Expectation Values, the Virial Theorem, and Scaling

The hydrogen wave functions are exact, so every quantity that a later correction needs is a definite number, not an estimate. Fine structure, hyperfine structure, the Stark and Zeeman shifts, and quantum defects all reduce, at first order, to expectation values of powers of $r$ taken in the unperturbed hydrogenic states. Relativistic kinetic energy weights $⟨ 1/ r^{2} ⟩$ ; spin-orbit coupling weights $⟨ 1/ r^{3} ⟩$ ; the Darwin and contact terms weight $∣ ψ (0) ∣^{2}$ . This lesson computes the family $⟨ r^{k} ⟩$ once and states the scaling laws that follow, and it establishes the virial balance that fixes the split of each energy into kinetic and potential parts.

Throughout, write the hydrogenic length scale as

a = \frac{a _{0}}{Z}, a_{0} = \frac{ℏ ^{2}}{μ k e ^{2}},

so that every radius carries $Z$ through $a$ and every energy through $E_{n} = - Z^{2} Ry / n^{2}$ .

The virial theorem

For a stationary state the expectation value of any operator that lacks explicit time dependence is constant, so its time derivative vanishes. Applying this to the virial operator $\hat{r} \cdot \hat{p}$ and using $d ⟨ \hat{G} ⟩ / d t = (i /ℏ) ⟨[\hat{H}, \hat{G}]⟩$ gives

0 = \frac{d}{d t} ⟨ \hat{r} \cdot \hat{p} ⟩ = \frac{i}{ℏ} ⟨ [\hat{H}, \hat{r} \cdot \hat{p}] ⟩ .

Evaluating the commutator with $\hat{H} = \overset{p}{^}^{2} /2 μ + V (\overset{r}{^})$ produces the kinetic energy on one side and the radial derivative of the potential on the other:

2 ⟨ T ⟩ = ⟨ \hat{r} \cdot \nabla V ⟩ .

The last line follows from $⟨ T ⟩ + ⟨ V ⟩ = E_{n}$ combined with $2 ⟨ T ⟩ = - ⟨ V ⟩$ . Every hydrogenic level therefore carries a kinetic energy equal to its binding energy and a potential energy twice as negative:

⟨ T ⟩_{n} = \frac{Z ^{2} Ry}{n ^{2}}, ⟨ V ⟩_{n} = - \frac{2 Z ^{2} Ry}{n ^{2}} .

The virial balance in hydrogen: the mean potential energy is 2E_n, the mean kinetic energy is −E_n, and their sum is the binding energy E_n < 0.

The mean inverse radius

The virial theorem gives $⟨ 1/ r ⟩$ with no further integration. Since $⟨ V ⟩ = - k Z e^{2} ⟨ 1/ r ⟩ = 2 E_{n}$ ,

⟨ \frac{1}{r} ⟩_{n ℓ} = - \frac{2 E _{n}}{k Z e ^{2}} = \frac{Z}{a _{0} n ^{2}} = \frac{1}{a n ^{2}} .

The result depends on $n$ alone, not on $ℓ$ . It reproduces the Bohr relation: the mean inverse radius equals the inverse of the $n$ -th Bohr radius $n^{2} a$ . This single expectation value already fixes the leading Coulomb energy through $⟨ V ⟩$ , and it recurs in every hyperfine and volume-shift estimate.

Kramers' recursion

The remaining moments follow from a recursion among consecutive powers, obtained by taking expectation values of the radial equation against $r^{s}$ and $r^{s - 1}$ . Kramers' relation connects three neighbouring moments:¹

\frac{s + 1}{n ^{2}} ⟨ r^{s} ⟩ - (2 s + 1) a ⟨ r^{s - 1} ⟩ + \frac{s}{4} [(2 ℓ + 1)^{2} - s^{2}] a^{2} ⟨ r^{s - 2} ⟩ = 0.

The relation is a finite-difference equation in the power $s$ ; seeded by $⟨ r^{0} ⟩ = 1$ and $⟨ 1/ r ⟩ = 1/ a n^{2}$ , it climbs to every positive moment.

Setting $s = 2$ in the same way delivers $⟨ r^{2} ⟩$ . The first positive moments are

⟨ r ⟩ = \frac{a}{2} [3 n^{2} - ℓ (ℓ + 1)], ⟨ r^{2} ⟩ = \frac{a ^{2} n ^{2}}{2} [5 n^{2} + 1 - 3 ℓ (ℓ + 1)] .

Both grow as $n^{2}$ and $n^{4}$ respectively — the electron of a highly excited state is far out and its position is broadly spread. The mild $ℓ$ -dependence subtracts a little for the low- $ℓ$ penetrating orbits, which reach closer to the nucleus on their eccentric excursions.

Feynman-Hellmann and the negative moments

Kramers' relation cannot start the negative tower on its own: at $s = 0$ the $⟨ r^{- 2} ⟩$ coefficient vanishes, so $⟨ 1/ r^{2} ⟩$ needs an independent input. The Feynman-Hellmann theorem supplies it. Treating the orbital quantum number $ℓ$ as a continuous parameter of the Hamiltonian, whose only $ℓ$ -dependence is the centrifugal term $ℏ^{2} ℓ (ℓ + 1) /2 μ r^{2}$ ,

\frac{\partial E _{n}}{\partial ℓ} = ⟨ \frac{\partial H ^}{\partial ℓ} ⟩ = \frac{ℏ ^{2} ( 2 ℓ + 1 )}{2 μ} ⟨ \frac{1}{r ^{2}} ⟩ .

Because $n = n_{r} + ℓ + 1$ , a shift in $ℓ$ at fixed radial number $n_{r}$ is a shift in $n$ , so $\partial E_{n} / \partial ℓ = \partial E_{n} / \partial n = 2 Z^{2} Ry / n^{3}$ . Solving,

⟨ \frac{1}{r ^{2}} ⟩_{n ℓ} = \frac{Z ^{2}}{a _{0}^{2} n ^{3} ( ℓ + \frac{1}{2} )} = \frac{1}{a ^{2} n ^{3} ( ℓ + \frac{1}{2} )} .

With $⟨ 1/ r^{2} ⟩$ in hand, Kramers' relation at $s = - 1$ closes the last one used by the fine structure. There the surviving terms give $a ⟨ 1/ r^{2} ⟩ = ℓ (ℓ + 1) a^{2} ⟨ 1/ r^{3} ⟩$ , so

⟨ \frac{1}{r ^{3}} ⟩_{n ℓ} = \frac{Z ^{3}}{a _{0}^{3} n ^{3} ℓ ( ℓ + \frac{1}{2} ) ( ℓ + 1 )} = \frac{1}{a ^{3} n ^{3} ℓ ( ℓ + \frac{1}{2} ) ( ℓ + 1 )}, ℓ > 0.

The $⟨ 1/ r^{3} ⟩$ moment diverges for $ℓ = 0$ , which is why the spin-orbit interaction — proportional to it — vanishes for s-states and the Darwin contact term takes over there instead.

The moment table and its scaling

The five moments and their scaling with $n$ , $ℓ$ , and $Z$ collect into one table. Each is written in units of the hydrogenic length $a = a_{0} / Z$ .

moment	value	$n$	$Z$
$⟨ r ⟩$	$\frac{a}{2} [3 n^{2} - ℓ (ℓ + 1)]$	$\sim n^{2}$	$\sim Z^{- 1}$
$⟨ r^{2} ⟩$	$\frac{a ^{2} n ^{2}}{2} [5 n^{2} + 1 - 3 ℓ (ℓ + 1)]$	$\sim n^{4}$	$\sim Z^{- 2}$
$⟨ 1/ r ⟩$	$\frac{1}{a n ^{2}}$	$\sim n^{- 2}$	$\sim Z$
$⟨ 1/ r^{2} ⟩$	$\frac{1}{a ^{2} n ^{3} ( ℓ + \frac{1}{2} )}$	$\sim n^{- 3}$	$\sim Z^{2}$
$⟨ 1/ r^{3} ⟩$	$\frac{1}{a ^{3} n ^{3} ℓ ( ℓ + \frac{1}{2} ) ( ℓ + 1 )}$	$\sim n^{- 3}$	$\sim Z^{3}$

The mean radius ⟨r⟩ grows almost quadratically with n; for fixed n it shrinks slightly as ℓ increases, since low-ℓ orbits are the eccentric ones.

The pattern is uniform. Positive powers scale as $Z^{- k}$ and grow with $n$ ; negative powers scale as $Z^{k}$ and fall with $n$ . A correction built from $⟨ 1/ r^{3} ⟩$ therefore scales as $Z^{3} / n^{3}$ and concentrates in the low shells of heavy ions, exactly where fine structure is largest.

Each perturbation of the hydrogen atom reduces at first order to one radial moment: ⟨1/r²⟩ for the relativistic term, ⟨1/r³⟩ for spin-orbit, and the density at the origin for the Darwin and hyperfine contact terms.

Length contraction with nuclear charge

The single factor $a = a_{0} / Z$ carries the entire $Z$ -dependence of the geometry. Increasing the nuclear charge from hydrogen to a one-electron ion contracts every orbital radius by $1/ Z$ and deepens every binding energy by $Z^{2}$ , without changing the shape of any wave function.

The hydrogenic length scale a₀/Z contracts every orbital as Z grows; the wave-function shape is unchanged, only rescaled inward.

Every moment in the table inherits the same scaling, and every correction built from those moments does too. The relativistic kinetic term, carrying $⟨ 1/ r^{2} ⟩ \sim Z^{2}$ together with the explicit $α^{2} Z^{2}$ of its prefactor, grows as $Z^{4}$ ; this is why fine-structure splittings, invisible in hydrogen at the eV scale, dominate the spectra of heavy elements. The moments computed here are the numbers those fine-structure corrections evaluate.

Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §4.2 and the Kramers-relation problem — the three-term recursion for $⟨ r^{s} ⟩$ , the virial theorem for the Coulomb potential, and the Feynman-Hellmann route to $⟨ 1/ r^{2} ⟩$ and $⟨ 1/ r^{3} ⟩$ . ↩