In
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the
Wigner 3-j symbols, also called 3
j or 3
-jm symbols,
are related to Clebsch–Gordan coefficients
through
Inverse relation
The inverse relation can be found by noting that
j1 -
j2 -
m3 is an integer number and making the substitution
Symmetry properties
The symmetry properties of 3
j symbols are more convenient than those of
Clebsch–Gordan coefficients. A 3
j symbol is invariant under an even
permutation of its columns:
An odd permutation of the columns gives a phase factor:
Changing the sign of the
quantum numbers also gives a phase:
Regge symmetries also give
Selection rules
The Wigner 3
j is zero unless all these conditions are satisfied:
Scalar invariant
The contraction of the product of three rotational states with a 3
j symbol,
is invariant under rotations.
Orthogonality relations