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MIT5_74s09_lec04_2

MIT5_74s09_lec04_2 - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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4-1 4.1. INTERACTION OF LIGHT WITH MATTER One of the most important topics in time-dependent quantum mechanics for chemists is the description of spectroscopy, which refers to the study of matter through its interaction with light fields (electromagnetic radiation). Classically, light-matter interactions are a result of an oscillating electromagnetic field resonantly interacting with charged particles. Quantum mechanically, light fields will act to couple quantum states of the matter, as we have discussed earlier. Like every other problem, our starting point is to derive a Hamiltonian for the light-matter interaction, which in the most general sense would be of the form H H H H = M + L + LM . (4.1) The Hamiltonian for the matter H M is generally (although not necessarily) 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 will start with a common semiclassical treatment of the problem. For this approach we treat the matter quantum mechanically, and treat the field classically. For the field we assume that the light only presents a time-dependent interaction potential that acts on the matter, but the matter doesn’t influence the light. (Quantum mechanical energy conservation says that we expect that the change in the matter to raise the quantum state of the system and annihilate a photon from the field. We won’t deal with this right now). We are just interested in the effect that the light has on the matter. In that case, we can really ignore H L , and we have a Hamiltonian that can be solved in the interaction picture representation: H H H t + ( ) M LM H V + ( ) (4.2) = 0 t Here, we’ll derive the Hamiltonian for the light-matter interaction, the Electric Dipole Hamiltonian. It is obtained by starting with the force experienced by a charged particle in an electromagnetic field, developing a classical Hamiltonian for this system, and then substituting quantum operators for the matter: Andrei Tokmakoff, MIT Department of Chemistry, 2/7/2008
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4-2 p → − i = ˆ (4.3) x x ˆ In order to get the classical Hamiltonian, we need to work through two steps: (1) We need to describe electromagnetic fields, specifically in terms of a vector potential, and (2) we need to describe how the electromagnetic field interacts with charged particles. Brief summary of electrodynamics Let’s summarize the description of electromagnetic fields that we will use. A derivation of the plane wave solutions to the electric and magnetic fields and vector potential is described in the appendix. Also, it is helpful to review this material in Jackson 1 or Cohen-Tannoudji, et al.
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