1. FermiGasy

2. W. Udo Schröder, 2005 Angular Momentum Coupling 2 Addition of Angular Momenta    

3. W. Udo Schröder, 2005 Angular Momentum Coupling 3

4. W. Udo Schröder, 2005 Angular Momentum Coupling 4 Constructing J Eigen States Can you show this??

5. W. Udo Schröder, 2005 Angular Momentum Coupling 5 Constructing J-1 Eigen States Normalization conditions leave open phase factors  choose asymmetrically ≥ 0 and ≤ 0 Condon-Shortley We have this state:

6. W. Udo Schröder, 2005 Angular Momentum Coupling 6 Clebsch-Gordan Coefficients

7. W. Udo Schröder, 2005 Angular Momentum Coupling 7 Recursion Relations

8. W. Udo Schröder, 2005 Angular Momentum Coupling 8 Recursion Relations for CG Coefficients Projecting on <j 1,j 2,m 1,m 2 | yields

9. W. Udo Schröder, 2005 Angular Momentum Coupling 9 Symmetries of CG Coefficients Triangular relation Condon-Shortley : Matrix elements of J 1z and J 2z have different signs

10. W. Udo Schröder, 2005 Angular Momentum Coupling 10 Explicit Expressions A. R. Edmonds, Angular Momentum in Quantum Mechanics

11. W. Udo Schröder, 2005 Angular Momentum Coupling 11 2 Particles in j Shell (jj-Coupling) Which J = j 1 +j 2 (and M) are allowed?  antisymmetric WF  JM Look for 2-part. wfs of lowest energy in same j-shell, V pair (r 1,r 2 ) < 0  spatially symmetric   j1 (r) =  j2 (r). Construct consistent spin wf. N = normalization factor

12. W. Udo Schröder, 2005 Angular Momentum Coupling 12 Symmetry of 2-Particle WFs in jj Coupling 1)j 1 = j 2 = j half-integer spins  J =even wave functions with even 2-p. spin J are antisymmetric wave functions with odd 2-p. spin J are symmetric jj coupling  LS coupling  equivalent statements 2) l 1 =l 2 =l integer orbital angular momenta  L wave functions with even 2-p. L are spatially symmetric wave functions with odd 2-p. L are spatially antisymmetric Antisymmetric function of 2 equivalent nucleons (2 neutrons or 2 protons) in j shell in jj coupling.

13. W. Udo Schröder, 2005 Angular Momentum Coupling 13 Tensor and Scalar Products Transforms like a J=0 object = number

14. W. Udo Schröder, 2005 Angular Momentum Coupling 14 Example: HF Interaction protons electrons only only

15. W. Udo Schröder, 2005 Angular Momentum Coupling 15 Wigner’s 3j Symbols

16. W. Udo Schröder, 2005 Angular Momentum Coupling 16 Explicit Formulas Explicit (Racah 1942): All factorials must be ≥ 0

17. W. Udo Schröder, 2005 Angular Momentum Coupling 17 Spherical Tensors and Reduced Matrix Elements  = Qu. # characterizing states Wigner-Eckart Theorem

18. W. Udo Schröder, 2005 Angular Momentum Coupling 18 Wigner-Eckart Theorem Take the simplest ME to calculate

19. W. Udo Schröder, 2005 Angular Momentum Coupling 19 Examples for Reduced ME

20. W. Udo Schröder, 2005 Angular Momentum Coupling 20 Reduced MEs of Spherical Harmonics Important for the calculation of gamma and particle transition probabilities

21. W. Udo Schröder, 2005 Angular Momentum Coupling 21 Isospin Charge independence of nuclear forces  neutron and proton states of similar WF symmetry have same energy  n, p = nucleons Choose a specific representation in abstract isospin space: Transforms in isospin space like angular momentum in coordinate space  use angular momentum formalism for isospin coupling.

22. W. Udo Schröder, 2005 Angular Momentum Coupling 22 2-Particle Isospin Coupling Use spin/angular momentum formalism: t  (2t+1) iso-projections

23. W. Udo Schröder, 2005 Angular Momentum Coupling 23 2-Particle Spin-Isospin Coupling Both nucleons in j shell  lowest E states have even J  T=1 ! For odd J  total isospin T = 0 3 states (M T =-1,0,+1) are degenerate, if what should be true (nn, np forces are same) Different M T states belong to different nuclei T 3 = (N-Z)/2

24. W. Udo Schröder, 2005 Angular Momentum Coupling 24 2-Particle Isobaric Analog (Isospin Multiplet) States Corresponding T=1levels in A=14 nuclei T 3 =-1 2n holes T 3 =+ 1 2n T 3 =0, pn

25. W. Udo Schröder, 2005 Angular Momentum Coupling 25

26. W. Udo Schröder, 2005 Angular Momentum Coupling 26 Separation of Variables: HF Interaction protons electrons only only

27. W. Udo Schröder, 2004 Nuclear Deform 27 Electric Quadrupole Moment of Charge Distributions |e|Z e  z arbitrary nuclear charge distribution with norm Coulomb interaction Point Charge Quadrupole moment Q  T 2 = Q 2 - ME in aligned state m=j Look up/calculate

28. W. Udo Schröder, 2005 Angular Momentum Coupling 28 Angular-Momentum Decomposition: Plane Waves Plane wave can be decomposed into spherical elementary waves z  Spherical Bessel function

29. W. Udo Schröder, 2005 Angular Momentum Coupling 29 j-Transfer Through Particle Emission/Absorption P T C N p+T 

30. W. Udo Schröder, 2005 Angular Momentum Coupling 30 Average Transition Probabilities f i If more than 1 initial state may be populated (e.g. diff. m)  average over initial states Sum over all components of T k  = total i  f T k transition probability

31. W. Udo Schröder, 2005 Angular Momentum Coupling 31

32. W. Udo Schröder, 2005 Angular Momentum Coupling 32

33. W. Udo Schröder, 2005 Angular Momentum Coupling 33 Translations x V(x) r V(r)