scholarly journals Unitary transformations depending on a small parameter

2011 ◽  
Vol 468 (2139) ◽  
pp. 685-700 ◽  
Author(s):  
F. Casas ◽  
J. A. Oteo ◽  
J. Ros
Keyword(s):  
Perturbation Theory ◽  
Quantum Mechanics ◽  
Small Parameter ◽  
Canonical Transformations ◽  
Magnus Expansion ◽  
Unitary Transformations ◽  
Standard Perturbation Theory

We formulate a unitary perturbation theory for quantum mechanics inspired by the Lie-Deprit formulation of canonical transformations. The original Hamiltonian is converted into a solvable one by a transformation obtained through a Magnus expansion. This ensures unitarity at every order in a small parameter. A comparison with the standard perturbation theory is provided. We work out the scheme up to order ten with some simple examples.

scholarly journals Exponential Perturbative Expansions and Coordinate Transformations

2020 ◽  
Vol 25 (3) ◽  
pp. 50
Author(s):  
Ana Arnal ◽  
Fernando Casas ◽  
Cristina Chiralt
Keyword(s):  
Perturbation Theory ◽  
Approximate Solutions ◽  
Time Dependent ◽  
Periodic Systems ◽  
Quantum Mechanical ◽  
Perturbation Techniques ◽  
Unified Approach ◽  
Magnus Expansion ◽  
Theory Lead ◽  
Standard Perturbation Theory

We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet–Magnus expansion for periodic systems, the quantum averaging technique, and the Lie–Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and it can be formulated for any time-dependent linear system of ordinary differential equations. All of the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

scholarly journals Convergent perturbation theory for lattice models with fermions

2016 ◽  
Vol 31 (13) ◽  
pp. 1650072 ◽  
Author(s):  
V. K. Sazonov
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Expansions ◽  
Lattice Model ◽  
Lattice Models ◽  
Convergent Series ◽  
Standard Perturbation Theory

The standard perturbation theory in QFT and lattice models leads to the asymptotic expansions. However, an appropriate regularization of the path or lattice integrals allows one to construct convergent series with an infinite radius of the convergence. In the earlier studies, this approach was applied to the purely bosonic systems. Here, using bosonization, we develop the convergent perturbation theory for a toy lattice model with interacting fermionic and bosonic fields.

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