Lecture 11 Lecture Notes

Lecture 11 Lecture Notes - 4.5 Maxwell Equations in Coulomb...

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4.4 Maxwell Equations in Lorentz gauge 4.4.1 Gauge conditions Lorentz gauge condition => remaining Maxwell equations become 'wave equations' d'Alembert Operator need to solve under constraint of Note: for given external and (satisfying ) Lorentz gauge can use gauge generated by to obtain different as long as Note: gauge not completely fixed by Lorentz gauge condition! 4.4.2 Special Lorentz gauge for charge free situation Consider situation without charges Maxwell in Lorentz gauge Note: The advantage of the Lorentz gauge is its compatibility with the theory of relativity Relationship fields potentials gauge transformation generated by Section about time development of gauge conditions withdrawn. Lec 11 - 28 Jan Lecture Notes Page 1
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choose gauge transform generated by as is one solution verify: valid choice? Then: for in this special choice: (remaining part of Lorentz gauge condition) (remaining part of continuity equation) Important gauge in Quantum Optics: easy way of quantizing fields
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Unformatted text preview: 4.5 Maxwell Equations in Coulomb Gauge 4.5.1 gauge condition => remaining Maxwell Equations (1) (2) special structure: Eqn (1) same as in electrostatics => only one differential equation! insert result into Eqn (2) 4.5.2: Achievability of Coulomb gauge => satisfies Lecture Notes Page 2 5. Vacuum Solutions of Maxwell Eqns. 5.1 Maxwell Equation vacuum => special case of Lorentz gauge possible gauge condition: (also Coulomb gauge!) monochromatic plane waves 5.4 Solution via mode decomposition: 5.4.1 Separation of variables Ansatz whenever continuity will take care of exceptions must be time independent! situations (see below) Note: positive choice would lead to unphysical Time contribution Spatial contribution Helmholtz Equation => each such solution is called a mode => find complete, orthonormal solutions Lecture Notes Page 3...
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Lecture 11 Lecture Notes - 4.5 Maxwell Equations in Coulomb...

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