselves to irreducible tensor operators. An operator matrix 0(r, K) that transforms according to the symmetry (T, K) obeys the relationship... 

The set of (2k + 1) Hermitian conjugated operators does not form an irreducible tensor, but from (2.21) we can easily see that the operators... 

Electron annihilation operators, as Hermitian conjugates of creation operators, are no longer the components of an irreducible tensor. According to (13.40), such a tensor is formed by (21 + l)(2s + 1) components of the operator... 

It follows from (15.49) that the tensors aiqls have the following property with respect to the operation of Hermitian conjugation ... 

Table 1. Behaviour of second-quantized 1-particle tensor operators under Hermitian conjugation, time reversal and their combinations in each order...

Since the density matrix is Hermitian, we obtain the property of polarization moments which is analogous to the classical relation (2.15) fq = (—1 ) (f-q) and tp = (—l) 3( g). The adopted normalization of the tensor operators (5.19) yields the most lucid physical meaning of quantum mechanical polarization moments fq and p% which coincides, with accuracy up to a normalizing coefficient that is equal for polarization moments of all ranks, with the physical meaning of classical polarization moments pq, as discussed in Chapter 2. For a comparison between classical and quantum mechanical polarization moments of the lower ranks see Table 5.1. 

Hermitian and unchanged by the operation. The notation used here conforms with the finite group irreducible tensor scheme 38, 39). We have already proved that cannot split the doublets and need therefore only discuss Now the states concerned are... 

The latter result (82) yields a quantum probability amplitude that, under Hermitian conjugation and time reversal, correctly equates to the corresponding amplitude for the time-inverse process of degenerate downconversion. To see this, we note that the matrix element for SHG invokes the tensor product Py (—2co co, ) p([/lC., where the brackets embracing two of the subscripts (jk) in the radiation tensor denote index symmetry, reflecting the equivalence of the two input photons. As shown previously [1], this allows the tensor product to be written without loss of generality as ( 2co co, co), entailing an index-symmetrized form of the molecular response tensor,... 

Each of the six terms of the hyperpolarizability tensor so formed transforms into one of the six counterpart terms in (2ca, go, — ), the tensor for degenerate downconversion, on performing the combined operations of Hermitian conjugation and time reversal (the radiation tensor for downconversion is also obtained by performing the same procedure on p,- ). For example, the last term of Pi(/ ) (—2 , ), in the order that logically follows from Eqs. (82) and (83), behaves as follows ... 

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. 

Consider next the term involving the third derivative tensor 4. Again we split the response vectors into Hermitian and anti-Hermitian contributions such that we obtain a sum of terms B BgBc where the vectors B., Bg and Be all have well-defined hermiticities and time reversal structures. In order to be able to use (235) we first analyze the vector structure obtained by contracting the third derivative tensor with two vectors ... 

The coefficient a, which is related to the displacement of charged particles in a solid, is called the electric polarizability of the medium. This coefficient is the second-rank Hermitian tensor for an anisotropic particle, but it reduces to a scalar for isotropic particles. Note that to avoid confusion with the absorption coefficient a, the polarizability has been noted by the symbol a (with a hat). From Eqs. (1.28) and (1.3) we obtain the following expression for the polarization P = aeoAE. Thus, in view of Eq. (1.2a), the permittivity e of an ensemble of particles can be written as... 

The j(qA ) form a set of independent coordinates if the u lk) are such a set since they are related by the unitary transformation defined by (2.11). We note, however, as a new feature, that the <(qA ) are complex. Both the kinetic energy T [from (2.14)] and the potential energy O [from (2.18)] are hermitian and therefore the Lagrangian L derived from these forms is hermitian as required. In tensor notation we obtain... 

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... 

The commutator of two Hermitian operators is an anti-Hermitian operator. From (2.3.2), we can therefore conclude that spin tensor operators are not in general Hermitian. Indeed, the only possible exception to this rule are the operators where M = 0, which may or may not be Hermitian. It is therefore of some interest to examine the Hermitian adjoints of the spin tensor operators. Taking the conjugate of the relations (2.3.1) and (2.3.2), we obtain ... 

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