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CRC Press eBooks

Time-Dependent Perturbation Theory (TDPT)

Steven M. Pascal

Published Jul 12, 2023• 0 citations
Publisher

Abstract

If the perturbation is turned on for a brief period of time and then turned off, the particle ends up in the original potential. But it may have been induced to transition from one eigenstate to another during the perturbation. The calculation of the probability of that transition is the topic of time-dependent perturbation theory (TDPT). Again, formula are presented rather than derived, and two examples are provided: a linear perturbation, and an oscillating perturbation. These examples, respectively, show how selection rules arise, and how light of a given frequency excites energy transitions of the same frequency. This latter phenomenon is called resonance .

DOI
10.1201/9781003356172-15

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Time-dependent perturbation theory is discussed within the quantum phase space representation as formulated by Torres-Vega and Frederick(T-F) and the quantum transition probability is studied.Time-dependent perturbation formula in T-F quantum phase space is established,in accordance with the calculation theory in T-F phase space.As an example,the one dimensional harmonic oscillator system,affected by an external electric field,is used to calculate in T-F phase space,the result indicate that the principle of the method is correct.Quantum transition probability of the system's first excited state in T-F phase space is discussed.

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Approximation Methods II: Time-Dependent Perturbation Theory

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This chapter discusses the time-dependent perturbation theory applicable for small time-dependent perturbation. The incident electromagnetic wave makes the electric and magnetic potentials to oscillate and the perturbation is usually very small compared with the static Hamiltonian. The theory of time-dependent perturbation was developed by Dirac in 1927. Approximation methods are developed for systems where the time dependence of the Hamiltonian is small, slowly varying or rapidly varying. These correspond to perturbation, adiabatic and sudden approximations. An interesting consequence of sudden approximation is that the system can undergo an adiabatic change due to the sudden approximation. The rate of spontaneous emission is proportional to the number of atoms in the excited state. In the stimulated emission, an incident radiation of appropriate frequency induces an atom in an excited state to emit radiation. In the case of time-dependent perturbations, the system undergoes transition from one state to another during the action of the perturbation.

CRC Press eBooks•Sep 29, 2022•0 citations
If the perturbation is turned on for a brief period of time and then turned off, the particle ends up in the original potential. But it may have been induced to transition from one eigenstate to another during the perturbation. The calculation of the probability of that transition is the topic of time-dependent perturbation theory (TDPT). Again, formula are presented rather than derived, and two examples are provided: a linear perturbation, and an oscillating perturbation. These examples, respectively, show how selection rules arise, and how light of a given frequency excites energy transitions of the same frequency. This latter phenomenon is called resonance .
How to deal with time-dependent potentials, what kinds of physical problems having time-dependent potentials can be solved exactly and which are to be solved with the use of perturbation theory, what are different possible forms of time-dependent perturbating fields, and semi-classical treatment of the interaction of fields with matter are the subject matters of this chapter. First order perturbation theory, computation of transition probability, transition rates and Fermi's Golden Rule for harmonic and constant perturbations, absorption, stimulated emission and dipole approximation when electromagnetic fields interact with matter, spontaneous emission and Einstein coefficients, and dipole selection rules are other topics which are dealt with in detail in the chapter. Discussions with properly justified mathematical treatment on time-dependent potentials and on the exactly solvable time-dependent two-state systems are the special foci of this chapter.
Abstract Although the discussion in the preceding two chapters concentrated on the stationary states of an electronic system, the need to consider perturbations which mix stationary states and thereby excite transitions between these states was noted. In this chapter we apply time-dependent perturbation theory to estimate the rates of such transitions, first by developing a general formula for the transition probability between initial |i〉 and final |f〇 states; if the initial state |i〉 is lower in energy than |f〇 an absorption transition is involved, whereas if |f〇 is the lower state emission takes place. We then consider in some detail how such transitions are caused by interactions between the electronic system and the radiation field.
Equivalent derivations of time-dependent pertubation theory; Fermi golden rule; the Born approximation for scattering from Fermi golden rule; survival probability of a state in a time-independent perturbation; positronium in static and oscillating magnetic fields; hydrogen atom in a time-dependent electric field; a model for inelastic scattering of a projectile with a target; semiclassical treatment of the electromagnetic field; ionization of the hydrogen atom by an electromagnetic wave; cross sections for stimulated absorption and emission in hydrogen; spontaneous emission and selection rules with an application to the 2p to 1s transition in hydrogen; theory of the line width; formal scattering theory; S- and T-operators.
Time-dependent perturbation theory is discussed within the quantum phase space representation as formulated by Torres-Vega and Frederick(T-F) and the quantum transition probability is studied.Time-dependent perturbation formula in T-F quantum phase space is established,in accordance with the calculation theory in T-F phase space.As an example,the one dimensional harmonic oscillator system,affected by an external electric field,is used to calculate in T-F phase space,the result indicate that the principle of the method is correct.Quantum transition probability of the system's first excited state in T-F phase space is discussed.
This chapter discusses the time-dependent perturbation theory applicable for small time-dependent perturbation. The incident electromagnetic wave makes the electric and magnetic potentials to oscillate and the perturbation is usually very small compared with the static Hamiltonian. The theory of time-dependent perturbation was developed by Dirac in 1927. Approximation methods are developed for systems where the time dependence of the Hamiltonian is small, slowly varying or rapidly varying. These correspond to perturbation, adiabatic and sudden approximations. An interesting consequence of sudden approximation is that the system can undergo an adiabatic change due to the sudden approximation. The rate of spontaneous emission is proportional to the number of atoms in the excited state. In the stimulated emission, an incident radiation of appropriate frequency induces an atom in an excited state to emit radiation. In the case of time-dependent perturbations, the system undergoes transition from one state to another during the action of the perturbation.