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classes:2009:fall:phys4101.001:lec_notes_1109

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classes:2009:fall:phys4101.001:lec_notes_1109 [2009/11/10 16:36] – x500_voukx002classes:2009:fall:phys4101.001:lec_notes_1109 [2009/11/10 18:17] (current) – x500_hakim011
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 With these polynomials there are  the 1st kind (<math>P_{l}^{m}(z) </math>) and the 2nd kind (<math>Q_{l}^{m}(z)</math>) With these polynomials there are  the 1st kind (<math>P_{l}^{m}(z) </math>) and the 2nd kind (<math>Q_{l}^{m}(z)</math>)
 In QM we don’t generally concern ourselves with polynomials of the second kind because these usually do not represent a physically realizable situation  (for E&M, however, this order is very applicable) In QM we don’t generally concern ourselves with polynomials of the second kind because these usually do not represent a physically realizable situation  (for E&M, however, this order is very applicable)
 +
 +//* we could have a general solution for E(θ) in electromagnetism that is a general sum of these two kinds of functions //<math>E(\theta)= \sum_n A_n P_{l}^{m}(z)+B_n Q_{l}^{m}(z)</math>
  
 ===Radial Wavefunction=== ===Radial Wavefunction===
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 We look for interesting characteristics, such  as We look for interesting characteristics, such  as
  
-*all terms, except for l=0, start at the origin+*all terms, except for l=0, start at the origin. //this term goes to 1//
  
 *the 0 order Bessel function approaches 1 as x->0 *the 0 order Bessel function approaches 1 as x->0
  
-*The polynomials listed in table above create a dampened oscillation+*The polynomials listed in table above create a dampened oscillation. //this is because of the (1/X), polynomial on the denominator, which is a common factor in all j's// 
 + 
 +//*When we are concerned with the behavior of the function at large x, the term with only one power of x is most important// 
  
 *Each term l=0,1,2,etc maintains the same wavelength, but note that for each increasing l-term, the phase shift is 90 degrees *Each term l=0,1,2,etc maintains the same wavelength, but note that for each increasing l-term, the phase shift is 90 degrees
 +
 +//*Also as l increases from 0 to a larger number, the amplitude of the function decrease//
 +
 +//*At the end of the class there was a question about// <math>B_{n}^{l}</math>//, the nth zero of the spherical Bessel function. for example// <math>P_{01}</math>// is the point where the Bessel function with l=1 crosses the x-axis for the first time (NOT counting the origin)//
 +
  
  
classes/2009/fall/phys4101.001/lec_notes_1109.1257892585.txt.gz · Last modified: 2009/11/10 16:36 by x500_voukx002

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