FermiGasy

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

W. Udo Schröder, 2005 Angular Momentum Coupling 3

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

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:

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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.

W. Udo Schröder, 2005 Angular Momentum Coupling 22 2-Particle Isospin Coupling Use spin/angular momentum formalism: t  (2t+1) iso-projections Both nucleons in j shell  lowest E states have even J  T=1 ! For odd J  total isospin T = 0

W. Udo Schröder, 2005 Angular Momentum Coupling 23 Isobaric Analog (Isospin Multiplet) States

W. Udo Schröder, 2005 Angular Momentum Coupling 24

W. Udo Schröder, 2004 Nuclear Deform 25 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

W. Udo Schröder, 2005 Angular Momentum Coupling 26 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

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W. Udo Schröder, 2005 Angular Momentum Coupling 29 Wigner-Eckart Theorem Know this for spherical harmonics

W. Udo Schröder, 2005 Angular Momentum Coupling 30 Spherical Tensors and Reduced Matrix Elements  = Qu. # characterizing state

W. Udo Schröder, 2005 Angular Momentum Coupling 31 More General Symmetries: Wigner’s 3j Symbols From before: Invariant under rotations

W. Udo Schröder, 2005 Angular Momentum Coupling 32 Translations x V(x) r V(r)