H R W R y y( ) ( )= (11.15) This system is the general case of a stationary state. This equation is also called the eigenvalue equation in which W is the eigenvalue of the system (constant) and y is the eigenfunction (wavefunction). If a possible value of observable A is defined as · Ò = =ÚA R t A R t dR R t A R tY Y Y Y* *( , ) ( , ) ( , ) ( , ) (11.16)substituting Eq. 11.16 in Eq. 11.15, we get A R t A R e iWt Y System ( , ) ( )= · Ò -y (11.17) Next, if the Hamiltonian operator is separated into unperturbed and perturbation parts as H H H = +0 ¢ (11.18) we will finally have a solution to the energy W and the wavefunction y as W W m H m k H m = + + - + π ¢ ¢ 2 (11.19) m k k H m W W = + - + π  ¢ (11.20) 11.3 Semi-Classical Quantum Mechanical TreatmentHere we will determine the dielectric response of matter under the light frequency by solving a time-dependent perturbation theory (Fig. 11.2).
H R W R y y( ) ( )= (11.15) This system is the general case of a stationary state. This equation is also called the eigenvalue equation in which W is the eigenvalue of the system (constant) and y is the eigenfunction (wavefunction). If a possible value of observable A is defined as · Ò = =ÚA R t A R t dR R t A R tY Y Y Y* *( , ) ( , ) ( , ) ( , ) (11.16)substituting Eq. 11.16 in Eq. 11.15, we get A R t A R e iWt Y System ( , ) ( )= · Ò -y (11.17) Next, if the Hamiltonian operator is separated into unperturbed and perturbation parts as H H H = +0 ¢ (11.18) we will finally have a solution to the energy W and the wavefunction y as W W m H m k H m = + + - + π ¢ ¢ 2 (11.19) m k k H m W W = + - + π  ¢ (11.20) 11.3 Semi-Classical Quantum Mechanical TreatmentHere we will determine the dielectric response of matter under the light frequency by solving a time-dependent perturbation theory (Fig. 11.2).
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.
The subject of perturbation theory in non-relativistic quantum mechanics is introduced. First, perturbation theory for stationary states in the absence of degeneracy is studied. The case of nearby levels, perturbations of a one-dimensional harmonic oscillator, the occurrence of degeneracy, Stark and Zeeman effects are then described in detail. After a brief outline of the variational method, the Dyson series for time-dependent perturbation theory is derived, with application to harmonic perturbations. The Fermi golden rule is also presented. The chapter ends with an assessment of four branches of the subject: regular perturbation theory, asymptotic perturbation theory, and summability methods, spectral concentration and singular perturbations. Approximate methods for stationary states It is frequently the case, in physical problems, that the full Hamiltonian operator H consists of an operator H 0 whose eigenvalues and eigenfunctions are known exactly, and a second term V resulting from a variety of sources (e.g. the interaction with an electric field, or a magnetic field, or the relativistic corrections to the kinetic energy, or the effects of spin). The problems and aims of perturbation theory for stationary states can be therefore summarized as follows. (i) Once the domain of (essential) self-adjointness of the ‘unperturbed’ Hamiltonian H 0 is determined, find on which domain the ‘sum’ H = H 0 + V represents an (essentially) self-adjoint operator (the reader should remember from appendix 4. A that in general, once some operators A and B are given, the intersection of their domains, on which their sum is defined, might be the empty set). […]
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.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
Introduction There are countless situations in physics when one is interested in calculating the response of a system to a small time-dependent perturbation acting on it. With some luck the response can be expanded in a power series of the strength of the perturbation, so that, to first order, it is a linear function of the latter. To compute this function is the objective of the linear response theory (LRT). Linear response theory has many important applications to the study of electronic matter. Virtually all interactions of electrons with experimental probes (electromagnetic fields, beams of particles) can be regarded as small perturbations to the system: if they were not, one would not be probing the system, but the system modified by the probe. Consequently, the results of these experiments can be expressed in terms of linear response functions, which are properties of the unperturbed system. In particular it will turn out that the analytic structure of these functions is entirely determined by the eigenvalues and eigenfunctions of the unperturbed system. Conversely, a measure of the linear response as a function of the frequency of the perturbation enables us to determine the excitation energies of the system. Beside being a cornerstone for the theory of single-particle properties to be developed in Chapter 8, the linear response functions also provide invaluable information in their own right. For example, as we will discuss in Chapter 5, the extent to which an external electrostatic potential is reduced by screening is controlled by the dynamical dielectric function which, in turn, is determined by the density–density response function.
This chapter begins with a review of the semi-classical model of the interaction of charged particles with light, and then proceeds to the full quantum theory by replacing the classical fields with the corresponding field operators. A presentation of the quantum Maxwell equations and their behavior under parity and time reversal transformations is followed by a discussion of stationary density operators. The notion of positive- and negative-frequency parts of field operators is extended to interacting fields and used to define multi-time correlation functions that describe experimental results. The perturbation expansion is formulated by means of the interaction picture and combined with the dipole approximation to calculate the Einstein A- and B-coefficients. The chapter ends with a discussion of spontaneous emission in a cavity and Raman scattering.