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