Introduction to Quantum Mechanics · Problem 6.24

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · vector_selection_rules

Problem

(a) Sandwich each of the six commutation relations in Equations 6.52–6.54 between

\langle n'\ell'm'|

and

|n\ell m\rangle

to obtain relations between matrix elements of

\hat{\mathbf{V}}

. As an example, Equation 6.52 with the upper signs gives

B^{m'}_{\ell'}\left\langle n'\,\ell'\left(m'-1\right)\left|\,V_+\,\right| n\,\ell\,m\right\rangle = A^m_\ell\left\langle n'\,\ell'\,m'\left|\,V_+\,\right| n\,\ell\left(m+1\right)\right\rangle.

(b) Using the results in Problem 6.23, show that the six expressions you wrote down in part (a) are satisfied by Equations 6.59–6.61.