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Coupling Charged Particles to the Electromagnetic Field: The Bohm Aharanov Effect, Magnetic Monopoles Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics University of California, Santa Cruz November 2010 Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo The Lagrangian for Particles in an Electromagnetic Field We can guess a lagrangian by demanding gauge invariance. Recalling that under gauge transformations: φ(~x , t) = φ0 (~x , t) − 1 ∂Λ(~x , t) c ∂t ~ ~x , t) = A ~ 0 (~x , t) + ∇Λ( ~ ~x , t) (1) A( the lagrangian X ma d ~xa 2 L= − 2 dt a Z (2) 1~ ~x , t) · ~J(~x , t) d 3 x −φ(~x , t)ρ(~x , t) + A( c is invariant, due to current conservation. Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo For a collection of point particles, X X ρ(~x , t) = qa δ(~x − ~xa (t)) ~J(~x , t) = qa~va δ(~x − ~xa (t)) (3) a a So, integrating over ~x , X ma qa ~ ~xa , t) . L= − qa Φ(~xa , t) + ~va · A( 2 c a (4) ~ are functions of ~xa and t. Note carefully that Φ and A Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo The Lorentz Force Law If all of this is correct, the Lorentz force law should emerge as the Euler-Lagrange equation for this system. It is convenient, to avoid getting confused about vector quantities, to work with Cartesian indices. We need to evaluate: ∂L j~i j i ~ . (5) = −q ∇ Φ − v ∇ A a a a a ∂xai After this, we drop the subscript a on the partial derivatives, to streamline the writing, i.e. ∂L ~ i Φ − vaj ∂ i Aj . = −q ∂ a ∂xai Physics 212 2010, Electricity and Magnetism (6) Coupling Charged Particles to the Electromagnetic Field: The Bo d ∂L qa i d i A . m ẋ + = a a dt ∂ ẋai dt c (7) To evaluate the time derivative, we need to keep in mind that this is the total derivative, i.e. dAi ∂Ai = + vj ∂j Ai . dt ∂t (8) So, now dropping the a subscript everywhere, which no longer can lead to any ambiguity: 1 q mẍ i = q(−∂i Φ − ∂t Ai ) + (+v j ∂i Aj − v j ∂j Ai ). c c (9) ~ the The first term we recognize immediately as giving us q E; ~ second is readily seen (using our favorite identity) to be q~v × B. So indeed this lagrangian gives us the Lorentz force for each charged particle! Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo The Hamiltonian: passing to quantum mechanics If we are interested in quantum mechanics, we need to construct the Hamiltonian. For this, we first need the canonical momentum. 1 ∂L (10) = mẋai + Ai (~xa , t). pai = i c ∂xa Now we compute: X H= qai pai − L = mẋai 2 + qa φ(~xa , t). (11) (We have written the argument of φ explicitly, again). Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo ~ does not appear in this expression. But the Not surprisingly, A Hamiltonian is not to be thought of as a function of the velocities, but of the coordinates and momenta. So X 1 2 ~ H= (~pa − qa A) + qa Φ (12) 2ma In passing to quantum mechanics, one takes ~a ~pa → −i~∇ (13) ~ appears in the Schrodinger equation. so A Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo Gauge invariance of the Schrodinger Equation What happened to gauge invariance? Consider HΨ = i ∂ Ψ. ∂t (14) The full gauge invariance is now: φ(~x , t) = φ0 (~x , t)− 1 ∂Λ(~x , t) c ∂t ~ ~x , t) = A ~ 0 (~x , t)+ ∇Λ( ~ ~x , t) (15) A( Ψ = ei qΛ ~ Ψ0 . as you can readily check. Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo The Aharanov-Bohm Effect ~ field, but Consider an infinite solenoid. Outside, there is no B ~ ~ inside, there is a net flux of B. As a result, the line integral of A around the loop is non-zero: I I ~ ~ d ` · A = dφAφ sin θr = ΦM . (16) ~ vanishes outside, this is a Despite the fact that B gauge-invariant statement. This follows from the fact that, under a gauge transformation, the left hand side is I I ~ ~ ~ · d ~` = Λ(φ + 2π) − Λ(φ). d ` · A = ∇Λ (17) So as long as the gauge transformation is periodic (single-valued), the integral is gauge invariant. Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo ~ must be a gauge Since ΦM is non-zero, on the other hand, A ~ ~ transformation of A = 0; in this way, the B field vanishes. Λ is not periodic, so it is not really a gauge transformation: ~ =∇ ~ Λ̃; A Λ̃ = ΦM φ. (18) Λ̃ is not a gauge transformation because it is not periodic. Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo You are have some familiar with the Feynman path integral. We want to consider the effect on the interference pattern, say in a double slit experiment, of the solenoid. In general, Feynman tells us that the amplitude to go from a point A to a point B along a particular classical path is eiS(A,B)/~ . We have seen ~ field. So if we how the action is modified in the presence of a A turn on the solenoid, we the amplitude changes to q eiS(A,B)/~ ei ~ RB A d~ `·~ A . (19) So if we compare to paths, one of which goes around one side of the solenoid, and the other around the other, ending up at the same point, they differ in phase (from their previous difference in phase) by ei H d~ `·~ A = eiΦM /~ . (20) When squaring the sum of these amplitudes, there are new (observable) interference effects.This is the Aharanov-Bohm effect. Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo In this light, one can understand the Dirac quantization condition for electric charge. We have seen that if monopoles exist, they are described by singular field configurations. This singularity is seemingly a gauge artifact. It can be chosen, for example, to lie in different directions by making a gauge transformation. But this gauge transformation will have observable consequences if it leads to modified interference patters for particles of charge q. The string is detectable, like the Bohm-Aharanov solenoid, in general. More precisely, the singularity can be written: ~ = ∇Λ ~ A iq e~ This means: H Λ = 2gφ. ~ d~ `·∇Λ = 1. 2gq = n. ~ Physics 212 2010, Electricity and Magnetism (21) (22) (23) Coupling Charged Particles to the Electromagnetic Field: The Bo This (with variants) is the only known explanation for the quantization of electric charge. Physics 212 2010, Electricity and Magnetism Coupling Charged Particles to the Electromagnetic Field: The Bo