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.
  

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>

First, we have <math> R®=\frac{U®}{r}</math> When angular momentum l=0 With U®=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®=A\frac{sin(kr)}{r}</math>, where <math>k^{2}=\frac{2mE}{\hbar^{2}}</math> So we find that U® 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®=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.

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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