724 In this problem you show that many matrix elements of the position operator

# 724 in this problem you show that many matrix

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7.24 In this problem you show that many matrix elements of the position operator x vanish when states of well defined l,m are used as basis states. These results will lead to selection rules for electric dipole radiation. First show that [ L 2 ,x i ] = i jk ǫ jik ( L j x k + x k L j ). Then show that L · x = 0 and using this result derive [ L 2 , [ L 2 ,x i ]] = i summationdisplay jk ǫ jik ( L j [ L 2 ,x k ] + [ L 2 ,x k ] L j ) = 2( L 2 x i + x i L 2 ) . (7 . 178) By squeezing this equation between angular-momentum eigenstates ( l,m | and | l ,m ) show that 0 = braceleftbig ( β β ) 2 2( β + β ) bracerightbig ( l,m | x i | l ,m ) , where β l ( l + 1) and β = l ( l + 1). By equating the factor in front of ( l,m | x i | l ,m ) to zero, and treating the resulting equation as a quadratic equation for β given β , show that ( l,m | x i | l ,m ) must vanish unless l + l = 0 or l = l ± 1.Explain why the matrix element must also vanish when

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