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AdditionOfAngularMomenta< The interaction representation | Index | NMR.Mrsimulator > These are private notes, not necessarily clear to other than the author. Addition of two angular momenta is physics jargon for the following situation. Immagine two particles each with quantized angular momentum, e.g. two spin S=1/2 particles. If they are isolated (e.g. in zero magnetic field, since an applied magnetic field would imply an external torque - the may or may not interact among themselves) the system must be rotationally invariant. From this invariance one can rewrite their Hilbert space, the tensor product of the Hilbert space of the individual spin with 4 dimensions (2x2), as the sum of the Hilbert spaces of the possible ways we can add the momenta: J=1/2+1/2=1, triplet, 3 dimensions, and J=1/2+1/2=0, singlet, 1 dimension. That is, we can write the eigenstates of the first basis as a linear combination of the eigenstates of the second basis. The first basis is |j1 j2 m1 m2>, the second basis is |J M>. The Cleibsch-Gordan therorem states that coefficient of the linear combination, the Cleibsch-Gordan coefficients are {$C^{1/2\, 1/2}_{m_1 m_2 \,J M}$}. We know that, using +,- as shorthand for +1/2, -1/2 |1 1> = |1/2 1/2 + +> |1 0> = (|1/2 1/2 + -> + |1/2 1/2 + ->)/ √2 |1 1> = |1/2 1/2 + +> |0 0> = (|1/2 1/2 + -> - |1/2 1/2 + ->)/ √2 Therefore the non-vanishing CG coefficients for two spin one half are {$C^{1/2\,1/2}_{+ +\, 1\, 1}=1=C^{1/2\, 1/2}_{- -\, 1\, -1} \qquad\qquad C^{1/2\, 1/2}_{+ -\, 1\, 0}= \frac {\sqrt2} 2=C^{1/2\, 1/2}_{- +\, 1\, 0}=C^{1/2\, 1/2}_{+ -\, 0\, 0}=-C^{1/2\, 1/2}_{- +\, 0\, 0} $}. In order two add three angular momenta, e.g. three s=1/2, one has to add two first, and then the third. Therefore it is like adding, e.g., a spin J=1 and a spin n s=1/2, plus a spin J=0 and spin s=1/2. The former yields JJ=3/2,1/2 and JJ=1/2 again, respectively. It is easy to check by construction that the addition of m s=1/2 yields the following scheme of half integer J=m/2<j<1/2 (m odd) or integer J=m/2<j<0 (m even), with multiplicity mJ j , along the scheme of a Tartaglia triangle
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