Lux aeterna luceat eis. Let there be light. Take the rest of the week off. Atomic energy levels deduced from optical spectra comprise one of the most precisely known sets of physical measurements that exist. However, the precision of the determinations of the relative oscillator strengths of these spectral lines from the relative intensities of spectral lines is much less precise. Fortunately, time-dependent methods for the study of the dynamics of the emission process exist (and are being developed) that permit the transition probability rates, oscillator strengths, and reaction rates to be determined with ever-increasing precision. In most elementary quantum mechanics textbooks, the section on the emission of radiation is the least satisfactory section of the book. Whereas the development of relationships among various spatial overlap integrals between state vectors for various operators (such as the electrostatic dipole moment) can be formulated in a very elegant ab initio manner, the connection of these matrix elements to the time dependence of the system often seems driven by a posteriori assumptions that are extended beyond their justifiable range of applicability. As Fermi observed in stating his Golden Rule , “the transition probability and energy perturbation can be calculated with the help of perturbation theory ( i.e. , there is no better way known).” The Weisskopf–Wigner approximation offers a scheme for making precise calculations, but does not provide the rigor that has characterized so many other areas of quantum mechanics.
Lux aeterna luceat eis. Let there be light. Take the rest of the week off. Atomic energy levels deduced from optical spectra comprise one of the most precisely known sets of physical measurements that exist. However, the precision of the determinations of the relative oscillator strengths of these spectral lines from the relative intensities of spectral lines is much less precise. Fortunately, time-dependent methods for the study of the dynamics of the emission process exist (and are being developed) that permit the transition probability rates, oscillator strengths, and reaction rates to be determined with ever-increasing precision. In most elementary quantum mechanics textbooks, the section on the emission of radiation is the least satisfactory section of the book. Whereas the development of relationships among various spatial overlap integrals between state vectors for various operators (such as the electrostatic dipole moment) can be formulated in a very elegant ab initio manner, the connection of these matrix elements to the time dependence of the system often seems driven by a posteriori assumptions that are extended beyond their justifiable range of applicability. As Fermi observed in stating his Golden Rule , “the transition probability and energy perturbation can be calculated with the help of perturbation theory ( i.e. , there is no better way known).” The Weisskopf–Wigner approximation offers a scheme for making precise calculations, but does not provide the rigor that has characterized so many other areas of quantum mechanics.
During the presence of the radiation perturbation ( ),ˆ tV the Hamiltonian of the system becomes ( )tVHH ˆˆˆ 0 += (37.4) and, since the state of the system varies with time, we must adopt the time-dependent Schrödinger equation to be a basis for the perturbation theory, that is ( ) ( ) ( )0ˆ ˆti H V tt ∂Ψ ⎡ ⎤= + Ψ⎣ ⎦∂= t (37.5) We assume that Eq.(37.5) has normalized solutions which can be expanded in terms of the as 0kΨ t a t t a t e ε−Ψ = Ψ = Ψ∑ ∑ = (37.6) where the normalization condition on ( )tΨ gives a t =∑ (37.7) that is, ( ) 2ka t gives the probability that the system is in the state at time t. Substituting Eq.(37.6) into Eq.(37.5) yields da t ii a t e V t a t e dt ε εε ε− −⎡ ⎤ ⎡ ⎤− Ψ = +⎢ ⎥ ⎣ ⎦⎣ ⎦∑ ∑ = == = which reduces to da t i e V t a t e dt ε− k ε−Ψ = Ψ∑ ∑= == (37.8) In view of orthogonality of the stationary state eigenfunctions, multiplication by ( ) followed by integration over the configuration space gives da t i e a t e V t dt ε εδ− −= Ψ Ψ∑ ∑= == (37.9) Denoting the matrix elements of the perturbation operator between the stationary states n and k by ( ) ( )0 ˆnk n kV t V t 0= Ψ Ψ (37.10) we can rewrite Eq.(37.9) as da t i a t V t e dt ε ε−= ∑ == (37.11) which is a set of simultaneous differential equations of the first order for the coefficients ( ).na t If we assume that the system is in a definite stationary state at some 0iΨ particular time 0,t = and then it is allowed to interact with the radiation field, we obtain from Eq.(37.6) that ( ) ( )0 00 0i k k aΨ = Ψ = Ψk∑ Multiplying by and integrating gives ( )∗Ψ 0k ( )0ka kiδ= (37.12) that is, and for .( ) 10 =ia ( ) 00 =ka ik ≠ If we consider a time interval t sufficiently short for the coefficients in Eqs.(37.11) to be approximated by the form (37.12), which means that the term gives the greatest contribution to the sum and all other terms can be neglected, we have ( )tak ik = ( ) ( ) ( )0 0 /n ii tn nida ti V t edt ε ε−= == , ( ) ( ) ( )i i iida ti a t Vdt == t (37.13) and on integration we obtain ( ) ( ) ( ) (0 0 / 0 ia t V t e dt n iε ε ′− ′= − ≠∫ == ) , ( ) ( ) tdtVita iii ′′−= ∫ 0 1 = (37.14) The probability (29.29) that the system is in the state at time t, in other words, that coincides with is given by ( )tΨ ,0nΨ ( ) ( ) ( )2 20in n nP t t a t= Ψ Ψ = (37.15) and this is taken to be the probability of direct transition from state to state in a time t, induced by the presence of the radiation field. Assuming a transition where a photon is emitted, that is, we recall the Bohr condition (28.19), which reads ,00 ni εε > (37.16) inni ωεε ==− 00 so that Eqs.(37.14) can be rewritten as ( ) ( ) ( ) 0 n ni ia t V t e dt n iω ′−′ ′= − ≠∫= (37.17) ( ) ( ) 0 1 t i ii ia t V t dt′ ′= − ∫= As we can see from the above, the time-dependent perturbation theory is concerned with transitions that result from perturbation, between the unperturbed levels of the system, while time-independent perturbation theory calculates the stationary shifts in the energy levels. 37.2. FERMI'S GOLDEN RULE Consider the case of a constant perturbation, switched on at time The integrals (37.17) become straightforward, and we obtain .0=t ( ) ( ) ( )1ini tnin in Va t e n iωω −= − ≠= (37.18) ( ) tVita iii =−=1 Substituting Eqs.(37.18) into Eq.(37.6) gives, to a first approximation, the wave function of the atomic system as Vt e t e tωω − − ≠ Ψ = Ψ + − Ψ∑= = (37.19) The probability (37.15) of observing the system in the state at time t, for the coefficients given by Eq.(37.18), reduces to 0 nΨ( )tan ( ) ( 2/sin4 222 2 t V ni in ωω== ) (37.20) which means that transitions from state to states only take place if the matrix elements do not vanish. The plot of niV ( )tPin as a function of inω , given in Figure 37.1 for a perturbation which is on for a time t, suggests that the probability behaves like a Dirac δ -function as t approaches infinity. We may represent the δ -function as ( ) dxe t t 2 1lim 2/ 2/sin lim ωππω ω ( ) ( ininxi dxe in ωδωδπ ω 22/2 1 2/ === ∫ ∞ ∞− ) (37.21) where use was made of a property derived from the normalization condition, written as ( ) ( ) ( ) ( ) ( )u k kukudkuduu δδδδ 1or1 === ∫∫ ∞ P t( ) ω4π6π t2π in t-t-t 2π 4π t 6π t0 Figure 37.1. Dependence of the transition probability on frequency It follows from Figure 37.1 that, in the large-t limit, we have ,0≅inω and hence ( )2sin / 2 / 1.in int tω ω⎡ →⎣ ⎤⎦ Thus, one obtains ( ) ( ) ( ) ( ) (2 24sin / 2 sin / 2 sin / 2lim lim / 2 2/ 2 / 2in in in in int t in inin t t t tt ω ω ω )π πδ ω πδ ωω πωω→∞ →∞ ⎡ ⎤= =⎢ ⎥⎣ ⎦ = so that the transition probability becomes ( ) ( ) (2 2 0 022 2lim in ni in ni i nt t tP t V Vπ π )δ ω δ ε→∞ = = == ε− (37.23) It follows that the transitions take place between states of equal unperturbed energies The transition probability is proportional to the time interval t during which the perturbation is on, that is .00 in εε → ( )tPin approaches infinity as .∞→t Since no experiment lasts for an infinite time, it is the transition probability per unit time, called the transition rate , which is of particular interest. Normally, is part of a continuum, and hence, the transition might end in any state which belongs to a finite range of N final states, rather than in a particular state. We may then define the transition rate as ( )dNtP t P intni ∫∞→→ = 1lim (37.24) The integral over the final states is usually transformed into one over energy, if we define a density of states ( ) ,ρ ε representing the number of continuum states per energy interval, by taking ( )0ndN d 0nρ ε ε= (37.25) so that, substituting Eqs.(37.23) and (37.25), Eq.(37.24) gives ( ) ( ) ( ) ( )20 0 0 0 01 2limi n in n n ni n i ntP P t d Vt πρ ε ε ρ ε δ ε ε→ →∞= =∫ = − (37.26) This result is known as Fermi's golden rule, which states that transitions may occur only between states of equal energies for which the matrix elements of the perturbation operator do not vanish, and that the transition rate is proportional to both the square modulus of these matrix elements and the density of final states. 37.3. EMISSION AND ABSORPTION OF RADIATION Consider the transitions between stationary states which are induced by a harmonic perturbation of the form ( ) ( ) ( ) ( ) ( )( )titi eeArVtArVtV ωωωωω −+== GG ˆcosˆ2ˆ (37.27) It is assumed that the harmonic perturbation corresponds to the interaction of either the electric or magnetic field components of the incoming radiation with the electric or magnetic moment of the atom. Thus, the coupling between the system and the perturbing field can be taken in the form of a field amplitude ( )ωA times a displacement of the system ( ).ˆ rV G On substitution of Eq.(37.27) into Eq.(37.10) one obtains ( ) ( ) ( )( ) ( )( )titinititiinni eeAVeeArVtV ωωωω ωω −− +=+ΨΨ= 00 ˆ G (37.28) It follows from Eq.(37.17) that n ni ia t V A e e dtω ω ω ωω ′ ′− − − +⎡ ⎤ ′= − +⎣ ⎦∫= (37.29) inin eeAV 0 ⎥⎦ ⎤⎢⎣ ⎡ ++−= ωωωω ω ωωωω V A e eω ω ω ωω ω ω ω ω − − − +⎡ ⎤− −= +⎢ ⎥− +⎢ ⎥⎣ ⎦= If the atomic system is bathed into a radiation field of frequency ,inni ωωω −== which implies that according to the Bohr condition (37.16), the transition from state to state describes a process of induced absorption, where the energy of the system is increased by energy transfer from the radiation field to the atom. If 00 in εε > ,ni inω ω ω= = − the second term on the right-hand side of Eq.(37.29) grows without limit, so that the first term can be neglected.
Backward, turn backward, O time in your flight. Make me a child again, just for tonight. –Elizabeth Akers Allen Since the time of Kirchhoff it has been known that, when light is passed through an atomic gas, those wavelengths are observed that would be emitted if the gas were incandescent. If the gas is sufficiently cold, then the wavelengths observed are limited to ground-state transitions. As the temperature of the sample is elevated, wavelengths corresponding to transitions between excited states become absorbing, and balances between emission and absorption occur. The study of the central wavelengths of emission lines or absorption notches is known as first-order spectroscopy, and it provides information on the energy-level structure of the atom. The study of the shape of these lines in frequency space is known as second-order spectroscopy, and provides information on the lifetime of the level and the collision rates and temperature of the gas. Thus, whereas first-order spectroscopy shows that emission and absorption measurements yield the same central wavelengths, second-order spectroscopy shows that the natural linewidth for emission and absorption are both specified by the level lifetime, and that the intensity of emission and absorption features both involve the linestrength factor (through the emission transition probability rate and the absorption oscillator strength). The connection between the lifetime and the linewidth can be made plausible by a simple semiclassical model. When an electron is excited to a specific orbit in an atom, its binding energy is established through the exchange of virtual photons with the effective central core. If the perturbations that eventually cause the electron to make a transition from that orbit are weak, the meanlife will be long.
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.
Abstract 1. The procedure usually adopted in the Quantum Theory for the estimation of the intensities of a spectrum line includes two stages : (1) the electric moment, p, of the distribution responsible for the spectrum line is calculated by means of Schrodinger’s integrals or some equivalent method ; (2) the rate of loss of energy due to radiation is calculated by the classical formula due to H. Hertz, viz., 2p2/3c3, just as if the distribution were concentrated in a dipole of precisely the same moment p. The first stage involves no difficulties beyond those met with in the evaluation of the integrals or matrices with which we are concerned, the only principles needed being those of the particular quantum theory adopted. But the second stage is open to grave objections and can only be justified on the ground of necessity owing to the absence of any completely satisfactory quantum theory of the electromagnetic field. Sometimes an appeal is made to Bohr’s Correspondence Principle, but this itself involves a somewhat hazardous extrapolation from infinitely large down to small quantum numbers and cannot take the place of a rigorous calculation. A logically consistent method is possible, based on Maxwell’s theory as represented by his electromagnetic equations and their solutions in terms of the classical retarded potential integrals. It may be objected that we do not know how far Maxwell’s equations can be applied to the interiors of atoms, in particular to distributions like those of Schrodinger, but precisely the same objection applies to the customary use of the classical expression for the radiation from an oscillating dipole. Moreover, the attempts made by Heisenberg and Pauli, Dirac and others to formulate a quantum theory of the electromagnetic field have all been based on the assumption that such a theory must be consistent with Maxwell’s equations. The direct application of the methods of classical electromagnetic theory to Schrodinger’s distributions can be justified at least as a rigorous attempt to determine their electromagnetic field and the radiation from them, and the validity of the procedure will have to be judged by the agreement, or otherwise, of the results obtained with experience. In this way we avoid the assumption that a spatially extended distribution can be treated as if its electric moment were concentrated in a point; in fact, the results obtained in this paper for the more important spectrum line series show definitely that the assumption is false, because it leads to results in many cases quite inconsistent with a rigorous application of Maxwell’s theory.
So far we have treated the energy levels of an atomic system as stable states with infinite lifetimes. The presence of an electromagnetic field changes this picture. Absorption of photons will excite atoms and ions to higher levels, while stimulated emission will make excited ones decay. Einstein (1917) showed, using purely statistical arguments, that an excited ion will decay through spontaneous emission in the absence of an electromagnetic field. This process can also be derived by using a rigorous quantum electrodynamic treatment, where the field is quantized. In this chapter we start by discussing transitions between atomic levels in a general sense, postulating the operators responsible for the decay. After the first few sections we will concentrate on the most important few multipoles or terms in the expansion of the electromagnetic field. We will introduce a number of different properties describing the transition, and eventually arrive at the measurable transition rate and, directly linked to it, the lifetime of an excited state. We will start, however, with the most basic property, the component strength. Our aim is to arrive at computationally useful expressions as soon as possible. For a more thorough treatment of the theory of radiative transitions we refer to the book by Sobelman (1972). For a brief review of the many factors affecting spectra of atoms, see Martin and Wiese (1996) .
The absorbed/emitted radiation will be treated classically, as a monochromatic plane wave, whereas the atomic system that can absorb/emit such radiation will be described quantum-mechanically. However, no incident radiation beam is needed, and the atom “falls down” after a certain time interval to the ground state without any external reason. In the spontaneous emission processes the emitting atom can be thought of as a classical oscillating dipole, whose amplitude decreases due to the energy loss through the radiation. If an ensemble of identical atoms is considered and if, for simplicity, each atom emits radiation with the same spectral shape and full width at half maximum, it is impossible to ascribe a definite spectral component with intensity to a definite atom. In this case, such a broadening is identical for all atoms in the ensemble. On the microscopic scale, absorption of radiation is reduced to the transitions between pairs of energy levels of a discrete atomic spectrum.