Problem.Set.1.soln.ver2

Problem.Set.1.soln.ver2 - Problem Set 1. (Fall 2006) The...

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Problem Set 1. (Fall 2006) The charge density for a hydrogenic p state can be written as r , 2 3 e a 0 3 64 r a 0 2 exp r a 0 P 0 cos P 2 cos  # e 4.8 10 10 statCoulomb , a 0 0.529 10 8 cm . (a) Determine the multipole moments of this charge distribution and give the potential for large r in terms of these moments. (b) Dtermine the potential, near the origin, correct to order r 2 . (c) Determine the interaction energy with a nuclear quadrupole moment, Q 10 24 cm 2 , located at at the origin. Solution: Part a : The multipole expansion for the potential for r r is in Gaussian units (see Jackson, Eq. 4.2): r , , l , m 4 2  1  r  r , , Y m , r 2 dr d Y m , r  1 where the multipole moment of the charge distribution, q lm , is given by: q lm  r  r , , Y m , r 2 dr d . We can write this in a general way by first noting that the charge distribution is r , , 2 3 e a 0 3 64 u 2 exp u 4 Y 00 , 4 5 Y 20 , where u r a 0 Thus the multipole moments are: q lm 2 3 a 0 e 64 u  4 exp u du  Y m , 4 Y 00 , 4 5 Y 20 , d . 2 3 a 0 e 64  4 ! 4 0 m 0 4 5 2 m 0 Thus only m 0 terms will be present, in particular only Y 00 , and Y 20 , .The non-zero moments are q 00  r , Y 00 , r 2 drd  e 4 and q 20  r , Y 20 , r 2 drd  65 a 0 2 e 4 The potential for large r , r  a 0 , is approximated by
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r , l , m 4 2  1 2 3 a 0 e 64  4 ! 4 0 m 0 4 5 2 m 0 Y m , r  1 4 2 3 e 64 4! 4 Y 00 , r 4 5 2 3 a 0 e 64 6! 4
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Problem.Set.1.soln.ver2 - Problem Set 1. (Fall 2006) The...

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