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Recurrence relation for harmonic oscillator wave functions

  1. Nov 18, 2017 #1
    1. The problem statement, all variables and given/known data
    I've been using a recurrence relation from "Adv. in Physics"1966 Nr.57 Vol 15 . The relation is :
    Screen Shot 2017-11-18 at 11.35.40 AM.png
    where Rnl are radial harmonic oscillator wave functions of form:

    Screen Shot 2017-11-18 at 11.35.49 AM.png

    The problem is that I can't prove the relation above with the form of Rnl given by the author(above). I've been trying to check this using Mathematica. The form in which I need to use this is the following:
    Screen Shot 2017-11-18 at 11.43.28 AM.png
    Screen Shot 2017-11-18 at 11.43.46 AM.png

    Here, I use the Rnl's as radial wave-functions for neutrons and protons and multiply the first relation with Rprotonnl and integrate over spherical coordinates.(P.S the R's where noted by g's ).

    My questions is why would a relation like this only hold if the R's(or g, respectively) are normalized? Should't this relation hold for any radial wavefunctions? Where does the normalization condition comes from?

    The above relations are equivalent with let's say

    $$ | \alpha \rangle = a |\beta \rangle + b|\gamma \rangle $$

    and by multiplying with the bra ## | \alpha \rangle ## to the left one gets

    $$ \langle \alpha | \alpha \rangle = a \langle \alpha | \beta \rangle + b \langle \alpha | \gamma \rangle $$

    This is kind of my case above, but from my reasoning it should hold with no dependence on the normalization of the functions.

    Thank you!

    2. Relevant equations


    3. The attempt at a solution
     
    Last edited: Nov 18, 2017
  2. jcsd
  3. Nov 20, 2017 #2

    DrClaude

    User Avatar

    Staff: Mentor

    The associated Laguerre polynomials are not orthogonal for different values of ##l##. It is the angular part that ensures the orthogonality of the wave functions for different ##l##s.
     
  4. Nov 20, 2017 #3
    I know that. My question was related to the fact that I can check the recurrence relation ONLY if I normalize the Rnl's first. If I try to use the relation with non-normalized wave-functions the relation doesn't hold and I can't understand why the relation doesn't hold. That recurrence relation, from my reasoning, should hold even for non-normalized functions.
     
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