Representations and Algebra FermiGasy Spherical Tensors Representations and Algebra

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 W. Udo Schröder, 2005

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 W. Udo Schröder, 2005

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 W. Udo Schröder, 2005

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 W. Udo Schröder, 2005

Example: Quadrupole Operator Spherical Tensors W. Udo Schröder, 2005

Example: 2p WF in p-Orbit V(r) 2l+1= 3 degenerate p states Spherical Tensors W. Udo Schröder, 2005

Addition of Angular Momenta q q1 q2 f2 Spherical Tensors W. Udo Schröder, 2005

Angular Momentum Coupling Spherical Tensors W. Udo Schröder, 2005

Constructing J Eigen States Can you show this?? Spherical Tensors W. Udo Schröder, 2005

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 W. Udo Schröder, 2005

Clebsch-Gordan Coefficients Spherical Tensors W. Udo Schröder, 2005

Recursion Relations Spherical Tensors W. Udo Schröder, 2005

Recursion Relations for CG Coefficients Projecting on <j1,j2,m1,m2| yields Spherical Tensors W. Udo Schröder, 2005

Symmetries of CG Coefficients Triangular relation Condon-Shortley : Matrix elements of J1z and J2z have different signs Spherical Tensors W. Udo Schröder, 2005

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

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 W. Udo Schröder, 2005

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 ! W. Udo Schröder, 2005

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 W. Udo Schröder, 2005

Tensor and Scalar Products Spherical Tensors Transforms like a J=0 object = number W. Udo Schröder, 2005

Example: HF Interaction Spherical Tensors protons electrons only only W. Udo Schröder, 2005

Wigner’s 3j Symbols Spherical Tensors W. Udo Schröder, 2005

Explicit Formulas Explicit (Racah 1942): Spherical Tensors W. Udo Schröder, 2005

Spherical Tensors and Reduced Matrix Elements a, b, g = Qu. # characterizing states Spherical Tensors Wigner-Eckart Theorem W. Udo Schröder, 2005

Wigner-Eckart Theorem Spherical Tensors W. Udo Schröder, 2005

Examples for Reduced ME Spherical Tensors W. Udo Schröder, 2005

RMEs of Spherical Harmonics Spherical Tensors Important for the calculation of gamma and particle transition probabilities W. Udo Schröder, 2005

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. W. Udo Schröder, 2005

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 W. Udo Schröder, 2005

Spherical Tensors W. Udo Schröder, 2005

Spherical Tensors W. Udo Schröder, 2005

Wigner-Eckart Theorem Spherical Tensors Know this for spherical harmonics W. Udo Schröder, 2005

Spherical Tensors and Reduced Matrix Elements a = Qu. # characterizing state Spherical Tensors W. Udo Schröder, 2005

More General Symmetries: Wigner’s 3j Symbols From before: Spherical Tensors Invariant under rotations W. Udo Schröder, 2005

Translations r V(r) x V(x) Spherical Tensors W. Udo Schröder, 2005