Question

Show that the angular momentum operator, Î_z = (ħ/i) d/dφ, is hermitian.

Hint: consider the wavefunction ψ(φ), where φ varies from 0 to 2π

Answer 1

We can show that LxLx is Hermitian by directly evaluating its adjoint and showing that it’s equal to LxLx, using the fact that the adjoint operator is antilinear and antidistributive:

L†x=(ypz−zpy)†=(ypz)†−(zpy)†=p†zy†−p†yz†=pzy−pyz=ypz−zpy=LxLx†=(ypz−zpy)†=(ypz)†−(zpy)†=pz†y†−py†z†=pzy−pyz=ypz−zpy=Lx

We have used the fact that (AB)†=B†A†(AB)†=B†A†.

Similarly Ly,LzLy,Lz are Hermitian.