Show that P 2 is hermitian, for hydrogen states with = 0. For such states is

Show that P² is hermitian, for hydrogen states with ℓ = 0.
For such states is independent of θ and ϕ, so

(Equation 4.13). Using integration by parts, show that

Check that the boundary term vanishes for Ψ_n00, which goes like

near the origin.
The case of P⁴ is more subtle. The Laplacian of 1/r picks up a delta function (see, for example, D. J. Griffiths, Introduction to Electrodynamics, 4th edn, Eq. 1.102). Show that

and confirm that P⁴ is hermitian.

Transcribed Image Text:

p² = Pz4 zd dr d (2²4) dr