QUANTUM ELECTRODYNAMICS *

Transcription

1 QUNTUM ELECTRODYNMCS * J.. N. F. GOMES, Facultade de Ciencias, Universidade do Porto, 4000 Porto, Portugal. J. C. PNGU, Departament de Química Física, Universitat de Barcelona, Martí i Franquès 1, E Barcelona, Spain. 1. ntroduction Quantization of the electromagnetic field Coherent states Electron creation and annihilation operators Time dependent perturbation theory nteraction between electromagnetic radiation and matter nteraction Hamiltonian Matrix elements Transition probabilities One-photon processes Multi-photon processes Exchange of virtual photons ppendix: Phonon creation and annihilation operators References *This document has been published as the first chapter of the book COMPUTTONL CHEMSTRY: Structure, nteractions and Reactivity, edited by S. Fraga. Studies in Physical and Theoretical Chemistry, Vol. 77 (B) 1992 Elsevier Science Publishers B. V. ll rights reserved.

2 J.. N. F. Gomes and J. C. Paniagua The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense. nd it agrees with experiment. So hope you can accept Nature as She is absurd. R. P. Feynman 1 1. ntroduction Elementary theories of the interaction between electromagnetic radiation and matter are often based on a semi-classical picture: the electromagnetic field is described classically, in the way devised in the 1870's by J. C. Maxwell, while a quantum mechanical description is used for matter. lthough incomplete, this picture, when supplemented by some ad hoc hypotheses, provides an adequate framework in which to model many phenomena of chemical interest. The alternative, consisting in using a quantized description of the field, requires the development of a more complicate formalism, but is more satisfactory from the theoretical point of view and indispensable for studying some phenomena that the semiclassical theory cannot explain. This approach to the problem is based on a theory known as "quantum electrodynamics" that developed soon after quantum mechanics, although relativistic inconsistencies were not corrected until the middle of the century. n this chapter we will introduce some notions of non-relativistic quantum electrodynamics that are needed for a quantum description of the electromagnetic radiation and its interaction with matter. n sections 2 and 3 the formalism of creation and annihilation operators for the electromagnetic radiation and for electronic systems is introduced, which is basic for the formulation of quantum electrodynamics. Section 4 is devoted to a mathematical tool necessary for the study of transition probabilities: time dependent perturbation theory. Finally, in section 5 the concepts introduced above are applied to the study of different processes that can result from the interaction between radiation and matter. S units have been used throughout. subject as vast and complex as this cannot be confined to a satisfactory and selfconsistent exposé of the size of this chapter. Our aim is only to introduce the interested reader to some of the methods in current use. t is hoped that the presentation will be provocative enough to make him find in the specialized literature the answers to the many questions that this text will leave open. Good introductions to the physical aspects of quantum electrodynamics may be 2

3 Quantum electrodynamics found in the books by Power 2, by Loudon 3 and by Healy 3, while the more demanding reader may be interested in other sources, eg references 5 to 8. The book of Moss 9 presents a chemist oriented introduction to the subject from a relativistic point of view, and in that of Graig and Thirunamachandran 10 a comprehensive discussion of molecular applications of the nonrelativistic theory is made. Much of the material presented here is developed further in this book. nstead of giving a detailed list of original papers, we refer the reader to some recent reviews, books and articles dealing with chemistry-related applications of the theory that include the main references on the subject. 2. Quantization of the electromagnetic field n the classical electromagnetic theory, the field created by a charge distribution of charge density ρ and current density j is described by Maxwell equations: E + B t = 0 (1) B = 0 (2) H D t = j (3) D = ρ (4) where the fields have been characterized by the usual symbols. lthough these equations provide a complete description of the electromagnetic field, they are not the best starting point from which to develop the quantum formulation. Electromagnetic fields are force fields and, in the same way as the quantum description of a mechanical system is built from the Hamiltonian rather than the Newtonian classical formulation, it is convenient to re-express eqns. (1) to (4) in terms of electromagnetic potentials as a previous step to quantization. To carry out this reformulation, we will start from the fact that eqn. (2) allows us to express B as the curl of a vector field (r,t), which is known as "vector potential": B =. (5) 3

4 J.. N. F. Gomes and J. C. Paniagua ntroducing this eqn. in (1) we obtain (E + / t) = 0, so that the expression enclosed between parentheses can be written as the gradient of a scalar field φ(r,t), known as "scalar potential": E = φ t. (6) Due to the differential character of definitions (5) and (6), the vector and scalar potentials are not uniquelly determined by given E and B values. n effect, it is easy to verify that any transformation of the type ' = f (7) φ' = φ + f (8) t where f is an arbitrary function of r and t, leaves the fields E and B invariant. Relationships (7) and (8) are known as "gauge transformation" and they express a degree of freedom in the choice of and φ that can be used to simplify the calculation of these potentials, as we will now see. The problem we are going to study is the propagation of the electromagnetic field in the absence of charges: j = 0 and ρ = 0. Since eqns. (1) and (2) are implicit in the definition of and φ, these potentials should be determined from the remaining two Maxwell equations (3) and (4), which, applied to vacuum, and taking into account that, for this state, E = D/ε 0 and B = Hµ 0, take the form E B c 2 t = 0, (9) E = 0, (10) where c 2 = ε 0 µ 0. ntroducing (5) - (6) into (9) - (10) and using the identity ( Α) = ( ) 2, we obtain c t 2 φ + c 2 t 2 φ + t = 0, (11) = 0. (12) 4

5 Quantum electrodynamics These equations can be significantly simplified by noting that the freedom implicit in the choice of and φ permits us to impose the condition = 0, (13) which, substituted in (11) and (12) leads to 2 2 c 2 t 2 φ c 2 t = 0, (14) 2 φ = 0. (15) The restriction (13), known as Coulomb gauge, does not completely remove the arbitrariness implicit in the choice of the potentials and, for the special case of vacuum, we can further impose the condition φ = 0, (16) so that we can describe the electromagnetic field in vacuum by using only the vector potential. This will be determined by eqn. (14), which now reduces to 2 2 c 2 = 0. (17) t 2 This is a well known wave equation, and its general real solution can be put into the form = π e i(k r ω kt) * * + π e i(k r ω k t), (18) where ω k = ck and the summation extends to every possible value of the wave vector k and, for each of these, to two λ values identifying complex orthonormal vectors perpendicular to the wave vector: π * k1 π k2 = 0, (19) 5

6 J.. N. F. Gomes and J. C. Paniagua π 2 = 1, k π = 0, λ = 1, 2. (20) Each term in the right hand side of (18) represents a plane monochromatic wave of frequency ω k propagating with velocity c in the direction of k. The two vectors π corresponding to a given wave vector represent different polarization states of the plane wave. n particular, if they have as components (1, 0, 0) and (0, 1, 0) in a Cartesian system with the z axis in the direction of k, the corresponding plane wave will be linearly polarized along the x and y axes respectively, while (1, i, 0) / 2 and (1, i, 0) / 2 represent two waves circularly polarized in opposite directions. n the general case, the vectors π represent elliptically polarized plane waves. Finally, are the arbitrary complex mixing coefficients of the plane waves. n problems with some specific symmetry it could be preferable to express the general solution of (17) as an expansion in components of a different kind (spherical waves, cylindrical waves, ). The Hamiltonian function associated to the electromagnetic field is, in general, H = ε 0 2 (E 2 + c 2 B 2 ) dv. (21) R 3 Substituting for (5) and (6) in (21) and introducing the expansion (18) one obtains the expression for the electromagnetic field in vacuum (i.e., the electromagnetic radiation) in terms of the plane components of the vector potential. The resulting expression is involved, but it can be greatly simplified by considering a field confined to an origin-centered cubic volume of side L and imposing periodicity conditions at the cube faces. This leads to the following restriction for the wave vector components: k i = 2π n i / L, n i = 0, ±1, ±2,, i = x, y, z. (22) ssuming that the value of L is high enough to make discretization in k negligible, this restriction will not significantly modify the problem. fter some algebraic manipulation one can express H as a sum of independent terms associated with each plane wave: with H = H (23) H = ε 0 V ω 2 * * k ( + ) (24) 6

7 Quantum electrodynamics where V = L 3 and the factor order has been preserved in the manipulations in order to obtain an expression formally equal to the quantum one, in which the operators associated to and * do not commute. Equation (24) can be re-expressed in terms of two real parameters Q and P defined as follows = (4 ε 0 V ω k 2 ) 1/2 ω k Q + i P, (25) in terms of which the Hamiltonian (24) associated to each plane component takes the form: H = 1 2 P 2 + ω 2 2 k Q. (26) This expression is formally identical to the Hamiltonian of a one-dimensional harmonic oscillator of unit mass and frequency ω k, P playing the role of the canonic conjugate moment of the variable Q. Hence, the problem of the electromagnetic radiation is mathematically equivalent to that of an infinity of independent one-dimensional unit-mass oscillators, which suggests introducing the quantum formalism for radiation in a parallel way to the quantum treatment of this particle system (see ppendix), that is, by associating to the classical variables Q and P quantum operators fulfilling the commutation relations Q, P k'λ' = i h - δ kk' δ λλ', Q, Q k'λ' = P, P k'λ' = 0 (27) and acting on vectors representing the radiation states. Extending to this problem the creation and annihilation operator formalism introduced in the ppendix, we can express the Hamiltonian operator of radiation in the form H = a a h- ω k, (28) where the creation and annihilation operators a and a have definitions analogous to (24) and (25). The radiation stationary states and the corresponding energies will be obtained by solving the time independent Schrödinger equation H n = n h- ω k n (29) 7

8 J.. N. F. Gomes and J. C. Paniagua where each occupation number n now refers to the excitation energy quantums of radiation (photons) in the state corresponding to the term of the plane wave decomposition (hereinafter referred to as the mode). The quantum operators corresponding to the classical variables (t) and (t) are readily obtained by comparing definitions (24) and (25) with eqn. (25) and its complex-conjugate: (t) = h - 2ε 0 Vω k 1/2 a (30) * (t) = h - 1/2 a 2ε 0 Vω k (31) so that the vector potential operator can be written in the form = h - 1/2 ( a π 2ε 0 Vω e ik r + a k * π e ik r ). (32) t should be noted that eqn. (29) leads, for the radiation ground state 0 = 0 0, to a non vanishing energy (which is in fact infinite) and, moreover, non vanishing values are also obtained for 0 E 2 0 and 0 B 2 0 from eqns. (5), (6), (16) and (32), even though 0 E 0 = 0 B 0 = 0. Therefore electromagnetic radiation exists even in the absence of photons, in sharp contrast with the classical picture, which explains, among other quantum electrodynamical effects, the spontaneous emission of radiation (see 5.3.1) Coherent states n the classical theory, the complex mixing coefficients appearing in the expansion of can be expressed as products of a real amplitude and a phase factor: = e iϕ (33) where 2 = determines the radiation intensity of the corresponding plane wave and ϕ its phase. n the quantum description we can proceed in a similar way by writing 8

9 Quantum electrodynamics a = (n + 1) 1/2 e iϕ (34) where (n + 1) = a a and ϕ are the quantum operators corresponding to the intensity and phase of the mode. t is easy to see that cosϕ and sinϕ do not commute with n, so that we cannot, in general, have radiation states with well defined phase and intensity. n an stationary state, for instance, the intensity is completely defined but the phase has a uniform probability distribution over (0, 2π). Moreover, this situation does not change for large n values so that, even in the high energy limit, we do not recover the classical situation in which intensity and phase can be determined with unlimited accuracy. t is thus interesting to find radiation states tending properly to the classical limit. The "coherent states", which are eigenstates of an annihilation operator, have this property. Let us consider the eigenvalue equation a α = α α (35) in the subspace expanded by the set of vectors { 0, 0, n, 0, }. ssuming α α = 1 one obtains α = e α 2 /2 n =0 n α (n!) 0, 0, n 1/2, 0,, (36) which shows that the states α are not stationary. The expectation value of n is α a a α = α 2 (37) and the dispersion of n is easily found to be n = α, so that n / n tends to zero for increasing intensities. On the other hand, putting α = α e iθ it can be seen that, for large n values, cosϕ sinθ / 2 α and similarly for sinϕ, so that intensity and phase relative dispersions vanish in the high energy limit. Coherent states are important for describing laser radiation, since this is highly monochromatic, intense and coherent. For low intensity sources, however, the coefficients of the expansion (36) decrease rapidly and coherent states can be well approximated by stationary states. 9

10 J.. N. F. Gomes and J. C. Paniagua 3. Electron creation and annihilation operators The creation and annihilation operator formalism introduced in the ppendix applies only to boson systems but, with some modifications, it can be used for studying fermion systems and, in particular, electron systems. ndeed the general formulation of quantum electrodynamics admits the possibility of creating and destroying electrons (and positrons) as well as photons and requires, therefore, both types of operators. Nevertheless changes in the number of electrons are non significant in chemistry, so that a simplified version of the theory is normally used in which conventional quantum mechanics is applied to the electrons and only the photon numbers are allowed to vary. ll the same, electron creation and annihilation operators, combined in such a way that the total number of electron is preserved, have gained widespread acceptance among quantum chemists as a tool for facilitating algebraic manipulations. We will thus give here a brief outline of the formalism, commonly known as "second quantization". fundamental requirement that polyelectronic systems must fulfil is the Pauli principle, which can be automatically ensured by writing the electronic wave function as a linear combination of Slater determinants. Given an orthonormal set of spin-orbitals {ϕ 1, ϕ 2,, ϕ N, } an N-electron Slater determinant built from them ϕ 1 (1) ϕ 2 (1) ϕ N (1) ϕ 1 (2) ϕ 2 (2) ϕ N (2) ϕ 1 (N) ϕ 2 (N) ϕ N (N) (38) can be expressed in the form b 1 b2 bn 0 (39) where 0 is the vacuum state (i.e., the state without electrons) and b i is an operator that creates an electron in the spin-orbital ϕ i. This operator is the adjoint of b i which destroys an electron in the same spin-orbital, and both types of operator must fulfil the anticommutation relations b i, b j + = δ ij, b i, b j + = b i, bj + = 0 (40) (where, B + B + B ) in order that the particles they create or destroy obey Fermi- 10

11 Quantum electrodynamics Dirac statistics. n effect, eqns. (40) guarantee that each spin orbital cannot be occupied by more than one electron: b i bj 0 = 1 2 b i, bj + 0 = 0 (41) and that the states are antisymmetric with respect to the interchange of electrons: b j bi 0 = bi bj 0. (42) The normalization of vector (39) follows from the assumption 0 0 = 1 by successively moving each b j to the left as far as bj in the following expression: 0 b N b 2 b 1 b 1 b2 bn 0 = 0 bn b N b2 b 2 b1 b 1 0 = 1. (43) Many properties of Slater determinants are readily deduced in the present formalism. Let us consider, for instance, the orthogonality between two different N-electron determinants, n i n' i i (b i ) 0 and i (b i ) 0, with i n i = i n' i = N. There must be at least two different indices in the sets {n i } and {n' i }, which can be put in the first two places: n 1 = 1, n 2 = 0, n' 1 = 0, n' 2 = 1. Noting that b i 0 = 0, we find: 0 (b 3 ) n3 (b 2 ) 0 (b 1 ) 1 (b 1 ) 0 (b2 ) 1 (b3 ) n' 3 0 = 0 (b 3 ) n3 (b 3 ) n' 3 (b 2 ) 0 (b 2 ) 1 (b1 ) 1 (b 1 ) 0 0 = 0. (44) ny mono- or bi-electron operator can be expressed in terms of the creation and annihilation operators corresponding to a complete set of spin-orbitals {ϕ i }: F = N i=1 f(i) = f kl b k bl kl (45) G = 1 N 2 g(i, j) i j = 1 2 klmn g klmn b k bl bn b m, (46) where f kl = ϕ k f ϕ l and g klmn = ϕ k (1) ϕ l (2) g(1, 2) ϕ m (1) ϕ n (2). These relationships can be verified by calculating the matrix elements of each member between arbitrary Slater 11

12 J.. N. F. Gomes and J. C. Paniagua determinants. For non-complete sets of spin-orbitals, the right hand side members of (45) and (46) represent the projections of the left hand side operators onto the corresponding subspace. When dealing with excited determinants of a many-electron system, a modified formalism is also used in which creation and annihilation operators indicate the differences in occupation numbers of a given determinant with respect to that of the ground state, which is thus taken as reference instead of the vacuum state. For the spin-orbitals occupied in that determinant the annihilation and creation particle operators are substituted by creation and annihilation hole operators, respectively: ξ i = bi, ξ i = b i, for i N, (47) while the operators b i and bi are kept for i > N. n this hole-particle formalism, the ground state determinant has no particles nor holes, and it is denoted by 0, so we have: ξ i 0 = 0 for i N and b i 0 = 0 for i > N. (48) Good expositions of the second quantization approach to quantum chemistry can be founded in the book by Jörgensen and Simons 21 and in that by Surján 22, while more general applications to physical chemistry can be found in the work of Paul Time dependent perturbation theory The applications of quantum electrodynamics to chemistry always turn on the interaction between electromagnetic radiation and matter. The usual way to tackle this problem consists in starting from an ideal description in which radiation and matter are assumed independent, and dealing with the interaction energy as a perturbation. This perturbation is, in general, time dependent, which makes it necessary to use a time-dependent perturbation formalism. Let us consider a system whose Hamiltonian can be split into two terms, one being time independent (H 0 ) and the other (H'(t)) being small with respect to the former: H(t) = H 0 + H'(t). (49) 12

13 Quantum electrodynamics The evolution of the system will be determined by the time dependent Schrödinger equation: H Ψ(t) = i h - Ψ(t) t. (50) f the perturbation were switched off, the solution of this equation would be Ψ(t) = e ih 0(t t 0 ) /h - Ψ(t 0 ). (51) Since we are interested in studying the effects due to the relatively small perturbation H', it is convenient to deduct from Ψ(t) the part of the evolution corresponding to H 0, which is achieved by making a unitary transformation to an "interaction picture" defined by Ψ (t) = e ih 0(t t 0 ) /h - Ψ(t) (52) where a subindex has been introduced to distinguish the interaction picture from the ordinary Schrödinger one. Using the fact that average values must be invariant under the change of picture one obtains the transformation law for operators: (t) = e ih 0(t t 0 ) /h - (t) e ih 0(t t 0 ) /h - (53) which, together with (52), allows transforming the Schrödinger equation to the new picture: H' (t) Ψ (t) = i h - Ψ (t) t. (54) The solution to this equation can be formally written as with Ψ (t) = U (t, t 0 ) Ψ (t 0 ) (55) t U (t, t 0 ) = 1 h - i dt' H' (t') U (t', t 0 ). (56) t 0 For small perturbations U will not differ much from the unit operator and we can get a good 13

14 J.. N. F. Gomes and J. C. Paniagua approximation to it by taking a few terms of the series obtained by iterative substitution in the right hand side of eqn. (56): with U n (t, t 0 ) = t i n 1n! h - dt 1 t 0 U (t, t 0 ) = 1 + U n (t, t 0 ) n=1 (57) t 1 t n 1 dt 2 dt n H' (t 1 ) H' (t 2 ) H' (t n ) (58) t 0 t 0 where the subindex has been omitted in U n and t 0 < t n < t n 1 < < t 1 < t. Equation (58) can be written in the simpler form U n (t, t 0 ) = t i n 1n! h - dt 1 t 0 t t 0 dt 2 t t 0 dt n P H' (t 1 ) H' (t 2 ) H' (t n ) (59) where P is the Dyson time ordering operator, which rearranges the factors to its right in the order of decreasing time arguments. The problem we are interested in consists in determining the probability of finding the system in the state Ψ b at time t provided that it was in the state Ψ a at t 0, Ψ a and Ψ b being different eigenstates of H 0, and assuming that the perturbation H'has been acting during the interval (t 0, t). The corresponding probability amplitude is c ba (t) = Ψ b U (t, t 0 ) Ψ a (t 0 ). (60) Since Ψ b is an eigenstate of H 0, its conversion to the interaction picture (eqn. 52) amounts to multiplying it by an irrelevant phase factor; on the other hand, Ψ a (t 0 ) = Ψ a (t 0 ), so we can put eqn. (60) in the form c ba (t) = Ψ b U (t, t 0 ) Ψ a (t 0 ) = b U (t, t 0 ) a. (61) ntroducing the expression (57) for U (t, t 0 ) we obtain: 14

15 Quantum electrodynamics where n=1 c ba (t) = c (0) ba (t) + c ba (n) (t) (62) c (n) ba (t) = b U n (t, t 0 ) a. (63) The zero order amplitude always vanishes, since it corresponds to taking only the identity on the right hand side of (57) and the eigenstates of H 0 can always be taken orthonormal. The first order contribution is c (1) ba (t) = b i t h - dt 1 H' (t 1 ) a t 0 = i t h - dt 1 t 0 = i t h - dt 1 t 0 b e ih 0(t 1 t 0 )/h - H'(t 1 ) e ih 0(t 1 t 0 )/h - a b H'(t 1 ) a e i(e b E a )(t 1 t 0 )/h -, (64) E a and E a being, respectively, the energies of states a and b. f the perturbation remains constant in the interval (t 0, t) c (1) ba (t) = ei(e b E a )(t t 0 )/h - 1 b H' a (65) E a E b and the first order contribution to the probability will be (1) P a b (t) = c (1) ba (t) 2 = 4 sin2 [(E b E a )(t t 0 )/2h - ] b H' a (E a E b ) 2 2. (66) This expression presents a sharp maximum for E a = E b and tends to zero rapidly for increasing E b E a values. Hence, the most probable transitions will be those preserving the total energy of the system, although slight violations of the exact conservation can be observed (in contrast with classical principles). These violations are less probable for large t t 0 values and, in fact, sin[(e they disappear in the limit of infinite interaction time, since b E a )(t t 0 )/2h - ] lim = π (E b E a ) δ(e b E a ). 15 (t t 0 )

16 J.. N. F. Gomes and J. C. Paniagua Later on, we will be interested in the probability of transitions from one state to any of a set of possible final states. For weak perturbations, the processes being considered can be assumed independent, so P a {b} will be equal to {b} P a b or {b} P a b ρ b de b (ρ b being the density of final states) depending on whether {b} is discrete or continuous. n the latter case (1) P a {b} (t) = 4 sin 2 [(E b E a )(t t 0 )/2h - ] 2 b H' a ρb de b (67) E (E a E b ) 2 b and, for (t t 0 ) >> 4πh - / E b, the quotient in the integral almost vanishes in the interval E b except for a narrow region close to E a, so we can write (1) P a {b} (t) 4 b H' a 2 ρb Eb = E a E b sin 2 [(E b E a )(t t 0 )/2h - ] (E a E b ) 2 de b 4 b H' a 2 ρb Eb = E a t t 0 2 h - sin 2 u u 2 = 2 π h - b H' a 2 ρb Eb = E a (t t 0 ). du (68) The probability per unit time or "transition rate" will then be (1) Γ a {b} dp (1) a {b} dt 2 π h - b H' a 2 ρb Eb = E a, (69) a result known as "Fermi Golden Rule". When b H' a vanishes, we will be interested in the second order contribution to the probability amplitude: c (2) ba (t) = b t i 2 12 h - dt 1 t 0 t t 0 dt 2 P {H' (t 1 ) H' (t 2 )} a (70) which, for a perturbation constant in (t 0, t) leads to 16

17 Quantum electrodynamics c (2) ba (t) = ei(e b E a )(t t 0 )/h - 1 b H' i i H' a E a E b E a E i i (71) This expression differs from the first order one (eqn. (65)) in containing, instead of a single matrix element, a sum over the H 0 eigenstates of terms containing two perturbation matrix elements, thus resulting in a quadratic dependence on H'. The treatment given to (65) can also be applied to (71) and, in particular, the following Fermi Golden Rule is obtained: (2) Γ a {b} = 2 π h - b H' i i H' a E a E i i 2 ρb E b = E a. (72) The third order contribution will include a double summation over the eigenstates of H 0, each term containing products of the form b H' i i H' j j H' a, and so on. 5. nteraction between electromagnetic radiation and matter 5.1 nteraction Hamiltonian s we anticipated at the beginning of the last section, the interaction between the electromagnetic radiation and matter will be studied by taking an unperturbed Hamiltonian in which the two parts are independent: H 0 = H mat + H rad (73) and introducing the interaction term as a perturbation: H' = H int. (74) The classical interaction Hamiltonian of a particle with charge q and mass m in an electromagnetic field (, φ) can be obtained by making the following substitutions in the Hamiltonian of the particle without field: 17

18 J.. N. F. Gomes and J. C. Paniagua H H qφ and p p q, (75) which, for the non-relativistic Hamiltonian, lead to H = qφ + p q 2 2m. (76) The suitability of the prescription (75) for describing the particle-field interaction can be verify by using the Hamilton equations of motion to deduce the equivalency between (76) and the experimental Lorentz law: F = q (E + v B). (77) The extension of the Hamiltonian (76) to a system with many particles interacting with each other through a potential V is immediate: H = q j φ j + p j q j 2 j 2m j j + V, (78) where φ j = φ(r j, t) and j = (r j, t). By developing this expression and grouping the terms explicitly depending on the field we obtain H = H mat + H int (79) with and H int = j H mat = p2 j + V (80) 2mj j q j φ j q j (p 2m j j + j p j ) + q j 2 j 2m j j 2. (81) or, since we can take a zero scalar potential for the electromagnetic radiation (eqn. 16), H int = j q j (p 2m j j + j p j ) + q j 2 j 2m j j 2. (82) The prescription used for obtaining this interaction Hamiltonian (eqn. (75)), known as 18

19 Quantum electrodynamics "minimal coupling", is not the only way of proceeding. The "multipolar coupling" is an equivalent alternative formalism that will not be considered here (in reference 10 this formalism is broadly used and the equivalence between both types of coupling is demonstrated). The quantum operator corresponding to interaction Hamiltonian (82) is: H int = j q j (p 2m j j + j p j ) + q j 2 j 2m j j 2. (83) n the semiclassical formalism only the particle variables are substituted by operators to obtain the interaction Hamiltonian, since radiation is not explicitly included in the treatment; therefore, this operator vanishes for the minimum energy state of radiation (zero field) and every eigenstate of H mat represents a true stationary state, in contrast with experimental evidence. Hence, the spontaneous emission cannot be explained and its rate coefficient can only be obtained by means of statistical arguments based on energy conservation in a closed cavity in equilibrium with radiation. However, in the full quantum picture radiation exists even in the vacuum state and spontaneous emission appears in a natural way, as we shall soon see. Since operator (83) has been built from a classical non-relativistic expression it does not include interaction terms involving spin, these appearing naturally in a quantum relativistic treatment. Here we will only consider one of them, the interaction between the spin magnetic moment and the radiation magnetic field, which is necessary for explaining magnetic resonance spectroscopies and some optical properties of chiral systems. ts general form is (m S ) j B j, (84) m S being the spin magnetic moment operator of particle j. The relationship between this and the spin angular moment operator depends on the g factor of the particle: for electrons and (m s ) j = g e e 2m e s j (85) (m ) j = g j e 2m p j (86) for nuclei (m p being the proton mass). The interaction Hamiltonian we will use will then be 19

20 J.. N. F. Gomes and J. C. Paniagua H int = j q j m j j p j + q j 2 2m j j 2 (µ S ) j ( j ), (87) where we have used eqn. (5) and the fact that p j and j commute when the Coulomb gauge (13) is adopted Matrix elements n order to apply perturbation theory to the process of interaction between radiation and matter we have to know the expressions of the perturbation Hamiltonian matrix elements, which now take the form a H' b = B (n ) b H int (n ) a, (88) where (n ) a (n ) a, and B being the initial and final states of matter. t is convenient to differentiate in eqn. (87) the terms linear and quadratic in : with and H 1 = j H int = H 1 + H 2 (89) q j m j j p j (µ S ) j ( j ) H 2 = q j 2 2mj j 2 j (90). (91) Since H 1 is linear in creation and annihilation operators (see eqn. (32)), it will only contribute to the matrix element (88) when the sets of occupation numbers {(n ) a } and {(n ) b } coincide except for one of them differing in 1, i.e., when the radiation states differ in one photon. n the same way, H 2 will only connect states differing in 0 or 2 photons, so it will never contribute to that same matrix elements as H 1, and so on. Let us first consider the case in which the initial and final state differ in one photon. Using (32) together with (18) and (19) and leaving aside, for the moment, the term involving the spin, one obtains 20

21 Quantum electrodynamics and B n 1 B n +1 q j mj j p j n = j q j mj j p j n = j n h - 1/2 q π B j 2ε 0 Vω k mj e ik r j p j (92) j (n +1)h - 2ε 0 Vω k 1/2 * q π B j mj e ik r j pj (93) j For radiation with wavelength in the UV region or longer, this is usually much greater than molecular dimensions and we can make the substitution e ±ik r j 1 (94) in the integrals in (92) and (93). This "long wavelength approximation" leads, after some manipulation, to and B n 1 H int n E1 = i B n +1 H int n E1 = i n 2ε 0 Vh - ω k 1/2 (EB E ) π B P, (95) n +1 1/2 2ε 0 Vh - * (EB E ω ) π B P, (96) k where P is the electric-dipole moment operator of the particle system: P = j q j r j (97) and the subindex E1 refers to "electric-dipole" contribution to the interaction matrix element. When the dipole integrals in (95) vanish we shall have to take the second term in the power series of e ±ik r, that is ±ik r, for calculating the main contribution to (92) and (93). This contribution can be split into two terms which, for the matrix element (92), are and B n 1 H int n E2 = 1 2 B n 1 H int n M1L = i n 2ε 0 Vh - ω k 1/2 (EB E ) π B Q k (98) n h - 2ε 0 Vω k 1/2 k π B m L, (99) where Q is the electric-quadrupole moment operator, with components 21

22 J.. N. F. Gomes and J. C. Paniagua j Q xy = q j (x j y j δ r j 2 xy 3 ), x, y = x, y, z, (100) m L is the orbital magnetic-dipole moment operator m L = q j 2mj l j, l j = r j p j j (101) and subindices E2 and M1L indicate the type of contribution being considered. The spin-dependent term in H 1 (eqn. (90)) yields, in the long wavelenght approximation, a contribution identical to (99) except for m L being substituted by m S = grouped into a single magnetic-dipole contribution: j (m S ) j, so both terms can be with B n 1 H int n M1 = i n h - 2ε 0 Vω k 1/2 k π B m, (102) m = m L + m S. (103) For matrix element (93) the E2 and M1 contributions are identical to (98) and (102) except for a change of sign and the substitution of n +1 for n on the left hand sides. Due to small value of k r in molecular systems, the electric-quadrupole and magneticdipole contributions are typically times smaller than the electric-dipole term, provided that this does not vanish by symmetry. When we consider two states differing by 0 or 2 photons in matrix element (88), the only possible contribution comes from the quadratic term H 2 and, in the long wavelength approximation, it vanishes unless the matter state does not change. s an example, we show the contribution corresponding to an element connecting states with the same number of photons: B n 1 n k'λ' +1 H int n n k'λ' h - 2ε 0 V n (n k'λ' +1) ω k ω k' 1/2 * q2 π π j k'λ' mj j B. (104) 22

23 Quantum electrodynamics 5.3. Transition probabilities The perturbation formalism developed in section 4 applies to perturbation Hamiltonians acting only during some time interval, while we have seen in 5.1 that the radiation-matter interaction term is constant in time. Notwithstanding, when a molecule in a stationary state is exposed to a photon flux, the experimental device can be arranged in such a way that the effects of the interaction Hamiltonian in the time elapsed between preparation and exposition and between this and measurement (i.e., the spontaneous emission) can be neglected, thus making the perturbation treatment applicable One-photon processes source Let us first consider an absorption of one photon in the mode from a single mode a =, 0 n, 0 b = B, 0 n 1, 0. (105) The initial stationary-state energy is E a = E + (n + 1/2) h - ω k and similarly for the final state. The first order contribution to the probability of the process will be found by introducing matrix element (95) into eqn. (66): (1) P a b (t) = 2 n ε 0 Vh - ω k ω 2 B sin 2 (ω B ω k )(t t 0 ) ω B ω k 2 π B P 2, (106) with ω B (E B E )/h -. This probability presents a narrow maximum at ω k = ω B and vanishes for ω B ω k = 2nπ/(t t 0 ), quickly tending to zero outside the first two nodes. This behaviour has been verified in spectroscopic measurements on molecular beams using highly monochromatic radiation. necessary condition for obtaining a non-vanishing probability is that the transition moment B P does not vanish, which leads to the electric-dipole selection rules. The probability also depends on the angle between the polarization vector and the transition moment, which can be important for polarized beams acting on crystalline samples. For the usual case of a many-mode radiation acting on a fluid sample, eqn. (106) has to be summed over the radiation modes and averaged over molecular rotations. When the transition is electric-dipole forbidden, we can still find probability contributions from the electric-quadrupole or the magnetic-dipole terms (eqns. (98) and (102)), provided that matrix 23

24 J.. N. F. Gomes and J. C. Paniagua elements B Q and B m do not both vanish, and so on. n practice, the radiation source always covers some frequency interval ω which is normally much larger than 4π/(t t 0 ); moreover, the radiation energy is indeed continuous (see discussion following eqn. (22)), thus making the Fermi Golden Rule (69) applicable: (1) Γ abs = π n ω B ε 0 V π B P 2 ρ(h - ωb ) (107) with k = ω B /c. To connect this result with experimental measurements in a fluid sample we have to sum it for the number of molecules in the initial state N and average it over rotations, which leads to (1) Γ abs = πn n ω B 3ε 0 V B P 2 ρ(h - ωb ). (108) The number of radiation states per unit energy interval ρ(h - ω) can be expressed in terms of the radiant energy per unit volume and unit angular frequency interval (ω) by noting that the radiation energy in a dω centered at ω is V dω = (ρ h - dω) (n h - ω), so that and (1) Γ abs ρ(h - ω) = = πn (ω B ) 3ε 0 h -2 (ω) V n(ω) h -2 ω B P (109) 2 = (1) N B B (ωb ) (110) (1) where B B is the Einstein coefficient for absorption: (1) B B = π B P 2 3 ε 0 h -2 (111) which can easily be related to the experimental molar absorption coefficient (see ref. 10). The preceding results can be extended to coherent radiation sources or, in general, to nonstationary initial states of the form a =, a a with a = n =0 c 0, 0, n, 0,. (112) 24

25 Quantum electrodynamics The usual problem consists in determining the probability of the transition B when radiation in the state a is used, whatever the final radiation state is. Proceeding along the same lines that led from eqn. (60) to eqn. (107) one obtains, for k = ω B /c, (1) Γ, a B = π n ω B ε 0 V π B P 2 ρ(h - ωb ), (113) a result identical to (107) except for the substitution of n = a n a for n. For a one-photon emission process the rate equations are very similar to those we have obtained in the absorption case but one important difference appears: the matrix element (96), which now replaces (95) in the probability amplitude, is proportional to (n +1) 1/2 and, therefore, does not vanish for zero photon flux. The emission probability can then be split into two terms, one of them being proportional to n and the other being independent of this number. The former is responsible for the stimulated or induced emission and the latter explains the spontaneous emission. Naming B and the initial and final molecular states, the rate of emission stimulated by radiation in the mode with k = ω B /c is deduced in the same way as the absorption rate, resulting, after averaging over rotations, (1) Γ stim = πn B (ω B ) 3ε 0 h -2 P B 2 = (1) NB B B (ωb ), (114) (1) where the Einstein coefficient for stimulated emission B B is identical to that of absorption (eqn. (111)). For spontaneous emission the deduction is somewhat different: the emerging photon can appear in any one of many different modes, all of them with a frequency close to ω B. This must be taken into account when evaluating the density of final states for applying the Fermi Golden Rule. The resulting rotational averaged transition rate is (1) Γ spont = N 3 B ω B 3πc 3 ε 0 h - P B 2 = (1) NB B, (115) (1) where B is the Einstein coefficient for spontaneous emission. t should be noted that in stimulated emission, photons are produced in the same mode as the inducing ones, whereas spontaneously emitted photons are randomly distributed in direction and polarization. The latter is the main emission mechanism in conventional sources, while stimulated emission dominates in lasers and masers. 25

26 J.. N. F. Gomes and J. C. Paniagua lthough the magnetic-dipole and electric-quadrupole contributions to the transition matrix element (eqn. (88)) are in general some orders of magnitude less than the electric-dipole contribution, they can produce observable effects due to the cross terms between the former contributions and the latter appearing when the probability amplitude is squared to obtain the transition rate (eqn. (69)). These cross terms may enhance or weaken the transition intensity and they can also cause new effects. One of these is the natural circular dichroism, consisting in a difference in absorption rates for left and right circularly polarized light passing through a chiral sample. For such a system, the cross term between the contributions E1 and M1 (eqns. (95) and (102)) has different sign for left and right circularly polarized light, leading to an enhancement of the electric dipolar absorption rate in one case and a weakening in the other. n effect, the electric and magnetic-dipole contributions (eqns. (95) and (102)) to the matrix element appearing in the Fermi Golden Rule (69) can be grouped, for π kl/r = (1, ±i, 0)/ 2, as follows: b H int a ωk =ω B = i n kl/rh - ω B 2ε 0 V 1/2 πkl/r B P + i c B m (116) (note that k π kl/r / k = (0, 0, 1) (1, ±i, 0)/ 2 = +iπ kl/r ). By comparing this expression with the electric-dipole contribution (95) it readily follows that the corresponding absorption rate for randomly oriented molecules will be: = π N (ω B ) 3ε 0 h -2 (1) Γ L/R B P = π N (ω B ) 3ε 0 h -2 B P + i c B m 2 2 ± 2 c m ( B P * B m ) + 1 c 2 B m 2 (117). Defining the optical rotatory strength as we finally have R B = m ( P B B m ) (118) Γ L (1) Γ R (1) = 4π N (ω B ) 3ε 0 h -2 c = N (ω B ) (B L B R B B R B ), (119) where we have introduced a differential absorption coefficient: 26

27 Quantum electrodynamics B L B B R B = 4π R B 3ε 0 h -2 c. (120) R B changes sign under inversion, so that this coefficient and the difference (119) will have opposite sign for enantiomers. Since the electric-quadrupole contribution to the interaction matrix element is of the same order as the magnetic-dipole one, we should have introduced it in the preceding treatment; however, its cross term with the electric-dipole contribution vanishes under rotational average and only needs to be considered when dealing with oriented molecules Multi-photon processes great variety of phenomena imply the interchange of several photons in a single radiation-matter interaction process. Here we shall only consider some representative cases, in order of increasing complexity. Two-photon processes include absorption, emission and scattering of radiation. t the end of 5.2 we saw that, in the long wavelength approximation, matrix elements connecting states differing in zero or two photons vanish unless the matter state remains unchanged. We shall, therefore, go to the second order of perturbation theory to obtain the rate of such processes. ccording to the second-order Fermi golden rule (72), the initial and final states can now be connected through intermediate states differing in one photon from each of those. For an absorption of two identical photons (2) Γ 2-abs with = 2π h - B n 2 H int n 1 E1 n 1 H int n E1 h - (ω ω k ) = 2 πω k 8ε 0 2 V 2 h - n (n 1) ij π i π j α ij B (ω k, ω k ) α ij B (±ω, ±ω') = 2 ρ(h - ωk ) P Bi P j ω +ω + P BjP i ω +ω' 2 ρ(h - ωk ) (121) (122) where the indices i and j run over Cartesian coordinates, stands for any molecular stationary state, P Bi = B P i, ω k = ω B /2 and it is assumed that a non-resonant frequency is used: ω k ω. fter rotation averaging one obtains 27

28 J.. N. F. Gomes and J. C. Paniagua (2) Γ 2-abs = πn ω k 2 120ε 0 2 V 2 h - n (n 1) 2 π π 2 1 α ii B (ω k, ω k ) α jj B* (ω k, ω k ) ij π π 2 3 α ij B (ω k, ω k ) α ji B* (ω k, ω k ) ρ(h - ω k ). (123) Contrary to the one-photon case, this equation strongly depends on the radiation polarization through π π, which takes the values 1 and 0 for linear and circular polarizations, respectively. On the other hand, selection rules are clearly different in both cases. For instance, the Laporte selection rule is inverted when passing from one- to two-photon absorption. The contributions to the transition probability amplitude (see the squared term in eqn. (121)) can be represented by a "time ordered diagram" in which the initial and final molecular states are represented at the bottom and the top of a vertical line with intermediate states lying between them, wavy lines stand for the interchanged photons and the point at which a wavy line reaches the vertical one corresponds to an interaction matrix element (see fig. 1 (a)). second diagram (fig. 1 (b)) should also be included representing the contribution from H 2 (eqn. (91)) in first order of perturbation theory; however, this term vanishes in the long wavelength approximation for the process being considered, since the initial and final states must necessarily be different. Of course one-photon processes can also be represented in the same way, but the interest of such diagrams lies mainly in multi-photon processes, where it provides a simple way of keeping track of the terms contributing to the transition probability. B B (a) (b) Figure 1. Two-identical-photon absorption diagrams. 28

29 Quantum electrodynamics B B k'λ' k'λ' Figure 2. Two-mode-photon absorption diagrams. n a two-photon absorption from different beams we can draw two different kinds of second-order diagrams (see fig. 2) depending on which photon is absorbed first (this, of course, is a manner of speaking, since there is no measurable time lapse between absorptions nor are intermediate states observed). The contributions of those diagrams to the squared term in the Fermi golden rule (72) are now: B n 1 n k'λ' 1 H int n 1 n k'λ' E1 n 1 n k'λ' H int n n k'λ' E1 h - (ω ω k ) and B n 1 n k'λ' 1 H int n n k'λ' 1 E1 n n k'λ' 1 H int n n k'λ' E1 h - (ω ω k' ) leading to the transition rate (2) Γ 2-abs = πω kω k' n n k'λ' 2ε 2 0 V 2 h - π i π k'λ'j α B ij (ω k, ω k' ) ij 2 ρ(h - ωk, h - ω k' ) (124) where ω k + ω k' = ω B, ω k ω and ω k' ω. Two-photon absorption techniques yield valuable information on the tensor α B by using convenient polarization states for the two beams. Two-photon emission can be treated in a similar way. s in the one-photon case, spontaneous as well as stimulated processes are possible. The former will be of interest mainly when the one-photon de-excitation of the first excited state having is forbidden. Nevertheless, it can also be observed in other exited states having a very low one-photon emission rate. This is the case for the transition 2s 1s of the hydrogen atom, whose two-photon transition rate is 29

30 J.. N. F. Gomes and J. C. Paniagua much larger than that of the sequence of one-photon processes (2s) 2 S 1/2 (2p) 2 P 1/2 (1s) 2 S 1/2, the first of which implies two very close energy levels (note that the spontaneous emission rate (115) is proportional to ω 3 B ). The two-photon emission can be stimulated by irradiation at an energy lower than that of the overall process (ω k < ω B ), the second photon carrying the excess energy (ω k' = ω B ω k ). Raman and Rayleigh scattering involve a change in photon mode as a consequence of the interaction with matter. n the former case there is an energy transfer between radiation and matter, while in the second the scattering is elastic, so that the wave-vector module remains unchanged. The corresponding diagrams are shown in fig. 3. B k'λ' B B k'λ' k'λ' Figure 3. Raman and Rayleigh scattering diagrams. n the Rayleigh case B= and, according to eqn. (104), the third diagram in fig. 3 can contribute in the long wavelength approximation. Transition probabilities in scattering processes are usually expressed in terms of differential cross sections, since the angular distribution of final states is an important source of information (specially in Rayleigh scattering). The resulting cross sections depend on the matrix elements of the frequency dependent polarizability tensor α ij (ω) = α ij (ω, ω). Raman scattering can be considered a spontaneous process inasmuch as the emerging photon modes were not initially occupied. By irradiating in one of these modes the scattering intensity can be greatly enhanced, this technique being known as stimulated Raman scattering. This effect can also be caused by a sufficiently intense Stokes radiation, which can make the scattered beam almost as intense as the incident one. variant of the stimulated Raman technique, in which measurements are made on the forward scattered radiation, is the inverse Raman spectroscopy. n this technique the sample is irradiated with two beams. One of them covers a frequency continuum (ω ω), leading to scattered radiation spread over a continuous range (ω ± ω B ). The other beam, which is intense 30

31 Quantum electrodynamics and monochromatic, stimulates the scattering at its own frequency (ω 0 ). This produces a strong intensity fall at the forward scattered frequency ω = ω 0 + ω B of the first beam, which can be measured in a similar way as is done in absorption spectroscopy. When a chiral transparent medium is traversed by linearly polarized light the polarization plane of the light is rotated, this phenomenon being known as optical rotation. This is a particular case of Rayleigh scattering in which only the radiation polarization state changes. The relevant elementary proces is, 0, n kx, 0 ky, 0,, 0, n kx 1, 1 ky, 0, (125) and the main contributing diagrams are those in fig. 3 with B= and k' = k. t can be seen that these contributions vanish in the electric-dipole approximation for a sample with randomly oriented molecules. However, when one includes the magnetic-dipole matrix element (102) in one of the vertices of each second order diagram (see fig. 4), non-vanishing contributions are obtained. The resulting probability amplitude depends on the optical rotatory strength defined by eqn. (118), and we can obtain the specific rotation φ (angle of rotation per unit path length) in terms of the number of molecules in the state per unit volume (c ): φ = 2 c 3ε 0 h - c 2 ω 2 R ω 2 2 k ω (126) ky ky M1 E1 kx E1 M1 ky E1 M1 kx M1 E1 ky kx kx Figure 4. Optical rotation diagrams Chiral molecules also have the property of scattering at different rates the left and right circularly polarized radiation. The technique known as differential Raman or Rayleigh spectroscopy consists in measuring the corresponding intensity difference in a direction 31

32 J.. N. F. Gomes and J. C. Paniagua perpendicular to the incident beam as a function of the scattered frequency. n order to explain this phenomenon one has to consider the magnetic-dipole and electric-quadrupole matrix elements, since the main diagrams contributing to the differential intensity are those shown in figure 5 together with the ones obtained by substituting E2 for M1. B k'λ B B k'λ B M1 E1 kl/r E1 M1 k'λ E1 M1 kl/r M1 E1 k'λ kl/r kl/r Figure 5. Differential Raman spectroscopy diagrams. chiral molecules can present circular dichroism when placed in a uniform static magnetic field. This effect is a consequence of the cross terms between first and second order perturbation coefficients appearing in the Fermi golden rule when including the diagrams represented in fig. 6 (the straight corresponds to the static field). B B B kl/r E1 M1 E1 kl/r M1 E1 kl/r Figure 6. Magnetic circular dichroism diagrams. Third and higher order diagrams explain a wide collection of phenomena, only feasible with intense laser sources, of which we will mention some representative examples. Frequency doubling or second harmonic generation is a forward three-photon elastic scattering in which two photons in the same mode are converted into one of double frequency, an effect often used to convert visible lasers to UV frequencies. Two representative diagrams contributing in the long wavelength approximation are shown in fig. 7 (a), and the resulting rate 32