Introduction to Quantum Mechanics · Problem 7.18

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · hermiticity_momentum_operators

Problem

Show that

p^2

is hermitian, for hydrogen states with

\ell = 0

. Hint: For such states

\psi

is independent of

\theta

and

\phi

, so

p^2 = -\frac{\hbar^2}{r^2} \frac{d}{dr}\left(r^2 \frac{d}{dr}\right)

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

\left\langle f \left| p^2 g \right\rangle = -4\pi\hbar^2 \left(r^2 f \frac{dg}{dr} - r^2 g \frac{df}{dr}\right)\bigg|_0^\infty + \left\langle p^2 f \left| g \right\rangle.\right.

Check that the boundary term vanishes for

\psi_{n00}

, which goes like

\psi_{n00} \sim \frac{1}{\sqrt{\pi}(na)^{3/2}} \exp(-r/na)

near the origin. The case of

p^4

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

\nabla^4\left[e^{-kr}\right] = \left(-\frac{4k^3}{r} + k^4\right)e^{-kr} + 8\pi k\,\delta^3(\mathbf{r}),

and confirm that

p^4

is hermitian.