By Y. C. Fung

Revision of a vintage textual content via a exotic writer. Emphasis is on challenge formula and derivation of governing equations. new version positive factors elevated emphasis on purposes. New bankruptcy covers long term alterations in fabrics less than tension.

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**Example text**

I,)3e Ae ([l i601,] r , t ) = A J ( r ,t ) + i68, [ - l i e A e ( r ,t ) + IfmrmaeA3(r,t ) ] . 55) Now we are ready to apply these ideas to the EM field. Angular Momentum Operator The angular momentum operator for the EM field is W= s d3r [ r x :( E L x B ) : ] ' . 56) This can be reduced to a more tractable form by expanding out the double cross product, 0' = = s / d3r tijerJc~kmEffmabdaAb d 3 r cz3erJ{ E p t A b - E l . 57) it is instructive to compute the commutator of R with A . 58) simplifies to Note that the right-hand side of this equation describes the infinitesimal rotation of the field around the ith axis [compare with Eq.

B) Repeat part (a) using the MeV as the fundamental unit of energy. Find a conversion factor between f and MeV. (c) An expression in natural units can always be converted uniquely into an expression in ordinary units (L,t,rn) by inserting h and c in the correct places. Give an argument describing precisely how to do this for any expression and give some examples showing the correctness of your argument. 2 The momentum operator of the string is Prove that this is the generator of translation in the r-direction.

This section may be omitted on a first reading. 39) where, for now, M is simply regarded as a real parameter. Later we will see that it can be interpreted as the mass of the particles which emerge from the quantization of the field. As before, the four-current j , is the source of the field, and it is assumed to be conserved. The equations of motion obtained from this Lagrangian are known as the Proca equations: t3,Fpu + M2A” = j ” . Taking the four-divergence of both sides and remembering that metric give M 2 &A” = a,j” = 0 .