Completeness Schrodinger eigenfunctions

But he was not at all certain that the principles considered so far in atomic theory could, in fact, be used for the realization of such a program. This was because the characteristic interaction of the chemical forces deviated completely from other familiar forces These forces seemed to "awake" after a previous "activation," and they suddenly vanished after the "exhaustion" of the available "valences." By making use of elementary symmetry considerations, it was known that the mode of operation of the homopolar valence forces could be mapped onto the symmetry properties of the Schrodinger eigenfunction of the atoms of the periodic system and could be interpreted as quantum mechanical resonance effects. This interpretation was formally equivalent to its chemical model, that is, it produced the same valence numbers and... 

The wave functions have the form (5.54), but since Pc does not commute with H, we cannot separate out a chi factor the Schrodinger equation is not separable, and we will try another method of dealing with the problem. We saw in Section 2.3 that the eigenvalues of an operator H can be found by expanding the unknown eigenfunctions in terms of some known complete orthonormal set [Pg.361]

Because of the spin-spin coupling term, (8.41) is not separable into the sum of Hamiltonians for the individual nuclei, and the corresponding Schrodinger equation is not separable. To deal with the problem, we shall use the method of expanding the unknown wave functions in terms of a known complete set of functions. The eigenvalues and eigenfunctions (eigenvectors) are obtained as the solutions of (2.68) [or (2.77)] and (2.67). 

It is generally true that the normalized eigenfunctions of an Hermitian operator such as the Schrodinger Ti constitute a complete orthonormal set in the relevant Hilbert space. A completeness theorem is required in principle for each particular choice of v(r) and of boundary conditions. To exemplify such a proof, it is helpful to review classical Sturm-Liouville theory [74] as applied to a homogeneous differential equation of the form... 

Here, /(S2) is any operator analytic function, % is the projection operator, and T ) is the complete eigenfunction of the studied system obeying the stationary Schrodinger equation ... 

We assume that the Schrodinger basis ) is locally complete, and this would permit a development of the eigenfunction Tt) of U from Eq. (17) as follows ... 

In order to evaluate 5s/ISv from Eq. (282), we further need the functional derivatives dqfjjdvs and ScpflSv. The stationary OPM eigenfunctions (< /r), = 1,..., oo) form a complete orthonormal t, and so do the time-evolved states unperturbed states, remembering that at t = ti the orbitals are held fixed with respect to variations in the total potential. We therefore start from t = ti, subject the system to an additional small perturbation (5i)s(r, t) and let it evolve backward in time. The corresponding perturbed wave functions [Pg.135]

The reality, of course, is that any complete set of square integrable functions, such as all the atomic orbitals of any atom, provide for an exact description of the eigenfunction solutions to the Schrodinger equation for a molecule. However, since such an approach is not practicable even on the largest modem computers, we settle for an approximation, which leads to good results when comparisons with experimental data are made. This is the essence of the LCAO-MO approximation and it would be normal to extend equation 6.2 and take linear combinations of valence atomic orbitals chosen from all... 

For fhe model operator Ho, the Schrodinger equation has a complete (and orthonormal) set of eigenfunctions... 

The perturbation-theory expansion of Y" is performed in the basis space of the eigenfunctions which form a complete orthonormal basis and are solutions of the Schrodinger equation Ho(t>i = where the S are the unperturbed eigenenergies. 

To describe completely the quantum mechanical behavior of electrons in solids it is necessary to calculate the many-electron eigenfunction for the system. In principle this may be obtained from the tune-independent Schrodinger equation, but in practice the potential experienced by each electron is dictated by the behavior of all the other electrons in the sohd. Of course, the influence of nearby electrons will be much stronger than that of far-away electrons since the interaction is electrostatic in nature, but the fact remains that the motion of any one electron is strongly coupled to the motion of aU other electrons in the system. 

The Schrodinger equation has not been solved exactly for electrons in molecules larger than the H2 ion the interactions of multiple electrons become too complex to handle. However, the eigenfunctions of the Hamiltonian operator provide a complete set of functions, and as mentioned in Sect. 2.2.1, a linear combination of such functions can be used to construct any well-behaved function of the same coordinates. This suggests the possibility of representing a molecular electronic wavefunction by a linear combination of hydrogen atomic orbitals centered at the nuclear positions. In principle, we should include the entire set of atomic orbitals... 

If the particle is completely confined in the box the potential energy is equal to a constant inside the box and is infinite outside the box. We set the constant equal to zero. The resulting time-independent Schrodinger equation can be solved by separation of variables, as described in Appendix F. The energy eigenfunction (coordinate wave function) is a product of three factors, each one of which is the same as a wave function for a particle in a one-dimensional box of length a, b, or c ... 

The wave function of a system at a given instant can be any function that obeys the proper boundary conditions. There is no requirement that it must obey the time-independent Schrodinger equation. However, since the energy eigenfunctions are a complete set, any wave function at a fixed time can be written as a linear combination of energy eigenfunctions, as in Eqs. (15.3-20) and (16.3-35) with t set equal to zero. 

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