The position-space representation of the radial component of the momentum operator is given by
p^r→iℏ(∂r∂+r1)
Show that for its expectation value to be real: ⟨ψ∣p^r∣ψ⟩=⟨ψ∣p^r∣ψ⟩∗, the radial wave function must satisfy the condition u(0)=0. Suggestion: Express the expectation value in position space in spherical coordinates and integrate by parts.