1. Representations and Algebra
FermiGasy
Spherical Tensors
Representations and Algebra

2. Rotational Matrices M M’ z
Arbitrary f
PJ operates in J space, keeps only components in J space
Effect of: (q,f)  (q’,f’) rotation
Spherical Tensors
“Spherical Tensor”
Transform among themselves under rotations
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3. Spherical Tensors k=0: scalar k=1: vector
Because of central potential, states of nucleus with different structure have different transformation properties under rotations  look for different rotational symmetries
Spherical tensor Tk (“rank” k) with 3k components
Irreducible tensor Tk of “degree” k with 2k+1 components transforms under rotations like spherical harmonics
k=0: scalar k=1: vector
Search for all irreducible tensors  find all symmetries/exc. modes.
Example tensor Tik of rank 2.
Spherical Tensors
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4. Irreducible Representations
1 Trace + 5 indep. symm + 3 indep. antisymm.= 9 components
Each set transforms separately: number, tensor, axial vector
Have different physical meaning
Spherical Tensors
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5. Example: Spherical Harmonics (Dipole)
Spherical harmonics , irreducible tensor degree k=1 (Vector)
Structure of generic irr. tensor of degree k=1 (Vector) in Cartesian coordinates:
Construct irr. representation from Cartesian coordinates Tx, Ty, Tz, like spherical harmonics. Then T will transform like a spherical harmonic
Spherical Tensors
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6. Example: Quadrupole Operator
Spherical Tensors
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7. Example: 2p WF in p-Orbit
V(r)
2l+1= 3 degenerate p states
Spherical Tensors
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8. Addition of Angular Momentaq q1 q2 f2
Spherical Tensors
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9. Angular Momentum Coupling
Spherical Tensors
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10. Constructing J Eigen States
Can you show this??
Spherical Tensors
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11. Constructing J-1 Eigen States
We have this state:
Spherical Tensors
Condon-Shortley
Normalization conditions leave open phase factors  choose asymmetrically <a|J1z|b> ≥ 0 and <a|J2z|b> ≤ 0
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12. Clebsch-Gordan Coefficients
Spherical Tensors
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13. Recursion Relations
Spherical Tensors
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14. Recursion Relations for CG Coefficients
Projecting on <j1,j2,m1,m2| yields
Spherical Tensors
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15. Symmetries of CG Coefficients
Triangular relation
Condon-Shortley : Matrix elements of J1z and J2z have different signs
Spherical Tensors
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16. Explicit Expressions
A. R. Edmonds, Angular Momentum in Quantum Mechanics
Spherical Tensors
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17. 2-(j1=j2) Particle j,m Eigen Function
Look for 2-part. wfs of lowest energy in same j-shell, Vpair(r1,r2) < 0  spatially symmetric  jj1(r) = jj2(r). Construct spin wf.
Which total spins j = j1+j2 (or = (L+S)) are allowed?
Exchange of particle coordinates. Spatially symmetric  spin antisymmetric jz m
Spherical Tensors
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18. 2-(j1=j2) Particle j,m Eigen Function
Which total spins j = j1+j2 (or = (L+S)) are allowed?
Exchange of particle jz m coordinates
Spherical Tensors
= antisymmetric !
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19. Exchange Symmetry of 2-Particle WF
j1 = j2 = half-integer total spins  states with even 2-p. spin j are antisymmetric
states with odd 2-p. spin j are symmetric
2) Orbital (integer) angular momenta l1= l2 
states with even 2-p. L are symmetric
states with odd 2-p. L are antisymmetric
Spherical Tensors
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20. Tensor and Scalar Products
Spherical Tensors
Transforms like a J=0 object = number
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21. Example: HF Interaction
Spherical Tensors
protons electrons only only
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22. Wigner’s 3j Symbols
Spherical Tensors
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23. Explicit Formulas Explicit (Racah 1942): Spherical Tensors
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24. Spherical Tensors and Reduced Matrix Elements
a, b, g = Qu. # characterizing states
Spherical Tensors
Wigner-Eckart Theorem
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25. Wigner-Eckart Theorem
Spherical Tensors
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26. Examples for Reduced ME
Spherical Tensors
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27. RMEs of Spherical Harmonics
Spherical Tensors
Important for the calculation of gamma and particle transition probabilities
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28. Isospin Formalism
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:
Spherical Tensors
Transforms in isospin space like angular momentum in coordinate space  use angular momentum formalism for isospin coupling.
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29. 2-Particle Isospin Coupling
Use spin/angular momentum formalism: t  (2t+1) iso-projections
Spherical Tensors
Both nucleons in j shell  lowest E states have even J  T=1 !
For odd J  total isospin T = 0
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30. Spherical Tensors
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31. Spherical Tensors
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32. Wigner-Eckart Theorem
Spherical Tensors
Know this for spherical harmonics
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33. Spherical Tensors and Reduced Matrix Elements
a = Qu. # characterizing state
Spherical Tensors
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34. More General Symmetries: Wigner’s 3j Symbols
From before:
Spherical Tensors
Invariant under rotations
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35. Translations r
V(r) x
V(x)
Spherical Tensors
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