Quantum Mechanics: Concepts and Applications · Problem 7.26

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · wigner-eckart-theorem-dipole-matrix-elements

Problem

(a) Calculate the reduced matrix element ⟨1∥Y1∥2⟩\langle 1\parallel Y_1\parallel 2\rangle. Hint: For this, you may need to calculate ⟨1, 0∣Y10∣2, 0⟩\langle 1,\,0\mid Y_{10}\mid 2,\,0\rangle directly and then from the Wigner–Eckart theorem. (b) Using the Wigner–Eckart theorem and the relevant Clebsch–Gordan coefficients from the table, calculate ⟨1, m∣Y1m′∣2, m′′⟩\langle 1,\,m\mid Y_{1m'}\mid 2,\,m''\rangle for all possible values of mm, m′m', and m′′m''. Hint: You may find the integral ∫0∞r3R21∗(r)R32(r) dr=64a0155(65)5\int_0^\infty r^3 R^*_{21}(r)R_{32}(r)\,dr = \frac{64a_0}{15\sqrt{5}}\left(\frac{6}{5}\right)^5 and the following coefficients useful: ⟨j, 1; m, 0∣(j−1), m⟩=−(j−m)(j+m)/[j(2j+1)]\langle j,\,1;\,m,\,0\mid (j-1),\,m\rangle = -\sqrt{(j-m)(j+m)/[j(2j+1)]}, ⟨j, 1; (m−1), 1∣(j−1), m⟩=(j−m)(j−m+1)/[2j(2j+1)]\langle j,\,1;\,(m-1),\,1\mid (j-1),\,m\rangle = \sqrt{(j-m)(j-m+1)/[2j(2j+1)]}, and ⟨j, 1; (m+1), −1∣(j−1), m⟩=(j+m)(j+m+1)/[2j(2j+1)]\langle j,\,1;\,(m+1),\,-1\mid (j-1),\,m\rangle = \sqrt{(j+m)(j+m+1)/[2j(2j+1)]}.

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