Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Galilean invariance of the Schrödinger equation in the adiabatic approximation: The central-field method

Alejandro López‐Castillo

Published Feb 1, 1996• 2 citations
Publisher

Abstract

In the present work we propose a method for solving the Schr\"odinger equation in the adiabatic approximation within the molecular context. We employ this method to solve problems with non-Galilean invariance, and an attempt is made to explain some theoretical aspects of it. Simple applications are also considered. This method, which we could call the central-field method (or linear combination of central fields), enables a better understanding of the problem of non-Galilean invariance. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.818

Related papers

- Approximation Methods I: Time-Independent Perturbation Theory

S. Rajasekar, R. Velusamy

The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.

CRC Press eBooks•Dec 11, 2014•0 citations

Quantum Mechanics Approximate Methods

Lukong Cornelius Fai

In quantum mechanics, there exist a lot of problems from the domain of atomic, molecular, solid-state physics, physics of magnetic phenomena, nuclear physics, etc. The number of problems which may be solved exactly are few such as some one-dimensional problems, problems in the rectangular well (barrier), the oscillator, problems of motion in a central field such as motion in a Coulomb field. These methods are well formulated and are used in evaluations to a greater approximation. However, in some cases, these may create problems that are very difficult, if not impossible, to be solved exactly. In such problems, the Hamiltonian cannot be exactly diagonalized in a simple manner. For such problems, we resort to approximate methods for finding the eigenvalues and eigenfunctions of the Hamiltonian. Besides, these approximate methods yield results closer to experimental ones. Amongst the approximate methods, we examine the variational method.

CRC Press eBooks•Apr 19, 2022•0 citations

On the use of the adiabaticity concept to obtain approximate solutions of the Schrödinger equation

Daniel A. Morales

In this paper the author discusses the use of the notion of adiabaticity to obtain approximate solutions of Schrödinger equation.

Journal of Chemical Education•Mar 1, 1990•0 citations

Adiabatic evolution of quantum-mechanical systems

Jorge Vidal, José Wudka

A description of the adiabatic approximation in terms of the time-evolution operator is presented. Corrections to the approximation are studied, and it is seen that these can be obtained in a simple way in the case of a rapidly oscillating Hamiltonian.

Physical Review A•Nov 1, 1991•12 citations

Radiative atomic processes and Galilean and gauge transformations

Alexander B Voitkiv

Radiative atomic processes, in which charged particles moving with velocities much less than the speed of light emit a photon, are normally considered by using the Schrödinger equation. By analysing the transformation properties of the radiative transition amplitudes under the change of a reference frame we establish that the total cross sections and decay rates, calculated with the Schrödinger equation, are Galilean invariant and that this invariance holds irrespective of the accuracy of wavefunctions describing the charged particles. By considering, within the scope of one reference frame, the radiative processes for two identical atomic systems which have different translational (centre-of-mass) velocities we show that the total cross sections and decay rates in general depend on this velocity if approximate wavefunctions are employed and that this is the direct consequence of the problem of gauge dependence. We also present a detailed discussion of the intimate interrelation between Galilean and gauge transformations which, if overlooked, can easily lead to the misinterpretation of the problem of gauge dependence as the 'problem of Galilean non-invariance'.

Journal of Physics B Atomic Molecular and Optical Physics•Oct 10, 2006•6 citations

On the adiabatic theorem in quantum mechanics

A. G. Chirkov

The adiabatic and postadiabatic approximations, as well as the adiabatic and nonstationary perturbation theories, are constructed using a canonical averaging method under the assumptions which are more general than those used in the existing theories. An asymptotic evaluation of the proximity of rigorous and approximate solutions is performed.

Technical Physics Letters•Feb 1, 2001•12 citations
In the present work we propose a method for solving the Schr"odinger equation in the adiabatic approximation within the molecular context. We employ this method to solve problems with non-Galilean invariance, and an attempt is made to explain some theoretical aspects of it. Simple applications are also considered. This method, which we could call the central-field method (or linear combination of central fields), enables a better understanding of the problem of non-Galilean invariance. \textcopyright{} 1996 The American Physical Society.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
In quantum mechanics, there exist a lot of problems from the domain of atomic, molecular, solid-state physics, physics of magnetic phenomena, nuclear physics, etc. The number of problems which may be solved exactly are few such as some one-dimensional problems, problems in the rectangular well (barrier), the oscillator, problems of motion in a central field such as motion in a Coulomb field. These methods are well formulated and are used in evaluations to a greater approximation. However, in some cases, these may create problems that are very difficult, if not impossible, to be solved exactly. In such problems, the Hamiltonian cannot be exactly diagonalized in a simple manner. For such problems, we resort to approximate methods for finding the eigenvalues and eigenfunctions of the Hamiltonian. Besides, these approximate methods yield results closer to experimental ones. Amongst the approximate methods, we examine the variational method.
Radiative atomic processes, in which charged particles moving with velocities much less than the speed of light emit a photon, are normally considered by using the Schrödinger equation. By analysing the transformation properties of the radiative transition amplitudes under the change of a reference frame we establish that the total cross sections and decay rates, calculated with the Schrödinger equation, are Galilean invariant and that this invariance holds irrespective of the accuracy of wavefunctions describing the charged particles. By considering, within the scope of one reference frame, the radiative processes for two identical atomic systems which have different translational (centre-of-mass) velocities we show that the total cross sections and decay rates in general depend on this velocity if approximate wavefunctions are employed and that this is the direct consequence of the problem of gauge dependence. We also present a detailed discussion of the intimate interrelation between Galilean and gauge transformations which, if overlooked, can easily lead to the misinterpretation of the problem of gauge dependence as the 'problem of Galilean non-invariance'.