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

CRC Press eBooks

Semiclassical Theory of Spectroscopic Transition

Abani K. Bhuyan

Published Feb 2, 2023• 0 citations
Publisher

Abstract

It has been postulated that the state function can be expressed as a linear superposition of eigenstates of Hermitian operators. When a system is subjected to an external impulse in the form of radiation, the probability of finding the system in different eigenstates of the state function changes. The total probability is always unity, which means the change in probabilities for different states occurs at the expense of each other. This change in probability of finding the system in different eigenstates brought about by system-radiation interaction is called the spectroscopic transition. Here, we use a two-level system to develop the details of a general spectroscopic transition, and work further to examine the effects of interruption in the coherent evolution of the state function. The physical meaning of absorption coefficient is discussed in detail. We derive the general forms of relaxation of the excited state, spectral line shape, and line broadening. The origin and the meaning of homogeneous and inhomogeneous line broadening are discussed. We describe the Doppler effect and its use in the interpretation of inhomogeneous line shape.

DOI
10.1201/9781003293064-3

Related papers

Excitation of transitions between atomic or molecular energy levels by monochromatic laser radiation - II. Excitation under conditions of strictly inhomogeneous line broadening and mixed homogeneous and inhomogeneous broadening

Robert A. L. Smith

Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly inhomogeneous line broadening due to such causes as doppler shift arising from translational velocity. The ratio of the number of molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width ∆, where E0 is the amplitude of the electric field in the incident radiation and is the dipole moment matrix element for the transition. Excitation functions obtained in a previous paper (I) for various values of the ratio γ/β, where γ is the half line-width in the absence of inhomo­geneous broadening, are used to obtain the variation of N2/N with time under conditions of mixed broadening for various values of the ratio γ/∆ and γ/β, when the exciting radiation is in exact resonance with the central frequency of the transition.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Jun 30, 1978•12 citations

Excitation of transitions between atomic or molecular energy levels by monochromatic laser radiation - I. Excitation under conditions of strictly homogeneous line broadening

Robert A. L. Smith

Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly homogeneous line-broadening. The ratio of the number of atoms or molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width γ. E0 is the amplitude of the electric field in the incident radiation and the dipole moment matrix element for the transition. Exact expressions for N2/N as a function of time are given for the situation when the frequency ω/2π of the radiation is exactly equal to that of the transition ω0/2π. Numerical calculations are made for the situation when the ‘detuning’ δ ═ ω — ω0 is not equal to zero. Approximate expressions are also derived in terms of δ for use in a later paper which deals with mixed homogeneous and inhomogeneous line-broadening.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Jun 30, 1978•17 citations

On the theory of homogeneous and inhomogeneous line broadening. An exactly solvable model

Ernst Walter Knapp, Sighart F. Fischer

The simultaneous influence of homogeneous and inhomogeneous line-broadening mechanisms are investigated for a reference system embedded in a medium. An analytically solvable model is constructed which includes the interaction of the reference mode with N strongly coupled environmental states (causing an inhomogeneity) and weakly coupled bath states. The modulation of the reference mode is caused by transitions between the strongly coupled perturber states. These may be switched on with a different rate than they are switched off. Consequently, effects of energy of activation and the change in entropy for environmental transitions on the line shape can be analyzed. The spectral function as well as the relaxation function are discussed as a function of the relevant parameters such as the number of perturbers, the lifetime T1, the temperature, and transition rates.

The Journal of Chemical Physics•Jan 1, 1981•70 citations

The quenching of resonance radiation and the breadths of absorption-lines

Robert William Ditchburn

In a paper by Slater a new theory of radiation is discussed and the following explanation of the breadths of spectral lines is given: “An atom in the ith. state has a probability Pi of suffering in unit time a transition. Thus there is a probability Pi that the vibrations of each of the oscillators will simultaneously cease. But we shall assume that, in addition to this probability Pi of ceasing its oscillation altogether, each oscillator has also an independent probability Pj of suffering an interruption in which it ceases its oscillation as if it were leaving the state, but immediately begins again as if it were entering the same state” (i.e. with an arbitrary phase difference). “This term Pj is the same as the probability that an atom in the jth state will leave that state.” The total probability of interruption of vibration is thus (Pi + Pj) which is symmetrical with regard to the two end states. This makes the breadths of absorption and emission lines equal and so satisfies Kirchhoff's law.

Mathematical Proceedings of the Cambridge Philosophical Society•Feb 1, 1926•1 citation

Non-Markovian nature of relaxation processes and its spectroscopic manifestations

Pavel A Apanasevich, Alexander P. Nizovtsev

An analysis is made of some manifestations of the non-Markovian nature of relaxation processes observed in the response of quantum systems to external radiation. It is shown that allowance for this nature enhances the saturation effect and the non-Lorentzian nature of spectral line wings, and gives rise to additional absorption (luminescence) lines due to transitions involving changes in the states of several particles.

Soviet Journal of Quantum Electronics•Apr 30, 1976•0 citations

Radiative Transitions between Discrete Quantum States

Greg J. Tallents

Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.

Cambridge University Press eBooks•Feb 20, 2018•0 citations
It has been postulated that the state function can be expressed as a linear superposition of eigenstates of Hermitian operators. When a system is subjected to an external impulse in the form of radiation, the probability of finding the system in different eigenstates of the state function changes. The total probability is always unity, which means the change in probabilities for different states occurs at the expense of each other. This change in probability of finding the system in different eigenstates brought about by system-radiation interaction is called the spectroscopic transition. Here, we use a two-level system to develop the details of a general spectroscopic transition, and work further to examine the effects of interruption in the coherent evolution of the state function. The physical meaning of absorption coefficient is discussed in detail. We derive the general forms of relaxation of the excited state, spectral line shape, and line broadening. The origin and the meaning of homogeneous and inhomogeneous line broadening are discussed. We describe the Doppler effect and its use in the interpretation of inhomogeneous line shape.
Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly inhomogeneous line broadening due to such causes as doppler shift arising from translational velocity. The ratio of the number of molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width ∆, where E0 is the amplitude of the electric field in the incident radiation and is the dipole moment matrix element for the transition. Excitation functions obtained in a previous paper (I) for various values of the ratio γ/β, where γ is the half line-width in the absence of inhomo­geneous broadening, are used to obtain the variation of N2/N with time under conditions of mixed broadening for various values of the ratio γ/∆ and γ/β, when the exciting radiation is in exact resonance with the central frequency of the transition.
Abstract The development with time of the excitation of a transition between two atomic or molecular energy levels under the influence of monochromatic laser radiation is examined under conditions of strictly homogeneous line-broadening. The ratio of the number of atoms or molecules, N2, in an excited state to the total number, N, is calculated for various ratios of the intensity parameter β ═ E0μ/ħ to the half line-width γ. E0 is the amplitude of the electric field in the incident radiation and the dipole moment matrix element for the transition. Exact expressions for N2/N as a function of time are given for the situation when the frequency ω/2π of the radiation is exactly equal to that of the transition ω0/2π. Numerical calculations are made for the situation when the ‘detuning’ δ ═ ω — ω0 is not equal to zero. Approximate expressions are also derived in terms of δ for use in a later paper which deals with mixed homogeneous and inhomogeneous line-broadening.
The simultaneous influence of homogeneous and inhomogeneous line-broadening mechanisms are investigated for a reference system embedded in a medium. An analytically solvable model is constructed which includes the interaction of the reference mode with N strongly coupled environmental states (causing an inhomogeneity) and weakly coupled bath states. The modulation of the reference mode is caused by transitions between the strongly coupled perturber states. These may be switched on with a different rate than they are switched off. Consequently, effects of energy of activation and the change in entropy for environmental transitions on the line shape can be analyzed. The spectral function as well as the relaxation function are discussed as a function of the relevant parameters such as the number of perturbers, the lifetime T1, the temperature, and transition rates.
In a paper by Slater a new theory of radiation is discussed and the following explanation of the breadths of spectral lines is given: “An atom in the ith. state has a probability Pi of suffering in unit time a transition. Thus there is a probability Pi that the vibrations of each of the oscillators will simultaneously cease. But we shall assume that, in addition to this probability Pi of ceasing its oscillation altogether, each oscillator has also an independent probability Pj of suffering an interruption in which it ceases its oscillation as if it were leaving the state, but immediately begins again as if it were entering the same state” (i.e. with an arbitrary phase difference). “This term Pj is the same as the probability that an atom in the jth state will leave that state.” The total probability of interruption of vibration is thus (Pi + Pj) which is symmetrical with regard to the two end states. This makes the breadths of absorption and emission lines equal and so satisfies Kirchhoff's law.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.