Operations, idempotent

We can now list some of the most important properties of the various types of operators and matrices Any hermitean, antihermitean, unitary, or idempotent operator has a spectral resolution where the eigenvectors form an ON-basis, so that... 

We write the antisymmetrizer as a properly idempotent operator for this discussion, contrary to the common practice that uses a x/i/2 prefactor. 

Idempotent Operations. A series of operations are termed idem-potent (11) if repeating the given operation results in no further change. Thus,... 

Idempotent operator, 174ff Identity element ( ), 3, 18, 51, 81 Infinite group, 70... 

Before explaining the representation of systems by means of Boolean functions an important property of Boolean variables must be presented, that of idempotence or the idempotent operation... 

There are two important properties of the antisymmetrizer. First, if it is applied to a wavefunction that is already properly antisymmetric, it will make no change. The implication of this statement is that if the antisymmetrizer is applied twice to an arbitrary function, the same result will be achieved as if it is applied only once the second application does not do anything. This is the condition of idempotency, and Aj is an idempotent operator. The operator equation that expresses this fact is... 

Introducing the HPHF wave-function expression (1) in (3), and taking into account the idempotency of operator A(s),the following equation may be obtained ... 

This means that the operators are mutually exclusive and that the operator is idempotent. Nevertheless, in three-dimensional Cartesian space the atoms do overlap, often even to a large extent. So they have no boundaries. 

The diagonal e-operators, efi, are special cases of what are called "primitive idempotents , which we now proceed to define. An operator p 6 U is called "idempotent if it satisfies... 

If we imagine an idempotent p expressed in terms of the e-operators, we see immediately that (21) must be satisfied separately for the parts belonging to each irreducible representation hence, we can treat the parts independently in deducing consequences from (21). Consider, then, an idempotent pM, belonging to the representation fW. We define a />-adapted basis for I M as follows ... 

Theorem 1 The operator Y defined by (39) is, apart from a multiplicative constant, a primitive idempotent of <3n. Y operators belonging to the same Young diagram belong to the same irreducible representation, while those belonging to different diagrams belong to different representations. 

Thus, symmetry projection need only be performed on the ket. Typically, projection operators are Hermitian and essentially idempotent cx in any... 

Symmetry adaptation ofVB wavefunctions Defining the (idempotent) projection operator... 

The Dirac density operator for the reference state is idempotent ... 

When the idempotent density operator p is constructed from orbital solutions of the Hartree-Fock equations, (Ti - ) = 0, it satisfies the commutator equation... 

When extended to include electronic correlation, for which an exact but implicit orbital functional was derived above, the TDHF formalism becomes a formally exact theory of linear response. In practice, some simplified orbital functional Ec[ 4>i ] must be used, and the accuracy of results is limited by this choice. The Hartree-Fock operator Ti is replaced by G = Ti + vc. Dirac defines an idempo-tent density operator p whose kernel is JA i(r) i (r/)- The Did. equations are equivalent to [0, p] = 0. The corresponding time-dependent equations are itijtP = [Q(t), p(t)]. Dirac proved, for Hermitian G, (hat the time-dependent equation ih i(rt) implies that p(l) is idempotent. Hence pit) corresponds to a normalized time-dependent reference state. 

Every single term in the sum in eq. (1.144) acts as the operator projecting on 4h(x) which are mutually orthogonal and normalized, so thus p p projects on the subspace spanned by the occupied spin-orbitals. The idempotency and hermiticity are checked immediately. So eq. (1.151) obviously coincides with the definition of an operator projecting to a subspace. The equation defining it reads ... 

The idempotence V2 = V is checked immediately as well as the fact that Sp V I. The Schrodinger equation for the projection operator V eq. (1.95) reads ... 

The (i, j) symbol in Eq. (21) stands for a binary interchange of the particles indicated. It will be observed that the particles operated upon in these operators are related closely to the way the particle labels occur in the tableau. As we have defined them, P and Af are strictly idempotent. 

Introducing two idempotent and mutually exclusive projectors P and Q for the spaces spanned by. and X, eq.(6.1.13) can be converted into an equivalent operator equation. Eq.(6.1.12) implies... 

In the calculations of the matrix elements concerned, three points are to be taken into account (i) the operator (projector) A is idempotent (ii) the Hamiltonian H is formulated according to the principles of quantum mechanics and as such it commutes with A as well as with the individual spin-space permutations (iii) the spin-dependent... 

Two other properties of the density operators follow from its definition (10.3). First, it is Hermitian, that is, p (Z) = p(Z). Second it is idempotent, that is, satisfies the property... 

We have already encountered the projection operator formalism in Appendix 9A, where an apphcation to the simplest system-bath problem—a single level interacting with a continuum, was demonstrated. This formalism is general can be applied in different ways and flavors. In general, a projection operator (or projector) P is defined with respect to a certain sub-space whose choice is dictated by the physical problem. By definition it should satisfy the relationship = P (operators that satisfy this relationship are called idempotent), but other than that can be chosen to suit our physical intuition or mathematical approach. For problems involving a system interacting with its equilibrium thermal environment a particularly convenient choice is the thermal projector. An operator that projects the total system-bath density operator on a product of the system s reduced density operator and the... 

The index of measurement statistics corresponding to a given preparation can be expressed in the form of a density operator 0. Some preparations result in states described by density operators that are pure (density matrices are idempotent), and some in states described by density operators that are mixed (density matrices are not Idempotent). In the context of the quantum mechanical postulates, the preceding sentence is all that need be said about any given preparation and, therefore, any given state. 

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