Fermion quasi-spherical harmonics
Abstract
Quasi-spherical harmonics, 0305-4470/32/5/011/img1 are derived and presented for half-odd-integer values of 0305-4470/32/5/011/img2 and m. The form of the 0305-4470/32/5/011/img3 factor is identical to that in the case of integer 0305-4470/32/5/011/img2 and m: 0305-4470/32/5/011/img5. However, the domain of these functions in the half-odd-integer case is 0305-4470/32/5/011/img6 rather than the domain 0305-4470/32/5/011/img7 in the case of integer 0305-4470/32/5/011/img2 and m (the true spherical harmonics). The form of the 0305-4470/32/5/011/img9 factor, 0305-4470/32/5/011/img10 (an associated Legendre function) is (as in the integer case) the factor 0305-4470/32/5/011/img11 multiplied by a polynomial in 0305-4470/32/5/011/img12 of degree 0305-4470/32/5/011/img13 (an associated Legendre polynomial). A substantial difference between the spherical (integer 0305-4470/32/5/011/img2 and m) and quasi-spherical (half-odd-integer 0305-4470/32/5/011/img2 and m) Legendre functions is that the latter have an irrational factor of 0305-4470/32/5/011/img16 whereas the factor of the truly spherical functions is an integer power of 0305-4470/32/5/011/img17. The domain of both the true and quasi-spherical associated Legendre functions is the same: 0305-4470/32/5/011/img18. A table of the associated Legendre functions is presented for both integer and half-odd-integer values of 0305-4470/32/5/011/img2 and m, for 0305-4470/32/5/011/img20, and for 0305-4470/32/5/011/img21. The table displays the similarity between the functions for integer 0305-4470/32/5/011/img2 and m (which are well known) and those for half-odd-integer 0305-4470/32/5/011/img2 and m (which have not been recognized previously).
- Publication:
-
Journal of Physics A Mathematical General
- Pub Date:
- February 1999
- DOI:
- 10.1088/0305-4470/32/5/011
- arXiv:
- arXiv:math-ph/9810001
- Bibcode:
- 1999JPhA...32..795H
- Keywords:
-
- Mathematical Physics;
- Mathematics - Quantum Algebra;
- 33C55;
- 33C80
- E-Print:
- 16 pages, 2 Tables. Submitted to J.Phys.A