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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

FermiGasy

Angular Momentum Coupling


Addition of angular momenta

f2

q2

q

q1

Addition of Angular Momenta

Angular Momentum Coupling


Angular momentum coupling

Angular Momentum Coupling

Angular Momentum Coupling


Constructing j eigen states

Constructing J Eigen States

Can you show this??

Angular Momentum Coupling


Constructing j 1 eigen states

Constructing J-1 Eigen States

We have this state:

Angular Momentum Coupling

Condon-Shortley

Normalization conditions leave open phase factors  chooseasymmetrically <a|J1z|b> ≥ 0 and <a|J2z|b> ≤ 0


Clebsch gordan coefficients

Clebsch-Gordan Coefficients

Angular Momentum Coupling


Recursion relations

Recursion Relations

Angular Momentum Coupling


Recursion relations for cg coefficients

Recursion Relations for CG Coefficients

Projecting on <j1,j2,m1,m2| yields

Angular Momentum Coupling


Symmetries of cg coefficients

Symmetries of CG Coefficients

Triangular relation

Condon-Shortley : Matrix elements of J1z and J2z have different signs

Angular Momentum Coupling


Explicit expressions

Explicit Expressions

A. R. Edmonds, Angular Momentum in Quantum Mechanics

Angular Momentum Coupling


2 particles in j shell jj coupling

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


Symmetry of 2 particle wfs in jj coupling

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


Tensor and scalar products

Tensor and Scalar Products

Angular Momentum Coupling

Transforms like a J=0 object = number


Example hf interaction

Example: HF Interaction

Angular Momentum Coupling

protons electrons only only


Wigner s 3j symbols

Wigner’s 3j Symbols

Angular Momentum Coupling


Explicit formulas

Explicit Formulas

Explicit (Racah 1942):

Angular Momentum Coupling

All factorials must be ≥ 0


Spherical tensors and reduced matrix elements

Spherical Tensors and Reduced Matrix Elements

a, b, g = Qu. # characterizing states

Angular Momentum Coupling

Wigner-Eckart Theorem


Wigner eckart theorem

Wigner-Eckart Theorem

Angular Momentum Coupling

Take the simplest ME to calculate


Examples for reduced me

Examples for Reduced ME

Angular Momentum Coupling


Reduced mes of spherical harmonics

Reduced MEs of Spherical Harmonics

Angular Momentum Coupling

Important for the calculation of gamma and particle transition probabilities


Isospin

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.


2 particle isospin coupling

2-Particle Isospin Coupling

Use spin/angular momentum formalism: t  (2t+1) iso-projections

Angular Momentum Coupling


2 particle spin isospin coupling

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


2 particle isobaric analog isospin multiplet states

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


Fermigasy

Further Applications

Tensors and

Angular Momentum Coupling

Angular Momentum Coupling


Separation of variables hf interaction

Separation of Variables: HF Interaction

Angular Momentum Coupling

protons electrons only only


Electric quadrupole moment of charge distributions

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


Angular momentum decomposition plane waves

Angular-Momentum Decomposition: Plane Waves

Plane wave can be decomposed into spherical elementary waves

q

z

Spherical Bessel function

Angular Momentum Coupling


J transfer through particle emission absorption

P

C N

T

j-Transfer Through Particle Emission/Absorption

p+T 

Angular Momentum Coupling


Average transition probabilities

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


Fermigasy

Angular Momentum Coupling


Fermigasy

Angular Momentum Coupling


Translations

V(x)

V(r)

x

r

Translations

Angular Momentum Coupling


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