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Revista Unifev: Ciência & Tecnologia

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

Michael de Melo, Angelo Rober Pulici

Published Sep 19, 2017• 0 citations
OpenAlex

Abstract

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.

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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.
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.
The basic phenomena of interaction between atoms and e.m. fields can be discussed in some detail without the need of a detailed model of the atom. Absorption and stimulated emission of radiation can be explained satisfactorily by the theory, where a classical electromagnetic wave provides a time-dependent perturbation to the energy of the atomic electrons. Laser radiation appears to a very good approximation to be a classical travelling wave of well-defined amplitude and phase. The radiation produced by a continuous laser is a good approximation to a coherent state. For conventional sources of radiation, the interaction energy is small compared with the energy an electron possesses by virtue of the attraction to the atomic nucleus, so perturbation theory produces quite acceptable results. The quantum mechanical analogy to the distortion of the classical charge cloud is given by the writing down of a general state function for the system, Ψ.
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.
We treat the interaction between an electromagnetic radiation bath and a material system in the semi-classical picture: the radiation is treated according to the Maxwell equations, but the process of emission/absorption is described through the concept of a photon; on the other hand, the material system is modelled according to quantum mechanics, assuming a discrete two-level energy spectrum. This very simple model allows one to elaborate the elementary theory of the absorption/emission coefficients, both for the spontaneous and for the stimulated processes which will be described within the electric dipole approximation. Selection rules for allowed transitions are eventually obtained and the nature of high-order transition is briefly outlined. Finally, a qualitative description of the light amplification by stimulated emission of radiation is presented.
We have discussed the problems of non linear interaction between electromagnetic radiations with atoms from semi-classical point of view. Time dependent Schrödinger equation for single electron system is solved by using perturbative technique to obtain transition probability. Higher order perturbation is also discussed which is used in multiple processes, in which two or more quanta are emitted instead of a single photon. The theory is based on assumption that the perturbation is small. From this transition probability ionization rate and absorption cross-section of hydrogen atom is calculated. Its variation with photon energy and field strength is analyzed which agrees very well with experimental observations.The Himalayan PhysicsVol. 3, No. 32012Page : 40-43
This chapter is devoted to an introduction to atomic physics which will be mainly concerned with one-electron atoms. After a brief discussion of the perturbation and variational methods in Section 14.1, in Section 14.2 we study the fine and hyperfine structure of the energy levels as well as the effect of a magnetic field on these levels. In Section 14.3 we examine the coupling of an atom to an electromagnetic field and important applications of this coupling such as the photoelectric effect and the rate of spontaneous emission. In Section 14.4 we give a brief introduction to a subject which has been expanding enormously in the last twenty years, the laser manipulation of atoms, and we discuss Doppler cooling and magneto-optical traps. Finally, Section 14.5 is devoted to a short discussion of two-electron atoms. Approximation methods Generalities In classical physics it is only in exceptional cases that it is possible to solve the Newton or Maxwell equations analytically given the initial conditions at time t = t 0 and, in the first case, the forces, or, in the second, the sources of electromagnetic field. In general, it is necessary to resort to an approximation method such as numerical integration of the equations, the perturbation method, or something else. The situation is no different in quantum physics: only in exceptional cases do we know how to “solve the Schrödinger equation” exactly, that is, how to obtain the time evolution of the state vector |ϕ( t 〉 as a function of its value |ϕ( t 0 〉 at initial time t = t 0 .