Small couplings must be compared with energy gaps

Write the Hamiltonian as H₀ plus a dimensionless bookkeeping parameter λ times a Hermitian operator V. Assume the chosen unperturbed eigenvalue is discrete and isolated from the rest of the relevant spectrum. Expand its energy and eigenstate in powers of λ, and insert both series into the eigenvalue equation. Matching the linear terms gives an equation for the first state correction and the first energy shift. Isolation permits division by the nonzero energy gaps that appear when this equation is projected onto other eigenstates.

Choose intermediate normalization: the component of the perturbed state along the original state is one, so the first correction is orthogonal to that original state. This choice organizes the algebra but does not keep the truncated state exactly normalized. Physical probabilities must use a normalized state or a consistent expansion of its norm. In infinite-dimensional problems one also needs suitable operator domains and perturbation regularity; a formally written sum need not converge simply because each denominator is nonzero.

Projecting the quadratic-order equation onto the original eigenstate gives the second-order energy as the perturbation acting on the first-order state correction. Substituting its expansion yields squared coupling magnitudes divided by signed energy differences. For a nondegenerate ground state, every other unperturbed energy is higher, so all these contributions are nonpositive. A vanishing first-order shift therefore does not mean there is no energy correction: off-diagonal matrix elements can lower the ground energy at second order.

For an excited state, lower levels contribute positive terms and higher levels contribute negative terms. They can compete or cancel. The sign should be checked term by term rather than replaced by an absolute denominator. Matrix elements squared have energy-squared units, so division by an energy gives the required energy shift. A symmetry can set a diagonal expectation to zero while leaving the off-diagonal couplings finite; that is precisely when looking only at the first-order energy is especially misleading.

A two-level Hamiltonian provides an exact benchmark. If its unperturbed energies are zero and Δ and the only perturbation is a real coupling g, its lower eigenvalue is Δ/2 minus the square root of Δ²/4+g². Expanding for |g| much smaller than Δ gives −g²/Δ. As Δ approaches the coupling scale, the state admixture grows and this expansion fails. The exact eigenvalues remain finite; the divergent perturbative coefficient diagnoses a poor choice of unperturbed description.

The quantum lab illustrates a small admixture geometrically. Reset and set both controls to mixing angle zero degrees and relative phase zero, predicting zero probability in basis state one. Reset again and set mixing angle 30 degrees and phase zero; predict probability sin²(15°), about 0.067. These are separately prepared pure states, not states obtained by applying a Hamiltonian perturbation. Playback adds relative-phase precession while the basis probabilities stay fixed; it does not calculate energy shifts or choose a physical coupling.