On nonintegral E corrections in perturbation theory: application to the perturbed Morse oscillator

1994 ◽  
Vol 72 (1-2) ◽  
pp. 80-85 ◽  
Author(s):  
Hafez Kobeissi ◽  
Majida Kobeissi ◽  
Chafia H. Trad
Keyword(s):  
Perturbation Theory ◽  
Good Accuracy ◽  
Morse Oscillator ◽  
Numerical Application ◽  
Correction Formula ◽  
Theory Application ◽  
Nonhomogeneous Differential Equation ◽  
Particular Solution ◽  
New Formulation ◽  
Schrödinger Perturbation

A new formulation of the Rayleigh–Schrödinger perturbation theory is applied to the derivation of the vibrational eigenvalues of the perturbed Morse oscillator (PMO). This formulation avoids the conventional projection of the Ψ corrections on the basis of unperturbed eigenfunctions [Formula: see text], or the projection of the nonhomogeneous Schrödinger equations on [Formula: see text], it gives simple expressions for each E correction [Formula: see text] free of summations and integrals. When the PMO is characterized by the potential U = UM + UP (where UM is the unperturbed Morse potential), the eigenvalue of a vibrational level ν is given by: [Formula: see text]. According to the new formulation the correction £, [Formula: see text] is given by [Formula: see text], where σp(r) is a particular solution of the nonhomogeneous differential equation y″ + f y = sp; here [Formula: see text], sp is well known for each p: for p = 0, [Formula: see text]; for [Formula: see text]. For the numerical application one single routine is used, that of integrating y″ + f y = s, where the coefficients are known as well as the initial values. An example is presented for the Huffaker PMO of the (carbon monoxide) CO-X1Σ+ state. The vibrational eigenvalues Eν are obtained to a good accuracy (with p = 4) even for high levels. This result confirms the validity of this new formulation and gives a semianalytic expression for the PMO eigenvalues.

"Nonintegral" expressions for the diatomic centrifugal distortion constants

1995 ◽  
Vol 73 (5-6) ◽  
pp. 339-343
Author(s):  
Hafez Kobeissi ◽  
Chafia H. Trad
Keyword(s):  
Model Potential ◽  
Centrifugal Distortion ◽  
Numerical Application ◽  
Lennard Jones ◽  
Particular Solution ◽  
New Formulation ◽  
Centrifugal Distortion Constants ◽  
Schrödinger Perturbation ◽  
Analytical Expressions ◽  
Nonhomogeneous Equation

The problem of the centrifugal distortion constants (CDC), Dν, Hν, … for a diatomic molecule is considered. It is shown that a new formulation of the standard Rayleigh–Schrödinger perturbation theory can give simple and compact analytical expressions of the CDC (up to any order). Thus, the constants e1 = Bν, e2 = −Dν, e3 = Hν,…, en,… are all of the form en = lim σn(r)/σ0(r) as r → ∞. σ0 is the particular solution of the nonhomogeneous equation y″ + k(Eν – U)y = s, with s = ψν, where (Eν, ψν) is the eigenvector corresponding to the rotationless potential U(r) and to the vibrational level ν; and where σ0(0) = σ′0(0) = 0. σn is the particular solution of the above equation, where s is known for each order of n. The numerical application to the standard Lennard–Jones model potential shows that good results are obtained for Dν, Hν, Lν,…,Oν, Pν, for ν = 0 to 22, which is only at 2 × 10−4 of the well depth. The program uses one routine (the integration of the equation y″ + fy = s) repeated for different s; it is quite simple and gives no difficulties at the boundaries and there is no need to use any mathematical or numerical artifices.

A REFORMULATED BOUNDARY PERTURBATION THEORY IN ELECTROMAGNETISM AND ITS APPLICATION TO A SPHERE

1967 ◽  
Vol 45 (5) ◽  
pp. 1729-1743 ◽  
Author(s):  
M. L. Burrows
Keyword(s):  
Electromagnetic Field ◽  
Perturbation Theory ◽  
Classical Method ◽  
Experimental Results ◽  
Boundary Perturbation ◽  
Conducting Sphere ◽  
Classical Formulation ◽  
Classical Class ◽  
New Formulation ◽  
Boundary Perturbations

The classical method of solving electromagnetic field problems involving boundary perturbations is reformulated in a way that is both more general and simpler. The new formulation makes it easier to apply the theory to the class of boundaries amenable to the classical formulation, and shows that it can also be applied to other boundary shapes. As an example, the perfectly conducting sphere with surface perturbations has been treated, using the methods appropriate only for boundaries in the classical class and also using those applicable to the larger class. Some experimental results which appear to support the theory are reported.

Applicability of the schrödinger perturbation theory to bound states

1964 ◽  
Vol 10 (1) ◽  
pp. 73 ◽  
Author(s):  
K. Hausmann ◽  
W. Macke ◽  
P. Ziesche
Keyword(s):  
Perturbation Theory ◽  
Bound States ◽  
Schrödinger Perturbation

scholarly journals Quantum square-well with logarithmic central spike

2018 ◽  
Vol 33 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Miloslav Znojil ◽  
Iveta Semorádová
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Closed Form ◽  
Wave Functions ◽  
Change Of Variables ◽  
State Dependent ◽  
Square Well ◽  
Strength Formula ◽  
Schrödinger Perturbation ◽  
Dependent Interaction

Singular repulsive barrier [Formula: see text] inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction [Formula: see text] in nonlinear Schrödinger equation. In the linearized case, Rayleigh–Schrödinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small [Formula: see text] or after an amendment of the unperturbed Hamiltonian. At any spike strength [Formula: see text], the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables [Formula: see text] which interchanges the roles of the asymptotic and central boundary conditions.

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