show that the Hamiltonian operator for the total energy of a system is hermitian provided that the eigenfunction is well behaved

Hi I need help on one of my physical chemistry questions and I don't know how to do it. Will you please help me solve for it thank you very much!

For the particle in the 1D box, the system approaches classical behavior as the spacing between energy levels (dE) for a change in level (dn) approaches zero, i.e. dE / dn = 0. Under what circumstances does this happen?

Consider the three spherical harmonics (a) Y0,0, (b) Y2,–1, and (c) Y3,+3. (a) For each spherical harmonic, substitute the explicit form of the function taken from Table 7F.1 into the left-hand side of eqn 7F.8 (the Schrödinger equation for a particle on a sphere) and confirm that the function is a solution of the equation; give the corresponding eigenvalue (the energy) and show that it agrees with eqn 7F.10. (b) Likewise, show that each spherical harmonic is an eigenfunction of lˆz = (ℏ/i)(d/dϕ) and give the eigenvalue in each case.