The physical question
Why can an approximation work far from a turning point yet fail sharply at the point itself?
A smooth turning point is a place where classical momentum vanishes and naive WKB amplitudes diverge. Here the linear-potential Schrödinger equation is exactly reducible to the Airy equation, so students can compare an independently integrated exact solution with connected WKB oscillatory and evanescent waves.
Prerequisites: the corresponding lecture-note treatment of eigenstates, phases, normalization and elementary differential equations. Work through the calculation on paper before exploring parameter sweeps.
Quantitative predictions
Live computed observables
Exact Airy solution and matched WKB
The oscillatory classically allowed side lies to the left. The decaying forbidden solution lies to the right. The vertical guide marks the shifted turning point.
Airy Ai wavefunction versus coordinate with valid WKB regions and a turning-point excluded region
WKB is deliberately omitted within |(x−xₜ)/ℓ|<0.65, exposing rather than hiding its singular approximation.
Error and validity
The absolute discrepancy is a quantitative reminder that even an asymptotically sound approximation has a local window of failure.
Absolute error of WKB connection formula versus dimensionless Airy argument
The plotted discrepancy uses the integrated Airy solution as reference; high positive coordinates may require more accurate special-function evaluation in research calculations.
Recover the mathematics
Derivation and physical meaning
Choose dimensionless ξ=(x−xₜ)/ℓ and the linear potential such that the stationary Schrödinger equation becomes y″(ξ)=ξy(ξ). The physical solution decreasing as ξ→+∞ is the Airy function Ai(ξ). It is regular at the classical turning point, where ξ=0. We calculate Ai numerically from the independently known initial values Ai(0)=0.355028053887817 and Ai′(0)=−0.258819403792807 by fourth-order Runge–Kutta integration.
Far into the classically forbidden region ξ>0, the decreasing WKB continuation is Ai(ξ)≈[2√π ξ1/4]−1 exp[−(2/3)ξ3/2]. In the allowed region ξ<0 it is Ai(ξ)≈[√π (−ξ)1/4]−1 sin[(2/3)(−ξ)3/2+π/4]. The π/4 is the Airy connection phase, obtained by matching across the turning point.
The local WKB validity test compares the spatial variation of the momentum to the local wavelength: |p′|≪p²/ℏ. For the linear problem it reduces to |ξ|3/2≫1. Near ξ=0 the predicted WKB magnitude diverges like |ξ|−1/4, while Ai(0) remains finite. The Airy equation acts as a uniform local description. A numerical solution that seems accurate only because the singular region was omitted would obscure the main lesson, so the exclusion is explicitly marked.
Translation xₜ and scaling ℓ change how the universal Airy profile appears in physical position but not the dimensionless connection formula. This universality explains why the same turning-point phase appears across widely different smooth one-dimensional potentials.
Check an exact limiting case
At large negative ξ the Airy solution oscillates with a slowly changing local wavelength. At ξ=0 its value is finite. For large positive ξ both exact and WKB tails decay rapidly; the relative error is more informative than absolute error where the amplitude becomes tiny.
Scope and validity of the model
Single smooth simple turning point, one-dimensional stationary Schrödinger equation, constant effective mass, no second nearby turning point and a linear local potential. Fourth-order integration is performed only on the finite ξ range displayed; at very large positive ξ exponential instability would require a specialized Airy routine.
Independent investigations
Explore and explain
- Find a failure. Move the probe from ξ=−4 toward zero and watch the WKB error increase. Identify which term in the asymptotic formula diverges.
- Recover the connection phase. Compare the oscillatory exact solution against the WKB sine including π/4. Explain why a naive sine without a connection phase cannot match the evanescent boundary condition.
- Test universality. Move xₜ and change ℓ. Recalculate ξ at the probe and explain why the same dimensionless Airy equation governs the curves.
Self-check: Why is the WKB prediction singular at a simple turning point?
For a deeper challenge, reproduce the displayed model independently with a symbolic or numerical notebook and ask an AI tutor for a critical review of each approximation, boundary condition and sign.
Synthesis and references
What the experiment establishes
The Airy turning point is a quantitative diagnostic of semiclassical approximations. Students can separately check the allowed-side phase, the forbidden decay, and the specific singular region in which local WKB breaks down, while seeing a regular exact solution throughout.
Lecture notes: Corresponding lecture-note chapter ↗.