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

Nonperturbative Approach to the Theory of Transition Probabilities

R. Wallace

Published Nov 1, 1970• 11 citations
Publisher

Abstract

A nonperturbative approach to the theory of transition probabilities is presented. The results of this theory are employed to investigate the validity of the assumptions inherent in the perturbative approach. For this purpose, the simple examples of absorption of radiation (first-order process) and Raman scattering (second-order process) are used. The method proposed, which is based upon finite-difference techniques, represents an accurate solution of the time-dependent Schr\"odinger equation in a given representation and has many advantages over methods based upon the perturbation series.

DOI
10.1103/physreva.2.1711

Related papers

Schrodinger Equation cannot Describe Transition Processes

Junhao, Z.

In this paper we prove the following: (1) The basic error of time-dependent perturbation theory is using the sum of first finite order of perturbed solutions to substitute the exact solution in the divergent interval of the series for calculating the transition probability. In addition quantum mechanics neglects the influence of the normality condition in the continuous case. In both cases Fermi golden rule is not a mathematically reasonable deductive inference from the Schrodinger equation. (2) The transition probability per unit time deduced from the exact solution of the Schrodinger equation is zero, which cannot be used to describe the transition processes.

arXiv:physics/0101049•Jan 9, 2001•0 citations

Transition between atomic states and semi-classical radiation theory using quantum perturbation

Michael de Melo, Angelo Rober Pulici

Some phenomena cannot be solved using the exact solution of the Schrödinger equation. However, using approximated methods such as perturbation theory we are able to obtain remarkable results considering terms up to second order. The physical phenomena studied here are the absorption and stimulated emission by the classical feature. In addition, we are going to discuss the physical meaning of perturbation theory when we treat the behaviour how initial state ket evolves as time goes on. We will evaluate the transition of probability between states, the rate and the energy involved in the atomic transitions. It will be introduced fundamental concepts of quantum mechanics, such as interaction picture, and a general perturbation theory using constant and sinusoidal time-dependent perturbation term. Concerning to these phenomena, we considered the perturbed term of the Hamiltonian of the interaction between the vector potential and electron momentum.

Revista Unifev: Ciência & Tecnologia•Sep 19, 2017•0 citations

Nonperturbative approach to the theory of transition probabilities by the stroboscopic method

Attila Aşkar

The exact differential equations with periodic coefficients characterizing the transition in a two-level quantum system are solved by the stroboscopic method which is a nonperturbative approach. This method is based on an averaging over the fluctuations in the system. The solution is unitary and is valid for large times also, as opposed to the short-time validity of the perturbative solution. It should also be noted that this method gives Rabi's well-known solution to the two-level system at resonance by the so-called transformation to the rotating coordinate frame.

Physical Review A•Dec 1, 1974•12 citations

Answer of the most important unsettled question of physical theory in 20th century

Junhao, Z.

Quantum mechanics take the sum of first finite order approximate solutions of time-dependent perturbation to substitute the exact solution. From the point of mathematics, it may be correct only in the convergent region of the time-dependent perturbation series. Where is the convergent region of this series? Quantum mechanics did not answer this problem. However it is relative to the question, can we use the Schrodinger equation to describe the transition processes? So it is the most important unsettling problem of physical theory. We find out the time-dependent approximate solution for arbitrary and the exact solution. Then we can prove that: (1) In the neighborhood of the conservation of energy, the series is divergent. The basic error of quantum mechanics is using the sum of the first finite orders approximate solutions to substitute the exact solution in this divergent region. It leads to an infinite error. So the Fermi golden rule is not a mathematically reasonable inference of the. Schrodinger equation (2) The transiton probability per unit time deduced from the exact solution of Schrodinger equation cannot describe the transition processes. This paper is only a prime discussion.

arXiv:physics/0101046•Jan 8, 2001•0 citations

A nonperturbative solution to induced transitions between atomic states

Lin Qiong-gu

An introduction is presented to a recently established nonperturbative approach to the transitions between atomic states induced by weak electromagnetic waves. A special case of the general solution is calculated and discussed in detail. Results of the nonperturbative approach reduce under perturbation condition to those obtained by the time- dependent perturbative method,but overcome the difficulty that the perturbative results are obviously unreasonable in the near resonant region. According to the nonperturbative solution,the electric- dipole selection rules are approximately recovered for short time. For large time,however,probabilities for forbidden transitions may become comparable with those for allowed ones.

College Physics•Jan 1, 2015•0 citations

General procedure of solving the time-dependent Schrödinger equation

C.Y. Chen

A nonperturbative procedure of solving the time-dependent Schrödinger equation, called the multi-projection approach or phase dynamics of quantum mechanics, is derived and illustrated. In addition to introducing a method with that time-dependent systems become solvable (under the assumption that corresponding time-independent systems are solvable), the new approach unveils several misconcepts related to the usual wavefunction expansion and the standard perturbation theory.

arXiv:quant-ph/0009015•Sep 4, 2000•0 citations
A nonperturbative approach to the theory of transition probabilities is presented. The results of this theory are employed to investigate the validity of the assumptions inherent in the perturbative approach. For this purpose, the simple examples of absorption of radiation (first-order process) and Raman scattering (second-order process) are used. The method proposed, which is based upon finite-difference techniques, represents an accurate solution of the time-dependent Schr"odinger equation in a given representation and has many advantages over methods based upon the perturbation series.
In this paper we prove the following: (1) The basic error of time-dependent perturbation theory is using the sum of first finite order of perturbed solutions to substitute the exact solution in the divergent interval of the series for calculating the transition probability. In addition quantum mechanics neglects the influence of the normality condition in the continuous case. In both cases Fermi golden rule is not a mathematically reasonable deductive inference from the Schrodinger equation. (2) The transition probability per unit time deduced from the exact solution of the Schrodinger equation is zero, which cannot be used to describe the transition processes.
Some phenomena cannot be solved using the exact solution of the Schrödinger equation. However, using approximated methods such as perturbation theory we are able to obtain remarkable results considering terms up to second order. The physical phenomena studied here are the absorption and stimulated emission by the classical feature. In addition, we are going to discuss the physical meaning of perturbation theory when we treat the behaviour how initial state ket evolves as time goes on. We will evaluate the transition of probability between states, the rate and the energy involved in the atomic transitions. It will be introduced fundamental concepts of quantum mechanics, such as interaction picture, and a general perturbation theory using constant and sinusoidal time-dependent perturbation term. Concerning to these phenomena, we considered the perturbed term of the Hamiltonian of the interaction between the vector potential and electron momentum.
The exact differential equations with periodic coefficients characterizing the transition in a two-level quantum system are solved by the stroboscopic method which is a nonperturbative approach. This method is based on an averaging over the fluctuations in the system. The solution is unitary and is valid for large times also, as opposed to the short-time validity of the perturbative solution. It should also be noted that this method gives Rabi's well-known solution to the two-level system at resonance by the so-called transformation to the rotating coordinate frame.
Quantum mechanics take the sum of first finite order approximate solutions of time-dependent perturbation to substitute the exact solution. From the point of mathematics, it may be correct only in the convergent region of the time-dependent perturbation series. Where is the convergent region of this series? Quantum mechanics did not answer this problem. However it is relative to the question, can we use the Schrodinger equation to describe the transition processes? So it is the most important unsettling problem of physical theory. We find out the time-dependent approximate solution for arbitrary and the exact solution. Then we can prove that: (1) In the neighborhood of the conservation of energy, the series is divergent. The basic error of quantum mechanics is using the sum of the first finite orders approximate solutions to substitute the exact solution in this divergent region. It leads to an infinite error. So the Fermi golden rule is not a mathematically reasonable inference of the. Schrodinger equation (2) The transiton probability per unit time deduced from the exact solution of Schrodinger equation cannot describe the transition processes. This paper is only a prime discussion.
An introduction is presented to a recently established nonperturbative approach to the transitions between atomic states induced by weak electromagnetic waves. A special case of the general solution is calculated and discussed in detail. Results of the nonperturbative approach reduce under perturbation condition to those obtained by the time- dependent perturbative method,but overcome the difficulty that the perturbative results are obviously unreasonable in the near resonant region. According to the nonperturbative solution,the electric- dipole selection rules are approximately recovered for short time. For large time,however,probabilities for forbidden transitions may become comparable with those for allowed ones.