Complete set of eigenfunctions

A. If the two operators act on different coordinates (or, more generally, on different sets of coordinates), then they obviously commute. Moreover, in this case, it is straightforward to find the complete set of eigenfunctions of both operators one simply forms a product of any eigenfunction (say fk) of R and any eigenfunction (say gn) of S. The function fk gn is an eigenfunction of both R and S ... 

If /f form the complete set of eigenfunctions of Ho with eigenvalues that is,... 

If the complete set of eigenfunctions for the unperturbed system includes a continuous range of functions, then the expansion of must include these functions. The inclusion of this continuous range is implied in the summation notation. The total eigenfunction tp for the perturbed system to first order in X is, then... 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the F >v matrix elements depend on the Cv,i LCAO-MO coefficients which are, in turn, solutions of the so-called Roothaan matrix Hartree-Fock equations- Zv F >v Cv,i = Zv S v Cvj. One should also note that, just as F (f>j = j (f>j possesses a complete set of eigenfunctions, the matrix Fp,v, whose dimension M is equal to the number of atomic basis orbitals used in the LCAO-MO expansion, has M eigenvalues j and M eigenvectors whose elements are the Cv>i- Thus, there are occupied and virtual molecular orbitals (mos) each of which is described in the LCAO-MO form with CV)i coefficients obtained via solution of... 

Exercise. For one-step processes W is a tridiagonal matrix. With the aid of (3.8) a similarity transformation can be constructed which makes the matrix symmetric, as in (V.6.15). Prove in this way that any finite one-step process has a complete set of eigenfunctions, and that its autocorrelation function consists of a sum of exponentials.510... 

Let us assume that zero-order Hamiltonian Ho is a Hermitian operator having a complete set of eigenfunctions... 

Polariton states can be used as a complete set of eigenfunctions, which is required when computing the trace51... 

The London theory uses second-order perturbation theory in its usual (R.S.) form an infinite sum over a complete basis set. The set taken for the composite system of two interacting molecules a and b consists of all products of the complete set of eigenfunctions of a and b separately. Thus, the London theory not only assumes that there is no overlap between the ground-state atoms but also that there is no overlap between any of the virtual atomic excited states. ... 

For class-1 states, a simple harmonic representation of U leads to a complete set of eigenfunctions ( ) this harmonic oscillator basis set is used to diagonalize equation (6). In this case, it is sufficient to construct U( 4>k) using a standard approach involving mass fluctuation (or nuclear ) coordinates and the corresponding electronic state dependent Hessian. The higher terms in the Taylor expansion define anharmonic contributions to the transition moments. These diabatic states are confining and only one stationary point in -space would be found for each... 

For a complete set of eigenfunctions to be simultaneously eigenfunctions of several operators, each operator must commute with every other operator. 

First, we show that if there exists a common complete set of eigenfunctions for two linear operators then these operators commute. Let A and B denote two linear operators that have a common complete set of eigenfunctions gi,... 

THEOREM 4. If the linear operators A and B have a common complete set of eigenfunctions, then A and B commute. 

Thus, when A and B commute, it is always possible to select a common complete set of eigenfunctions for them. For example, consider the hydrogen atom, where the operators and H were shown to commute. If we desired, we could take the phi factor in the eigenfunctions of H as sin m and cos m4> (Section 6.6). If we did this, we would not have eigenfunctions of except for w = 0. However, the linear combinations... 

Extension of the above proofs to the case of more than two operators shows that for a set of Hermitian operators A,B,C,... there exists a common complete set of eigenfunctions if and only if every operator commutes with every other operator. 

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