Molecular Quantum Electrodynamics

By D. P. Craig and T. Thirunamachandran

This systematic introduction to quantum electrodynamics focuses on the interaction of radiation with outer electrons and nuclei of atoms and molecules, answering the long-standing need of chemists and physicists for a comprehensive text on this highly specialized subject.
Geared toward postgraduate students in the chemical sciences who require an understanding of quantum electrodynamics as applied to the interpretation of optical experiments on atoms and molecules, the text offers a detailed explanation of the quantum theory of electromagnetic radiation and its interaction with matter. It features formal derivations of the quantized field matrix elements for an amazing number of laser-molecule interaction effects: one- and two-photon absorption and emission; Rayleigh and Raman scattering; dispersion forces in a radiation field; radiation-induced chiral discrimination; both linear and nonlinear optical processes such as Coherent Anti-Stokes Raman Scattering (CARS) and laser-induced optical rotation; self-energy; and the Lamb shift.
Virtually a one-volume encyclopedia, this self-contained book starts with first principles, making it useful both for students and experts in the field. Molecular physicists, quantum chemists, chemical physicists, and theoretical chemists will find essential calculation techniques explained with the greatest clarity.

• Language: English
• Publisher: Dover Publications
• Release date: Nov 13, 2012
• ISBN: 9780486135632

Book preview

SCIENCES

CHAPTER 1

Introduction

1.1 The Nature of Electrodynamics

Quantum electrodynamics is the most precise and widely applicable theory so far found for calculating the interaction of electromagnetic radiation with atomic and molecular matter, and the interactions between molecules. It gives a comprehensive account of both sets of phenomena in a single theoretical framework, and as well as agreeing closely with experiment it affords clear physical insights that are of great value.

Broadly the subject matter of electrodynamics concerns the motion of charged particles and the dynamics of electromagnetic fields in mutual interaction. Where the particles are the constituents of atoms and molecules, external fields exert forces additional to those of the internal fields responsible for atomic or molecular structure and cause perturbations on the states of the bound systems. If the systems are ions, carrying a net charge, external fields also cause ionic currents to flow. In classical electrodynamics the field strengths can have arbitrary values from zero upwards. The classical theory is sufficient to describe the motion of charged particles in those cases where the de Broglie wavelength is much less than any length significant in the experiment. Examples are electron motions in thermionic valves, and trajectories of heavy charged particles in accelerators.

On the atomic and molecular scale, where quantum mechanics has to be used for the charged particles the electromagnetic fields are also subject to quantum conditions, leading to quantum electrodynamics. A feature of the theory refined in this way is that the electric and magnetic field strengths cannot be identically zero. The state of lowest energy is the zero-point state (vacuum field), and in it there are fluctuations in the electric and magnetic field which are intimately connected with processes such as the Lamb shift.

Molecular quantum electrodynamics covers applications to systems broadly within the scope of chemical physics and theoretical chemistry, involving bound electrons of low binding energies and non-relativistic velocities. Problems to be discussed in this book include the absorption and emission of radiation, Rayleigh and Raman scattering, optical activity, and higher order processes such as two-photon absorption and the hyper-Raman effect which have become important with the advent of lasers. Applications to interactions between molecules, dealt with by the same methods, include resonance coupling, dispersion forces, chiral discriminations and induced optical activity.

1.2 Maxwell’s Equations for the Macroscopic Field

So far as is known, the facts of large-scale electromagnetic phenomena, including the discoveries of Coulomb, Ampere, Faraday, Biot, Hertz and others, as well as more recent work on the properties of electromagnetic radiation over the whole accessible range of wavelengths, are all compatible with Maxwell’s equations. These equations (1.2.1)–(1.2.4) give the relation between the four electromagnetic field vectors E, B, D and H and their sources, namely the distribution of charges given by the charge density ρ, and the distribution of current J. E and B are the electric and magnetic field strengths, and D and H the auxiliary fields. In the older literature B is called the magnetic induction, H the magnetic field, and D the electric displacement. E and D transform like polar vectors; B and H transform like axial vectors. If we know the scalar field ρ, namely the charge density, and the vector field J, the current density, at every point in the domain, we are able in principle to find the electromagnetic field vectors likewise at every point subject to boundary conditions.

(1.2.1)

(1.2.2)

(1.2.3)

(1.2.4)

The density ρtrue is the true charge density; it excludes charges forming part of the structure of atoms and molecules, which are internally compensated. Jtrue is the analogous true current. Two of the field vectors, the electric field E and the magnetic field B, can exist even in source-free space [see Eqns (1.2.2) and (1.2.3)]. They are the fundamental fields and, if every particle is included individually in ρ and J, they give a complete description of the electromagnetic field (see Section 1.3 on Microscopic Fields). The auxiliary fields D and H are introduced when not all charged particles are included as sources. These fields however contain implicitly parts of the sources, namely the bound charge and current densities, in the form of polarization fields and currents. The bound particles are regarded as present as matter, not included in ρ and J, but as part of a material background in which the fields are present. The auxiliary fields include the response of the matter (the electric polarization and the magnetization) to the applied fields; they are connected with E and B by relations depending on the constitution of the matter. In the simplest constitutive relations D is parallel and proportional to E, and H parallel and proportional to B, according to D = εE and H = B/µ. ε and µ are respectively the electric permittivity and magnetic permeability. If the values in free space are ε0 and µ0 we can use the dimensionless ratios ke = ε/ε0, km = µ/µ0 to characterize the medium, namely the dielectric constant (relative permittivity) and relative permeability. In anisotropic materials the constants ke and km are replaced by second rank tensors, because D is in general not parallel to E nor H to B. In strong fields, such as those produced by lasers, the relationships must take account of nonlinearity, so that ke and km are to be interpreted as functions of E and B respectively.

A leading feature of Maxwell’s equations, marking the decisive advance over earlier theories, is the following. Equation (1.2.4) shows that curl H has two sources: one is the convective current J, and the other the time derivative of the electric displacement ∂D/∂t: the latter stands on exactly the same footing as a source as does the convective current, and is for that reason given the name displacement current, even though it does not refer entirely to the motion of charged particles.

Maxwell’s equations are the equations of motion of the electromagnetic field, in which the charged particles appear as sources. The equations of motion of the charged particles on the other hand (in classical dynamics) are Newton’s laws with a force term for the effect of the electromagnetic fields. This force is the Lorentz force (1.2.5)

(1.2.5)

where q is the charge and ν the velocity of the free electron or ion. The electric force qE is parallel to the electric field; the component of the magnetic force q(ν × B) in the direction of E is evidently determined by the velocity and magnetic field in the plane perpendicular to E.

Thus the charges and currents which are taken to be the sources of the fields have their motions modified by the same fields. The problem in which all these interactions are treated in a self-consistent way has proved too difficult to solve except for simple cases and models. In quantum electrodynamics it is usual to apply perturbation theory: in the first order the particles of which the motion is being studied do not affect the fields, which thus appear as driving fields. In higher orders, the dynamics of particles and fields are inextricably mixed, as will be seen in later chapters.

1.3 The Microscopic Field Equations

The charge density ρtrue used in Eqn (1.2.1) is the charge ΔQ in a volume ΔV, the volume being large enough compared with atomic or molecular dimensions to allow the charge density to be treated as a continuous function of position. Likewise in the current density Jtrue we have for the current ΔI across a surface of area ΔA is the unit vector normal to the surface, and ΔA is an area large on the atomic scale, but small on the macroscopic scale. The microscopic electrical structure of matter is thus too fine to be included in the macroscopic fields of Eqns (1.2.1)–(1.2.4), which are local spatial averages (cellular averages) over microscopic fields to which every charged particle makes a contribution. In going to the microscopic field equations (Maxwell-Lorentz equations) the idea of a continuous distribution of charge is given up and each elementary charge has to be considered separately. We are then faced with the need to define a charge density for a collection of point charges. Whereas for a continuous charge distribution Q(r), the limit of ΔQ/ΔV could be taken giving

(1.3.1)

with the total charge given by

(1.3.2)

we see that for a charge distribution that consists of separate point charges Q(r) is a discontinuous function, zero everywhere except at the positions of the charges. It is not possible to define the density field as in (1.3.1). Instead the desired density must be zero everywhere except at the charges, where it is infinite. Such a density can be constructed with the help of the Dirac delta function defined buy

(1.3.3)

and

(1.3.4)

The delta function has a strong singularity at r = q, giving a sufficient contribution to the result (1.3.4) in an integration over a volume including the point q. We define

(1.3.5)

(1.3.6)

the sums are over all particles α, carrying charges eα, located at points qα. The microscopic field equations are expressed in terms of vectors of the microscopic field which, here and throughout, are written in lower case symbols:

(1.3.7)

(1.3.8)

(1.3.9)

(1.3.10)

where ε0 and µ0 are the vacuum permittivity and the magnetic permeability. In this book the quantity µ0 is suppressed in favour of ε0 and c through ε0µ0 = 1/c². The equations (1.3.7)–(1.3.10), in contrast to the macroscopic Maxwell’s equations (1.2.1)–(1.2.4), are expressed in terms only of the fundamental fields e and b, and the sources ρ and j. All charged particles in the system contribute to ρ and j and there are no auxiliary fields.

The microscopic equations (1.3.7)–(1.3.10) can be converted to the macroscopic equations (1.2.1)–(1.2.4), following a method of averaging first used by Lorentz. The latter equations together with (1.2.5) are the fundamental equations of motion in macroscopic electrodynamics.

1.4 The Electromagnetic Potentials

In quantum electrodynamics both the particles and the electromagnetic field are to be subject to quantum conditions. Quantization of the field, which will be discussed in Chapter 2, cannot easily be done in terms of the fields e and b which have the character of fields of force, but requires the use of potentials.

A familiar example of the relation of a force field to its potential is the following. An electrostatic field E is expressible as the gradient of an electric potential ξ,

(1.4.1)

Inasmuch as the potential is to be found as the integral of the field it is not unique, being determined only up to a constant of integration: the electromagnetic potentials are in an analogous way not unique, being determined up to an additive gauge function.

Two results from vector analysis important for the treatment of the electromagnetic potentials are: (i) If ∇. V, the divergence of a vector field V, vanishes the field can be expressed as the curl of a vector field W,

(1.4.2)

since ∇. (∇ × W) = 0 for any vector field. (ii) If ∇ ×V, the curl of a vector field V, vanishes the field may be expressed as the gradient of a scalar field φ,

(1.4.3)

since ∇ × (∇φ) = 0 for any scalar field.

To introduce the definitions of the electromagnetic potentials we first apply (i) to the magnetic field b, which has zero divergence [Eqn (1.3.8)], and set it equal to the curl of the vector potential a.

(1.4.4)

Evidently a cannot be uniquely defined as an integral of the field and, as already indicated, is defined only up to the addition of the gradient of a scalar function χ, since

(1.4.5)

χ is known as the gauge function. Substitution of (1.4.4) into (1.3.9) gives

(1.4.6)

where the order of space and time differentiations has been interchanged. Equation (1.4.6) can be rewritten

(1.4.7)

so that using the second of the two results on vector fields, Eqn (1.4.3), we can write

(1.4.8)

where φ is the scalar potential. The choice of the negative sign is convenient for later work. This equation, together with (1.4.4), defines the potentials a and φ. The freedom introduced by the gauge is expressed in the transformations (1.4.9), taken in conjunction,

(1.4.9)

Thus the fields, and Maxwell’s equations, are unaffected by these substitutions, which are known as gauge transformations; there is a family of (a, φ) potential pairs giving the same fields (e, b). This invariance is described as the gauge invariance of the fundamental field vectors.

We now consider the remaining two Maxwell’s equations in order to relate the potentials a and φ to the sources. Equation (1.3.7), together with (1.4.8) gives

(1.4.10)

That is,

(1.4.11)

From (1.3.10), (1.4.4) and (1.4.8) we get

(1.4.12)

With the help of the vector identity

(1.4.13)

Eqn (1.4.12) may be written

(1.4.14)

1.5 Lorentz and Coulomb Gauges

Equations (1.4.11) and (1.4.14) relate the potentials to the sources. The equations may be simplified by special choice of the gauge function, taking advantage of the invariance to gauge of the field vectors e and b. One choice is implied by setting

(1.5.1)

Such a choice is always possible. For suppose the potentials a and φ did not satisfy equation (1.5.1). Then a gauge function χ can be found to define new potentials (1.4.9) satisfying

i.e.

(1.5.2)

Thus there exists an (a, φ) pair obeying (1.5.1). After substitution for ∇. a from Eqn (1.5.1), (1.4.11) becomes

(1.5.3)

and with the same substitution (1.4.14) becomes

(1.5.4)

By the adoption of (1.5.1), the Lorentz gauge, the scalar potential is related to the charges and the vector potential to the currents. The Lorentz gauge is used in the covariant formulation of quantum electrodynamics appropriate to problems of fast moving particles which must be treated relativistically.

Another choice, the Coulomb gauge, is the more useful for the non-covariant theory developed in this book, having particular advantages for slow-moving particles in bound states. It is implemented by separating fields which have zero curl (irrotational or longitudinal fields) from those with zero divergence (solenoidal or transverse fields). This separation enables one class of field to be described by the vector potential a alone, and the other by the scalar potential φ alone. The Coulomb gauge is defined by the condition

(1.5.5)

It is always possible to find a vector potential a with zero divergence. Let us suppose that ∇. a ≠ 0 for some choice of a. We may transform a to a′ by adding the gradient of a scalar χ according to

(1.5.6)

and a′ can be made divergence-free

(1.5.7)

by choosing χ to be a solution of the Poisson’s equation (1.5.8)

(1.5.8)

With a having zero divergence equations (1.4.11) and (1.4.14) become

(1.5.9)

and

(1.5.10)

Equation (1.5.9) shows that the scalar potential is that for an instantaneous distribution of charges, so that this gauge allows separation of the static and dynamic aspects of the sources of the electromagnetic field. Also, for the field in regions free of charges, the scalar potential is zero, and there are the following free field equations easily soluble for a, to be discussed in Chapter 2

(1.5.11)

A valuable insight is possible through the concept of longitudinal (curl-free) and transverse (divergence-free) fields to be elaborated in Chapter 3. Any vector field V can be decomposed into a perpendicular or transverse part V⊥ and a parallel or longitudinal part V according to

(1.5.12)

with the properties

(1.5.13)

and

(1.5.14)

Based on the decomposition (1.5.12), let us put

(1.5.15)

(1.5.16)

Since ∇.b = 0 [Eqn (1.3.8)], b must be purely transverse, and b = b⊥. Maxwell’s equations (1.3.7) and (1.3.9) become

(1.5.17)

(1.5.18)

Equation (1.3.10) divides into separate equations for transverse and longitudinal components. For the transverse components:

(1.5.19)

and for the longitudinal

(1.5.20)

By taking the divergence of the last equation and using (1.5.17) we get the well-known equation of continuity

(1.5.21)

Also since in Coulomb gauge the vector potential is transverse, a = a⊥, and ∇φ is irrotational, Eqn (1.4.8) can be separated into longitudinal and transverse terms,

(1.5.22)

and

(1.5.23)

Equation (1.5.10) can now be written in terms of transverse quantities only, giving the equation (1.5.24) for a. The equation (1.5.25) for φ is unchanged from (1.5.9):

(1.5.24)

(1.5.25)

Thus the Coulombic fields are completely separated from the solenoidal fields; the former are described by the scalar potential (which is also the electrostatic potential in this case) and the latter are described by the vector potential; hence the use of the term Coulomb gauge.

Gauge transformations do not change the transverse part of the vector potential a: according to (1.4.5) the addition of the gradient of a scalar field to a, in (1.4.9), can add only a longitudinal component. Thus the effect of gauge transformations is to vary the way in which e , the longitudinal part of the electric field, is made up of contributions from the potentials a and φ. a⊥in all gauges. In Lorentz gauge a to e (Eqn (1.4.8)). In Coulomb gauge the vector potential contributes only to e⊥.

The usefulness of Coulomb gauge for problems of radiation coupled to atoms and molecules is a consequence of the fact that the Coulomb potential separates out. On account of this separation the total Hamiltonian for field plus particles (see Eqn (1.7.1) and Chapter 3) includes the Coulomb potential in the same way as in standard particles-only quantum mechanics. Solutions of the appropriate particles-only Schrödinger equations can be used as bases for a perturbation theory of the complete problem including the radiation field. Only the radiation field is quantized, giving a transverse (solenoidal) field. The photons that result from quantization of the transverse field are referred to as transverse photons. They are the photons involved in all radiative processes. In Lorentz gauge both the vector and scalar potentials obey wave equations (1.5.3) and (1.5.4), and in the passage to quantum field theory both are quantized, to give quantized scalar, longitudinal and transverse fields. Thus scalar, longitudinal and transverse photons appear.

Although the coupled equations of motion for the particles and fields cannot be solved exactly, the two interesting limiting cases of a system of charges only, and of the field in free space without sources, can be studied in detail. For the systematic development these extreme cases form a useful starting point for the solution of the more general problem. The particles-only model is discussed in the next section and the free field model in the next chapter.

1.6 Quantum Mechanics of a System of Charges

In the quantum mechanics of atoms and molecules the Hamiltonian operator H in the Schrödinger equation Hψ = Eψ is found from the classical Hamiltonian function by promoting the dynamical variables to quantum operators. Where, as is common, the classical potential and kinetic energies are given in cartesian coordinates x,y,z, and momenta px, py, pz, the momentum px is replaced by the operator − i ∂/∂x and the displacement x by the operator x. The necessity that the classical Hamiltonian be expressed in terms of canonical variables in this case passes unnoticed, in that the linear momenta px are the conjugates of the displacements x, and are the only momenta that appear. Even in some elementary problems however care is needed. For example in polar coordinates the momentum operator conjugate to the radial displacement is not of the form suggested by the canonically conjugate variables in cartesians. In the quantization of the electromagnetic field the same procedure is used, but it is not obvious how to express the Hamiltonian function in appropriate conjugate variables. We must then proceed in a formal way through a Lagrangian function, being guided by the requirement that it must be chosen to lead to the correct equations of motion. An outline of this path to the equations of motion is given and its application illustrated in the familiar case of a system of particles in the absence of fields.

. The configuration of the system at time t is represented by a point in the many-dimensional configuration space spanned by the generalized coordinates. The changes in the configuration with time correspond to the motion of the point in the configuration space. The curve traced, called the path of the system, must not be confused with the trajectory of a particle; it represents the change in configuration of the total system with time. The equations of motion are found from Hamilton’s Principle which states that, between two times t1 and t2, of all possible paths joining the initial and final points in configuration space, the path taken by the system is that for which the action integral

(1.6.1)

has its variational minimum; that is the path for which

(1.6.2)

For a conservative system, the Lagrangian L in (1.6.1) and (1.6.2) is given by T-V, T being the kinetic energy, and V the potential energy.

By effecting the variation (1.6.2) it is possible to obtain the equations of motion. Suppose q(t) is the function for which the action S is a minimum. Let us change the path from q(t) to q(t)+δq(t) where δq(t) is small everywhere in the time interval of interest, and suppose that in the variation the endpoints of the path are unaffected, that is,

(1.6.3)

If q(t) changes by δq(tchanges by (d/dt) δq(t). Hence

(1.6.4)

Integrating the second term in (1.6.4) by parts and using (1.6.3), we get

(1.6.5)

Since δS = 0 the integral must vanish for all δq, and it follows that

(1.6.6)

For a system with N degrees of freedom, there are N such equations :

(1.6.7)

These Lagrange’s equations of motion hold for particle systems. For fields the equation is slightly modified as discussed in Chapter 2.

Instead of the N second-order Eqn (1.6.7) it is possible in an alternative formulation to describe the motion by 2N first-order equations, in terms of the Hamiltonian. The dynamical variables are then