Consider the parity operator in three dimensions. (a) Show that ^(r) = '(r) = (-r) is equivalent to a mirror reflection followed by a rotation.

Consider the parity operator in three dimensions.
(a) Show that ^∏Ψ(r) = Ψ'(r) = Ψ(-r)  is equivalent to a mirror reflection followed by a rotation.
(b) Show that, for Ψ expressed in polar coordinates, the action of the parity operator is

(c) Show that for the hydrogenic orbitals,

That is, Ψnℓm is an eigenstate of the parity operator, with eigenvalue (-1)^ℓ.

y (r,0,0) = (r, - 0, 0+).