Question

Not sure if anyone can help, but how does one go about explicitly proving that both L^2 and L_z are Hermitian operators?

Answer 1

for example, the matrices of Pauli, of \(S_x,S_y,S_z\) are the generators of a rotation in the spin space, and such generator can be expressed by hermitian Matrix, please see the discussion on the book of P.A.M.Dirac "The Principles of Quantum Mechanics"
Now we can prove that the operator:
\[{S^2} = S_x^2 + S_y^2 + S_z^2\]
is also hermitian using the hermitian proerty of \[{S_x},{S_y},{S_z}\]

Answer 2

property*

Answer 3

for example, we have:
\[\left\langle {S_x^2\alpha } \right|\left. \beta \right\rangle = \left\langle \alpha \right|\left. {S_x^2\beta } \right\rangle \]
similarly for
\[{S_y},{S_z}\]