Unperturbed eigenfunctions

The relation (2.13 7) clearly shows how the perturbation first order of the first order energy correction is calculated based on the free system eigenfunctions, unperturbed. Meanwhile, here will be generalized the treatment to the orbital d-case (2/+l)-degenerated in the atomic orbitals bases... 

If the unperturbed system is degenerate, so that several linearly independent eigenfunctions correspond to the same energy value, then a more complicated procedure must be followed. There can always be found a set of eigenfunctions (the zeroth order eigenfunctions) such that for each the perturbation energy is given by equation 9 and the perturbation theory provides the... 

The system to be treated consists of two nuclei A and B and two electrons 1 and 2. In the unperturbed state two hydrogen atoms are assumed, so that the zeroth-order energy is 2WH-If the first electron is attached to nucleus A and the second to nucleus B, the zeroth-order eigenfunction is ip (1)

zeroth-order energy, so that the system... 

The Helium Molecule and Molecule-ion.—The simplest example of a molecule containing a three-electron bond is the helium molecule-ion, in which a Is eigenfunction for each of two identical atoms is involved. The two unperturbed states of equal energy are He He+ and He-+ He. The formation of this molecule might be represented by the equation He Is2 >5 + He+ Is 5 —>- He (Is + ls) 2 Three dots in a horizontal line placed between the two atomic symbols may be used to designate a three-electron bond He He+. 

Perhaps the most straightforward method of solving the time-dependent Schrodinger equation and of propagating the wave function forward in time is to expand the wave function in the set of eigenfunctions of the unperturbed Hamiltonian [41], Hq, which is the Hamiltonian in the absence of the interaction with the laser field. 

The quantity k > is the unperturbed Hamiltonian operator whose orthonormal eigenfunctions and eigenvalues are known exactly, so that... 

The operator k is called the perturbation and is small. Thus, the operator k differs only slightly from and the eigenfunctions and eigenvalues of k do not differ greatly from those of the unperturbed Hamiltonian operator k The parameter X is introduced to facilitate the comparison of the orders of magnitude of various terms. In the limit A 0, the perturbed system reduces to the unperturbed system. For many systems there are no terms in the perturbed Hamiltonian operator higher than k and for convenience the parameter A in equations (9.16) and (9.17) may then be set equal to unity. 

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... 

We next expand the function xp in terms of the complete set of unperturbed eigenfunctions xpj ... 

The Hamiltonian operator for the unperturbed harmonic oscillator is given by equation (4.12) and its eigenvalues and eigenfunctions are shown in equations (4.30) and (4.41). The perturbation H is... 

To find the perturbation corrections to the eigenvalues and eigenfunctions, we require the matrix elements for the unperturbed harmonic... 

Each of the eigenfunctions is orthogonal to all the other unperturbed eigenfunctions for k n, but is not necessarily orthogonal to the other eigenfunctions for E Any linear combination of the members of the set... 

The first-order equations (9.22) and (9.24) apply here provided the additional index a and the correct unperturbed eigenfunctions are used... 

To find E l we multiply equation (9.62) by, the complex conjugate of one of the initial unperturbed eigenfunctions belonging to the degenerate eigenvalue E and integrate over all space to obtain... 

Since A and commute, they have simultaneous eigenfunctions. Therefore, we may select, Xi, , Xg the initial set of unperturbed eigenfunctions... 

The first-order perturbation correction to the ground-state energy is obtained by evaluating equation (9.24) with (9.80) as the perturbation and (9.82) as the unperturbed eigenfunction... 

As a normalized trial function 0 for the determination of the ground-state energy by the variation method, we select the unperturbed eigenfunction r2) of the perturbation treatment, except that we replace the atomic number Zby a parameter Z ... 

Since equation (10.43) with F = 0 is already solved, we may treat V as a perturbation and solve equation (10.43) using perturbation theory. The unperturbed eigenfunctions S H q) are the eigenkets n) for the harmonic oscillator. The first-order perturbation correction to the energy as given by equation (9.24) is... 

Assume that the perturbation, XH is small compared with H°, where X is a parameter. As X - 0, the eigenvalues and eigenfunctions are those of the unperturbed system, as given by... 

To establish the solution 1(or higher orders) the function is expanded in the basis set of the unperturbed eigenfunctions. That is... 

In this interpretation Q is the number of linearly independent eigenfunctions of the unperturbed Hamiltonian in the interval AE. From (36) the microcanonical average of an observable represented by the operator A in an arbitrary basis, is... 

E. A. Hylleraas, Z Phys. 65 (1930), 209 note 6, p. 279. Note that if(2) can alternatively be expressed as an infinite expansion in the unperturbed eigenfunctions but the Hylleraas variation-perturbation expression (1.5d) is generally more useful for practical numerical applications. 

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