Define a quadrupole operator \(Q_{20}=r^{2} Y_{20}(\theta, \phi)\). The quadrupole moment \(Q\) for a state of good angular momentum \(|j mangle\) is conventionally defined as the expectation value of this operator in the substate \(|j, m=jangle\), multiplied by a factor \(\sqrt{16 \pi / 5}\),

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16 Q = = (j,mjQ20j, m = j). 5 Use the Wigner-Eckart theorem to show that 167 (jm| Q20 |jm) = 5 3m - j(j + 1)) j(2j - 1) Q.