Schrodinger’s perturbational theory

P.-O. Lowdin, Studies in perturbation theory. IX. Connection between various approaches in the recent development-evaluation of upper bounds to energy eigenvalues in Schrodinger s perturbation theory. J. Math. Phys. 6, 1341-1353 (1965)... 

Studies in Perturbation Theory. IX. Connection Between Various Approaches in the Recent Development - Evaluation of Upper Bounds to Energy Eigenvalues in Schrodinger s Perturbation Theory P.-O. Lowdin... 

The treatment of Schrodinger s perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. It is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. 

As we shall see below, the main mathematical function of unlinked clusters is to cancel the normalization terms in the denominator of Eq. (37.) This cancellation of unlinked clusters - plays an important role in infinite many-body problems, because — co as N — Rayleigh-Schrodinger (R.S.) perturbation theory but it must be recognized in variational methods by augmenting the trial functions with imlinked clusters. > ... 

Dalgarno, A., Proc. Phys. Soc. A69, 784, "Application of the Rayleigh-Schrodinger perturbation theory to the hydrogen atom." Extension of Wigner s (1954) and Tree s (1956) work. 

Note that the choice of non-orthogonal versus orthogonal basis functions has no consequence for the numerical variational solutions (cf. Coulson s treatment of He2, note 76), but it undermines the possibility of physical interpretation in perturbative terms. While a proper Rayleigh-Schrodinger perturbative treatment of the He- He interaction can be envisioned, it would not simply truncate at second order as assumed in the PMO analysis of Fig. 3.58. Note also that alternative perturbation-theory formulations that make no reference to an... 

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... 

Transition probabilities. The interaction of quantum systems with light may be studied with the help of Schrodinger s time-dependent perturbation theory. A molecular complex may be in an initial state i), an eigenstate of the unperturbed Hamiltonian, Jfo I ) = E 10- If the system is irradiated by electromagnetic radiation of frequency v = co/2nc, transitions to other quantum states /) of the complex occur if the frequency is sufficiently close to Bohr s frequency condition,... 

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