===== Nov 09 (Mon) more on Radial wave equation =====
** Responsible party: Andromeda,Hydra ** 

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Please try to include the following

  * main points understood, and expand them - what is your understanding of what the points were.
    * expand these points by including many of the details the class discussed.
  * main points which are not clear.  - describe what you have understood and what the remain questions surrounding the point(s).
    * Other classmates can step in and clarify the points, and expand them. 
  * How the main points fit with the big picture of QM.  Or what is not clear about how today's points fit in in a big picture.
  * wonderful tricks which were used in the lecture.\\
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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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