Connect semiclassical waves across a simple turning point

For a one-dimensional stationary Schrödinger equation, define positive local wavenumber k in an allowed region where energy exceeds potential. Substituting an exponential phase ansatz and retaining leading semiclassical orders gives phases equal to plus or minus the integral of k, with amplitude proportional to one over the square root of k. The amplitude factor is needed for approximately constant probability current in a traveling branch; a constant-amplitude plane wave is generally inconsistent when the local momentum changes.

The practical requirement is that k changes little over one reduced wavelength, so |k′|/k² is small, together with adequate smoothness for the neglected derivative terms. In a forbidden region replace k by a positive decay constant κ. Exponentially increasing and decreasing branches then replace the oscillatory ones. These statements are local approximations away from endpoints. A small value of ħ by itself is not enough if the potential has a discontinuity or the chosen observation point approaches a turning point as ħ changes.

Consider a simple right turning point xₜ, with an allowed region to its left, a forbidden region to its right, and positive nonzero potential slope there. Linearizing the potential reduces the local Schrödinger equation to the Airy equation after rescaling distance. Its decaying Airy solution on the forbidden side has an oscillatory asymptotic form on the allowed side. Matching these asymptotic forms fixes a doubled oscillatory coefficient and a phase shift of plus π/4 in the sine convention used below.

Define the allowed phase by integrating k from the observation point up to the turning point, and the forbidden action by integrating κ outward from that turning point. A purely decaying forbidden branch determines the displayed allowed solution to leading order. This directional use matters: a tiny error in an assumed oscillatory solution can contain a growing forbidden component that eventually dominates. The approximate connection is not a numerically stable instruction to continue every oscillatory estimate into a pure decaying tail.

The Airy length is the cube root of ħ² divided by twice mass times the turning-point slope magnitude. Within a distance comparable to that scale, use a uniform Airy description rather than the divergent outer prefactor. The potential must be nearly linear across the matching region for this elementary formula to apply. A double turning point, a barrier top with zero slope, and a hard wall require other local analyses or boundary conditions; they do not inherit the same connection merely by sharing the word reflection.

The function lab can illustrate a potential threshold without pretending to construct WKB waves. Reset and set both controls to coefficient a=0.5 and position x=1, predicting y=0.5. Reset again and set a=0.5 and x=2, predicting y=2. For a dimensionless energy of one, the first point is allowed and the second forbidden in the illustrative potential V=x²/2. The lab only draws a parabola, tangent, and signed area. It has no momentum, wavefunction, turning-point matching, or Airy-function solver.