Hydrogen expectation values can be evaluated directly from radial integrals, recursively through the Kramers-Pasternack relation, or by differentiating the energy with respect to a Hamiltonian parameter.
Useful closed forms include
Choose a parameter γ whose derivative isolates the desired operator.
Differentiating the Coulomb Hamiltonian with respect to e reproduces \(\langle r^{-1}\rangle=1/(n^2a_0)\). Treating l as a parameter in the radial Hamiltonian yields
If \(H(\gamma)|n\rangle=E_n(\gamma)|n\rangle\), differentiating and using normalization cancels derivatives of the state, leaving \(dE_n/d\gamma=\langle\partial H/\partial\gamma\rangle\). The method converts knowledge of the energy spectrum into an expectation value. The parameter must be treated consistently in both Hamiltonian and energy.
The result \(\langle1/r\rangle=1/(n^2a_0)\) depends only on n, consistent with the virial theorem for a Coulomb potential. By contrast, \(\langle r\rangle\) and \(\langle r^2\rangle\) also depend on l because the centrifugal barrier shifts probability outward. Dimensional analysis checks every formula: \(\langle r^q\rangle\) must carry length to the power q.
Use the following chain when working with Hydrogen expectation values. First identify the symmetry, prepared state and observable being discussed; this determines which basis and quantum numbers are meaningful. Next write the operator or differential equation before substituting eigenvalues. Apply boundary conditions, commutators or normalization one step at a time, keeping dimensions and allowed quantum-number ranges visible. Only then simplify to the highlighted result. Finally translate the mathematics into a measurement statement: specify which outcomes are possible, which quantities remain uncertain, how degeneracy is counted, or where probability is concentrated. This sequence prevents a correct formula from becoming disconnected from the physical assumptions that justify it.
Self-check: Radial moments, Kramers-Pasternack relations and Feynman-Hellmann methods for extracting observables. The final answer should explicitly connect this objective to at least one equation on the page and one experimentally meaningful prediction.
Direct integration is transparent for low states; recurrence relations generate families of moments; Feynman-Hellmann is fastest when the target operator appears as a parameter derivative.
This page is designed to support short guided exercises on: Radial moments, Kramers-Pasternack relations and Feynman-Hellmann methods for extracting observables.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.