Completeness relation

The completeness relation for a multi-dimensional wave function is given by equation (3.32). However, this expression does not apply to the wave functions vs,A for a system of identical particles because vs,a are either symmetric or antisymmetric, whereas the right-hand side of equation (3.32) is neither. Accordingly, we derive here the appropriate expression for the completeness relation or, as it is often called, the closure property for vs,a- 

For compactness of notation, we introduce the 4A-dimensional vector Q with components q, for / = 1, 2,. .., N. The permutation operators P are allowed to operate on Q directly rather than on the wave functions. Thus, the expression P P(1, 2, N) is identical to I (.PQ). In this notation, equation (8.32) takes the form 

We begin by considering an arbitrary function /(Q) of the 4A-dimensional vector Q. Following equation (8.33), we can construct from /(Q) a function F(Q) which is either symmetric or antisymmetric by the relation 

Since F(Q) is symmetric (antisymmetric), it may be expanded in terms of a complete set of symmetric (antisymmetric) wave functions v(Q) (we omit the subscript S, A) 

We now introduce the reciprocal or inverse operator P to the permutation operator P (see Section 3.1) such that 

Accordingly, we derive here the appropriate expression for the completeness relation or, as it is often called, the closure property for 

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These three projeetion matriees play important roles in what follows. B. Completeness Relation or Resolution of the Identity... 

This expression, known as the completeness relation and sometimes as the closure relation, is valid only if the set of eigenfunctions is complete, and may be used as a mathematical test for completeness. Notice that the completeness relation (3.31) is not related to the choice of the arbitrary function /, whereas the criterion (3.30) is related. 

The completeness relation for the multi-variable case is slightly more complex. When expressed explicitly in terms of its variables, equation (3.29) is... 

Thus, the expression (3.43) is related to the completeness criterion (3.30) and is called, therefore, the completeness relation. 

Parseval s theorem is also known as the completeness relation and may be used to verify that the set of functions e " for —oo oo are complete, as (fiscussed in... 

Applying equation (C.5e), we obtain the completeness relation for the functions [Pg.300]

For each coordinate 2 in the full space, we may define a covariant basis vector 0R /02 and a contravariant basis vector 02 /0R, which obey orthogonality and completeness relations... 

If the set 3N contravariant Cartesian vectors given by the / a vectors and K m vectors form a complete basis for 3N space of Cartesian vectors, which we will hereafter assume to be true, then they must also obey a completeness relation ... 

The biorthogonality and completeness relations presented above do not uniquely define the reciprocal basis vectors and mi a list of (3N) scalar components is required to specify the 3N components of these 3N reciprocal basis vectors, but only (3N) —fK equations involving the reciprocal vectors are provided by Eqs. (2.186-2.188), leaving/K more unknowns than equations. The source of the resulting arbitrariness may be understood by decomposing the reciprocal vectors into soft and hard components. The/ soft components of the / b vectors are completely determined by the equations of Eq. (2.186). Similarly, the hard components of the m vectors are determined by Eq. (2.187). These two restrictions leave undetermined both the fK hard components of the / b vectors and the Kf soft components of the K m vectors. Equation (2.188) provides another fK equations, but still leaves fK more equations than unknowns. Equation (2.189) does not involve the reciprocal vectors, and so is irrelevant for this purpose. We show below that a choice of reciprocal basis vectors may be uniquely specified by specifying arbitrary expressions for either the hard components of the b vectors or the soft components of m vectors (but not both). 

Biorthogonality conditions (2.186-2.189) and completeness relation (2.190) are equivalent to the biorthogonality and completeness relations (2.5) and (2.6) obeyed by partial derivatives in the full space, if we identify... 

By repeating the reasoning applied in Section VI to the dynamical reciprocal vectors, we may confirm that any vectors so defined will satisfy Eqs. (2.186)-(2.189). It will hereafter be assumed that (except for pathological choices of S v) they also satisfy completness relation (2.190). A few choices for the tensors S v and T yield useful reciprocal vectors and projection tensors, for which we introduce special notation ... 

The reciprocal vectors defined in Section VIII may be used to construct a more direct derivation. Let b and m refer to reciprocal basis vectors defined using Spv and in Eqs. (2.207) and (2.208), respectively. By substituting definitions (2.207) and (2.208) into completeness relation (2.190), we find that... 

Physically, this means that when you project onto the components of a vector in these three directions, you don t lose any of the vector. This happens because our vectors are orthogonal and complete. The completeness relation means that any vector in this three-dimensional space can be written in terms of v(l), v(2), and v(3) (i.e., we can use v(l),v(2),v(3) as anew set of bases instead of ei,e2,e3). 

This is how we will most commonly make use of the completeness relation as it pertains to the eigenvectors of Hermitian matrices. 

The problem with these equations is that they correspond to infinite different Hamiltonians so that the solutions for different electronic quantum numbers are incommensurate. To do away with these objections, use instead the complete set of functions rendering the kinetic energy operator Kn diagonal. The set, within normalization factors, is fk(Q) exp(ik Q) k is a vector in nuclear reciprocal space. Including the system in a box of volume V, the reciprocal vectors are discrete, ki, and the functions f (Q) = (1/Vv) exp(iki Q) form an orthonormal set with the completeness relation 8(Q-Q ) = Si fi(Q) fi(Q )- The direct product set ( )j(q)fki(Q) is complete. The matrix elements of eq. (8) reads ... 

Again the situation is much simpler when only asymptotic states containing stable particles are considered. Then unstable particles enter neither into the completeness relation nor into the unitary relations of the theory.5 However, in the intermediate states unstable particles may appear. They manifest themselves as poles exactly as in Eq, (16). We may then describe such poles by various approximate formulas of the Breit-Wigner type. But again this approach is severely limited. By definition we have to exclude the production or destruction processes involving unstable particles. It is even not easily seen how this can be done in a consistent manner. 

Let the eigenvalues be —A., and the corresponding eigenfunctions Pk(q)- The completeness relation states... 

The summation over final states /> has been carried out by the completeness relation, or equivalently, by matrix multiplication. Thus, the zeroth moment is just an equilibrium average of the square of the perturbation amplitude. The first moment can likewise be expressed as an equilibrium property. 

The remaining task is to show that the set of normal modes found is complete. For this purpose we choose the normalization C = /ijn and prove the completeness relation. First we take m > 0, n > 0 and prove... 

From the completeness relations we now obtain the spectral representation... 

In order to work with the on-shell (physical) scattering T matrix, we must consider the kinetic equation in the space of the asymptotic states which is the direct sum of the channel subspaces In this space we have the following completeness relation... 

From Eq. (C.2) we conclude that the square integrable solution / contains the independent solution spectral function p(E) to be used in the completeness relation and the eigenfunction expansion. The former gives... 

The strong reduction is caused by the presence, at the valence and conduction band edges, of confined, flat, states completely related to the Siio-Si02 interface. Actually neither the isolated, H-passivated cluster (see Figure 11, central panel), nor the pure SiC>2 matrix (see Figure 11, right panel) show these states, whereas deep inside the valence and conduction... 

This is a very important relation that we will be using over and over again in the following. The operator Pqi = qi) qi is called the projection operator for the ket qi). Equation (F.7), which is called the completeness relation, or closure relation, expresses the identity operator as a sum over projection operators. The relation is true for any orthonormal basis we may choose. 

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