1. Angular Momentum Coupling
FermiGasy
Angular Momentum Coupling

2. Addition of Angular Momentaq q1 q2 f2
Angular Momentum Coupling
W. Udo Schröder, 2005

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

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
W. Udo Schröder, 2005

6. Clebsch-Gordan Coefficients
Angular Momentum Coupling
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7. Angular Momentum Coupling
Recursion Relations
Angular Momentum Coupling
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8. Recursion Relations for CG Coefficients
Projecting on <j1,j2,m1,m2| yields
Angular Momentum Coupling
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9. Symmetries of CG Coefficients
Triangular relation
Condon-Shortley : Matrix elements of J1z and J2z have different signs
Angular Momentum Coupling
W. Udo Schröder, 2005

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

11. 2 Particles 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
W. Udo Schröder, 2005

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 =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) l1=l2=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
Angular Momentum Coupling
W. Udo Schröder, 2005

13. Tensor and Scalar Products
Angular Momentum Coupling
Transforms like a J=0 object = number
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14. Example: HF Interaction
Angular Momentum Coupling
protons electrons only only
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15. Angular Momentum Coupling
Wigner’s 3j Symbols
Angular Momentum Coupling
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16. Angular Momentum Coupling
Explicit Formulas
Explicit (Racah 1942):
Angular Momentum Coupling
All factorials must be ≥ 0
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17. Spherical Tensors and Reduced Matrix Elements
a, b, g = Qu. # characterizing states
Angular Momentum Coupling
Wigner-Eckart Theorem
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18. Wigner-Eckart Theorem
Angular Momentum Coupling
Take the simplest ME to calculate
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19. Examples for Reduced ME
Angular Momentum Coupling
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20. Reduced MEs of Spherical Harmonics
Angular Momentum Coupling
Important for the calculation of gamma and particle transition probabilities
W. Udo Schröder, 2005

21. Angular Momentum Coupling
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:
Angular Momentum Coupling
Transforms in isospin space like angular momentum in coordinate space  use angular momentum formalism for isospin coupling.
W. Udo Schröder, 2005

22. 2-Particle Isospin Coupling
Use spin/angular momentum formalism: t  (2t+1) iso-projections
Angular Momentum Coupling
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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 MT states belong to different nuclei T3 = (N-Z)/2
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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
W. Udo Schröder, 2005

25. Angular Momentum Coupling
Further Applications
Tensors and
Angular Momentum Coupling
Angular Momentum Coupling
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26. Separation of Variables: HF Interaction
Angular Momentum Coupling
protons electrons only only
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27. Electric Quadrupole Moment of Charge Distributions
arbitrary nuclear charge distribution with norm
|e|Z e q z
Coulomb interaction
Point Charge
Quadrupole moment Q  T2= Q2 -ME in aligned state m=j
Nuclear Deform
Look up/calculate
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28. Angular-Momentum Decomposition: Plane Waves
Plane wave can be decomposed into spherical elementary waves q z
Spherical Bessel function
Angular Momentum Coupling
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29. j-Transfer Through Particle Emission/Absorption
C N
p+T 
Angular Momentum Coupling
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30. Average Transition Probabilitiesf i
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
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31. Angular Momentum Coupling
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32. Angular Momentum Coupling
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33. Angular Momentum Coupling
Translations r
V(r) x
V(x)
Angular Momentum Coupling
W. Udo Schröder, 2005