Slow following requires a smooth path and an isolated eigenstate

Consider a finite-dimensional Hermitian family H(s) on a fixed parameter interval from zero to one, with twice continuously differentiable matrix entries and a selected nondegenerate eigenvalue separated from all others by a uniform positive gap. Traverse this same path with s=t/T and begin in its selected eigenstate. Under the standard smoothness and gap hypotheses, leakage from the associated instantaneous eigenspace tends to zero as total duration T grows. The size of a quantitative bound also depends on derivatives of the path and its endpoint behavior.

Holding the path fixed is essential. Adding faster oscillations while increasing T changes the family and can defeat conclusions drawn from a simple total-duration comparison. In infinite-dimensional systems one needs further domain and spectral hypotheses; an isolated finite-matrix argument cannot be applied automatically to a continuum. If the selected gap closes, this elementary theorem no longer applies. Some symmetry-protected situations may still have simple exact dynamics, but that requires a separate analysis rather than an unqualified appeal to slow evolution.

Differentiate the instantaneous eigenvalue equation and project onto another eigenstate. Orthogonality removes the energy derivative term and gives the off-diagonal eigenvector derivative as the matrix element of the Hamiltonian derivative divided by a signed gap. In an instantaneous-eigenstate expansion of the Schrödinger equation, these derivative terms couple different branches. Their rapidly winding dynamical phases suppress transfer when the path is suitably slow and gapped. This derivation explains why a small absolute change in the Hamiltonian is not the relevant criterion by itself.

A common local diagnostic is ħ times the magnitude of an off-diagonal Hamiltonian time derivative divided by gap squared. Small values help identify plausible adiabatic regimes and locate difficult narrow-gap portions of a schedule. They are not a universal necessary-and-sufficient theorem or a guaranteed transition-probability bound. Long resonant accumulation, rapid higher derivatives, or an inadequately controlled path can require global estimates. Conversely, a large diagonal derivative need not cause a transition when the instantaneous eigenvectors do not rotate.

The leading followed state acquires a dynamical phase from the integrated instantaneous energy and a geometric phase from the changing eigenvector convention. Their combined physical prediction is gauge consistent. Remaining in an instantaneous eigenspace does not mean that every population in a fixed laboratory basis remains constant. A spin whose eigenvector slowly rotates from the z direction to the x direction changes those fixed-basis probabilities while staying in its ground branch. Relative phases become observable when branches or paths interfere.

The quantum lab displays this geometric distinction through prepared states. Reset and set both controls to θ=0° and φ=0°, predicting basis-one probability zero. Reset again and set θ=90° and φ=0°, predicting probability one half. These can represent endpoint ground states of a Hamiltonian whose field direction rotates from z to x. Changing the controls prepares a fresh model at the retained clock; it does not evolve a physical ramp. The lab has no energy-gap, schedule-duration, leakage, or Berry-phase calculation, and playback only adds its fixed relative-phase precession.