The addition theorem for binomial coefficients, namely
gives
( a) b�c� a! = I: (a-b-s)! a-c-s !(b+ c-a+ s)!s!ea-b) ��a-c) !
s
if x and y are positive ;
(b ) I: -( 1 ) s s!(b-s)!(c-s)!(a-s) ! = (-1 )c ...._ ... ______• I s if y is negative and b > a > c ) O ; or if y is negative and a > b ) O, a ) c
>
O; ( c ) b!c! a ! = I: (-1 ) s �a-c) !(s-a+ b-1) ! (b-a-1!
s!(a-c-s)!(b-a+c+ s)! s if x is negative.= L �a-t) � t (b-t)!(c-t !(d-t)!(e+t)!t! L t , s by ( a) -�---�----· L -r-:---"""'""'...,.-,:-------.--,....,...�--.--. ... ---.--..--.- b+c+ e-a+ s . s also by ( a) . Equivalently ( a+s ) ! L (b+s)!(c+s)!(d-s)!(e-s)!s! s
= (a-b) � (a-c) � d! e!
t
(a-b-t)!(e+c-t)!(d+c-t)!(b-c+t)!t! (d+e+c-t) !II. = by = L (-1 ) s s!(c-s)!(d-s)!(e-s)! (a-&) !(b-s) � s L s , t ( a) ; L t e . by (b).
.
III. L (a-t ) ! t (b-t)!(c-t)!(d-t)!(e+t)!t! �a-c) ! ! (d+e ! (b-a-1) ! Ls a-b negative by ( c) and (a) . L s Equivalently (a-c) !(a-d) !b! e! (e+d-b-1-t�! ; (e+d-t)!(c-d+t)!(d-b- -t)!(a-c-t)! 144
LITERATURE
1. G. R acah, Phys. R ev . .'.Z§, 1 352 (19 49 ).
2. G. R acah, "Group Theory and Spectroscopy'' Egarb. der exakten Naturwiss, Vol. 37, S pringer-Verlag, Berlin (1965).
3. B. R . Judd, " Group Theory in Atomic S pectroscopy", in " Group Theory and its Applications" E.M. Loebel, Ed. ,
Academic Press Inc., New York (1968). 4. J.D. Louck, Am. J. Phys. 38, 3 (19 70).
5.
B. R . Judd, "Operator Techniques in Atomic S pectroscopy" McGraw-Hill Book Co. , Inc., New York, 1963.6. B.R . Judd, "S econd Quantization and Atomic Spectroscopy" Johns Hopkins Press, Baltimore, Md. (1967).
7.
B.G. Wybourne, "S ymmetry Principles and Atomic S pect roscopy" Wiley-Interscience, New York (19 70).8. A. O. Barut and H.M. Kleinert, Phys. R ev. 156, 1541 (1967), Phys. R ev.
12.2,
1180 (1967), Phys. R ev. 160,1 1 49 (1 967).
9.
L. Armstrong, Jr, Abstracted in "Theorie de laS tructure Atomique" Colloque du Centre de la R echerche S cientifique (19 70).
10. M.J. Cunningham, J. Math. Phys., to be published (19 70).
12.
1 .3.
1 46 L. Armstrong Jr, and B.R . Judd, Proc. R oy. S oc. A JI ��
2'7 -'37, J q-1?� (/'170)
M.J. Cunningham and B.G. Wybourne, J. Math. Phys. 10, 2149 (1 969 ).
14. M.J. Cunningham and B.G. Wybourne, J. Math. Phys. 1 1 , 1 288 (1 9 70).
15. J. S chwinger, " On Angular Momentum", Technical Inform ation S ervice, Oak R idge, Tennessee (R ept W-2309 1, No.
NY0-.3071 ), 1 952. To be found in '1Quantum Theory of Angular Momentum" edited by L.C. Biedenharn and H. van Dam. Academic Press, New York and London, 1965.
160 V. Bargmann, Commun. Pure Appl. Math. 14, 19 8 (1 961 ).
17. M. Moshinsky, Nucl. Phys. 31 , 384 (1 962) ; R ev. M0d. Phys. 34, 81 .3 (1 962) ; J. Math. Phys.
1 ,
1 1 28 (196.3). 18. G.E. Baird and L.C. Biedenharn , J. Math. Phys.1 ,
4.36(1 963) ;
1,
1 449 (1 963) ;2,
1 723 (1964) ;2,
1 730 (1 964).19 . M. Moshinsky and V. S yamala Devi, J. Math. Phys. 10,
455 (1 969 ).
20. M. Moshinsky, " Group Theory and the Many-body Problem" to be found in "Many-body Problems and Other S elected Topics in Theoretical Physics". Edited by M.
Moshinsky, T.A. Brody and G. Jacob. Gordon and Breach, New York (1966).
21 . J. Flores , E. Chacon, P.A. Mello and M. De Llano , "Studies on Nuclear Structure in the 2s-1 d Shell , I and II " to be found in "Many-body Problems " , ref. 20. 22. H.A. Jahn , Pro c. Roy. Soc. A205 , 192 (1951).
23. D.E. Littlewood, "The Theory of Group Characters and Matrix Representations of Groups". Oxford University Press ( 1 940 ).
24. F . D . Murnaghan , "The Unitary and Rotation Groups" , Spartan Books , Washington , D.C. (1962) .
25. C.W. Nielson and G.F. Ko ster , "Spectroscopic
Coeffici ent s for the pn, dn and fn Configurations"
M.I.T. Press , Cambridge , Massachusetts ( 1963).
260 P.J. Redmond , Proc. Roy. Soc. (London) A222 , 84 (1 954).
27. A . Hassitt , Proc. Roy. Soc. (London) A229 , 110 (1955).
28. M. Horie , J. Phys. Soc . Japan
.12,
1783 ( 1 964) . 29. F . Innes , J. Math. Phys. �' 816 (1967 ) .30. B.R. Judd , Proc. Roy. Soc. (London) A250 , 562 (1959) .
31. P. Nutter and C . Nielson, Techni cal Memorandum , T-330 ,
Research Dividion, Raytheon Company (1962).
32. S.A. Williams and O . L . Pursey , J. Math. Phys.
2,
1 230 (1968).33. J. Vizbaraite , A. Savukynas and A. Jucys , Lit. Fiz.
1 48
34. A. Jucys , J. Levinsonas and V. Vanagas , "Mathematical Apparatus of the Theory of Angular Momentum" , (Vilna , 1960 ) .
35. P.H. Butler , Thesis , University of Canterbury (1970)
36. B.R. Judd , Phys. Rev. 1 62 , 28 (1967 ).
37. L . Armstrong Jr , and B.R. Judd , "Quasiparticles in
Atomic Shell Theory" , Proc. Roy. Soc. (London)
38. R . D . Mattuck , "A Guide to Feynman Diagrams in the Many-Body Problem" McGraw-Hill Publishing Co . , New York ( 1 967 ).
39. J.P. Elliott , Proc. Roy. Soc. (London) A245 , 1 28
( 1958) .
40. P.H. Butler and B.G. Wybourne , J. Phys. 30 , 655 ( 1969).
41. H. Weyl , Math. Z. Q, 271 (1925).
42. B.G. Wybourne, J. Chem. Phys. 48 , 2596 (1968).
43. S. Feneuille , J. Phys. 30 , 923 (1969) .
44. S. Feneuille , J. Phys. 28 , 61 (1967 ).
45. P.H. Butler and B.G. Wybourne , J. Math. Phys.
1.::!,
2 51 2 ( 1970 ) •
46. J.C. Morrison, J. Math. Phys. 1 0 , 1431 (1969 ).
47. A.R. Edmonds , "Angular Momentum in Quantum Mechani cs" , Princeton University Press , Princeton, N.J. (1960) .
48. S. Pasternack and R.M. Sternheimer , J. Math. Phys.
2,
49 . N.V.V.J. Swamy , R .G. Kulkarni and L.C. Biedenharn, J. Math. Phys.
11,
1165 (19 70).50. V. Bargmann ,
Ann.
Math. 48 , 568 (19 47).51. A.O. Barut and C. Fronsdal , Proc. R oy. S oc. (London) A287, 532 (1965).
52. H.M. Kleinert , Forts. Phys. 16 , 1 (1968).
53. L. Infeld and T.E. Hull, R ev. Mod. Phys. 23 , 21 (1951). 54.
55.
A. Messiah , "Quantum Mechanics of One- and Two
Electron Atoms " Academic Press Inc., New York (1957). V. Bargmann , R ev. Mod. Phys. 34 , 829 (1962).
56. B.L. van de Waerden , 11Die gruppentheoretische Methode
in der Quentenmechanik" (Verlag Julius S pringer,
Berlin , Germany (19 32).
57. G. R acah , Phys. R ev. 62 , 438 (19 42).
58. Bethe and S alpeter , "Quantum Mechanics of One- and Two-Electron Atoms". Academic Press , Inc., New York
(1957).
59 . L.C. Biedenharn , Phys. R ev. 126, 845 (1962).
60. A. Crubellier and S. Feneuille, Private Communication. 61. H. Bacry and J.L. R ichard, J. Math. Phys. �' 2230
(1967).
62. A . O . Barut and G.L. Bornzin, Private Communication.
63. J. Lanik , Czech. J. Phys. B19 , 1540 (1969 ).
150 64. H. B.G. Casimir, "On the Interaction between Atomic
Nuclei and Electrons " Freemann and Co. , San Francisco (1963).
65. D. Layzer, Ann. Phys. 8, 271 (1959).
PUBLICATIONS
pi.,t
A largeAof this thesis has either been published or accepted for publication in the Journal of Mathematical Physics, namely Chapters III and IV which were done with Professor Wybourne, and Chapter I. Chapter V has been accepted for publication while Chapter VI has been sub mitted.
In detail :
Chapter I : "Second Quantized Atomic Wavefunctions with Definite Unitary and Rotational Symmetry", J. Math. Phys. 11, 2781 (1970).
Chapter III : "Quasiparticle Formalism and Atomic Shell Theory", J. Math. Phys. 10, 2149 ( 1969).
Chapter IV: "Quasiparticle Formalism and Atomic Shell Theory II. Mixed Configurations", J. Math. Phys •
.11,
1288 (1970).Chapter V: "Radial Matrix Elements in the Radial-Angular Factorised Hydrogen Atom", accepted for publication, J.
Math. Phys., May.
Chapter VI: "An 0(2,1) x 0(3) Solution to a Generalised Quantum Mechanical Kepler Problem", submitted to J. Math. Phys.