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6. INTERACTION of LIGHT and MATTER 6.1. Introduction

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6. INTERACTION of LIGHT and MATTER 6.1. Introduction 6-1 6. INTERACTION OF LIGHT AND MATTER 6.1. Introduction One of the most important topics in time-dependent quantum mechanics is the description of spectroscopy, which refers to the study of matter through its interaction with electromagnetic radiation. Classically, light–matter interactions are a result of an oscillating electromagnetic field resonantly interacting with charged particles in the matter, most often bound electrons. We observe these processes either through changes to the light induced by the matter, such as absorption or emission of new light fields, or by light-induced changes to the matter, such as ionization and photochemistry. By studying such processes as a function of the control variables for the light field (amplitude, frequency, polarization, phase, etc.), we can deduce properties of the samples. The term “spectroscopy” comes from the Latin “spectron” for spirit or ghost and the Greek “” for to see. These roots are telling because in molecular spectroscopy you use light to interrogate matter, but you actually never see the molecules, only their influence on the light. Different types of spectroscopy give you different perspectives. This indirect contact with the microscopic targets means that the interpretation of spectroscopy requires a model, whether it is stated or not. Modeling and laboratory practice of spectroscopy are dependent on one another, and spectroscopy is only as useful as its ability to distinguish different models. This makes an accurate theoretical description of the underlying physical process governing the interaction of light and matter important. Quantum mechanically, we will treat spectroscopy as a perturbation induced by the light which acts to couple quantum states of the charged particles in the matter, as we have discussed earlier. Our starting point is to write a Hamiltonian for the light–matter interaction, which in the most general sense would be of the form HH M HLLM H (6.1) Although the Hamiltonian for the matter may be time-dependent, we will treat the Hamiltonian for the matter H M as time-independent, whereas the electromagnetic field H L and its interaction with the matter H LM are time-dependent. A quantum mechanical treatment of the light would describe the light in terms of photons for different modes of electromagnetic radiation, which we will describe later. We begin with a semiclassical treatment of the problem, which describes the matter quantum mechanically and the light field classically. We assume that a light field described by a time-dependent vector potential acts on the matter, but the matter does not influence the light. (Strictly, energy conservation requires that any change in energy of the matter be matched with an equal and opposite change in the light field.) For the moment, we are just interested in the effect 6-2 that the light has on the matter. In that case, we can really ignore H L , and we have a Hamiltonian for the system that is HHMLM H t (6.2) HVt0 which we can solve in the interaction picture. We will derive an explicit expression for the Hamiltonian HLM in the Electric Dipole Approximation. Here, we will derive a Hamiltonian for the light–matter interaction, starting with the force experienced by a charged particle in an electromagnetic field, developing a classical Hamiltonian for this interaction, and then substituting quantum operators for the matter: p i ˆ (6.3) x xˆ In order to get the classical Hamiltonian, we need to work through two steps: (1) describe electromagnetic fields, specifically in terms of a vector potential, and (2) describe how the electromagnetic field interacts with charged particles. 6.2. Classical Light–Matter Interactions Classical plane electromagnetic waves As a starting point, it is helpful to first summarize the classical description of electromagnetic fields. A derivation of the plane wave solutions to the electric and magnetic fields and vector potential is described in the appendix in Section 6.6. Also, it is helpful to review this material in Jackson1 or Cohen-Tannoudji et al.2 Maxwell’s equations describe electric and magnetic fieldsEB, ; however, to construct a Hamiltonian, we must use the time-dependent interaction potential (rather than a field). To construct the potential representation of E and B , you need a vector potential Art, and a scalar potential rt, . For electrostatics we normally think of the field being related to the electrostatic potential through E , but for a field that varies in time and in space, the electrodynamic potential must be expressed in terms of both A and . In general, an electromagnetic wave written in terms of the electric and magnetic fields requires six variables (the x, y, and z components of E and B). This is an over determined problem; Maxwell’s equations constrain these. The potential representation has four variables ( Ax ,,AAyz, and ), but these are still not uniquely determined. We choose a constraint—a representation or 1 Jackson, J. D. Classical Electrodynamics (John Wiley and Sons, New York, 1975). 2 Cohen-Tannoudji, C., Diu, B. & Lalöe, F. Quantum Mechanics (Wiley-Interscience, Paris, 1977), Appendix III. 6-3 gauge—that allows us to uniquely describe the wave. Choosing a gauge such that 0 (Coulomb gauge) leads to a unique description of E and B : 1 2 Art , 2 Art,0 (6.4) ct22 A 0 (6.5) This wave equation for the vector potential gives a plane wave solution for charge free space and suitable boundary conditions: ikr t * ikr t Art, A00ˆˆee A (6.6) This describes the wave oscillating in time at an angular frequency and propagating in space in the direction along the wave vector k , with a spatial period 2 k . Writing the relationship between k, , and in a medium with index of refraction n in terms of their values in free space n0 2 n knk0 (6.7) c 0 The wave has an amplitude A0 , which is directed along the polarization unit vector ˆ . Since A 0 , we see that k ˆ 0 or k ˆ . From the vector potential we can obtain E and B A E t (6.8) ikr t ikr t iA0 ˆ ee BA ikr t ikr t (6.9) ikˆ A0 e e If we define a unit vector along the magnetic field polarization as bkˆˆˆˆ kk , we see that the wave vector, the electric field polarization and magnetic field polarization are mutually orthogonal kbˆˆˆ . Also, by comparing eqs. (6.6) and (6.8) we see that the vector potential oscillates as cos(t), whereas the electric and magnetic fields oscillate as sin(t). If we define 1 EiA (6.10) 2 00 1 B ikA (6.11) 2 00 6-4 then, Ert ,sin E0 ˆ kr t (6.12) ˆ B rt,sin Bb0 k r t (6.13) Note, EB00 / k c. We will want to express the amplitude of the field in a manner that is experimentally accessible. The intensity I, the energy flux through a unit area, is most easily measured. It is the time-averaged value of the Poynting vector 2 SEB 0c (6.14) 1 I ScE 2 (W/m2) (6.15) 2 00 An alternative representation of the amplitude that is useful for describing quantum light fields is the energy density I 1 UE 2 (J/m3) (6.16) c 2 00 Classical Hamiltonian for radiation field interacting with charged particle Now, we obtain a classical Hamiltonian that describes charged particles interacting with a radiation field in terms of the vector potential. Start with Lorentz force3 on a particle with charge q: FqEvB (6.17) Here v is the velocity of the particle. Writing this for one direction (x) in terms of the Cartesian components of E , v , and B , we have: FqEvBvBx xyzzy (6.18) In Lagrangian mechanics, this force can be expressed in terms of the total potential energy U as UdU Fx (6.19) x dt vx Using the relationships that describe E and B in terms of A and (eqs. (6.10) and (6.11), inserting into eq. (6.18), and working it into the form of eq. (6.19), we can show that Uq qvA (6.20) 3 See Schatz and Ratner, pp. 82–83. 6-5 This is derived elsewhere4 and is readily confirmed by replacing it into eq. (6.19). Now we can write a Lagrangian in terms of the kinetic and potential energy of the particle L TU (6.21) 1 LmvqvAq2 (6.22) 2 The classical Hamiltonian is related to the Lagrangian as HpvL (6.23) 1 2 p vmvqvAq2 L Recognizing p mv qA (6.24) v 1 we write vpqAm (6.25) Now substituting eq. (6.25) into eq. (6.23), we have 112 q Hmm ppqA2m pqA pqAA q (6.26) 1 2 HpqArtqrt,, (6.27) 2m This is the classical Hamiltonian for a particle in an electromagnetic field. In the Coulomb gauge 0 , the last term is dropped. We can write a Hamiltonian for a single particle in a bound potential V0 in the absence of an external field as p2 HVr (6.28) 002m and in the presence of the EM field, 1 2 HpqArtVr, 0 (6.29) 2m 2 q q 2 Expanding we obtain HH0 pAAp Art, (6.30) 22mm Generally the last term which goes as the square of A is small compared to the cross term, which is proportional to first power of A. This term should be considered for extremely high field strength, which is non-perturbative and significantly distorts the potential binding molecules together, i.e., 4 Cohen-Tannoudji et al. App. III, p. 1492. 6-6 when it is similar in magnitude to V0.
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