hw4_p5406_s06 - HW 4 Phys5406 S06 due 2/23/06 1)We will use...

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Unformatted text preview: HW 4 Phys5406 S06 due 2/23/06 1)We will use the solution for the empty box with V(x,y) = V at z=c to get the Green's function G D for all problems with this geometry. a) To see the connection between r 2 G D ( r ; r ) = 4 ( r r ) and a sine series, expand the one dimensional delta function in the range (0,a) as below, ( x x ) = X j [ A j sin j x=a sin j x =a ] ; (1) and nd the coe cients A j . This is called the completeness relation for the sine series, indicating any function f ( x ), x in the range (0,a), can be expressed as, f ( x ) = X m B m sin m x=a: (2) You should now write the 3-d delta function in a sum of the product of sines. b) Now expand the sinh function below for z in the range (0,c) in a sine series, sinh jk z sinh jk c = X m [ A m sin m z=c ] ; (3) and obtain the coe cients A m . In the above equation, the LHS doesn't vanish at z =c, but the RHS does vanish. Thus the representation as a Fourier series allows you to get accuracy arbitrarily close to the boundary, but perhaps not exactly at the boundary. Also note that,arbitrarily close to the boundary, but perhaps not exactly at the boundary....
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hw4_p5406_s06 - HW 4 Phys5406 S06 due 2/23/06 1)We will use...

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