Endpoint phases quantize a closed orbit

Consider a smooth one-dimensional confining potential with exactly two simple turning points x₁ and x₂ at the energy of interest. Between them the momentum is real; outside them a normalizable bound state must decay. Starting from each decaying tail gives two allowed-region sine forms with quarter-cycle shifts. They describe the same state only when the phase accumulated between the turning points satisfies a discrete condition. The resulting half-integer offset is not an arbitrary correction fitted to a known spectrum.

Integrating momentum from the left turning point to the right gives half the classical closed-orbit action. The two Airy matches require that integral to be πħ times n plus one half, where n is a nonnegative integer counting the bound states in this simple well. The formula is a leading semiclassical approximation in general. It is most naturally controlled when the wavelength is short on the potential scale; its exact success for every harmonic-oscillator level is a special property of that potential.

Write the full classical action as an integral around one closed orbit in phase space. For the standard one-dimensional soft well, each turning-point caustic contributes one to the Maslov index, so the total index is two. This notation packages the matching phase into the closed-orbit quantization rule. It does not remove the need to inspect the geometry: multi-dimensional caustics, singular coordinate reductions, and imposed boundary conditions can change the phase bookkeeping.

A useful independent check comes from differentiating action with respect to energy. For a one-dimensional bound orbit this derivative equals its classical period. Neighboring quantized actions differ by 2πħ, so the leading level spacing is approximately 2πħ divided by that period. Long-period motion therefore produces more densely spaced semiclassical levels. Near a separatrix the period can become large and the simple approximation can require uniform treatment; one should not extend a regular isolated-orbit estimate through every critical energy.

An infinite hard wall imposes a wavefunction zero directly. There is no narrow smooth region where a linear potential and Airy matching describe its reflection. Two Dirichlet walls give pL=Nπħ for a constant-potential interior, with positive integer N. The N=0 sine is the identically zero function and is not a state. Reusing the soft-well half-integer rule would therefore predict the wrong square-well ground energy. A mixed hard-wall and soft-turning-point problem has yet another offset determined by their combined phases.

The oscillator lab checks the classical period that enters the action derivative. Reset and set both controls to damping ratio zero and natural frequency one; predict a period 2π in its time units. Reset again and set damping zero and frequency two; predict period π. The model has unit mass and starts at displacement one with zero velocity. It computes a classical oscillator trajectory, not a quantum spectrum, a wavefunction, or a Maslov index. Its role here is to test the classical period relation used in the semiclassical reasoning.