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FermiGasy

FermiGasy. Angular Momentum Coupling. f 2. q 2. q. q 1. Addition of Angular Momenta. Angular Momentum Coupling. Constructing J Eigen States. Can you show this??. Constructing J-1 Eigen States. We have this state:. Condon-Shortley.

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FermiGasy

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  1. FermiGasy Angular Momentum Coupling

  2. f2 q2 q q1 Addition of Angular Momenta Angular Momentum Coupling

  3. Angular Momentum Coupling Angular Momentum Coupling

  4. Constructing J Eigen States Can you show this?? Angular Momentum Coupling

  5. Constructing J-1 Eigen States We have this state: Angular Momentum Coupling Condon-Shortley Normalization conditions leave open phase factors  choose asymmetrically <a|J1z|b> ≥ 0 and <a|J2z|b> ≤ 0

  6. Clebsch-Gordan Coefficients Angular Momentum Coupling

  7. Recursion Relations Angular Momentum Coupling

  8. Recursion Relations for CG Coefficients Projecting on <j1,j2,m1,m2| yields Angular Momentum Coupling

  9. Symmetries of CG Coefficients Triangular relation Condon-Shortley : Matrix elements of J1z and J2z have different signs Angular Momentum Coupling

  10. Explicit Expressions A. R. Edmonds, Angular Momentum in Quantum Mechanics Angular Momentum Coupling

  11. 2Particles in j Shell (jj-Coupling) Look for 2-part. wfs of lowest energy in same j-shell, Vpair(r1,r2) < 0 spatially symmetric  jj1(r) = jj2(r). Construct consistent spin wf. N = normalization factor Which J= j1+j2 (and M)are allowed?  antisymmetric WF yJM Angular Momentum Coupling

  12. Symmetry of 2-Particle WFs in jj Coupling Antisymmetric function of 2 equivalent nucleons (2 neutrons or 2 protons) in j shell in jj coupling. • j1 = j2= j half-integer spins J =evenwavefunctions with even 2-p. spinJ are antisymmetric • wave functions with odd 2-p. spinJ are symmetric • jj coupling  LS coupling  equivalent statements • 2)l1=l2=lintegerorbital angular momenta L • wave functions with even 2-p. L are spatially symmetric • wave functions with odd 2-p. L are spatiallyantisymmetric Angular Momentum Coupling

  13. Tensor and Scalar Products Angular Momentum Coupling Transforms like a J=0 object = number

  14. Example: HF Interaction Angular Momentum Coupling protons electrons only only

  15. Wigner’s 3j Symbols Angular Momentum Coupling

  16. Explicit Formulas Explicit (Racah 1942): Angular Momentum Coupling All factorials must be ≥ 0

  17. Spherical Tensors and Reduced Matrix Elements a, b, g = Qu. # characterizing states Angular Momentum Coupling Wigner-Eckart Theorem

  18. Wigner-Eckart Theorem Angular Momentum Coupling Take the simplest ME to calculate

  19. Examples for Reduced ME Angular Momentum Coupling

  20. Reduced MEs of Spherical Harmonics Angular Momentum Coupling Important for the calculation of gamma and particle transition probabilities

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

  22. 2-Particle Isospin Coupling Use spin/angular momentum formalism: t  (2t+1) iso-projections 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 (MT=-1,0,+1) are degenerate, if what should be true (nn, np forces are same) Angular Momentum Coupling Different MTstates belong to different nuclei T3 = (N-Z)/2

  24. 2-Particle Isobaric Analog (Isospin Multiplet) States Corresponding T=1levels in A=14 nuclei Angular Momentum Coupling T3=+1 2n T3=-1 2n holes T3=0, pn

  25. Further Applications Tensors and Angular Momentum Coupling Angular Momentum Coupling

  26. Separation of Variables: HF Interaction Angular Momentum Coupling protons electrons only only

  27. z e |e|Z q Electric Quadrupole Moment of Charge Distributions arbitrary nuclear charge distribution with norm Coulomb interaction Point Charge Quadrupole moment Q  T2= Q2 -ME in aligned state m=j Nuclear Deform Look up/calculate

  28. Angular-Momentum Decomposition: Plane Waves Plane wave can be decomposed into spherical elementary waves q z Spherical Bessel function Angular Momentum Coupling

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

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

  31. Angular Momentum Coupling

  32. Angular Momentum Coupling

  33. V(x) V(r) x r Translations Angular Momentum Coupling

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