Title:

      Asymptotic approximations to Clebsch-Gordan coefficients
      from a tight-binding model


      Abstract:
      The talk is intended to be interesting to students and
      teachers of quantum mechanics. The results are not
      particularly new, but the picture presented makes them
      seem more intuitive.

      The recurrence relations of the angular momentum vector
      addition coefficients are interpreted as a tight-binding
      model of a one-dimensional potential. From this model we derive their
      semi-classical limits in a simple manner, treating separately
      large J approx L+S, (corresponding to the ground and
      low-lying states) and small J approx |L-S|, (corresponding to
      the highest lying states). The first case leads to oscillator
      wave functions;  the second case gives the well known
      approximation of Edmonds [`Angular Momentum in Quantum
      Mechanics', Princeton, (1957)] in terms of symmetric top wave
      functions. The resulting picture makes their qualitative
      behaviour transparent to beginners, without the use of
      advanced concepts. The case |L| = |S| and M=0 is especially
      simple; non-zero M slightly more complicated, and |S| < |L|
      doable.  [to appear in Am. J. Phys. (2009)]

      In recent work, T.A. Heim, J. Hinze and A.R.P. Rau, [to
      appear in J. Phys. A, (2009)] have discussed the so-called
      `non-trivial' zeroes of the 3-j symbols. These are cases
      where there is no ``obvious" symmetry principle which causes
      the symbol to vanish. The asymptotic relations we have
      established in terms of oscillator and symmetric top wave
      functions provide an explanation, as they correspond to wave
      function nodes.