Time-independent projection operators

Neuhauser D and Baer M 1990 A new accurate (time independent) method for treating three-dimensional reactive collisions the application of optical potentials and projection operators J. Chem. Phys. 92 3419... 

The starting point for Lowdin s PT [1-6] and Eeshbach s projection formalism [7-9] is the fragmentation of the Hilbert space H = Q V, of a given time-independent Hamiltonian H, into subspaces Q and V by the action of projection operators Q and P, respectively. The projection operators satisfy the following conditions ... 

The two projection operators are chosen to be time-independent. Thus they commute with the differential operator on the left-hand side of Eq. (14). As a consequence, by applying to Eq. (14) both the operator P and the operator Q, and by applying the property p(t) (P I Q)p(t) = pj(t) + p2(t) as well, we split this equation into the following coupled equations... 

Note that the time-independent nature of the projection operator P [already exploited to derive Eq. (3.5)] allows us to use the property... 

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However, due to the lack of Hermiticity, the spectrum of H includes, in addition to the normal spectrum of the ordinary Hamiltonian H, the possibility of complex eigenvalues. The occurence of such then causes the general operator exp(-iHt/h) to be undefined unless one considers it in a projected subspace (42). Within such a subspace comprised of all physical solutions to the time dependent Schrodinger equation, equation (82) is perfectly well behaved. Let us consider again the time independent equation... 

A set of functions A, B and this definition of scalar product defines a space in which the functions can be thought of as vectors and operators transform these vectors. The length of vector A is defined as (A,A ) and we note one important feature of this vector space since (A(t), A (t)) is independent of time, any time-displacement operator can rotate the vector but must leave its length unchanged. In particular, exp(iLt) is a generalized rotation operator. Also, Cy (t) is a measure of the component of A(t) parallel to A(0) i.e., it is the projection of A(t) onto A(0). This suggests that we define a projection operation P which, when it acts on an arbitrary vector B, projects B onto A. Thus,... 

As in the Zwanzig approach to nonequilibrium processes, we partition the vector pt) correspondtag to the density operator p t) of some system into relevant and irrelevant parts by using orthogonal projection operators. However, the projection operators P t) and Q t) used to accomplish this task are time dependent rather than time independent. Nonetheless, P t) and Q t) possess the usual properties P(0+6(0=/, Htf = P t 6(O = 0(O, and Pp)QiO = QiOPiO = 0 of orthogonal projection operators, where 6 is the null operator. 

If the system is close to thermal equilibrium, the projection operator P(t) assumes the time-independent form... 

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The dependence of the used orbital basis is opposite in first and second quantization. In first quantization, the Slater determinants depend on the orbital basis and the operators are independent of the orbital basis. In the second quantization formalism, the occupation number vectors are basis vectors in a linear vector space and contain no reference to the orbitals basis. The reference to the orbital basis is made in the operators. The fact that the second quantization operators are projections on the orbital basis means that a second quantization operator times an occupation number vector is a new vector in the Fock space. In first quantization an operator times a Slater determinant can normally not be expanded as a sum of Slater determinants. In first quantization we work directly with matrix elements. The second quantization formalism represents operators and wave functions in a symmetric way both are expressed in terms of elementary operators. This... 

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