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Taylor & Francis eBooks

Scattering Theory

Donald Gary Swanson

Published Aug 9, 2006• 0 citations
Publisher

Abstract

In Section 5.4.2 where we treated time-dependent perturbation theory, we took the perturbation to be of the general form Hˆ(1)(r, t) = 2Hˆ(1)(r) cosωt (9.1) but for absorption and emission of photons, we want the perturbation to be an electromagnetic field of the form E = E0 cosωt eˆz (9.2) so that the perturbing potential is V (r, t) = − ∫ E dz = −qE0z cosωt (9.3) and because of the factor of 2 in Equation (9.1), the matrix element is Hˆ(1)kj = − 12E0〈k|℘z|j〉 (9.4) where ℘z ≡ qz is the z-component of the dipole moment. In the derivation of Equation (5.48) from Equation (5.47), we integrated over the energy E = ~ω, whereas if we instead integrate over ω, the result is Rj→k = 2pi ~2 |Hˆ(1)kj |2 , (9.5) where the δ-function in Equation (5.48) tells us to evaluate the expression at ω = ωkj . The energy density in an electric field is given by and we let u→ ρ(ω)dω so that Rj→k = pi ²0~2 |〈k|℘z|j〉|2ρ(ω) , (9.6) and There is nothing special about the z-direction in evaluating the matrix element, so it is common to use the magnitude instead of the single component, where by symmetry, |℘|2 = |℘x|2 + |℘y|2 + |℘z|2 = 3|℘z|2 so the final result is expressed as Rj→k = pi 3²0~2 |〈k|℘|j〉|2ρ(ω) . (9.7) If we imagine a closed system in equilibrium, where there are atoms in both the lower state (ψk) and in the upper state (ψj), we can be assured that there is a balance between the rates of emission and absorption. If we define the spontaneous emission rate to be A∗ so that if there are Nj atoms in the upper state, there will be NjA transitions per second from state j to state k, as illustrated in Figure 5.2c. There are also transitions from state j to state k by stimulated emission as illustrated in Figure 5.2a. This rate is proportional to the energy density in the electromagnetic field of the wave and is given by Bjkρ(ω), where Bjk is proportional to Rk from Equation (5.48) and ρ(ω) is the energy density in the electric field of the wave at frequency ω. The total number of transitions per second from state j to state k due to stimulated emission is therefore NjBjkρ(ω). If there is an equilibrium, there must be an equal number of transitions from state k to state j by absorbing photons as illustrated in Figure 5.2b. The total number of transitions per second from state k to state j is then given by NkBkjρ(ω) where Nk is the number of atoms in the lower state (ψk) and Bkj is the transition rate for the absorption process. The rate equation is therefore dNj dt = −NjA−NjBjkρ(ω) +NkBkjρ(ω) . (9.8) In equilibrium, there is a steady state, so dNj/dt = 0, so that the energy density in the wave must be given by ρ(ω) = NjA NkBkj −NjBjk = A (Nk/Nj)Bkj −Bjk . (9.9) Equilibrium also implies, however, that there is a temperature T and that the relative probability of finding an atom in one state compared to another is given by the Boltzmann factor of Equation (7.26), which depends on the energies of the two states such that Nk Nj = e(Ej−Ek)/KT , (9.10) but with Ej − Ek = ~ω, this becomes ρ(ω) = A e~ω/KTBkj −Bjk . (9.11) We wish to relate this energy density expression to the black-body distribution since both are equilibrium expressions, but Equation (9.11) is in terms of the energy density of the electromagnetic field, while our black-body expression of Equation (1.3), which is the same as Equation (7.31) without the zero-point term, is given in terms of the intensity, I(λ, T ). In order to relate these two expressions, we revisit the expression for the density of states given by Equation (7.4) where dxdy dz = V , dpxdpydpz = 4pip2 dp, and p = ~k. Then we have N dxdy dz dpxdpydpz = 4piV h3 ~3k2 dk = V 2pi2 k2 dk . (9.12) Then, since k = ω/c, and since there are two independent photon states for each energy (spin 1 and spin −1), the density of photons is Nω V = ω2 pi2c3 dω e~ω/KT − 1 (9.13) where we have included the Bose-Einstein factor since photons have integral spin, and the energy density is therefore ρ(ω)dω = ~ω Nω V = ~ω3 pi2c3 dω e~ω/KT − 1 . (9.14) Comparing this result with Equation (9.11), we first conclude that Bkj = Bjk, which means that the transition rates are the same for the upward and downward transitions. Solving for A, we find for the spontaneous emission rate, A = ~ω3 pi2c3 Bkj , (9.15) and since, from Equation (9.7) Bkj = pi|〈k|℘|j〉|2 3²0~2 , (9.16) the spontaneous rate is given by A = ω3|〈k|℘|j〉|2 3pi²0~c3 . (9.17) and The lifetime in an excited state is the reciprocal of the spontaneous rate, although if there is more than one final state, the lifetime is the reciprocal of the sum of the various rates.

DOI
10.1201/b15900-11

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ATOMIC RADIATION

Mircea S. Rogalski, Stuart B. Palmer

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.

CRC Press eBooks•Jan 8, 2014•0 citations

Light-Matter Interactions

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•Aug 8, 2011•0 citations

Allowed and Forbidden Transitions

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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) .

Routledge eBooks•Jan 27, 2022•1 citation

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

Elementary approach to scattering theory

Kurt Haller

Expressions for the time-dependent scattering wave function, S-matrix elements, and the transition rate are derived using simple time-dependent perturbation theory as it is presented in elementary quantum-mechanics texts.

American Journal of Physics•Sep 1, 1984•1 citation

Quantum mechanical interaction of light with atoms

Peter Blood

Abstract The quantum mechanical description of the interaction of light with atoms treats absorption, gain, and recombination as upward and downward induced transitions of electrons between two states by perturbation due to the optical electric field. This provides the basis for a quantitative theory of diode lasers. The chapter begins with Einstein’s description of absorption, stimulated emission, and spontaneous emission, setting the scene for calculation of the transition probability by solving the time-dependent Schrödinger equation for a coherent superposition of the wavefunctions of two states. The resulting absorption spectrum has a Lorentzian lineshape due to dephasing of these wavefunctions. Expressions for the optical absorption cross section and Einstein coefficients are derived in terms of the dipole matrix element, and the dipole is illustrated using a hypothetical one-dimensional atom. Fermi’s Golden Rule for transitions to a continuum of states (as in quantum wells) is introduced in terms of the momentum matrix element.

•Oct 1, 2015•0 citations
In Section 5.4.2 where we treated time-dependent perturbation theory, we took the perturbation to be of the general form Hˆ(1)(r, t) = 2Hˆ(1)(r) cosωt (9.1) but for absorption and emission of photons, we want the perturbation to be an electromagnetic field of the form E = E0 cosωt eˆz (9.2) so that the perturbing potential is V (r, t) = − ∫ E dz = −qE0z cosωt (9.3) and because of the factor of 2 in Equation (9.1), the matrix element is Hˆ(1)kj = − 12E0〈k|℘z|j〉 (9.4) where ℘z ≡ qz is the z-component of the dipole moment. In the derivation of Equation (5.48) from Equation (5.47), we integrated over the energy E = ω, whereas if we instead integrate over ω, the result is Rj→k = 2pi 2 |Hˆ(1)kj |2 , (9.5) where the δ-function in Equation (5.48) tells us to evaluate the expression at ω = ωkj . The energy density in an electric field is given by and we let u→ ρ(ω)dω so that Rj→k = pi ²02 |〈k|℘z|j〉|2ρ(ω) , (9.6) and There is nothing special about the z-direction in evaluating the matrix element, so it is common to use the magnitude instead of the single component, where by symmetry, |℘|2 = |℘x|2 + |℘y|2 + |℘z|2 = 3|℘z|2 so the final result is expressed as Rj→k = pi 3²02 |〈k|℘|j〉|2ρ(ω) . (9.7) If we imagine a closed system in equilibrium, where there are atoms in both the lower state (ψk) and in the upper state (ψj), we can be assured that there is a balance between the rates of emission and absorption. If we define the spontaneous emission rate to be A∗ so that if there are Nj atoms in the upper state, there will be NjA transitions per second from state j to state k, as illustrated in Figure 5.2c. There are also transitions from state j to state k by stimulated emission as illustrated in Figure 5.2a. This rate is proportional to the energy density in the electromagnetic field of the wave and is given by Bjkρ(ω), where Bjk is proportional to Rk from Equation (5.48) and ρ(ω) is the energy density in the electric field of the wave at frequency ω. The total number of transitions per second from state j to state k due to stimulated emission is therefore NjBjkρ(ω). If there is an equilibrium, there must be an equal number of transitions from state k to state j by absorbing photons as illustrated in Figure 5.2b. The total number of transitions per second from state k to state j is then given by NkBkjρ(ω) where Nk is the number of atoms in the lower state (ψk) and Bkj is the transition rate for the absorption process. The rate equation is therefore dNj dt = −NjA−NjBjkρ(ω) +NkBkjρ(ω) . (9.8) In equilibrium, there is a steady state, so dNj/dt = 0, so that the energy density in the wave must be given by ρ(ω) = NjA NkBkj −NjBjk = A (Nk/Nj)Bkj −Bjk . (9.9) Equilibrium also implies, however, that there is a temperature T and that the relative probability of finding an atom in one state compared to another is given by the Boltzmann factor of Equation (7.26), which depends on the energies of the two states such that Nk Nj = e(Ej−Ek)/KT , (9.10) but with Ej − Ek = ω, this becomes ρ(ω) = A eω/KTBkj −Bjk . (9.11) We wish to relate this energy density expression to the black-body distribution since both are equilibrium expressions, but Equation (9.11) is in terms of the energy density of the electromagnetic field, while our black-body expression of Equation (1.3), which is the same as Equation (7.31) without the zero-point term, is given in terms of the intensity, I(λ, T ). In order to relate these two expressions, we revisit the expression for the density of states given by Equation (7.4) where dxdy dz = V , dpxdpydpz = 4pip2 dp, and p = ~k. Then we have N dxdy dz dpxdpydpz = 4piV h3 3k2 dk = V 2pi2 k2 dk . (9.12) Then, since k = ω/c, and since there are two independent photon states for each energy (spin 1 and spin −1), the density of photons is Nω V = ω2 pi2c3 dω eω/KT − 1 (9.13) where we have included the Bose-Einstein factor since photons have integral spin, and the energy density is therefore ρ(ω)dω = ω Nω V = ω3 pi2c3 dω eω/KT − 1 . (9.14) Comparing this result with Equation (9.11), we first conclude that Bkj = Bjk, which means that the transition rates are the same for the upward and downward transitions. Solving for A, we find for the spontaneous emission rate, A = ω3 pi2c3 Bkj , (9.15) and since, from Equation (9.7) Bkj = pi|〈k|℘|j〉|2 3²02 , (9.16) the spontaneous rate is given by A = ω3|〈k|℘|j〉|2 3pi²0c3 . (9.17) and The lifetime in an excited state is the reciprocal of the spontaneous rate, although if there is more than one final state, the lifetime is the reciprocal of the sum of the various rates.
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.
We start the discussion by a formal theory of perturbation, which will be further elucidated by straightforward calculations. 2.1.1 The Golden Rule Assume that H0 is a time-independent Hamiltonian and all solutions of the eigenvalue problem H0|k〉 = Ek|k〉 are known; it is straightforward to construct the solution of the equation of motion i ∂|ψ(t)〉 ∂t = H0|ψ(t)〉 (2.1) which coincides at t = 0 with the given initial state |ψ(0)〉. Under these circumstances, the state vector |ψ(t)〉 = e−iH0t/|ψ(0) (2.2) represents the general solution of the equation of motion, Eq. 2.1. Since∑ |k〉〈k| = 1 we obtain |ψ(t)〉 = ∑ k e−iH0t/|k〉〈k|ψ(0)〉 = ∑ k e−iEkt/|k〉〈k|ψ(0)〉 (2.3) We now calculate transition amplitudes between the relevant unperturbed eigenstates due to the appearance of an external perturbation V at t = 0. We transform the equation of motion in the Schro¨dinger picture i ∂|ψ(t)〉 ∂t = (H0 + V )|ψ(t)〉 (2.4) to the interaction picture by a time-dependent unitary operator |ψˆ(t)〉 = exp (iH0t/)|ψ(t)〉 (2.5) so that in the interaction picture, the equation of motion for the state is i ∂|ψˆ(t)〉 ∂t = Vˆ (t)|ψˆ(t)〉 (2.6) where the new interaction operator is given by Vˆ (t) = exp (iH0t/)V exp (−iH0t/) (2.7) Next we introduce a time development operator Tˆ (t) such that |ψˆ(t)〉 = Tˆ (t)|ψˆ(0)〉 (2.8) The time development operator satisfies the integral equation Tˆ (t) = 1− i Vˆ (t)Tˆ (t′)dt′ (2.9) The relationship between the operator expressions in the two pictures is Tˆ (t) = exp (iH0t/)T (t) exp (−iH0t/) (2.10) The transition matrix elements of the time development operator between eigen- states of the unperturbed Hamiltonian are 〈q|Tˆ (t)|k〉 = δ(q − k)− i 〈q|Vˆ (t)Tˆ (t′)|k〉dt′ = δ(q − k)− i 〈q|Vˆ (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.11) By Eq. 2.10, and note H0|k〉 = Ek|k〉 and H0|q〉 = Eq|q〉, 〈q|Tˆ (t)|k〉 = δ(q − k)− i ei(Eq−Ek′ )t ′/〈q|V (t′)|k′〉〈k′|Tˆ (t′)|k〉dt′ (2.12) The strategy of time-dependent perturbation theory is to assume that V is small and proceed with iterating Eq. 2.9 as a power series in terms of Vˆ Tˆ (t) = 1− i Vˆ ′dt′ − 1 2 Vˆ ′′dt′′ + · · · (2.13) and the transition amplitude from initial state |k〉 to final state |q〉 in the perturbation expansion becomes 〈q|Tˆ (t)|k〉 = δ(q − k)− i − 1 2 × ∫ t′ 0 ei(Ek′−Ek)t ′′/〈k′|V (t′′)|k〉dt′dt′′ + · · · (2.14) It is equivalent to obtain the above equation by iteratively replacing the term 〈k′|Tˆ (t′)|k〉 on the right side of Eq. 2.12 by the whole right-side expression of Eq. 2.12, i.e., ∑ k′ = ∫ t′ 0 + ∑ k′ =k ei(Eq−Ek′ )t ′′/〈q|V (t′′)|k′〉〈k′|Tˆ (t′′)|k〉dt′′ (2.15) By Eq. 2.12 〈k|Tˆ (t′)|k〉 = 1− i ei(Ek−Ek′′ )t ′′/〈k|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.16) The first term on the right side of the above equation results in the second term on the right side of Eq. 2.14. For k′ = k and by Eq. 2.12, the second term on the right side of Eq. 2.15 becomes 〈k′|Tˆ (t′)|k〉 = − i ei(Ek′−Ek′′ )t ′′/〈k′|V (t′′)|k′′〉〈k′′|Tˆ (t′′)|k〉dt′′ (2.17) We use Eq. 2.15 to separate the term k′′ = k from k′′ = k in the above equation, which becomes the third term on the right side of Eq. 2.14.
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) .
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.
Abstract The quantum mechanical description of the interaction of light with atoms treats absorption, gain, and recombination as upward and downward induced transitions of electrons between two states by perturbation due to the optical electric field. This provides the basis for a quantitative theory of diode lasers. The chapter begins with Einstein’s description of absorption, stimulated emission, and spontaneous emission, setting the scene for calculation of the transition probability by solving the time-dependent Schrödinger equation for a coherent superposition of the wavefunctions of two states. The resulting absorption spectrum has a Lorentzian lineshape due to dephasing of these wavefunctions. Expressions for the optical absorption cross section and Einstein coefficients are derived in terms of the dipole matrix element, and the dipole is illustrated using a hypothetical one-dimensional atom. Fermi’s Golden Rule for transitions to a continuum of states (as in quantum wells) is introduced in terms of the momentum matrix element.