8sup - MIT Department of Chemistry 5.74 Spring 2004...

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MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Andrei Tokmakoff Non-Lecture Review of Free Electromagnetic Field Maxwell’s Equations (SI): (1) ∇ ⋅ B = 0 (2) ∇ ⋅ E = ρ / 0 B (3) ∇ × E =− t E (4) ∇ × B = µ 0 J +∈ 0 0 t E : electric field; B : magnetic field; J : current density; : charge density; 0 : electrical permittivity; 0 : magnetic permittivity We are interested in describing E and B in terms of a scalar and vector potential. This is required for our interaction Hamiltonian. Generally: A vector field F assigns a vector to each point in space, and: (5) ∇ ⋅ F = F x x + F y y + F z z is a scalar For a scalar field φ (6) φ = x x ˆ + y y ˆ + z z ˆ is a vector where x ˆ 2 + y ˆ 2 + z ˆ 2 = r ˆ 2 unit vector Also: x ˆ y ˆ z ˆ (7) ∇ × F = x y z F x F y F z Some useful identities from vector calculus are: (8) ∇ ⋅ ( ∇ × F ) = 0 (9) ∇× ∇ ( ) = 0
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2 (10) ∇× ( ∇ × F ) = ∇ ∇⋅ F () − ∇ 2 F We now introduce a vector potential A ( r ,
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8sup - MIT Department of Chemistry 5.74 Spring 2004...

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