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Quantum Mechanics with Basic Field Theory by Bipin R. Desai

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29 Irreducible tensors and Wigner–Eckart theorem

Irreducible spherical tensors are like spherical harmonics that under rotations transform into one another. A typical irreducible spherical tensor depends on two quantities, usually designated as k and q, which play a role parallel to j and m of the angular momentum. The matrix elements of these tensors between different angular momentum states j′, m′ and j, m are often quite complicated but ttey can be simplified considerably due to a theorem by Wigner and Eckart. This allows one to separate the matrix element into two factors: one which is just the Clebsch-Gordan coefficient involved in combining k with j to give j′. This is called the geometrical factor and the other is a single term that is ...

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