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# 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 ϕ are mathematical constructs that can be written in various representations (gauges). We choose a gauge such that ϕ = 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 ω 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 i k r + c.c. (oscillates as sin ω t) t i k r ( ⋅ −ω t ) B = ∇× A = i k ×∈ ) A e + c.c 0 ( ±²³ ˆ b k

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p. 55 so we see that k ˆ ˆ ∈ ⊥ n ˆ ˆ
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8 - p 54 MIT Department of Chemistry 5.74 Spring 2004...

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