Please help with the QM question below regarding raising and lowering operators for angular momentum.

In class we showed that L: raise and lower the L z quantum number by one unit of 5 while leaving the L2 eigenvalue unchanged. However, this does not imply that LIV, m) is a normalized state, assuming that Mm) is a normalized state. Here, the L2 eigenvalue is 523(6 | l) and the L; eigenvalue is m5. (a) Use the relation between L2, LI, and L2 to compute both L_L+|,m) and L+L_|, m). (b) Use your result from above to show that L+,m) = (+1)m(m+1) L_|,m) = h E(+ 1) m(m l)|,m 1). 13,911+ 1) (2) (0) Explain why the results above make sense in terms of satisfying the requirements for the highest and lowest weight states under L2. ((1) Let's now use these results to reconstruct the spherical harmonics. Find the positionspace wave function 11,9(9, (b) (i.e., the appropriate spherical harmonic) for the state |, E) by requiring that it is annihilated by the raising operator in position space. Solve the resulting differential equation by separation of variables and normalize the state. Then, use the result above to nd the spherical harmonic Yg'g_1