Schrodinger equation unperturbed

This equation contains the time-dependent Schrodinger equation for the unperturbed states... 

The solutions for the unperturbed Hamilton operator from a complete set (since Ho is hermitian) which can be chosen to be orthonormal, and A is a (variable) parameter determining the strength of the perturbation. At present we will only consider cases where the perturbation is time-independent, and the reference wave function is non-degenerate. To keep the notation simple, we will furthermore only consider the lowest energy state. The perturbed Schrodinger equation is... 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

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 K-matrix method is essentially a configuration interaction (Cl) performed at a fixed energy lying in the continuum upon a basis of "unperturbed funetions that (at the formal level) includes both diserete and eontinuous subsets. It turns the Schrodinger equation into a system of integral equations for the K-matrix elements, which is then transformed into a linear system by a quadrature upon afinite L basis set. 

In reality, this term is not small in comparison with the other terms so we should not expect the perturbation technique to give accurate results. With this choice for the perturbation, the Schrodinger equation for the unperturbed Hamiltonian operator may be solved exactly. 

Note that the subscript (/x) of / /, peitJ is the name of the electron described by the unperturbed Schrodinger equation,... 

We consider the problem of s-state energy shift according to the perturbation theory. Such analysis was performed for the pionic hydrogen in Ref. (Lyubovitskji and Rusetsky, 2000). Let Ho + Hc be the unperturbed Hamiltonian, whereas V is considered as a perturbation. The ground-state solution of the unperturbed Schrodinger equation in the center of mass (CM) system frame (E — Ho — Hc) To(0)) = 0, with E = M + m + E, is given by... 

The solution of the unperturbed Hamiltonian operator forms a complete orthonormal set. The perturbed Schrodinger equation is given by... 

The formal similarity between Eq. (10) and the time-dependent Schrodinger equation is striking, and we shall indeed develop methods which are very reminiscent of quantum mechanics. In particular, we may calculate the eigenfunctions and eigenvalues of the unperturbed Liouville operator L0. We look for solutions of ... 

Omitting all discussion of the mathematical properties and the subtleties with regard to the continuum of final state energies, we will hop to the perturbation expansion in three short equations. The initial state describing the unperturbed reactant (DA) system describes the solution to the zeroth order Schrodinger equation ... 

MBPT) and our Hamiltonian (76). If we assume that we know the solution of the unperturbed Schrodinger equation... 

To obtain workable expressions for the perturbative corrections to the wavefunction /k, the full Schrodinger equation is first projected against all of the unperturbed eigenstates lj> other than the state d>k whose perturbative corrections are sought ... 

If the solutions (energies and wave functions P ) of the Schrodinger equation for the unperturbed system Tf(°) P = F,1,01 4/jl°l are known, and the operator form of the perturbation, Hp, can be specified, the Rayleigh-Schrodinger perturbation theory will provide a description of the perturbed system in terms of the unperturbed system. Thus, for the perturbed system, the SE is... 

Equation ( .96) is just the Schrodinger equation for the unperturbed system. Equation... 

If the solutions (energies E and wave functions XP ) of the Schrodinger equation for the unperturbed system //(°)xp(°) = are known, and the operator form of the... 

Equation (A.96) is just the Schrodinger equation for the unperturbed system. Equation (A.97) is the first-order equation. Multiplying each term of equation (A.97) on the left by and integrating yield... 

Equations (3.23) and (3.24) are valid also for a model space containing several unperturbed energies, e.g. several atomic configurations. These equations will form the basis for our many-body treatment. The generalized Bloch equation is exact and completely equivalent to the Schrodinger equation for the states considered. 

By this, the expansion coefficients uffl are themselves of the 0-eth order in A. The restriction l / k indicates that the correction is orthogonal to the unperturbed vector. In order to get the corrections to the /c-th vector, we find the scalar product of the perturbed Schrodinger equation for it written with explicit powers of A with one of the eigenvectors of the unperturbed problem p (j k). For the first order in A we get ... 

Let us consider the situation when the eigenvalues of the unperturbed Hamiltonian II 0) are respectively gi, -fold degenerate. In this case the unperturbed Schrodinger equation reads ... 

The point is that the vectors k 4 satisfying the unperturbed Schrodinger equation, if used to expand 44 make the right hand side disappear and the equation becomes a uniform one. The only thing we can do is to use it to determine the proper expansion coefficients of the zeroth order wave function b p in terms of the degenerate subspace as well as the first order energy. (The first order wave function is usually not calculated/considered in the degenerate case.)... 

We now turn to a completely different method for solving the time dependent Schrodinger equation (6.2.1). The central idea of the method is to neglect transitions between states that correspond to continuum states of the unperturbed SSE Hamiltonian, but to take transitions from the bound space into the continuum (and back) accurately into account. We start with the unperturbed Hamiltonian Hq defined in (6.1.22). Its bound spectrum is given by... 

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]

Up to this point we are still dealing with undetermined quantities, energy and wave function corrections at each order. The first-order equation is one equation with two ImEnowns. Since the solutions to the unperturbed Schrodinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrodinger perturbation theory, and the equation in (4.32) becomes... 

Using the normal product unperturbed Hamiltonian, the zero-order Schrodinger equation becomes... 

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