8 - p. 54 MIT Department of Chemistry 5.74, Spring 2004:...

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MIT Department of Chemistry p. 54 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Andrei Tokmakoff Interaction of Light with Matter We want to derive a Hamiltonian that we can use to describe the interaction of an electromagnetic field with charged particles: Electric Dipole Hamiltonian. Semiclassical: matter treated quantum mechanically Field: classical Brief outline of electrodynamics : See nonlecture handout. Also, see Jackson, Classical Electrodynamics , or Cohen-Tannoudji, et al., Appendix III. > Maxwell’s Equations describe electric and magnetic fields ( E , B ) . > For Hamiltonian, we require a potential. > To construct a potential representation of E and B , you need a vector potential A r , t () and a scalar potential ϕ F , t . > A and are mathematical constructs that can be written in various representations (gauges). We choose a gauge such that = 0 (Coulomb gauge) which leads to plane-wave description of E and B : +∈ 0 µ 0 2 A r , t −∇ 2 A r , t = 0 t ∇ ⋅ A = 0 This wave equation allows the vector potential to be written as a set of plane waves: = A 0 e i ( k r ω t * ˆ i ( k r t ) A r , t ˆ ) + A 0 e (oscillates as cos ω t) since ∇ ⋅ A = 0, k ˆ where ˆ ∈= 0 k ˆ is the polarization direction of the vector potential. i ˆ ( ⋅−ω t ) E =− A = ω A 0 e ik r + c . c . (oscillates as sin ω t) t t ) B =∇× A = i k ×∈ ) A e + c .
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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8 - p. 54 MIT Department of Chemistry 5.74, Spring 2004:...

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