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--- classes:2009:fall:phys4101.001:lec_notes_1109 [2009/11/07 21:43] – yk
+++ classes:2009:fall:phys4101.001:lec_notes_1109 [2009/11/10 18:17] (current) – x500_hakim011
@@ Line 19: / Line 19: @@
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+===Legendre continuation from Friday===
+As a carry-on from Friday’s lecture we have that the Legendre polynomial is normalized at z=1, and that only integers converge.
+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)
+//* 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===
+First, we have <math> R(r)=\frac{U(r)}{r}</math>
+When angular momentum l=0
+With U(r)=Asin(kr)+Bcos(kr) as a very general solution, but we can eliminate the B term because this diverges.
+By using the two equations above we find that <math>R(r)=A\frac{sin(kr)}{r}</math>, where <math>k^{2}=\frac{2mE}{\hbar^{2}}</math>
+So we find that U(r) is continuous at r=0 , but U’( r) is discontinuous, so with the infinite square well we eliminate the U’ term.
+(If we were dealing with a non-infinite square well, with a small potential, we would find that U’(  r) is ≈continuous.  It is in the cases of larger potentials that we find the discontinuity)
+With <math>R(r)=A\frac{sin(kr)}{r}</math> we have “A” as unknown, and “k” as unknown, but it is more important for us to find “k” because it has more meaning ( information on the Energy!) than the normalization constant “A”.
+So we solve <math>A\frac{sin(kr)}{r}=0</math> so ka=n(pi), and we solve easily for k.
+===Spherical Bessel Function===
+Using Figure 4.2 from the book ( page 143)
+We look for interesting characteristics, such  as
+*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 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
+//*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)//

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